Mathematical Problems in Engineeringdownloads.hindawi.com/journals/specialissues/395987.pdf · Mathematical Problems in Engineering Theory, Methods, and Applications Guest Editors:

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Guest Editors: Antonio F. Bertachini A. Prado, Maria Cecilia Zanardi, Tadashi Yokoyama, Silvia Maria Giuliatti Winter, and Josep J. Masdemont

Special IssueMathematical Methods Applied to the Celestial Mechanics of Artificial Satellites

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Mathematical Methods Applied tothe Celestial Mechanics ofArtificial Satellites

Mathematical Problems in Engineering

Mathematical Methods Applied tothe Celestial Mechanics ofArtificial Satellites

Guest Editors: Antonio F. Bertachini A. Prado,

Maria Cecilia Zanardi, Tadashi Yokoyama,

Silvia Maria Giuliatti Winter, and Josep J. Masdemont

Copyright q 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Mathematical Problems in Engineering.” All articles are open access articles distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Editorial BoardEihab Abdel-Rahman, Canada

Rashid K. Abu Al-Rub, USA

Salvatore Alfonzetti, Italy

Igor Andrianov, Germany

Sebastian Anita, Romania

W. Assawinchaichote, Thailand

Erwei Bai, USA

Ezzat G. Bakhoum, USA

Jose Manoel Balthazar, Brazil

Rasajit K. Bera, India

Jonathan N. Blakely, USA

Stefano Boccaletti, Spain

Daniela Boso, Italy

M. Boutayeb, France

Michael J. Brennan, UK

John Burns, USA

Salvatore Caddemi, Italy

Piermarco Cannarsa, Italy

Jose E. Capilla, Spain

Carlo Cattani, Italy

Marcelo M. Cavalcanti, Brazil

Diego J. Celentano, Chile

Mohammed Chadli, France

Arindam Chakraborty, USA

Yong-Kui Chang, China

Michael J. Chappell, UK

Xinkai Chen, Japan

Kui Fu Chen, China

Kue-Hong Chen, Taiwan

Jyh Horng Chou, Taiwan

Slim Choura, Tunisia

Swagatam Das, India

Filippo de Monte, Italy

M. do R. de Pinho, Portugal

Antonio Desimone, Italy

Yannis Dimakopoulos, Greece

Baocang Ding, China

Joao B. R. Do Val, Brazil

Daoyi Dong, Australia

Balram Dubey, India

Horst Ecker, Austria

M. Onder Efe, Turkey

Elmetwally Elabbasy, Egypt

Alex Elias-Zuniga, Mexico

Anders Eriksson, Sweden

Vedat Suat Erturk, Turkey

Qi Fan, USA

Moez Feki, Tunisia

Ricardo Femat, Mexico

Rolf Findeisen, Germany

R. A. Fontes Valente, Portugal

C. R. Fuerte-Esquivel, Mexico

Zoran Gajic, USA

Ugo Galvanetto, Italy

Xin-Lin Gao, USA

Furong Gao, Hong Kong

Behrouz Gatmiri, Iran

Oleg V. Gendelman, Israel

Didier Georges, France

Paulo Batista Goncalves, Brazil

Oded Gottlieb, Israel

Quang Phuc Ha, Australia

Muhammad R. Hajj, USA

Thomas Hanne, Switzerland

Katica R. Hedrih, Serbia

C. Cruz Hernandez, Mexico

M.I. Herreros, Spain

Wei-Chiang Hong, Taiwan

J. Horacek, Czech Republic

Chuangxia Huang, China

Gordon Huang, Canada

Yi Feng Hung, Taiwan

Hai-Feng Huo, China

Asier Ibeas, Spain

Anuar Ishak, Malaysia

Reza Jazar, Australia

Zhijian Ji, China

J. Jiang, China

J. J. Judice, Portugal

Tadeusz Kaczorek, Poland

Tamas Kalmar-Nagy, USA

Tomasz Kapitaniak, Poland

Hamid Reza Karimi, Norway

Metin O. Kaya, Turkey

Nikolaos Kazantzis, USA

Farzad Khani, Iran

Kristian Krabbenhoft, Australia

Jurgen Kurths, Germany

Claude Lamarque, France

F. Lamnabhi-Lagarrigue, France

Marek Lefik, Poland

Stefano Lenci, Italy

Jian Li, China

Shihua Li, China

Shanling Li, Canada

Tao Li, China

Ming Li, China

Teh-Lu Liao, Taiwan

P. Liatsis, UK

Shueei M. Lin, Taiwan

Jui-Sheng Lin, Taiwan

Yuji Liu, China

Wanquan Liu, Australia

Bin Liu, Australia

Fernando Lobo Pereira, Portugal

Paolo Lonetti, Italy

Vassilios C. Loukopoulos, Greece

Junguo Lu, China

Chien-Yu Lu, Taiwan

Alexei Mailybaev, Brazil

Manoranjan Maiti, India

Oluwole D. Makinde, South Africa

Rafael Martinez-Guerra, Mexico

Bohdan Maslowski, Czech Republic

Driss Mehdi, France

Roderick Melnik, Canada

Xinzhu Meng, China

Yuri Vladimirovich Mikhlin, Ukraine

Gradimir V. Milovanovic, Serbia

Ebrahim Momoniat, South Africa

Trung Nguyen Thoi, Vietnam

Hung Nguyen-Xuan, Vietnam

Ben T. Nohara, Japan

Anthony Nouy, France

Sotiris K. Ntouyas, Greece

Gerard Olivar, Colombia

Claudio Padra, Argentina

Francesco Pellicano, Italy

Vu Ngoc Phat, Vietnam

A. Pogromsky, The Netherlands

Seppo Pohjolainen, Finland

Stanislav Potapenko, Canada

Sergio Preidikman, USA

Carsten Proppe, Germany

Hector Puebla, Mexico

Justo Puerto, Spain

Dane Quinn, USA

Ruben R. Garcıa, Spain

K. R. Rajagopal, USA

Gianluca Ranzi, Australia

Sivaguru Ravindran, USA

G. Rega, Italy

Pedro Ribeiro, Portugal

J. Rodellar, Spain

R. Rodriguez-Lopez, Spain

A. J. Rodriguez-Luis, Spain

Ignacio Romero, Spain

Hamid Reza Ronagh, Australia

Carla Roque, Portugal

Mohammad Salehi, Iran

Miguel A. F. Sanjuan, Spain

Ilmar Ferreira Santos, Denmark

Nickolas S. Sapidis, Greece

Bozidar Sarler, Slovenia

Andrey V. Savkin, Australia

Massimo Scalia, Italy

Mohamed A. Seddeek, Egypt

Alexander P. Seyranian, Russia

Leonid Shaikhet, Ukraine

Cheng Shao, China

Daichao Sheng, Australia

Tony Sheu, Taiwan

Zhan Shu, UK

Dan Simon, USA

Luciano Simoni, Italy

Grigori M. Sisoev, UK

Christos H. Skiadas, Greece

Davide Spinello, Canada

Sri Sridharan, USA

Rolf Stenberg, Finland

Changyin Sun, China

Jitao Sun, China

Xi-Ming Sun, China

Andrzej Swierniak, Poland

Allen Tannenbaum, USA

Cristian Toma, Romania

Irina N. Trendafilova, UK

Alberto Trevisani, Italy

Jung-Fa Tsai, Taiwan

Kuppalapalle Vajravelu, USA

Victoria Vampa, Argentina

Josep Vehi, Spain

Stefano Vidoli, Italy

Xiaojun Wang, China

Dan Wang, China

Youqing Wang, China

Yongqi Wang, Germany

Moran Wang, USA

Yijing Wang, China

Cheng C. Wang, Taiwan

Gerhard-Wilhelm Weber, Turkey

Jeroen A.S. Witteveen, USA

Kwok-Wo Wong, Hong Kong

Zheng-Guang Wu, China

Ligang Wu, China

Wang Xing-yuan, China

X. Frank XU, USA

Xuping Xu, USA

Jun-Juh Yan, Taiwan

Xing-Gang Yan, UK

Suh-Yuh Yang, Taiwan

Mahmoud T. Yassen, Egypt

Mohammad I. Younis, USA

Huang Yuan, Germany

S. P. Yung, Hong Kong

Ion Zaballa, Spain

Arturo Zavala-Rio, Mexico

Ashraf M. Zenkour, Saudi Arabia

Yingwei Zhang, USA

Xu Zhang, China

Liancun Zheng, China

Jian Guo Zhou, UK

Zexuan Zhu, China

Mustapha Zidi, France

Contents

Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites,Antonio F. Bertachini A. Prado, Maria Cecilia Zanardi, Tadashi Yokoyama,Silvia Maria Giuliatti Winter, and Josep J. MasdemontVolume 2012, Article ID 979752, 7 pages

A Numerical Study of Low-Thrust Limited Power Trajectories between Coplanar CircularOrbits in an Inverse-Square Force Field, Sandro da Silva Fernandes,Carlos Roberto Silveira Filho, and Wander Almodovar GolfettoVolume 2012, Article ID 168632, 24 pages

Application of the Hori Method in the Theory of Nonlinear Oscillations,Sandro da Silva FernandesVolume 2012, Article ID 239357, 32 pages

IMU Fault Detection Based on x2-CUSUM, Elcio Jeronimo de Oliveira, Helio Koiti Kuga,and Ijar Milagre da FonsecaVolume 2012, Article ID 740752, 15 pages

Low-Thrust Orbital Transfers in the Two-Body Problem, A. A. Sukhanov and A. F. B. A. PradoVolume 2012, Article ID 905209, 20 pages

Four-Impulsive Rendezvous Maneuvers for Spacecrafts in Circular Orbits Using GeneticAlgorithms, Denilson Paulo Souza dos Santos, Antnio Fernando Bertachini de Almeida Prado,and Guido ColasurdoVolume 2012, Article ID 493507, 16 pages

Semianalytic Integration of High-Altitude Orbits under Lunisolar Effects, Martin Lara,Juan F. San Juan, and Luis M. LopezVolume 2012, Article ID 659396, 17 pages

Error Modeling and Analysis for InSAR Spatial Baseline Determination of Satellite FormationFlying, Jia Tu, Defeng Gu, Yi Wu, and Dongyun YiVolume 2012, Article ID 140301, 23 pages

Analysis of Attitude Determination Methods Using GPS Carrier Phase Measurements,Leandro Baroni and Helio Koiti KugaVolume 2012, Article ID 596396, 10 pages

Numerical Analysis of Constrained, Time-Optimal Satellite Reorientation, Robert G. MeltonVolume 2012, Article ID 769376, 19 pages

Low-Thrust Out-of-Plane Orbital Station-Keeping Maneuvers for Satellites, Vivian M. Gomesand Antonio F. B. A. PradoVolume 2012, Article ID 532708, 14 pages

An Adaptive Remeshing Procedure for Proximity Maneuvering Spacecraft Formations,Laura Garcia-Taberner and Josep J. MasdemontVolume 2012, Article ID 429479, 14 pages

Closed Relative Trajectories for Formation Flying with Single-Input Control, Anna Guerman,Michael Ovchinnikov, Georgi Smirnov, and Sergey TrofimovVolume 2012, Article ID 967248, 20 pages

Modified Chebyshev-Picard Iteration Methods for Station-Keeping of Translunar Halo Orbits,Xiaoli Bai and John L. JunkinsVolume 2012, Article ID 926158, 18 pages

Assessment of the Ionospheric and Tropospheric Effects in Location Errors of Data CollectionPlatforms in Equatorial Region during High and Low Solar Activity Periods,

Aurea Aparecida da Silva, Wilson Yamaguti, Helio Koiti Kuga, and Claudia Celeste CelestinoVolume 2012, Article ID 734280, 15 pages

Study of Stability of Rotational Motion of Spacecraft with Canonical Variables,William Reis Silva, Maria Cecılia F. P. S. Zanardi, Regina Elaine Santos Cabette,and Jorge Kennety Silva FormigaVolume 2012, Article ID 137672, 19 pages

Analysis of Filtering Methods for Satellite Autonomous Orbit Determination Using Celestialand Geomagnetic Measurement, Xiaolin Ning, Xin Ma, Cong Peng, Wei Quan,and Jiancheng FangVolume 2012, Article ID 267875, 16 pages

The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance,Jarbas Cordeiro Sampaio, Rodolpho Vilhena de Moraes, and Sandro da Silva FernandesVolume 2012, Article ID 405870, 22 pages

Application of Periapse Maps for the Design of Trajectories Near the Smaller Primary inMulti-Body Regimes, Kathleen C. Howell, Diane C. Davis, and Amanda F. HaapalaVolume 2012, Article ID 351759, 22 pages

Unscented Kalman Filter Applied to the Spacecraft Attitude Estimation with Euler Angles,Roberta Veloso Garcia, Helio Koiti Kuga, and Maria Cecilia F. P. S. ZanardiVolume 2012, Article ID 985429, 12 pages

Optimal Two-Impulse Trajectories with Moderate Flight Time for Earth-Moon Missions,Sandro da Silva Fernandes and Cleverson Maranhao Porto MarinhoVolume 2012, Article ID 971983, 34 pages

Comparison between Two Methods to Calculate the Transition Matrix of Orbit Motion,Ana Paula Marins Chiaradia, Helio Koiti Kuga,and Antonio Fernando Bertachini de Almeida PradoVolume 2012, Article ID 768973, 12 pages

Higher-Order Analytical Attitude Propagation of an Oblate Rigid Body underGravity-Gradient Torque, Juan F. San-Juan, Luis M. Lopez, and Rosario LopezVolume 2012, Article ID 123138, 15 pages

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 979752, 7 pagesdoi:10.1155/2012/979752

EditorialMathematical Methods Applied to theCelestial Mechanics of Artificial Satellites

Antonio F. Bertachini A. Prado,1 Maria Cecilia Zanardi,2Tadashi Yokoyama,3 Silvia Maria Giuliatti Winter,2and Josep J. Masdemont4

1 INPE-DMC, Avenida dos Astronautas 1758, 12227-010 Sao Jose dos Campos, SP, Brazil2 Universidade Estadual Paulista (UNESP), Campus de Guaratingueta, CEP 12516-410 Guaratingueta,SP, Brazil

3 Universidade Estadual Paulista, Campus de Guaratingueta, Caixa Postal 178, 13500-970 Rio Claro,SP, Brazil

4 IEEC and Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647,08028 Barcelona, Spain

Correspondence should be addressed to Antonio F. Bertachini A. Prado, [email protected]

Received 9 May 2012; Accepted 9 May 2012

Copyright q 2012 Antonio F. Bertachini A. Prado et al. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Celestial mechanics is a science that comes from the more general field of astronomy. It is

devoted to the study of the motion of the planets, moons, asteroids, comets, and other celestial

bodies. Some of the main researchers involved in this field are well-known names in history,

like Johannes Kepler and Isaac Newton, who wrote the basic laws that govern these motions.

With the advances of technology, the world entered the so-called “Space Age,” where

artificial artifacts started to be built and launched into space. To perform those tasks, the

same laws used to describe the motion of the natural celestial bodies can be used to study the

motion of those manmade spacecrafts. Although it is not the beginning of the space activities,

the launch of the satellite Sputnik in 1957, by the former Soviet Union, is an important mark

of this age. This launch was then followed by the United States of America (USA) that, among

other space activities, landed a man on the Moon in 1969.

Nowadays, after those first historical events, the activities related to artificial satellites

are one of the most important fields in science and technology. It includes several important

applications of engineering that improved the life of everybody in the entire world.

Communications by satellite are well established for many years, and it is hard to imagine

a world without this capability. Another field that has been helped by the space activities is

the field related to the exploration of resources on Earth. Images from satellites can generate

important maps to find their locations, and this can make the difference for a sustainable use

of our resources. The use of location devices based on the GPS constellation is also everyday

2 Mathematical Problems in Engineering

more common. Several other similar services will be soon provided by Europe, Russia, and

China. The applications of this type of service include the monitoring of valuables under

transport and the search for the fast way to reach a destination. Artificial satellites are also

used to make weather forecast, which can contribute to the very important task of natural

disaster prediction. It is not only possible to help agricultural development by predicting the

behavior of the weather, but it is also possible to anticipate large-scale phenomenon such as

the occurrence of tsunamis. These activities is extremely important for everybody on Earth.

The “Mathematical Methods Applied to the Celestial Mechanics of Artificial Satel-

lites,” the title of the present issue of the journal, has a key role in those activities. One of the

tasks to be performed when planning a satellite mission is to find the most adequate orbit to

place this satellite. After this first step, it is also necessary to study practical problems related

to the maintenance of its nominal orbit and how much and when it will be modified by the

natural forces present in nature. These questions can be addressed by analyzing the position

of the spacecraft. There are also several questions regarding the orientation of the spacecraft.

It is necessary to know its orientation and how to control it in order to plan the space missions.

Those topics are all covered in the list of papers published in the present volume.

In that scope, this special issue is focused on the recent advances in mathematical

methods applied to the celestial mechanics of artificial satellites. It has a total of twenty two

papers briefly described below.

Eight of them are concerned with the problems of optimal maneuvers. “Low-thrustorbital transfers in the two-body problem” is written by A. A. Sukhanov and A. F. B. A. Prado.

Although several methods for optimization of multirevolution orbital transfers are available

in the literature, most of them are rather complicated, with limited usage due to a variety

of assumed hypothesis. In this paper, based on a convenient linearization near a reference

orbit, the authors propose a very simple method for low-thrust transfers (power limited).The method is free from any averaging concepts, and no limit on the number of orbits around

the attracting center nor any constraint on the thrust direction is assumed. Several additional

advantages are outlined in the paper.

“Four-impulsive rendezvous maneuvers for spacecraft in circular orbits using genetic algo-rithms” is written by D. P. S. dos Santos et al.. In this paper, the problem of rendezvous is

considered, which is necessary to transfer a spacecraft from one orbit to another, but with

the extra constraint of meeting another spacecraft when reaching the final orbit. This work

analyzes optimal rendezvous maneuvers between two coplanar circular orbits, seeking to

perform this transfer with low fuel consumption, by assuming a time-free problem and using

four burns during the process. The genetic algorithm represents a new alternative that can

be used for comparisons with the results obtained by standard procedures available in the

literature. The results show that this technique brings good results for the proposed four-

impulsive rendezvous maneuvers, when compared with the ones obtained by the traditional

impulsive method. Therefore, this approach can be used in real cases, especially when a bi-

impulsive transfer is not possible due to the limitations of the engine of the spacecraft.

Another paper concerned with the spacecraft maneuvers is “Low-thrust out-of-planeorbital station-keeping maneuvers for satellites,” written by V. M. Gomes and A. F. B. A. Prado.

This paper considers the problem of-out-of plane orbital maneuvers for station keeping of

satellites. The main idea is to consider an Earth satellite in a disturbed orbit, and it is necessary

to perform a small amplitude orbit correction to return the satellite to the nominal orbit,

to keep it performing its mission. Optimal control is used to build an algorithm to search

for solutions for the problem of minimum fuel consumption to make the required orbital

Mathematical Problems in Engineering 3

maneuvers. The problem takes into account the accuracy in the constraint’s satisfaction.

The adjustments made in the algorithm and the used parameters allow the convergence in

most of the analyzed cases. The results show that it is possible to reduce the costs by

exploring tolerable errors in the constraint’s satisfaction. It can also be observed that the

increasing of the number of propulsion arcs decreases the fuel costs. The reason for this

fact is that increasing this number causes an increase in the degree of freedom available for

the optimization technique. Since the maneuvers have small amplitudes, this increase in the

burning arcs has a limit in the savings, so the fuel consumption reaches constant values after

a certain value of the number of arcs.

“An adaptive remeshing procedure for proximity maneuvering spacecraft formations” is

written by L. Garcia-Taberner and J. J. Masdemont. This paper searches for an optimal way to

obtain reconfigurations of spacecraft formations. It uses a discretization of the time interval

and obtains local solutions by applying variational method. Applications are made for

libration point orbits, trying to find optimal meshes using an adaptive remeshing procedure

and on the determination of the parameter that governs it.

Next, “Closed relative trajectories for formation flying with single-input control” is written

by A. Guerman et al.. The authors study the problem of formation shape control under a

constrained thrust direction. It is considered a formation with two satellites and a linear

model for the relative motion (including J2) is used. It is also assumed that the satellites

are equipped with a passive attitude control system that provides one-axis stabilization.

The propulsion system consists of one or two thrusters oriented along the stabilized axis.

Based on those assumptions, the authors prove that, in both cases (passive magnetic attitude

stabilization and spin stabilization) and for all initial relative positions and velocities of the

satellites, there are controls that guaranty their periodic relative motion.

Another paper of this topic is “Modified Chebyshev-Picard iteration methods for station-keeping of translunar halo orbits,” written by X. Bai and J. L. Junkins. The authors present a new

modified Chebyshev-Picard iteration method for the station-keeping of a halo orbit around

the L2 libration point of the Earth-Moon system. Compared to other competing methods,

the authors emphasize some advantages of the present technique: it does not require weight

turning and computation of the state transition matrix so that, computationally, it is clearly

very efficient. Also, the final results are given in an orthogonal polynomial form.

Then, the paper “A numerical study of optimal low-thrust limited power trajectories betweencoplanar circular orbits in an inverse-square force field” is written by S. Fernandes et al.. Motivated

by the use of the low-thrust propulsion system in space missions, S. Fernandes et al. presented

a numerical analysis of optimal low-thrust limited power trajectories for simple transfers

between circular coplanar orbits in an inverse square force field. In this paper, two power-

limited variable ejection velocity systems are analyzed by considering the fuel consumption

as the performance criteria. The numerical results showed a good agreement with the

analytical results derived from the linear theory. Therefore, the linear theory can be used

as a good approximation for the transfer problem with small amplitudes.

Another paper of this sequence is “Optimal two-impulse trajectories with moderate flighttime for earth-moon missions,” written by S. S. Fernandes and C. M. P. Marinho. The paper

considers two-impulse trajectories with moderate flight time for Earth-Moon missions where

the goal is to optimize the fuel consumption, expressed by the total characteristic velocity.

The well-known patched-conic approximation is used as the dynamical model. After that,

two versions of the planar circular restricted three-body problem are also considered. Two

parameters are optimized in the two-body version: the initial phase angle of the space vehicle


and the first impulse. When considering the three-body dynamics, the parameters to be

optimized are four: initial phase angle of the space vehicle, the flight time, and the first and

the second impulses.

Another topic considered in this issue is related to the determination, simulation, and

propagation of trajectories of spacecraft under several different circumstances. Concerning

this topic is the paper “Analysis of filtering methods for satellite autonomous orbit determinationusing celestial and geomagnetic measurement,” written by X. Ning et al.. Orbit determination

of a satellite is a complex process which evolves the precise estimation of the ephemeris

of the satellite. In the paper by Xiaolin et al., the performance of three filters methods,

extended Kalman filter (EKF), unscented Kalman filter (UKF), and unscented particle filter

(UPF), is analyzed under different conditions. Their simulation results showed that the UPF

performs the best while the EKF gives the worst orbit determination (OD) accuracy under the

same conditions. The computation cost analysis demonstrates that the UPF-based OD system

provides the best orbit determination accuracy.

Another paper of this topic is “The orbital dynamics of synchronous satellites: irregularmotions in the 2 : 1 resonance,” written by J. C. Sampaio et al.. It analyses the dynamical behavior

of a synchronous satellite trapped in a 2 : 1 resonance with the Earth under the effects of the

zonal, J20 and J40 coefficients, and the tesseral harmonics, J22 and J42 coefficients. Through a

sample of Mathieu transformations, the order of the dynamical system was reduced. Three

critical angles associated to the 2 : 1 resonance were analyzed by numerically simulating the

reduced system. Their results showed that for several different values of the initial conditions

the system presented a chaotic motion, which was confirmed by calculating the Lyapunov

exponents.

Another one in this category is “Application of periapse maps for the design of trajectoriesnear the smaller primary in multi-body regimes,” written by K. C. Howell et al.. This paper has

the goal of incorporating multibody dynamics into the preliminary design of a mission. This

technique adds flexibility to the problem and, in some cases, generates impact in the cost

of the maneuver. Attention is given on the development and application of design tools to

help preliminary trajectory design considering a multibody environment. Using the model

given by the circular restricted three-body problem, a trajectory near the smaller primary is

studied. Periapse Poincare maps are then used to predict both short- and long-term trajectory

behaviors. Several trajectories are computed using this approach.

The next paper is entitled “Comparison between two methods to calculate the transitionmatrix of orbit motion,” written by A. P. M. Chiaradia et al.. The goal of this paper is to

compare and choose the most suited method to calculate the transition matrix used to

propagate the covariance matrix of the position and velocity of the state estimator, within

the procedure of the artificial orbit determination. Two methods are evaluated according to

accuracy, processing time, and handing complexity of the equations for both circular and

elliptical orbits. The first method is an approximation of the Keplerian motion, providing

an analytical solution which is calculated numerically by solving the Kepler equation. It

is also optimized for an elliptical orbit. The second one is a local numerical approximation

that includes the effect of the Earth oblateness; it has no singularity and no restriction about

the kind of orbit. By analyzing the results, it is possible to observe that for short periods of

time, and when more accuracy is necessary, it is recommended the use of the second method,

since the CP time does not overload excessively the computer during the orbit determination

procedure. However, for large intervals of time and when one expects more stability on the

calculations, the use of the first method is recommended.


Another paper of this category is “Semianalytic integration of high-altitude orbits underlunisolar effects,” written by M. Lara et al.. In this paper, the long-term behavior of high-

altitude orbits with lunisolar perturbations and about a noncentral Earth (including J2 and J3

potential terms) is approached in a semianalytical way using canonical perturbation theory.

The authors take advantage of the different scales of time to do the averaging in the extended

phase space. In addition, up to second-order terms in the Earth’s oblateness coefficient,

the averaging has been computed in closed form of the eccentricity, and therefore, the

semianalytical integration can be applied to an orbit of arbitrary eccentricity.

The next set of papers is related to the attitude of the satellites, including maneuvering,

determination, and propagation of attitude. A paper related to that is “Analysis of attitudedetermination methods using GPS carrier phase measurements,” written by L. Baroni and H. K.

Kuga. Global navigation satellite systems (GNSS) can be used to determine the orientation

of a platform. By using at least three GPS antennas, properly mounted on a platform, the

determination of the baseline vectors, formed between the antennas, is the prerequisite for

the attitude determination of this platform. L. Baroni and H. K. Kuga implemented and tested

two algorithms (LSAT and LAMBDA) in order to obtain an accurate attitude determination in

real time, by using data provided by the GPS receivers. As a result, they found that although

both algorithms had a similar performance, the processing time of the LSAT method was

larger. Although the inclusion of the quaternions increases the processing time, they are

necessary to avoid singularities.

Another one of this topic is “Numerical analysis of constrained, time-optimal satellitereorientation,” written by R. G. Melton. The author studies on time-optimal satellite slewing

maneuvers with one satellite axis required to obey multiple path constraints. The paper

considers four cases in which this axis is either forced to follow a finite arc of the constraint

boundary or has initial and final directions that lie on the constraint boundary. It is shown

that the precession created by the control torques, moving the axis away from the constraint

boundary, provides faster slewing maneuvers. The paper also proposes a two-stage process

for the generation of optimal on-board solutions in less time.

Another one is the paper “Study of stability of rotational motion of spacecraft with canonicalvariables,” written by W. R. Silva et al.. In this work, the main problem of the stability of

the rotational motion of a satellite is revisited. The most adequate set of variables (Andoyer

canonical set) and Kovalev-Savchenko theorem are adopted to analyze the stability of the

equilibrium points. Some important optimization on the techniques involved (determination

of the equilibrium points, normalization and application of Kovalev-Savchenko theorem) is

obtained. Examples considering real satellites with very large number of equilibrium points

are analyzed.

Then, the paper “Unscented Kalman filter applied to the spacecraft attitude estimation withEuler angles” is written by R. V. Garcia et al.. In this paper, an algorithm is used to make real-

time estimation of the attitude of an artificial satellite. Some real data obtained by the CBERS-

2 satellite (China-Brazil Earth Resources Satellite) is used. The estimator used for attitude

determination is the unscented kalman filter, which is a new alternative to the extended

Kalman filter. This algorithm is capable of carrying out estimation of the states of nonlinear

systems, and it does not require the linearization of the functions present in the model. It uses

a transformation to generate a set of vectors that, after a nonlinear transformation, preserves

the same mean and covariance of the random variables obtained before the transformation.

A comparison between the unscented Kalman filter and the extended Kalman filter is

performed using real data.


Another of this category is the paper “Higher-order analytical attitude propagation of anoblate rigid body under gravity-gradient torque,” written by J. F. San-Juan et al.. This paper is

related to the rotation of an oblate rigid body under gravity-gradient torque, a nonintegrable

problem that may be analyzed through the perturbation theory. The Andoyer variables, used

to describe the rotational motion, simplify the solution of the partial differential equations

that provide the generating function of the Lie transforms. It is shown that when a higher-

order perturbation theory is used to study the rotation of a uniaxial satellite under gravity-

gradient torque, this shows that known special configurations of the attitude dynamics at

which the satellite rotates, on average, as in a torque-free state, are only the result of an early

truncation of the secular frequencies of motion. The higher order of the analytical solution

can be compared to the long-term numerical integration of the equations of motion.

Other four papers are related to specific applications, including error analysis of

space missions, locations of platforms, and solutions of mathematical equations. One of

this type is the paper “Error modeling and analysis for InSAR spatial baseline determination ofsatellite formation flying,” written by J. Tu et al.. Close formation flying satellites equipped

with synthetic aperture radar (SAR) antenna have become the focus of space technology.

The analysis of the errors introduced by spatial baseline measurement is important for the

performance of the SAR satellite system. J. Tu et al. carefully analyzed these errors and

classified them into two groups: the errors related to the baseline transformation and

those related to the GPS measurements. Through a set of simulations, they verified that

the errors related to the GPS measurements are the principal influence on the space baseline

determination, while the carrier phase noise of the GPS observation and fixing error of the

GPS receiver antenna are the main factors of the errors related to the GPS measurements.

Another one is “IMU fault detection based on χ2-CUSUM,” written by E. J. de Oliveira et

al.. Combining chi-square cumulative sum and media filter techniques, an efficient algorithm

is presented for the problem involving fault detection and isolation on inertial measurement

units. In particular, the success and feasibility of the method is clearly demonstrated in the

case of low-level step fault.

The paper “Assessment of the ionospheric and tropospheric effects in location errors of datacollection platforms in equatorial region during high and low solar activity periods,” written by

A. A. da Silva et al., provides an assessment of ionospheric effects using IRI (international

reference ionosphere) and IONEX (IONosphere map exchange) models and tropospheric

delay compensation using climatic data provided by the National Climatic Data Center—

NOAA Satellite and Information Service. Two selected DCPs (Data Collection System) are

used in conjunction with the Brazilian Data Collection Satellite (SCD2) during high- and

low-solar activity periods. Results show that the ionospheric effects on transmission delays

are significant in equatorial region and should be considered to reduce the DCP location

errors, mainly in high solar activity periods. It is also shown that the platform location errors

can be reduced when the ionospheric and the tropospheric effects are properly considered.

Another is the paper “Application of the Hori method in the theory of nonlinear oscillations,”written by S. S. Fernandes. In this paper, the Hori method is applied to the theory of nonlinear

oscillations. Two algorithms to determine and generate functions and the new system of

differential equations are derived, based on a more general algorithm proposed by Sessin

(1983). The solutions are not uniquely determined because the algorithms involve arbitrary

functions. Different choices of them can be made to simplify the new system of differential

equations and to define near-identity transformations. These algorithms are then applied to

determine second-order asymptotic solutions of two well-known equations in the theory of

nonlinear oscillations: van der Pol and Duffing equations.


Acknowledgments

The Guest Editors would like to thanks all the reviewers, the authors, and all the staff of this

journal involved in the preparation of this special issue for the opportunity to publish the

papers related to this topic.

Antonio F. BertachiniA. Prado

Maria Cecilia ZanardiTadashi Yokoyama

Silvia Maria Giuliatti WinterJoseph J. Masdemont


Research ArticleA Numerical Study of Low-Thrust Limited PowerTrajectories between Coplanar Circular Orbits inan Inverse-Square Force Field

Sandro da Silva Fernandes,1 Carlos Roberto Silveira Filho,2and Wander Almodovar Golfetto3

1 Departamento de Matematica, Instituto Tecnologico de Aeronautica, 12228-900 Sao Jose dos Campos, SP,Brazil

2 EMBRAER S. A., Divisao de Ensaio em Voo, 12227-901 Sao Jose dos Campos, SP, Brazil3 Subdepartamento Tecnico do Departamento de Ciencia e Tecnologia Aeroespacial, 12228-900 Sao Jose dosCampos, SP, Brazil

Correspondence should be addressed to Sandro da Silva Fernandes, [email protected]

Received 15 November 2011; Accepted 19 January 2012

Academic Editor: Silvia Maria Giuliatti Winter

Copyright q 2012 Sandro da Silva Fernandes et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A numerical study of optimal low-thrust limited power trajectories for simple transfer (norendezvous) between circular coplanar orbits in an inverse-square force field is performed bytwo different classes of algorithms in optimization of trajectories. This study is carried out bymeans of a direct method based on gradient techniques and by an indirect method based on thesecond variation theory. The direct approach of the trajectory optimization problem combines themain positive characteristics of two well-known direct methods in optimization of trajectories:the steepest-descent (first-order gradient) method and a direct second variation (second-ordergradient) method. On the other hand, the indirect approach of the trajectory optimizationproblem involves two different algorithms of the well-known neighboring extremals method.Several radius ratios and transfer durations are considered, and the fuel consumption is takenas the performance criterion. For small-amplitude transfers, the results are compared to the onesprovided by a linear analytical theory.

1. Introduction

The main purpose of this work is to present a numerical study of optimal low-thrust limited

power trajectories for simple transfers (no rendezvous) between circular coplanar orbits in an

inverse-square force field. This study has been motivated by the renewed interest in the use

of low-thrust propulsion systems in space missions verified in the last two decades due to the

development and the successes of space missions powered by ionic propulsion; for instance,


Deep Space One and SMART 1 missions. Several researchers have obtained numerical and

sometimes analytical solutions for a number of specific initial orbits and specific thrust

profiles [1–6]. Averaging methods are also used in such researches [7–11].Two idealized propulsion models have most frequently been used in the analysis of

optimal space trajectories [12]: the limited power variable ejection velocity systems—LP

systems—are characterized by a constraint concerning with the power (there exists an upper

constant limit for the power), and the constant ejection velocity-limited thrust systems—

CEV systems—are characterized by a constraint concerning with the magnitude of the thrust

acceleration which is bounded. In both cases, it is usually assumed that the thrust direction

is unconstrained. The utility of these idealized models is that the results obtained from them

provide good insight into more realistic problems.

In the study presented in this paper only LP systems are considered. The fuel con-

sumption is taken as the performance criterion and it is calculated for various radius ratios

ρ = rf/r0, where r0 is the radius of the initial circular orbit O0, and rf is the radius of the

final circular orbit Of and for various time of flight tf − t0. The optimization problem associ-

ated to the space transfer problem is formulated as a Mayer problem of optimal control with

Cartesian elements—components of position and velocity vectors—as state variables. Trans-

fers with small, moderate, and large-amplitudes are studied, and the numerical results are

compared to the results provided by a linear theory given in terms of orbital elements [12–

15].Two different classes of algorithms are applied in determining the optimal trajectories.

They are computed through a direct approach of the trajectory optimization problem based

on gradient techniques, and through an indirect approach based on the solution of the two-

point boundary value problem obtained from the set of necessary conditions for optimality.

The direct approach involves a gradient-based algorithm which combines the main

positive characteristics of the steepest-descent (first-order gradient) method and of a direct

method based upon the second variation theory (second-order gradient method). This

algorithm has two distinct phases: in the first one, it uses a simplified version of the steepest-

descent method developed for a Mayer problem of optimal control with free final state

and fixed terminal times, in order to get great improvements of the performance index in

the first iterates with satisfactory accuracy. In the second phase, the algorithm switches to

a direct method based upon the second variation theory developed for a Bolza problem

with fixed terminal times and constrained initial and final states, in order to improve the

convergence as the optimal solution is approached. This kind of algorithm for determining

optimal trajectories is well known in the literature [16], and the version used in this paper is

quite simple, since it uses a simplified version of the steepest-descent method, as mentioned

before, with terminal constraints added to the performance index by using a penalty function

method (see Section 2.2). This procedure simplifies the algorithm, providing a solution with

satisfactory accuracy, and can avoid some of typical divergence troubles of the classical

steepest-descent method as discussed in McDermott and Fowler [17].The indirect approach involves the solution of the two-point boundary value problem

through two different algorithms of the neighboring extremals method. The formulation of

the neighboring extremals method, as presented herein, is associated with a Bolza optimal

control problem with fixed initial and final times, fixed initial state and constrained final

state [18, 19]. Basically, the method consists in iteratively determining the initial values of the

adjoint variables and the Lagrange multipliers associated to the final constraints. It involves

the linearization, about an extremal solution, of the nonlinear two-point boundary value


problem defined by the application of Pontryagin Maximum Principle [20] to the optimi-

zation problem. The linearized problem has been solved through the state transition matrix,

and through the generalized Riccati transformation [16, 27]. The algorithms based on the

state transition matrix and the Riccati transformation are well known in the literature, and

the version used in this paper has a slight modification as described in da Silva Fernandes

and Golfetto [15].A brief description of the versions of the algorithms used in this paper can be found in

da Silva Fernandes [21]. Finally, note that the results presented herein complete and extend

the results previously obtained [15, 21, 22].

2. Optimal Low-Thrust Limited Power Trajectories

In this section, the optimization problem concerning with optimal low-thrust limited power

trajectories is formulated. Application of each one of the proposed algorithms is also present-

ed. For completeness, a very brief description of the linear theory is included.

2.1. Formulation of the Optimization Problem

Low-thrust limited power propulsion systems are characterized by low-thrust acceleration

level and a high specific impulse [12]. The ratio between the maximum thrust acceleration

and the gravity acceleration on the ground, γmax/g0, is between 10−4 and 10−2. For such

system, the fuel consumption is described by the variable J defined as

J =1

2

∫ tt0

γ2dt, (2.1)

where γ is the magnitude of the thrust acceleration vector γ , used as control variable. The

consumption variable J is a monotonic decreasing function of the instantaneous mass m of

the space vehicle:

J = Pmax

(1

m− 1

m0

), (2.2)

where Pmax is the maximum power, and m0 is the initial mass. The minimization of the final

value Jf is equivalent to the maximization ofmf or the minimization of the fuel consumption.

The optimization problem concerning with simple transfers (no rendezvous) between

coplanar orbits is formulated as: at time t, the state of a space vehicle M is defined by the

radial distance r from the center of attraction, the radial and circumferential components of


the velocity, u and v, and the fuel consumption J . In the two-dimensional formulation, the

state equations are given by [23]:

du

dt=v2

r− μ

r2+ R,

dv

dt= −uv

r+ S,

dr

dt= u,

dJ

dt=

1

2

(R2 + S2

),

(2.3)

where μ is the gravitational parameter,R and S are the radial and circumferential components

of the thrust acceleration vector, respectively. The optimization problem is stated as: it is

proposed to transfer a space vehicle M from the initial state at the time t0 = 0:

u(0) = 0 v(0) = 1 r(0) = 1 J(0) = 0, (2.4)

to the final state at the prescribed final time tf :

u(tf)= 0 v

(tf)=

√μ

rfr(tf)= rf , (2.5)

such that Jf is a minimum, that is, the performance index is defined by:

IP = J(tf). (2.6)

Equations (2.4) and (2.5) are given in canonical units, and they define the initial and final

circular orbits. u(tf), v(tf), and r(tf) denote the state variables at the prescribed final time

tf , and 0,√μ/rf , and rf are the prescribed values defining the final circular orbit. Similar

definition applies at the initial time t0 = 0 (see (2.4)). For LP system, it is assumed that there

are no constraints on the thrust acceleration vector [12].In the formulation of the optimization problem described above, the variables are

written in canonical units, such that the gravitational parameter μ is equal to 1.

2.2. Application of the Gradient-Based Algorithm

As described in da Silva Fernandes [21], the first phase of the gradient-based algorithm

involves a simplified version of the steepest-descent method, which has been developed for

a Mayer problem of optimal control with free final state and fixed terminal times. So, the

optimal control problem defined by (2.3)–(2.6) must be transformed into a new optimization

problem with final state completely free. In order to do this, the penalty function method


[24, 25] is applied. The new optimal control problem is then defined by (2.3) and (2.4), with

the new performance index obtained from (2.5) and (2.6):

IP = J(tf)+ k1

(u(tf))2 + k2

(v(tf)− 1√rf

)2

+ k3

(r(tf)− rf)2, (2.7)

where k1, k2, k3 � 1. The penalty function method involves the progressive increase of the

penalty constants; but, for simplicity, they are taken fixed in the gradient-based algorithm,

since the steepest-descent is used only to provide a convex nominal solution as starting

solution for the second order gradient method.

According to the algorithm of the simplified version of the steepest-descent method,

the adjoint variables λu, λv, λr , and λJ are introduced, and the HamiltonianH is formed using

(2.3) [21, 26]:

H = λu

(v2

r− μ

r2+ R

)+ λv(−uvr

+ S)+ λru +

1

2λJ(R2 + S2

). (2.8)

From the Hamiltonian H, one finds the adjoint equations:

dλudt

=v

rλv − λr,

dλvdt

= −2v

rλu +

u

rλv,

dλrdt

=

(v2

r2− 2

μ

r3

)λu −

uv

r2λv,

dλJ

dt= 0,

(2.9)

and, from the performance index defined by (2.7), we get the terminal conditions for the

adjoint equations:

λu(tf)= −2k1u

(tf),

λv(tf)= −2k2

(v(tf)− 1√rf

),

λr(tf)= −2k3

(r(tf)− rf),

λJ(tf)= −1.

(2.10)


The algorithm also requires the partial derivatives of the Hamiltonian H with respect

to the control variables. These partial derivatives are given by:

∂H

∂R= λu + RλJ,

∂H

∂S= λv + SλJ. (2.11)

The second phase of the gradient-based algorithm involves the second order gradient

method, developed for a Bolza problem with fixed terminal times and constrained initial

and final states, which requires the computation of the first order derivatives of the vector

function ψ containing the terminal constraints and the scalar function Φ, corresponding to

the augmented performance index, and the computation of the second order derivatives of

the Hamiltonian H with respect to all arguments. The partial derivatives of the Hamiltonian

function are given in a matrix form by:

Hαα =

[λJ 0

0 λJ

],

Hλα =

⎡⎢⎢⎢⎢⎢⎣1 0

0 1

0 0

R S

⎤⎥⎥⎥⎥⎥⎦,

Hxα = 0(4 × 2 null matrix),

Hλx =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 2v

r−v

2

r2+ 2

μ

r30

−vr

−ur

uv

r20

1 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,

Hxx =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 −λvr

v

r2λv 0

−λvr

2λur

−2v

r2λu +

u

r2λv 0

λvv

r2−2

v

r2λu +

u

r2λv

(2v2

r3− 6

μ

r4

)λu − 2

uv

r3λv 0

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

(2.12)

α denotes the control vector αT = [R S], and it has been introduced to avoid confusion with

the state variable u, x is the state vector xT = [u v r J], and λ is the adjoint vector λT =[λu λv λr λJ].


From (2.5) and (2.6), one defines the functions ψ and Φ:

ψ =

⎡⎢⎢⎢⎣u(tf)

v(tf)− 1√rf

r(tf)− rf

⎤⎥⎥⎥⎦, (2.13)

Φ = J(tf)+ Λ1u

(tf)+ Λ2

(v(tf)− 1√rf

)+ Λ3

(r(tf)− rf), (2.14)

where Λi, i = 1, 2, 3 are Lagrangian multipliers associated to the final constraints defined by

(2.13). The partial derivatives of ψ and Φ are then given by:

ψx =

⎡⎢⎢⎣1 0 0 0

0 1 0 0

0 0 1 0

⎤⎥⎥⎦,Φxx = 0. (4 × 4 null matrix).

(2.15)

The results of the gradient-based algorithm to the optimization problem described

above are presented in Section 3.

2.3. Application of the Neighboring Extremals Algorithms

Let us to consider the Hamiltonian function defined by (2.8). Following the Pontryagin Max-

imum Principle [20], the control variables R and S must select from the admissible controls

such that the Hamiltonian function reaches its maximum along the optimal trajectory. Thus,

R∗ = −λuλJ, S∗ = −λv

λJ. (2.16)

The adjoint variables λu, λv, λr , and λJ must satisfy the adjoint differential equations

and the transversality conditions. Therefore, from (2.3)–(2.5) and (2.16) one finds the

following two-point boundary value problem for the transfer problem defined by (2.3)–(2.6):

du

dt=v2

r− μ

r2− λuλJ,

dv

dt= −uv

r− λvλJ,


dr

dt= u,

dJ

dt=

1

2λ2J

(λ2u + λ

2v

),

dλudt

=v

rλv − λr,

dλvdt

= −2v

rλu +

u

rλv,

dλrdt

=

(v2

r2− 2

μ

r3

)λu −

uv

r2λv,

dλJ

dt= 0,

(2.17)

with the boundary conditions:

u(0) = 0,

v(0) = 1,

r(0) = 1,

J(0) = 0,

u(tf)= 0,

v(tf)=

√μ

rf,

r(tf)= rf ,

λJ(tf)= −1.

(2.18)

The neighboring extremals algorithms are based on the solution of a linearized two-

point boundary value problem that involves the derivatives of the right-hand side of (2.3)with respect to the state and adjoint variables [16, 21, 27]. These equations can be put in the

following form:

dδx

dt= Aδx + Bδλ,

dδλ

dt= Cδx −ATδλ, (2.19)


with δx(t) = xn+1(t) − xn(t) and δλ(t) = λn+1(t) − λn(t), where n denotes the iterate, and A, B,

and C are matrices given by:

A =

⎡⎢⎢⎢⎢⎢⎢⎣0

2v

r−a 0

−vr

−ur

uv

r20

1 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− 1

λJ0 0

λu

λ2J

0 − 1

λJ0

λv

λ2J

0 0 0 0

λu

λ2J

λv

λ2J

0 − c

λ3J

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0λvr

−vλvr2

0

λvr

−2λur

b 0

−vλvr2

b d 0

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,

(2.20)

where

a =v2

r2− 2

μ

r3,

b =2v

r2λu −

u

r2λv,

c = λ2u + λ

2v,

d =

(−2v2

r3+

6μ

r4

)λu +

2uv

r3λv.

(2.21)

The results of the neighboring extremals algorithms to the optimization problem de-

scribed above are presented in Section 3.

2.4. Linear Theory

For completeness, a very brief description of a first-order analytical solution for the problem

of optimal simple transfer (no rendezvous) between close quasicircular coplanar orbits in


an inverse-square force field is presented. This approximate solution, also referred as linear

theory, is expressed in nonsingular orbit elements, and it is valid for orbits with very small

eccentricities. According to Marec [12] or da Silva Fernandes and Golfetto [15], for transfers

between circular orbits, only Δα is imposed, and assuming that the initial and final positions

of the vehicle in orbit are symmetric with respect to x-axis of the inertial reference system,

that is, f = −0 = Δ/2, the linear solution can be written as:

u = na(h sin − k cos

),

v = na(

1 + h cos + k sin ),

r =a

1 + h cos + k sin ,

J(t) =1

2

√√√a5

μ3

{4( − 0

)λ2α + 8

(sin − sin 0

)λαλh − 8

(cos − cos 0

)λαλk

+[

5

2

( − 0

)+

3

4

(sin 2 − sin 20

)]λ2h −

3

2

(cos 2 − cos 20

)λhλk

+[

5

2

( − 0

)− 3

4

(sin 2 − sin 20

)]λ2k

},

(2.22)

with

α(t) = α0 + 4

√√√a5

μ3

{( − 0

)λα +

(sin − sin 0

)λh −

(cos − cos 0

)λk}, (2.23)

h(t) = h0 +

√√√a5

μ3

{4(

sin − sin 0

)λα +

[5

2

( − 0

)+

3

4

(sin 2 − sin 20

)]λh

−3

4

(cos 2 − cos 20

)λk

},

(2.24)

k(t) = k0 +

√√√a5

μ3

{−4(

cos − cos 0

)λα −

3

4

(cos 2 − cos 20

)λh

+[

5

2

( − 0

)− 3

4

(sin 2 − sin 20

)]λk

},

(2.25)

λα =1

2

√μ3

a5

⎧⎪⎨⎪⎩Δα(

5Δ + 3 sinΔ)

10Δ2+ 6Δ sinΔ − 64 sin2

(Δ/2

)⎫⎪⎬⎪⎭, (2.26)


λh = −√μ3

a5

⎧⎪⎨⎪⎩ 8Δα sinΔ/2

10Δ2+ 6Δ sinΔ − 64 sin2

(Δ/2

)⎫⎪⎬⎪⎭, (2.27)

λk = 0, (2.28)

where α = a/a, h = e cosω, k = e sinω, where a is the semimajor axis, e is the eccentricity, ω

is the argument of the pericenter, = 0 + n(t − t0), and n =√μ/a3 is the mean motion. The

overbar denotes the reference orbit O about which the linearization is done.

The optimal thrust acceleration Γ∗ during the maneuver is expressed by:

Γ∗ =1

na

{(λh sin − λk cos

)er + 2

(λα + λh cos + λk sin

)es}, (2.29)

where er and es are unit vectors extending along radial and circumferential directions in a

moving reference frame, respectively.

The linear theory is applicable only for orbits which are not separated by large radial

distance. If the reference orbit is chosen in the conventional way, that is, with the semimajor

axis as the radius of the initial orbit, the radial excursion to the final orbit will be maximized

[14]. A better reference orbit is defined with a semimajor axis given by an intermediate value

between the values of semimajor axes of the terminal orbits. In this study, a is taken as a =(a0 + af)/2 in order to improve the accuracy in the calculations.

In the next section, the results of this linear theory are compared to the ones provided

by the proposed algorithms.

3. Results

The results of a numerical analysis for optimal low-thrust limited power simple transfers

(no rendezvous) between coplanar circular orbits in an inverse-square force field, obtained

through the analytical and numerical methods described in the preceding sections, are

presented for various radius ratios ρ = rf/r0 and for various time of flight tf − t0 presented in

Tables 1–8. All results are presented in canonical units as described in Section 2. A preliminary

analysis of some interplanetary missions considering transfers from Earth to Venus, Mars,

asteroid belt, Jupiter, and Saturn, which correspond to ρ = 0.727, 1.523, 2.500, 5.203, and

9.519, respectively, is presented. In this preliminary analysis of interplanetary missions, the

following assumptions are considered:

(1) the orbits of the planets are circular;

(2) the orbits of the planets lie in the plane of the ecliptic;

(3) the flight of the space vehicle takes place in the plane of the ecliptic;

(4) only the heliocentric phase is considered, that is, the attraction of planets on the

spacecraft is neglected.


Table 1: Consumption variable J(ρ > 1) for transfers with small time of flight.

ρ tf − t0 Janal Jgrad JNeigh1 JNeigh2 drel 1 drel 2 drel 3

1.0250

2.03.04.05.0

3.5856 × 10−4

8.4459 × 10−5

3.1226 × 10−5

1.7138 × 10−5

3.5855 × 10−4

8.4462 × 10−5

3.1233 × 10−5

1.7147 × 10−5

3.5854 × 10−4

8.4456 × 10−5

3.1230 × 10−5

1.7143 × 10−5

3.5854 × 10−4

8.4456 × 10−5

3.1230 × 10−5

1.7143 × 10−5

0.000.000.010.03

0.000.010.010.02

0.000.000.000.00

1.0500

2.03.04.05.0

1.4463 × 10−3

3.4169 × 10−4

1.2533 × 10−4

6.7541 × 10−5

1.4463 × 10−3

3.4166 × 10−4

1.2538 × 10−4

6.7611 × 10−5

1.4459 × 10−3

3.4164 × 10−4

1.2537 × 10−4

6.7598 × 10−5

1.4459 × 10−3

3.4164 × 10−4

1.2537 × 10−4

6.7598 × 10−5

0.030.010.030.08

0.030.010.000.02

0.000.000.000.00

1.1000

2.03.04.05.0

5.8778 × 10−3

1.3977 × 10−3

5.0619 × 10−4

2.6374 × 10−4

5.8741 × 10−3

1.3970 × 10−3

5.0666 × 10−4

2.6453 × 10−4

5.8716 × 10−3

1.3969 × 10−3

5.0664 × 10−4

2.6451 × 10−4

5.8716 × 10−3

1.3969 × 10−3

5.0664 × 10−4

2.6451 × 10−4

0.110.060.090.29

0.040.000.000.01

0.000.000.000.00

1.2000

2.03.04.05.0

2.4187 × 10−2

5.8370 × 10−3

2.0813 × 10−3

1.0260 × 10−3

2.4097 × 10−2

5.8200 × 10−3

2.0845 × 10−3

1.0346 × 10−3

2.4097 × 10−2

5.8199 × 10−3

2.0844 × 10−3

1.0345 × 10−3

2.4097 × 10−2

5.8199 × 10−3

2.0844 × 10−3

1.0345 × 10−3

0.370.290.150.82

0.000.000.000.00

0.000.000.000.00

Table 2: Consumption variable J(ρ < 1) for transfers with small time of flight.

ρ tf − t0 Jlinear Jgrad JNeigh1 JNeigh2 drel 1 drel 2 drel 3

0.8000

2.03.04.05.0

2.0951 × 10−2

4.9040 × 10−3

2.0703 × 10−3

1.3838 × 10−3

2.0842 × 10−2

4.9173 × 10−3

2.1047 × 10−3

1.4198 × 10−3

2.0842 × 10−2

4.9172 × 10−3

2.1046 × 10−3

1.4197 × 10−3

2.0842 × 10−2

4.9172 × 10−3

2.1046 × 10−3

1.4197 × 10−3

0.520.271.632.53

0.000.000.000.00

0.000.000.000.00

0.9000

2.03.04.05.0

5.4740 × 10−3

1.2771 × 10−3

5.0063 × 10−4

3.0496 × 10−4

5.4672 × 10−3

1.2772 × 10−3

5.0198 × 10−4

3.0653 × 10−4

5.4671 × 10−3

1.2771 × 10−3

5.0198 × 10−4

3.0652 × 10−4

5.4671 × 10−3

1.2771 × 10−3

5.0198 × 10−4

3.0652 × 10−4

0.130.000.270.51

0.000.010.000.00

0.000.000.000.00

0.9500

2.03.04.05.0

1.3958 × 10−3

3.2649 × 10−4

1.2451 × 10−4

7.2585 × 10−5

1.3955 × 10−3

3.2649 × 10−4

1.2459 × 10−4

7.2671 × 10−5

1.3955 × 10−3

3.2647 × 10−4

1.2458 × 10−4

7.2667 × 10−5

1.3955 × 10−3

3.2647 × 10−4

1.2458 × 10−4

7.2667 × 10−5

0.020.010.060.11

0.000.010.010.01

0.000.000.000.00

0.9750

2.03.04.05.0

3.5225 × 10−4

8.2555 × 10−5

3.1120 × 10−5

1.7765 × 10−5

3.5231 × 10−4

8.2560 × 10−5

3.1126 × 10−5

1.7772 × 10−5

3.5223 × 10−4

8.2554 × 10−5

3.1124 × 10−5

1.7771 × 10−5

3.5223 × 10−4

8.2553 × 10−5

3.1124 × 10−5

1.7771 × 10−5

0.010.000.010.03

0.020.010.010.00

0.000.000.000.00

Tables 1–4 show the values of the consumption variable J for small-amplitude trans-

fers computed through the different approaches and the absolute relative difference in per-

cent between the numerical and analytical results, according to the following definition:

drel 1 =

∣∣∣∣∣(Jneigh1 − Jlinear

)Jneigh1

∣∣∣∣∣ × 100%,

drel 2 =

∣∣∣∣∣(Jneigh1 − Jgrad

)Jneigh1

∣∣∣∣∣ × 100%,

drel 3 =

∣∣∣∣∣(Jneigh1 − Jneigh2

)Jneigh1

∣∣∣∣∣ × 100%.

(3.1)


Table 3: Consumption variable J(ρ > 1) for transfers with moderate time of flight.


1.0250

20.030.040.050.0

3.7722 × 10−6

2.5221 × 10−6

1.8855 × 10−6

1.5066 × 10−6

3.7733 × 10−6

2.5226 × 10−6

1.8859 × 10−6

1.5072 × 10−6

3.7730 × 10−6

2.5230 × 10−6

1.8860 × 10−6

1.5070 × 10−6

3.7714 × 10−6

2.5209 × 10−6

1.8841 × 10−6

1.5049 × 10−6

0.020.040.030.03

0.010.020.000.01

0.040.080.100.14

1.0500

20.030.040.050.0

1.4520 × 10−5

9.7411 × 10−6

7.2599 × 10−6

5.8158 × 10−6

1.4536 × 10−5

9.7482 × 10−6

7.2659 × 10−6

5.8199 × 10−6

1.4533 × 10−5

9.7480 × 10−6

7.2660 × 10−6

5.8200 × 10−6

1.4531 × 10−5

9.7460 × 10−6

7.2636 × 10−6

5.8182 × 10−6

0.090.070.080.07

0.020.000.000.00

0.010.020.030.03

1.1000

20.030.040.050.0

5.4007 × 10−5

3.6278 × 10−5

2.7003 × 10−5

2.1653 × 10−5

5.4168 × 10−5

3.6390 × 10−5

2.7083 × 10−5

2.1719 × 10−5

5.4167 × 10−5

3.6389 × 10−5

2.7078 × 10−5

2.1718 × 10−5

5.4165 × 10−5

3.6387 × 10−5

2.7077 × 10−5

2.1716 × 10−5

0.300.300.270.30

0.000.000.020.00

0.000.010.000.01

1.2000

20.030.040.050.0

1.8980 × 10−4

1.2543 × 10−4

9.4416 × 10−5

7.5157 × 10−5

1.9172 × 10−4

1.2695 × 10−4

9.5396 × 10−5

7.5976 × 10−5

1.9154 × 10−4

1.2693 × 10−4

9.5391 × 10−5

7.5928 × 10−5

1.9154 × 10−4

1.2693 × 10−4

9.5390 × 10−5

7.5927 × 10−5

0.911.181.021.02

0.090.020.010.06

0.000.000.000.00

Table 4: Consumption variable J(ρ < 1) for transfers with moderate time of flight.


0.8000

20.030.040.050.0

3.4529 × 10−4

2.2973 × 10−4

1.7196 × 10−4

1.3736 × 10−4

3.5015 × 10−4

2.3313 × 10−4

1.7467 × 10−4

1.3978 × 10−4

3.5011 × 10−4

2.3316 × 10−4

1.7465 × 10−4

1.3959 × 10−4

3.5010 × 10−4

2.3311 × 10−4

1.7465 × 10−4

1.3959 × 10−4

1.371.471.541.59

0.010.010.010.13

0.000.020.000.00

0.9000

20.030.040.050.0

7.3862 × 10−5

4.8663 × 10−5

3.6467 × 10−5

2.9218 × 10−5

7.4146 × 10−5

4.8851 × 10−5

3.6588 × 10−5

2.9317 × 10−5

7.4146 × 10−5

4.8852 × 10−5

3.6589 × 10−5

2.9316 × 10−5

7.4144 × 10−5

4.8850 × 10−5

3.6587 × 10−5

2.9314 × 10−5

0.380.390.330.33

0.000.000.000.00

0.000.000.000.01

0.9500

20.030.040.050.0

1.7023 × 10−5

1.1240 × 10−5

8.4569 × 10−6

6.7519 × 10−6

1.7042 × 10−5

1.1251 × 10−5

8.4642 × 10−6

6.7581 × 10−6

1.7040 × 10−5

1.1249 × 10−5

8.4640 × 10−6

6.7580 × 10−6

1.7038 × 10−5

1.1247 × 10−5

8.4620 × 10−6

6.7561 × 10−6

0.100.080.080.09

0.010.020.000.00

0.010.020.020.03

0.9750

20.030.040.050.0

4.0858 × 10−6

2.7075 × 10−6

2.0361 × 10−6

1.6230 × 10−6

4.0869 × 10−6

2.7081 × 10−6

2.0366 × 10−6

1.6234 × 10−6

4.0870 × 10−6

2.7080 × 10−6

2.0360 × 10−6

1.6230 × 10−6

4.0847 × 10−6

2.7059 × 10−6

2.0349 × 10−6

1.6216 × 10−6

0.030.020.000.00

0.000.000.030.02

0.060.080.050.09

Table 5: Consumption variable J for Earth-Venus transfers.


0.7270

2.03.04.05.0

3.7654 × 10−2

8.9269 × 10−3

4.0482 × 10−3

2.8941 × 10−3

3.7299 × 10−2

9.0261 × 10−3

4.2133 × 10−3

3.0573 × 10−3

3.7298 × 10−2

9.0259 × 10−3

4.2131 × 10−3

3.0572 × 10−3

3.7298 × 10−2

9.0259 × 10−3

4.2131 × 10−3

3.0572 × 10−3

0.951.103.915.33

0.000.000.000.00

0.000.000.000.00

20.030.040.050.0

7.2355 × 10−4

4.8236 × 10−4

3.6177 × 10−4

2.8942 × 10−4

7.4857 × 10−4

4.9863 × 10−4

3.7385 × 10−4

2.9903 × 10−4

7.4856 × 10−4

4.9862 × 10−4

3.7384 × 10−4

2.9901 × 10−4

7.4856 × 10−4

4.9862 × 10−4

3.7383 × 10−4

2.9897 × 10−4

3.343.263.223.20

0.000.000.000.01

0.000.000.000.01


Table 6: Consumption variable J for Earth-Mars transfers.


1.5236

2.03.04.05.0

1.7743 × 10−1

4.4947 × 10−2

1.6051 × 10−2

7.2498 × 10−3

1.7434 × 10−1

4.4067 × 10−2

1.5889 × 10−2

7.3352 × 10−3

1.7434 × 10−1

4.4066 × 10−2

1.5889 × 10−2

7.3351 × 10−3

1.7434 × 10−1

4.4066 × 10−2

1.5889 × 10−2

7.3351 × 10−3

1.771.991.021.16

0.000.000.000.00

0.000.000.000.00

20.030.040.050.0

8.6591 × 10−4

5.7537 × 10−4

4.2991 × 10−4

3.4273 × 10−4

9.3232 × 10−4

6.1074 × 10−4

4.5311 × 10−4

3.6096 × 10−4

9.3151 × 10−4

6.1071 × 10−4

4.5296 × 10−4

3.6093 × 10−4

9.3158 × 10−4

6.1073 × 10−4

4.5299 × 10−4

3.6095 × 10−4

7.025.795.095.04

0.090.000.030.01

0.010.000.010.01

Table 7: Consumption variable J for interplanetary transfers with large-amplitude.

ρ tf − t0 JNeigh1 JNeigh2

2.500

20.030.040.050.060.0

3.7736 × 10−3

2.3900 × 10−3

1.7541 × 10−3

1.3859 × 10−3

1.1454 × 10−3

3.7733 × 10−3

2.3900 × 10−3

1.7541 × 10−3

1.3859 × 10−3

1.1453 × 10−3

5.203

20.030.040.050.060.0

1.3746 × 10−2

7.5307 × 10−3

5.0533 × 10−3

3.7103 × 10−3

2.9897 × 10−3

1.3746 × 10−2

7.5309 × 10−3

5.0533 × 10−3

3.7100 × 10−3

2.9896 × 10−3

9.519

60.070.080.090.0100.0

5.9485 × 10−3

4.6970 × 10−3

3.9003 × 10−3

3.3295 × 10−3

2.8858 × 10−3

5.9392 × 10−3

4.6980 × 10−3

3.9009 × 10−3

3.3285 × 10−3

2.8857 × 10−3

The results provided by the neighboring extremals algorithm based on the state transition

matrix (denoted by number 1) have been chosen as the exact solution for each maneuver, in

view of the accuracy obtained in fulfillment of the terminal constraints.

Similar results for interplanetary transfers are presented in Tables 5, 6 and 7. Results

for large-amplitude transfers with long time of flight are presented in Table 8. In both cases,

the transfers are only computed through the neighboring extremals algorithms.

From the results presented in Tables 1–8, major comments are as follows:

(1) The linear theory provides a very good approximation for the fuel consumption

considering small-amplitude transfers with |ρ − 1| ≤ 0.100, that is, for transfers

between close circular coplanar orbits. For the most of the maneuvers, drel 1 < 0.5%;

(2) For transfers with small time of flight (tf − t0 = 2.0, 3.0, 4.0, 5.0 time units), Tables

1, 2, 5 and 6 show that the maximum absolute relative difference drel 1 occur for the

most of the transfers with tf − t0 = 5. This maximum value of drel 1 is about 2% for

ρ > 1 and 5.5% for ρ < 1;

(3) For transfers with moderate time of flight (tf − t0 = 20.0, 30.0, 40.0, 50.0 time units),Tables 3, 4, 5, and 6 show that the maximum absolute relative difference drel 1 is

about 7% for ρ > 1, and 3.5% for ρ < 1;

(4) In all cases described above, the maximum absolute relative differences drel 1 occur

for transfers with large radial excursion;


Table 8: Consumption variable J for large transfers.

ρ tf − t0 JNeigh1 JNeigh2

2.500

100.0125.0150.0175.0200.0

6.7827 × 10−4

5.4241 × 10−4

4.5154 × 10−4

3.8651 × 10−4

3.3812 × 10−4

6.7818 × 10−4

5.4241 × 10−4

4.5154 × 10−4

3.8688 × 10−4

3.3812 × 10−4

3.750

100.0125.0150.0175.0200.0

1.1975 × 10−3

9.5018 × 10−4

7.8787 × 10−4

6.7324 × 10−4

5.8793 × 10−4

1.1966 × 10−3

9.4987 × 10−4

7.8784 × 10−4

6.7325 × 10−4

5.8794 × 10−4

5.000

100.0125.0150.0175.0200.0

1.6105 × 10−3

1.2678 × 10−3

1.0441 × 10−3

8.8948 × 10−4

7.7584 × 10−4

1.6106 × 10−3

1.2678 × 10−3

1.0441 × 10−3

8.8947 × 10−4

7.7584 × 10−4

(5) For transfers between close orbits with small time of flight, the fuel consumption

can be greatly reduced if the duration of the transfer increases: for instance, the fuel

consumption for transfers with time of flight tf − t0 = 2.0 (time units) is approx-

imately ten times the fuel consumption for transfers with time of flight tf − t0 =4.0 (time units), and, it is approximately hundred times the fuel consumption for

transfers with time of flight tf − t0 = 20.0 (time units), considering any value of ρ;

(6) For transfers with moderate time of flight, the fuel consumption decreases almost

linearly with the time of flight;

(7) For transfers with moderate amplitude (ρ = 0.727 and ρ = 1.523), Tables 5 and 6

show that 7.0% > drel 1 > 1.0%;

(8) Tables 1–6 show that the results obtained through the numerical algorithms—

gradient and neighboring extremals—are very quite similar, regardless the

amplitude of maneuver and the time of flight;

(9) Table 7 shows that an Earth-asteroid belt mission with tf − t0 = 20.0 time units

(approximately, 3.2 years) and an Earth-Jupiter mission with tf − t0 = 50.0 time

units (approximately, 8.0 years) spend almost the same quantity of fuel. This result

is closely related to the concept of transversals (payoff curves) in a field of extremals

introduced by Edelbaum [7] in the study of optimal limited-power transfers in

strong gravity field. Low-thrust limited power transfers with different amplitude

ρ and different time of flight tf − t0 can be performed with the same amount of

fuel as shown in Figures 7, 8, and 9. From Figure 7, one finds that an Earth-Venus

mission with tf − t0 = 30.0 time units (approximately, 4.8 years) and an Earth-Venus

mission with tf − t0 = 36.5 time units (approximately, 5.8 years) also spend almost

the same quantity of fuel, J = 4.98 × 10−4 (canonical units).

In order to follow the evolution of the optimal thrust acceleration vector during the

maneuver, it is convenient to plot the locus of its tip in the moving frame of reference. Figures

1, 2, 3, and 4 illustrate these plots for ρ = 0.727, 0.950, 0.975, 1.025, 1.050, and 1.523, with

tf − t0 = 2.0, 3.0, 30.0 and 50.0. Note that the agreement between the numerical and analytical


−0.04 −0.02 0 0.02 0.04

Radial acceleration

−0.03

−0.01

0

−0.02

0.01C

ircu

mfe

ren

tial

acc

eler

ati

on

Thrust acceleration-ρ = 0.975

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.04 −0.02 0 0.02 0.04

−0.01

0

0.01

0.02

0.03Thrust acceleration-ρ = 1.025

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−0.06

−0.04

−0.02

0


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.08 −0.04 0 0.04 0.08

−0.02

0

0.02

0.04


Cir

cum

fere

nti

al

acc

eler

ati

on

Radial acceleration

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.4

−0.3

−0.2

−0.1

0

0.1

Linear theory

Gradient-based algorithm

Neighboring extremals


Cir

cum

fere

nti

al

acc

eler

ati

on

Radial acceleration

Linear theory



−0.8 −0.4 0 0.4 0.8

−0.2

0

0.2

0.4

0.6 Thrust acceleration-ρ = 1.523

Figure 1: Thrust acceleration for tf − t0 = 2.0.


−0.008 −0.004 0 0.004 0.008

−0.016

−0.012

−0.008

−0.004

0


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.008 −0.004 0 0.004 0.008

−0.004

0

0.004

0.008

0.012


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.02 −0.01 0 0.01 0.02

−0.01

0

0.01

0.02


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.02 −0.01 0 0.01 0.02

−0.03

−0.02

−0.01

0


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.08 −0.04 0 0.04 0.08

−0.16

−0.12

−0.08

−0.04

0Thrust acceleration-ρ = 0.727

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

Linear theory



−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.1

0

0.1

0.2


Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

Linear theory





−1.2E−0

05

−8E−0

06

−4E−0

06 0

4E−0

06

8E−0

06

1.2E−0

05

−0.00045

−0.00044

−0.00043

−0.00042

−0.00041

−0.0004Thrust acceleration-ρ = 0.975

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−2E−0

05

−1E−0

05 0

1E−0

05

2E−0

05

0.00036

0.00038

0.0004

0.00042

0.00044

0.00046

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−8E−0

06

−4E−0

06 0

4E−0

06

8E−0

06

−0.00088

−0.000875

−0.00087

−0.000865

−0.00086

−0.000855

−0.00085

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−6E−0

05

−4E−0

05

−2E−0

05 0

2E−0

05

4E−0

05

6E−0

05

0.00072

0.00076

0.0008

0.00084

0.00088

0.00092

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−0.0

002

−0.0

001 0

0.0

001

0.0

002

0.0

003

−0.0062

−0.006

−0.0058

−0.0056

−0.0054

−0.0052

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


Linear theory


−0.0

008

−0.0

004 0

0.0

004

0.0

008

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


Linear theory




−0.0

002

−0.0

001 0

0.0

001

0.0

002

0.0032

0.0034

0.0036

0.0038

0.004

0.0042

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

Linear theory



−2E−0

05

−1E−0

05 0

1E−0

05

2E−0

05

0.00044

0.00046

0.00048

0.0005

0.00052

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−6E−0

06

−4E−0

06

−2E−0

06 0

2E−0

06

4E−0

06

6E−0

06

0.000236

0.00024

0.000244

0.000248

0.000252

0.000256

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−4E−0

06

−2E−0

06 0

2E−0

06

4E−0

06

−0.000264

−0.00026

−0.000256

−0.000252

−0.000248

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−1.5E−0

05

−1E−0

05

−5E−0

06 0

5E−0

06

1E−0

05

1.5E−0

05

−0.00055

−0.00054

−0.00053

−0.00052

−0.00051

−0.0005

−0.00049

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−8E−0

05

−4E−0

05 0

4E−0

05

8E−0

05

0.0

0012

−0.0037

−0.0036

−0.0035

−0.0034

−0.0033

−0.0032

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

Linear theory





−0.0004 −0.0002 0 0.0002 0.0004

0.0032

0.0034

0.0036

0.0038

0.004

0.0042

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

−0.002 −0.001 0 0.001 0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on

Neighboring extremals Neighboring extremals

Thrust acceleration-ρ = 2.500, and tf−t0 = 100.0Thrust acceleration-ρ = 5.000, and tf−t0 = 100.0

Figure 5: Thrust acceleration for large-amplitude transfers with tf − t0 = 100.0.

−8E−0

05

−4E−0

05 0

4E−0

05

8E−0

05

0.0

0012

0.0017

0.00175

0.0018

0.00185

0.0019

0.00195

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


−0.0

008

−0.0

004 0

0.0

004

0.0

008

0.002

0.0024

0.0028

0.0032

0.0036

Radial acceleration

Cir

cum

fere

nti

al

acc

eler

ati

on


Thrust acceleration-ρ = 2.500, and tf−t0 = 200.0Thrust acceleration-ρ = 5.000, and tf−t0 = 200.0

Figure 6: Thrust acceleration for large-amplitude transfers with tf − t0 = 200.0.

results is better for small-amplitude transfers. For moderate- and large-amplitude transfers,

this difference increases with the duration of the maneuvers.

Similarly, Figures 5 and 6 show the evolution of the optimal thrust acceleration vector

for large-amplitude transfers with long time of flight. Note that the magnitude of the thrust

acceleration becomes smaller as the time of flight increases.

Figures 3, 4, and 5 show that for transfers with long time of flight the circumferential

thrust acceleration is the main component of the optimal thrust acceleration. As the time of


20 30 40 50

Time of flight

0.0002

0.0004

0.0006

0.0008

0.001

Consumption for Earth-Venus and Earth-Mars transfers

Earth-Venus transfer

Earth-Mars transfer

Co

nsu

mp

tio

n v

ari

ab

le J

Figure 7: Consumption for Earth-Venus and Earth-Mars transfers with moderate time of flight.

Time of flight

Earth-asteroid belt transfer

Earth-Jupiter transfer

20 30 40 50 60

0

0.004

0.008

0.012

0.016

Co

nsu

mp

tio

n v

ari

ab

le J

Consumption for interplanetary transfers

Figure 8: Consumption for Earth-asteroid belt and Earth-Jupiter transfers.


100 120 140 160 180 200

Time of flight

0

0.0004

0.0008

0.0012

0.0016

0.002

Co

nsu

mp

tio

n v

ari

ab

le J

Consumption for large transfers

ρ = 2.500

ρ = 3.750

ρ = 5.000

Figure 9: Consumption for large transfers.

flight increases the contribution of the radial component of the optimal thrust acceleration

decreases and the optimal thrust acceleration tends to the circumferential acceleration.

Figures 7 and 8 show the consumption variable for interplanetary transfers with

moderate flight of time, and Figure 9 shows the consumption variable for large-amplitude

transfers with long time of flight. Note that transfers with different amplitude ρ and different

time of flight tf − t0 can be performed with the same amount of fuel (see Comment 9).

4. Conclusion

In this paper, a gradient-based algorithm and two different algorithms of the neighboring

extremals method are applied to the analysis of optimal low-thrust limited power transfers

between circular coplanar orbits in an inverse-square force field. The numerical results given

by these algorithms have been compared to the analytical results obtained by a linear theory.

The good agreement between these results shows that the linear theory provides a good

approximation for the solution of the transfer problem concerned with small-amplitudes,

that is, for transfers between close circular coplanar orbits. The linear theory can be used in

preliminary mission analysis involving such kind of transfers. The results presented in the

paper also show that the fuel consumption can be reduced if the time of flight of the transfer

increases. For transfers with long time of flight, the circumferential thrust acceleration

becomes the main component of the optimal thrust acceleration. A preliminary analysis of

some interplanetary missions is presented using the neighboring extremals algorithms.

Acknowledgment

This paper has been supported by CNPq under Contract 302949/2009-7.


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[20] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelideze, and E. F. Mishchenko, The Mathematical Theoryof Optimal Control Processes, John Wiley & Sons, New York, NY, USA, 1962.

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[22] S. da Silva Fernandes and W. A. Golfetto, “Numerical computation of optimal low-thrust limited-power trajectories—transfers between coplanar circular orbits,” Journal of Brazilian Society ofMechanical Sciences, vol. 27, no. 2, pp. 177–184, 2005.

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Research ArticleApplication of the Hori Method in the Theory ofNonlinear Oscillations

Sandro da Silva Fernandes

Departamento de Matematica, Instituto Tecnologico de Aeronautica, 12228-900,Sao Jose dos Campos, SP, Brazil



Academic Editor: Antonio F. Bertachini A. Prado

Copyright q 2012 Sandro da Silva Fernandes. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Some remarks on the application of the Hori method in the theory of nonlinear oscillations arepresented. Two simplified algorithms for determining the generating function and the new systemof differential equations are derived from a general algorithm proposed by Sessin. The vectorfunctions which define the generating function and the new system of differential equationsare not uniquely determined, since the algorithms involve arbitrary functions of the constantsof integration of the general solution of the new undisturbed system. Different choices of thesearbitrary functions can be made in order to simplify the new system of differential equationsand define appropriate near-identity transformations. These simplified algorithms are appliedin determining second-order asymptotic solutions of two well-known equations in the theory ofnonlinear oscillations: van der Pol equation and Duffing equation.

1. Introduction

In da Silva Fernandes [1], the general algorithm proposed by Sessin [2] for determining the

generating function and the new system of differential equations of the Hori method for

noncanonical systems has been revised considering a new approach for the integration theory

which does not depend on the auxiliary parameter t∗ introduced by Hori [3, 4].In this paper, this new approach is applied to the theory of nonlinear oscillations for

a second-order differential equation and two simplified versions of the general algorithm are

derived. The first algorithm is applied to systems of two first-order differential equations

corresponding to the second-order differential equation, and the second algorithm is applied

to the equations of variation of parameters associated with the original equation. According

to these simplified algorithms, the determination of the unknown functions T(m)j and

Z∗(m)j , defined in the mth-order equation of the algorithm of the Hori method, is not


unique, since these algorithms involve at each order arbitrary functions of the constants

of integration of the general solution of the new undisturbed system. Different choices of

the arbitrary functions can be made in order to simplify the new system of differential

equations and define appropriate near-identity transformations. The problem of determining

second-order asymptotic solutions of two well-known equations in the theory of nonlinear

oscillations—van der Pol and Duffing equations—is taken as example of application of the

simplified algorithms. For van der Pol equation, two generating functions are determined:

one of these generating functions is the same function obtained by Hori [4], and, the

other function provides the well-known averaged equations of variation of parameters in

the theory of nonlinear oscillations. For Duffing equation, only one generating function

is determined, and the second-order asymptotic solution is the same solution obtained

through Krylov-Bogoliubov method [5], through the canonical version of Hori method [6]or through a different integration theory for the noncanonical version of Hori method [7].For completeness, brief descriptions of the noncanonical version of the Hori method [4] and

the general algorithm proposed by Sessin [2] are presented in the next two sections.

2. Hori Method for Noncanonical Systems

The noncanonical version of the Hori method [4] can be briefly described as follows.

Consider the differential equations:

dzj

dt= Zj(z, ε), j = 1, . . . , n, (2.1)

where Zj(z, ε), j = 1, . . . , n, are the elements of the vector function Z(z, ε). It is assumed that

Z(z, ε) is expressed in power series of a small parameter ε:

Z(z, ε) = Z(0)(z) +∑m=1

εmZ(m)(z). (2.2)

The system of differential equations described by Z(0)(z) is supposed to be solvable.

Let the transformation of variables (z1, . . . , zn) → (ζ1, . . . , ζn) be generated by the

vector function T(ζ, ε). This transformation of variables is such that the new system:

dζj

dt= Z∗

j (ζ, ε), j = 1, . . . , n, (2.3)

is easier to solve or captures essential features of the system. Z∗j (ζ, ε), j = 1, . . . , n, are the

elements of the vector function Z∗(ζ, ε), also expressed in power series of ε:

Z∗(ζ, ε) = Z∗(0)(ζ) +∑m=1

εmZ∗(m)(ζ). (2.4)


It is assumed that the vector function T(ζ, ε), that defines a near-identity transformation, is

also expressed in powers series of ε:

T(ζ, ε) =∑m=1

εmT (m)(ζ). (2.5)

Following Hori [4], the transformation of variables (z1, . . . , zn) → (ζ1, . . . , ζn) gener-

ated by T(ζ, ε) is given by

zj = ζj +∑k=1

1

k!DkTζj = e

DT ζj , j = 1, . . . , n. (2.6)

For an arbitrary function f(z), the expansion formula is given by

f(z) = f(ζ) +∑k=1

1

k!DkTf(ζ) = e

DT f(ζ). (2.7)

The operator DT is defined by

DTf(ζ) =n∑j=1

Tj∂f

∂ζj,

DnTf(ζ) = D

n−1T

⎛⎝ n∑j=1

Tj∂f

∂ζj

⎞⎠.

(2.8)

According to the algorithm of the perturbation method proposed by Hori [4], the

vector functions Z and T are obtained, at each order in the small parameter ε, from the

following equations:

order 0: Z(0)j = Z∗(0)

j , (2.9)

order 1:[Z(0), T (1)

]j+ Z(1)

j = Z∗(1)j , (2.10)

order 2:[Z(0), T (2)

]j+

1

2

[Z(1) + Z∗(1), T (1)

]j+ Z(2)

j = Z∗(2)j ,

...

(2.11)

j = 1, . . . , n, where []j stands for the generalized Poisson brackets

[Z, T]j =n∑k=1

[Tk∂Zj

∂ζk− Zk

∂Tj

∂ζk

]. (2.12)

Z∗(0), Z(m), Z∗(m), and T (m) are written in terms of the new variables ζ1, . . . , ζn.


The mth-order equation of the algorithm can be put in the general form:

[Z∗(0), T (m)

]j+ Ψ(m)

j = Z∗(m)j , j = 1, . . . , n, (2.13)

where the functions Ψ(m)j are obtained from the preceding orders.

3. The General Algorithm

The determination of the functions Z∗(m)j and T

(m)j from (2.13) is based on the following

proposition presented in da Silva Fernandes [1].

Proposition 3.1. Let F be a n × 1 vector function of the variables ζ1, . . . , ζn, which satisfy the systemof differential equations:

dζj

dt= Z∗(0)

j (ζ) + R∗j (ζ; ε), j = 1, . . . , n, (3.1)

where Z∗(0) describes an integrable system of differential equations:

dζj

dt= Z∗(0)

j (ζ), j = 1, . . . , n, (3.2)

a general solution of which is given by

ζj = ζj(c1, . . . , cn, t), j = 1, . . . , n, (3.3)

being c1, . . . , cn arbitrary constants of integration; then

[F,Z∗(0)

]j=∂Fj

∂t−

n∑k=1

∂Z∗(0)j

∂ζkFk, j = 1, . . . , n. (3.4)

A corollary of this proposition can be stated.

Corollary 3.2. Consider the same conditions of Proposition 3.1 with the general solution of (3.2)given by

ζj = ζj(c1, . . . , cn−1,M), j = 1, . . . , n, (3.5)

being c1, . . . , cn−1 arbitrary constants of integration andM = t + τ , where τ is an additive constant;then

[F,Z∗(0)

]j=∂Fj

∂M−

n∑k=1

∂Z∗(0)j

∂ζkFk, j = 1, . . . , n. (3.6)


Now, consider (2.13). According to Proposition 3.1, this equation can be put in the

form:

∂T(m)j

∂t−

n∑k=1

∂Z∗(0)j

∂ζkT(m)k

= Ψ(m)j − Z∗(m)

j , j = 1, . . . , n, (3.7)

with Ψ(m)j written in terms of the general solution (3.3) of the undisturbed system (3.2),

involving n arbitrary constants of integration—c1, . . . , cn. Z∗(m)j and T

(m)j are unknown

functions.

Equation (3.7) is very similar to the one presented by Hori [4], which is written in

terms of an auxiliary parameter t∗ through an ordinary differential equation, that is,

dT(m)j

dt∗−

n∑k=1

∂Z∗(0)j

∂ζkT(m)k

= Ψ(m)j − Z∗(m)

j , j = 1, . . . , n. (3.8)

To determine Z∗(m)j and T

(m)j , j = 1, . . . , n, Hori [4] extends the averaging principle

applied in the canonical version: Z∗(m)j are determined so that the T

(m)j are free from secular

or mixed secular terms. However, this procedure is not sufficient to determine Z∗ such that

the new system of differential equations (2.3) becomes more tractable, and, a tractability

condition is imposed

[Z(0), Z∗

]j= 0, j = 1, . . . , n. (3.9)

This condition is analogous to the condition {F(0), F∗} = 0 in the canonical case, which

provides the first integral F0(ξ, η) = const, where ξ, η denotes the new set of canonical

variables, and, F(0) and F∗ are the undisturbed Hamiltonian and the new Hamiltonian,

respectively, and {} stands for Poisson brackets [3, 4].In the next paragraphs, the general algorithm for determining Z

∗(m)j and T

(m)j , j =

1, . . . , n, proposed by Sessin [2] and revised in da Silva Fernandes [1] is briefly presented.

Introducing the n × 1 matrices:

T (m) =(T(m)j

), Ψ(m) =

(Ψ(m)j

), Z∗(m) =

(Z

∗(m)j

), j = 1, . . . , n, (3.10)

and the n × n Jacobian matrix

J(t) =

⎛⎝∂Z∗(0)j

∂ζk

⎞⎠, j, k = 1, . . . , n, (3.11)

the system of partial differential equations (3.7) can be put in the following matrix form:

∂T (m)

∂t− J(t)T (m) = Ψ(m) − Z∗(m). (3.12)


The vector functions Z∗(m) and T (m) are determined from the following equations:

Z∗(m) = Δζ∂

∂t

{Δ−1ζ

[ΔζD

(m) + Δζ

∫Δ−1ζ Ψ(m)dt

]s

}, (3.13)

T (m) =[ΔζD

(m) + Δζ

∫Δ−1ζ Ψ(m)dt

]p

, (3.14)

where Δζ = [∂ζj(c1, . . . , cn, t)/∂ck] is the Jacobian matrix associated to the general solution

(3.3) of the undisturbed system (3.2), s denotes the secular or mixed secular terms, and p

denotes the remaining part. D(m) is the n × 1 vector, D(m) = (D(m)j ), which depends only on

the arbitrary constants of integration c1, . . . , cn of the general solution (3.3). The choice ofD(m)

is arbitrary. Recall that in the integration process, the arbitrary constants of integration of the

general solution (3.3) are taken as parameters.

Equations (3.13) and (3.14) assure that T (m) is free from secular or mixed secular terms.

Moreover, these equations provide the tractability condition (3.9) as it will be shown in the

case of nonlinear oscillation problems presented in the next section.

Finally, it should be noted that D(m) can be chosen at each order to simplify the

generating function T and the new system of differential equations (2.3). This aspect is

discussed thoroughly in the examples of Section 5.

4. Simplified Algorithms in the Theory of Nonlinear Oscillations

In this section two simplified algorithms will be derived from the general algorithm in the

case of nonlinear oscillations described by a second-order differential equation of the general

form:

x + x = εf(x, x). (4.1)

The first algorithm is applied to the system of first-order differential equations with

x and x as elements of the vector z, and, the second algorithm is applied to the system of

equations of variation of parameters associated to the differential equation with c′ and θ′ as

elements of the vector z; c′ and θ′are defined in (4.26).

4.1. Simplified Algorithm I

For completeness, we present now the first simplified algorithm [1]. Additional remarks are

included at the end of section.

Introducing the variables z1 = x and z2 = x, (4.1) can be put in the form:

z1 = z2, z2 = −z1 + εf(z1, z2). (4.2)


According to the notation introduced in (2.1) and (2.2):

Z(0) =[z2

−z1

], Z(1) =

[0

f(z1, z2)

]. (4.3)

Following the algorithm of the Hori method for noncanonical systems, one finds from

zero-th-order equation, (2.9), that

Z∗(0) =[ζ2

−ζ1

]. (4.4)

Applying Proposition 3.1, it follows that the undisturbed system (3.2) is given by

ζ1 = ζ2, ζ2 = −ζ1, (4.5)

general solution of which can be written in terms of the exponential matrix as [2]:

[ζ1

ζ2

]= eEt

[c1

c2

], (4.6)

where

eEt =[

cos t sin t

− sin t cos t

], (4.7)

and E is the symplectic matrix:

E =[

0 1

−1 0

], (4.8)

and ci, i = 1, 2, are constants of integration. The Jacobian matrix Δζ associated with the

solution (4.6) is then given by

Δζ = eEt, (4.9)

with inverse Δ−1ζ

= e−Et = ΔTζ

, since Δζ is an orthogonal matrix.

In view of (4.6), the functions Ψ(m)j defined at each order of the algorithm (see (2.13)

or (3.7)) are expressed by Fourier series with multiples of t as arguments such that the vector

function Ψ(m)I can be written as

Ψ(m)I =

∞∑k=0

⎡⎢⎣a(m)k,I

cos kt + b(m)k,I

sin kt

c(m)k,I

cos kt + d(m)k,I

sin kt

⎤⎥⎦, (4.10)


where the coefficients a(m)k,I

, b(m)k,I

, c(m)k,I

, and d(m)k,I

are functions of the constants ci, i = 1, 2. The

Fourier series can also be put in matrix form:

Ψ(m)I =

∞∑k=0

(eEktA

(m)k,I

+ e−EktB(m)k,I

), (4.11)

with

A(m)k,I

=1

2

⎡⎢⎣a(m)k,I

− d(m)k,I

b(m)k,I

+ c(m)k,I

⎤⎥⎦, B(m)k,I

=1

2

⎡⎢⎣a(m)k,I

+ d(m)k,I

c(m)k,I

− b(m)k,I

⎤⎥⎦. (4.12)

The subscript I is introduced to denote the first simplified algorithm.

Substituting (4.9) and (4.11) into (3.13), one finds

Z∗(m)I = eEt

∂

∂t

{e−Et[teEtA

(m)1,I + periodic terms

]s

}= eEtA(m)

1,I . (4.13)

On the other hand,

⟨e−EtΨ(m)

I

⟩= A(m)

1,I , (4.14)

where 〈〉 stands for the mean value of the function.

Therefore, from (4.9), (4.13), and (4.14), it follows that

Z∗(m)I = eEt

⟨e−EtΨ(m)

I

⟩= Δζ

⟨Δ−1ζ Ψ(m)

I

⟩. (4.15)

The second equation of the general algorithm, (3.14) can be simplified as described

bellow.

From (3.14), (4.14), and (4.15), one finds

teEtA(m)1,I = Δζ

∫⟨Δ−1ζ Ψ(m)

I

⟩dt =

⌊ΔζD

(m)I + Δζ

∫Δ−1ζ Ψ(m)

I dt

⌋s

. (4.16)

On the other hand,

ΔζD(m)I + Δζ

∫Δ−1ζ Ψ(m)

I dt =⌊ΔζD

(m)I + Δζ

∫Δ−1ζ Ψ(m)

I dt

⌋p

+ teEtA(m)1,I . (4.17)

Thus, introducing (4.16) and (4.17) into (3.14), one finds

T(m)I =

⌊ΔζD

(m)I + Δζ

∫Δ−1ζ Ψ(m)

I dt

⌋p

= ΔζD(m)I + Δζ

∫⌊Δ−1ζ Ψ(m)

I −⟨Δ−1ζ Ψ(m)

I

⟩⌋dt. (4.18)


Equations (4.15) and (4.18) define the first simplified form of the general algorithm

applicable to the nonlinear oscillations problems described by (4.1) with x and x as elements

of vector z.

Finally, we note that (4.15) satisfies the tractability condition (3.9) up to order m. In

order to show this equivalence, one proceeds as follows. Since, from (4.14), 〈Δ−1ζΨ(m)I 〉 does

not depend explicitly on the time t, it follows that

∂Z∗(m)

∂t=∂Δζ

∂t

⟨Δ−1ζ Ψ(m)

I

⟩= JΔζ

⟨Δ−1ζ Ψ(m)

I

⟩= JZ∗(m). (4.19)

Using Proposition 3.1 and taking into account that J = ∂Z∗(0)j /∂ζk, this equation can be put in

the following form:

[Z∗(m), Z(0)

]j=∂Z

∗(m)j

∂t−

n∑k=1

∂Z∗(0)j

∂ζkZ

∗(m)k

= 0, j = 1, 2, (4.20)

which is the tractability condition (3.9) up to order m.

Remark 4.1. It should be noted that (4.15) and (4.18) for determining the vector functions

Z∗(m)I and T

(m)I , respectively, are invariant with respect to the form of the general solution

of the undisturbed system described by Z∗(0). This means that if the general solution of the

undisturbed system is written in terms of a second set of constants of integration, for instance,

if this solution is given by

ζ1 = c cos(t + θ),

ζ2 = −c sin(t + θ),(4.21)

where c and θ denote new constants of integration, then Z∗(m)I and T

(m)I are determined

through (4.15) and (4.18), with the Jacobian matrix Δζ given by

Δζ =[

cos(t + θ) −c sin(t + θ)− sin(t + θ) −c cos(t + θ)

]. (4.22)

This result can be proved as follows. The two sets of constants of integration (c1, c2) and (c, θ)are related through the following transformation:

c2 = c21 + c

22,

tan θ = −c2

c1.

(4.23)

In view of this transformation, the Jacobian matrix Δ1ζ

can be written in terms of the Jacobian

matrix Δ2ζ

as

Δ1ζ = Δ2

ζΔC, (4.24)


where the superscripts 1 and 2 are introduced to denote the form of the general solution of

the undisturbed system described by Z∗(0) with respect to the set of constants of integration

(c1, c2) and (c, θ), respectively. ΔC is the Jacobian matrix of the transformation. Since ΔC does

not depend on the time t, it follows from (4.15) and (4.18) that

Z∗(m)I = Δ1

ζ

⟨(Δ1ζ

)−1Ψ1I

(m)⟩

= Δ2ζΔC

⟨Δ−1C

(Δ2ζ

)−1Ψ2I

(m)⟩

= Δ2ζΔCΔ−1

C

⟨(Δ2ζ

)−1Ψ2I

(m)⟩

= Δ2ζ

⟨(Δ2ζ

)−1Ψ2I

(m)⟩,

T(m)I = Δ1

ζD1I

(m)+ Δ1

ζ

∫[(Δ1ζ

)−1Ψ1I

(m) −⟨(

Δ1ζ

)−1Ψ1I

(m)⟩]

dt

= Δ2ζΔCD

1I

(m)+ Δ2

ζΔC

∫[Δ−1C

(Δ2ζ

)−1Ψ2I

(m) −⟨Δ−1C

(Δ2ζ

)−1Ψ2I

(m)⟩]

dt

= Δ2ζΔCD

1I

(m)+ Δ2

ζΔCΔ−1C

∫[(Δ2ζ

)−1Ψ2I

(m) −⟨(

Δ2ζ

)−1Ψ2I

(m)⟩]

dt

= Δ2ζD

2I

(m)+ Δ2

ζ

∫[(Δ2ζ

)−1Ψ2I

(m) −⟨(

Δ2ζ

)−1Ψ2I

(m)⟩]

dt.

(4.25)

Finally, we note that the general solution given by (4.21) is more suitable in practical

applications than the general solution given by (4.6), that, in turn, is more suitable for

theoretical purposes.

4.2. Simplified Algorithm II

In this section, a second simplified algorithm is derived from the general one. Introducing the

transformation of variables (x, x) → (c′, θ′) defined by the following equations

x = c′ cos(t + θ′

),

x = −c′ sin(t + θ′

),

(4.26)

equation (4.1) is transformed into

dc′

dt= −εf

(c′ cos

(t + θ′

),−c′ sin

(t + θ′

))sin(t + θ′

),

dθ′

dt= −ε 1

c′f(c′ cos

(t + θ′

),−c′ sin

(t + θ′

))cos(t + θ′

).

(4.27)

These differential equations are the well-known variation of parameters equations associated

to the second-order differential equation (4.1). Equation (4.27) define a nonautonomous

system of differential equations.

The sets (c, θ) and (c′, θ′), defined, respectively in (4.21) and (4.26), have different

meanings in the theory: in (4.21), c and θ are constants of integration of the general solution

of the new undisturbed system described by Z∗(0)(ζ1, ζ2); in (4.26), c′ and θ′ are new

variables which represent the constants of integration of the general solution of the original

undisturbed system described by Z(0)(z1, z2) in the variation of parameter method. These

sets, (c, θ) and (c′, θ′), are connected through a near identity transformation.


Remark 4.2. It should be noted that a second transformation of variables involving a fast

phase, (x, x) → (c′, φ′), can also be defined. This second transformation is given by

x = c′ cos φ′,

x = −c’sinφ′.(4.28)

In this case, (4.1) is transformed into

dc′

dt= −εf

(c’cosφ′,−c’sinφ′) sinφ′,

dφ′

dt= 1 − ε 1

c′f(c’cosφ′,−c’sinφ′) cosφ′.

(4.29)

These equations define an autonomous system of differential equations. In what follows, the

first set of variation of parameters equations, (4.27), will be considered.

Now, introducing the variables z1 = c′ and z2 = θ′, one gets from (4.27) that

Z(0) =[

0

0

],

Z(1) =

⎡⎢⎢⎣−f(z1 cos(t + z2),−z1 sin(t + z2)) sin(t + z2)

− 1

z1f(z1 cos(t + z2),−z1 sin(t + z2)) cos(t + z2)

⎤⎥⎥⎦.(4.30)

Applying Proposition 3.1, it follows that the undisturbed system (3.2) is given by

ζ1 = 0, ζ2 = 0, (4.31)

and its general solution is very simple,

ζi = ci, i = 1, 2, (4.32)

where ci, i = 1, 2, are constants of integration. The Jacobian matrix Δζ associated with this

general solution is also very simple, and it is given by

Δζ = I, (4.33)

where I is the identity matrix.

Substituting (4.33) into (3.13) and (3.14), it follows that

Z∗(m)II =

∂

∂t

{[D

(m)II +

∫Ψ(m)II dt

]s

},

T(m)II =

[D

(m)II +

∫Ψ(m)II dt

]p

.

(4.34)

The subscript II is introduced to denote the second simplified algorithm.


Equation (4.34) can be put in a more suitable form as follows. In view of (4.30), the

functions Ψ(m)j defined at each order of the algorithm (see (2.13) or (3.7)) are expressed by

Fourier series with multiples of t + z2 as arguments such that the vector function Ψ(m)II can be

written as

Ψ(m)II =

∞∑k=0

⎡⎢⎣a(m)k,II

cos k(t + c2) + b(m)k,II

sin k(t + c2)

c(m)k,II

cos k(t + c2) + d(m)k,II

sin k(t + c2)

⎤⎥⎦, (4.35)

where the coefficients a(m)k,II

, b(m)k,II

, c(m)k,II

, and d(m)k,II

are functions of the constant c1. The vector

function Ψ(m)II can also be put in matrix form:

Ψ(m)II =

∞∑k=0

(eEk(t+c2)A

(m)k,II

+ e−Ek(t+c2)B(m)k,II

), (4.36)

with

A(m)k,II

=1

2

⎡⎢⎣a(m)k,II

− d(m)k,II

b(m)k,II

+ c(m)k,II

⎤⎥⎦, B(m)k,II

=1

2

⎡⎢⎣a(m)k,II

+ d(m)k,II

c(m)k,II

− b(m)k,II

⎤⎥⎦. (4.37)

Note that Ψ(m)II is very similar to Ψ(m)

I , defined by (4.11). They represent different forms of

Fourier series of Ψ(m), but they are not the same, since they involve different sets of arbitrary

constants of integration.

Thus, it follows from (4.36) that

D(m)II +

∫Ψ(m)II dt = D

(m)II +

(A

(m)0,II + B

(m)0,II

)t + periodic terms, (4.38)

with the periodic terms given by

∞∑k=1

((Ek)−1eEk(t+c2)A

(m)k,II

+ (−Ek)−1e−Ek(t+c2)B(m)k,II

). (4.39)

Therefore,

[D

(m)II +

∫Ψ(m)II dt

]s

= D(m)II +

(A

(m)0,II + B

(m)0,II

)t,[

D(m)II +

∫Ψ(m)II dt

]p

=∞∑k=1

((Ek)−1eEk(t+c2)A

(m)k,II

+ (−Ek)−1e−Ek(t+c2)B(m)k,II

).

(4.40)


Substituting (4.40) into (4.34), one finds

Z∗(m)II = A(m)

0,II + B(m)0,II =

⟨Ψ(m)II

⟩, (4.41)

T(m)II = D(m)

II +∫(

Ψ(m)II −

⟨Ψ(m)II

⟩)dt. (4.42)

Note that D(m)II depends only on c1 = ζ1.

It should be noted that (4.41) and (4.42) can be straightforwardly obtained from (3.12)by applying the averaging principle if D

(m)II is assumed to be zero, since in this second

approach:

J(t) =

⎛⎝∂Z∗(0)j

∂ζk

⎞⎠ = O, (4.43)

where O denotes the null matrix. Thus, the general algorithm defined by (3.13) and (3.14) is

equivalent to the averaging principle usually applied in the theory of nonlinear oscillations

[5, 7].

Remark 4.3. Equations (4.41) and (4.42) are also obtained, if the second set of variation of

parameters equations is considered (see Remark 4.2). In this case, the undisturbed system

(3.2) is given by

ζ1 = 0, ζ2 = 1, (4.44)

with general solution defined by

ζ1 = c1, ζ1 = t + c2, (4.45)

and Jacobian matrix Δζ = I.

Finally, we note that (4.41) is the tractability condition (3.9) up to orderm. Since 〈Ψ(m)II 〉

does not depend explicitly on the time t, it follows that

∂Z∗(m)j

∂t=[Z∗(m), Z(0)

]j= 0, j = 1, 2, (4.46)

which is the tractability condition (3.9) up to order m.

5. Application to Nonlinear Oscillations Problems

In order to illustrate the application of the simplified algorithms, two examples are presented.

The noncanonical version of the Hori method will be applied in determining second-order

asymptotic solutions for van der Pol and Duffing equations. For the van der Pol equation,


two different choices of the vector D(m) will be made, and two generating functions T (m)

will be determined, one of these generating functions is the same function obtained by Hori

[4] through a different approach, and, the other function gives the well-known averaged

variation of parameters equations in the theory of nonlinear oscillations obtained through

Krylov-Bogoliubov method [5]. It should be noted that the solution presented by Hori defines

a new system of differential equations with a different frequency for the phase in comparison

with the solution obtained by Ahmed and Tapley [7] and by Nayfeh [5], using different

perturbation methods. For the Duffing equation, only one generating function is determined,

and the second simplified algorithm gives the same generating function obtained through

Krylov-Bogoliubov method.

The section is organized in two subsections: in the first subsection, the asymptotic solu-

tions are determined through the first simplified algorithm, and, in the second subsection,

they are determined through the second simplified algorithm.

5.1. Determination of Asymptotic Solutions through Simplified Algorithm I

5.1.1. Van der Pol Equation

Consider the well-known van der Pol equation:

x + ε(x2 − 1

)x + x = 0. (5.1)

Introducing the variables z1 = x and z2 = x, this equation can be written in the form:

dz1

dt= z2,

dz1

dt= −z1 − ε

(z2

1 − 1)z2. (5.2)

Thus

Z(0) =[z2

−z1

], (5.3)

Z(1) =[

0

−(z2

1 − 1)z2

]. (5.4)

As described in preceding paragraphs, two different choices of D(m) will be made, and

two generating functions T (m) will be determined. Firstly, we present the solution obtained

by Hori [4].

(1) First Asymptotic Solution: Hori’s [4] Solution

Following the simplified algorithm I defined by (4.15) and (4.18), the first-order terms Z∗(1)

and T (1) are calculated as follows.


Introducing the general solution given by (4.21) of the undisturbed system described

by Z∗(0)(ζ1, ζ2) into (5.4), with ζ replacing z, one gets

Z(1) =

⎡⎣ 0(−c + 1

4c3

)sin(t + θ) +

1

4c3 sin 3(t + θ)

⎤⎦. (5.5)

Computing Δ−1ζZ(1),

Δ−1ζ Z

(1) =

⎡⎢⎢⎣1

2c

(1 − 1

4c2

)− 1

2c cos 2(t + θ) +

1

8c3 cos 4(t + θ)

1

2

(1 − 1

2c2

)sin 2(t + θ) − 1

8c2 sin 4(t + θ)

⎤⎥⎥⎦, (5.6)

and taking its secular part, one finds

⟨Δ−1ζ Z

(1)⟩=

⎡⎣1

2c

(1 − 1

4c2

)0

⎤⎦. (5.7)

From (4.15) and (4.22), it follows that Z∗(1) is given by

Z∗(1) =

⎡⎢⎢⎣1

2c

(1 − 1

4c2

)cos(t + θ)

−1

2c

(1 − 1

4c2

)sin(t + θ)

⎤⎥⎥⎦. (5.8)

In view of (4.21), Z∗(1) can be written explicitly in terms of the new variables ζ1 and ζ2 as

follows:

Z∗(1) =

⎡⎢⎢⎣1

2ζ1

(1 − 1

4

(ζ2

1 + ζ22

))1

2ζ2

(1 − 1

4

(ζ2

1 + ζ22

))⎤⎥⎥⎦. (5.9)

To determine T (1), the indefinite integral∫�Δ−1

ζZ(1) − 〈Δ−1

ζZ(1)〉�dt is calculated:

∫[Δ−1ζ Z

(1) −⟨Δ−1ζ Z

(1)⟩]dt =

⎡⎢⎢⎣−1

4c sin 2(t + θ) +

1

32c3 sin 4(t + θ)

−(

1

4− 1

8c2

)cos 2(t + θ) +

1

32c2 cos 4(t + θ)

⎤⎥⎥⎦. (5.10)


Thus, from (4.18), it follows that T (1) is given by

T (1) =

⎡⎢⎢⎣−1

4c

(1 − 1

4c2

)sin(t + θ) − 1

32c3 sin 3(t + θ)

1

4c

(1 − 1

4c2

)cos(t + θ) − 3

32c3 cos 3(t + θ)

⎤⎥⎥⎦ + ΔζD(1). (5.11)

In view of (4.21), T (1) can be written explicitly in terms of the new variables ζ1 and ζ2 as

follows:

T (1) =

⎡⎢⎢⎣1

4ζ2

(1 +

1

8

(ζ2

1 + ζ22

))− 1

8ζ3

2

1

4ζ1

(1 +

7

8

(ζ2

1 + ζ22

))− 3

8ζ3

1

⎤⎥⎥⎦ + ΔζD(1), (5.12)

with ΔζD(1) put in the form:

ΔζD(1) =

⎡⎢⎣d(1)1 ζ1 + d

(1)2 ζ2

d(1)1 ζ2 − d(1)

2 ζ1

⎤⎥⎦, (5.13)

being d(1)i = d(1)

i (c), i = 1, 2,D(1)1 = cd(1)

1 , andD(1)2 = d(1)

2 . The auxiliary vector d(1) is introduced

in order to simplify the calculations, and, it is calculated in the second-order approximation

as described below.

Following the algorithm of the Hori method described in Section 2, the second-order

equation, (2.11), involves the term Ψ(2) given by

Ψ(2) =1

2

[Z(1) + Z∗(1), T (1)

]. (5.14)

The determination of Ψ(2) is very arduous. The generalized Poisson brackets must be

calculated in terms of ζ1 and ζ2 through (2.12), and, the general solution of the undisturbed

system, defined by (3.1), must be introduced. It should be noted that d(1)i , i = 1, 2, in (5.12)

are functions of the new variables ζ1 and ζ2 through c2 = ζ21 + ζ

22. So, their partial derivatives


must be considered in the calculation of the generalized Poisson brackets. After lengthy

calculations performed using MAPLE software, one finds

(Δ−1ζ Ψ(2)

)1=(

7

64c3 − 3

128c5

)sin 2(t + θ)

− 1

32c3 sin 4(t + θ) − 1

128c5 sin 6(t + θ)

+ d(1)1

(−1

4c3 +

1

8c3 cos 4(t + θ)

)+ d(1)

2

(1

2c sin 2(t + θ) − 1

4c3 sin 4(t + θ)

)

+dd

(1)1

dζ1

(−1

4

(c2 − 1

4c4

)cos(t + θ)

)

+dd

(1)1

dζ2

(3

4

(c2 − 1

4c4

)sin(t + θ) − 1

8c4 sin 3(t + θ)

),(

Δ−1ζ Ψ(2)

)2= −1

8+

3

16c2 − 11

256c4 −(

1

32c2 +

1

64c4

)cos 2(t + θ)

+(− 1

32c2 +

1

128c4

)cos 4(t + θ) − 1

128c4 cos 6(t + θ)

+ d(1)1

(−1

4c2 sin 2(t + θ) − 1

8c2 sin 4(t + θ)

)+ d(1)

2

((1

2− 1

4c2

)cos 2(t + θ) − 1

4c2 cos 4(t + θ)

)

+dd

(1)2

dζ1

(−1

4

(c − 1

4c3

)cos(t + θ)

)

+dd

(1)2

dζ2

(3

4

(c − 1

4c3

)sin(t + θ) − 1

8c3 sin 3(t + θ)

).

(5.15)

In order to obtain the same result presented by Hori [4] for the new system of

differential equations and the near-identity transformation, the following choice is made for

the auxiliary vector d(1). Taking

d(1) =

⎡⎣ 0

−1

4+

1

16c2

⎤⎦, (5.16)

it follows from (5.15) that

⟨Δ−1ζ Ψ(2)

⟩=

⎡⎣ 0

−1

8+

1

8c2 − 7

256c4

⎤⎦. (5.17)


From (4.15), (4.21), and (5.17), one finds

Z∗(2) =

⎡⎢⎢⎣−1

8ζ2

(1 −(ζ2

1 + ζ22

)+

7

32

(ζ2

1 + ζ22

)2)

1

8ζ1

(1 −(ζ2

1 + ζ22

)+

7

32

(ζ2

1 + ζ22

)2)⎤⎥⎥⎦. (5.18)

In view of the choice the auxiliary vector d(1), (5.12) can be simplified, and T (1) is then

given by

T (1) =

⎡⎢⎢⎣1

32ζ2

(3ζ2

1 − ζ22

)1

32ζ1

(16 − 7ζ2

1 + 5ζ22

)⎤⎥⎥⎦. (5.19)

Computing the indefinite integral∫�Δ−1

ζΨ(2) − 〈Δ−1

ζΨ(2)〉�dt and substituting the

general solution of the new undisturbed system, it follows that T (2) is given by

T (2) =

⎡⎢⎢⎣1

16ζ1 −

5

64ζ3

1 +13

768ζ5

1 +1

96ζ3

1ζ22 +

11

768ζ1ζ

42

− 1

16ζ2 +

3

64ζ2

1ζ2 +1

16ζ3

2 −29

768ζ2ζ

41 −

5

192ζ2

1ζ32 −

7

768ζ5

2

⎤⎥⎥⎦ + ΔζD(2), (5.20)

with ΔζD(2) put in the form:

ΔζD(2) =

⎡⎢⎣d(2)1 ζ1 + d

(2)2 ζ2

d(2)1 ζ2 − d(2)

2 ζ1

⎤⎥⎦, (5.21)

being d(2)i = d(2)

i (c), i = 1, 2, D(2)1 = cd(2)

1 , and D(2)2 = d(2)

2 . D(2) is obtained from the third-order

approximation.

In order to get the same result presented by Hori [4], one finds, repeating the

procedure described in the preceding paragraphs, that the auxiliary vector d(2) must be taken

as follows:

d(2) =

⎡⎢⎣− 1

16+

15

256c2 − 7

512c4

0

⎤⎥⎦. (5.22)

Accordingly, T (2) is given by

T (2) =

⎡⎢⎢⎣5

1536ζ5

1 −13

768ζ3

1ζ22 +

1

1536ζ1ζ

42 −

5

256ζ3

1 +15

256ζ1ζ

22

− 35

1536ζ5

2 −41

768ζ2

1ζ32 −

79

1536ζ2ζ

41 +

31

256ζ3

2 +27

256ζ2

1ζ2 −1

8ζ2

⎤⎥⎥⎦. (5.23)


The new system of differential equations and the generating function are given, up to

the second-order of the small parameter, by

dζ1

dt= ζ2 + ε

1

2ζ1

(1 − 1

4

(ζ2

1 + ζ22

))− ε2 1

8ζ2

(1 −(ζ2

1 + ζ22

)+

7

32

(ζ2

1 + ζ22

)2),

dζ2

dt= −ζ1 + ε

1

2ζ2

(1 − 1

4

(ζ2

1 + ζ22

))+ ε2 1

8ζ1

(1 −(ζ2

1 + ζ22

)+

7

32

(ζ2

1 + ζ22

)2),

(5.24)

T1 = ε1

32ζ2

(3ζ2

1 − ζ22

)+ ε2

(5

1536ζ5

1 −13

768ζ3

1ζ22 +

1

1536ζ1ζ

42 −

5

256ζ3

1 +15

256ζ1ζ

22

),

T2 = ε1

32ζ1

(16 − 7ζ2

1 + 5ζ22

)+ ε2

(− 35

1536ζ5

2 −41

768ζ2

1ζ32 −

79

1536ζ2ζ

41 +

31

256ζ3

2 +27

256ζ2

1ζ2 −1

8ζ2

).

(5.25)

These results are in agreement with the ones obtained by Hori [4] using a different approach.

Following da Silva Fernandes [1], the Lagrange variational equations—equations of

variation of parameters—for the noncanonical version of the Hori method are given by

dC

dt= Δ−1

ζ R∗, (5.26)

where R∗ =∑

m=1 εmZ∗(m), and C is the n × 1 vector of constants of integration of the

general solution of the new undisturbed system (3.3). In view of (4.15), Lagrange variational

equations can be put in the form:

dC

dt=∑m=1

εm⟨Δ−1ζ Ψ(m)

I

⟩. (5.27)

Accordingly, the Lagrange variational equations for the new system of differential equations,

(5.24), are given by

dc

dt=εc

2

(1 − 1

4c2

), (5.28a)

dθ

dt= ε2

(−1

8+

1

8c2 − 7

256c4

). (5.28b)

The solution of the new system of differential equations can be obtained by

introducing the solution of the above variational equations into (4.21).The originalvariables x and x are calculated through (2.6), and the second-order

asymptotic solution is

x = ζ1 + ε1

32ζ2

(3ζ2

1 − ζ22

)+ ε2

(− 43

6144ζ5

1 +29

3072ζ3

1ζ22 −

59

6144ζ1ζ

42 +

1

256ζ3

1 +9

256ζ1ζ

22

),

x = ζ2+ε1

32ζ1

(16−7ζ2

1+5ζ22

)+ε2

(− 155

6144ζ5

2−35

3072ζ3

2ζ21−

715

6144ζ2ζ

41+

29

256ζ3

2+53

256ζ2ζ

21−

1

8ζ2

).

(5.29)

Equations (5.29) define exactly the same solution presented by Hori.


(2) Second Asymptotic Solution

Now, let us to consider a different choice of the auxiliary vector d(1). Taking d(1) as a null

vector, it follows straightforwardly from (5.15) that

⟨Δ−1ζ Ψ(2)

⟩=

⎡⎣ 0

−1

8+

3

16c2 − 11

256c4

⎤⎦. (5.30)

Thus, from (4.15), (4.21), (5.28a), and (5.28b), one finds

Z∗(2) =

⎡⎢⎢⎣−1

8ζ2

(1 − 3

2

(ζ2

1 + ζ22

)+

11

32

(ζ2

1 + ζ22

)2)

1

8ζ1

(1 − 3

2

(ζ2

1 + ζ22

)+

11

32

(ζ2

1 + ζ22

)2)⎤⎥⎥⎦. (5.31)

Since d(1) is a null vector, (5.12) simplifies, and T (1) is given by

T (1) =

⎡⎢⎢⎣1

4ζ2

(1 +

1

8

(ζ2

1 + ζ22

))− 1

8ζ3

2

1

4ζ1

(1 +

7

8

(ζ2

1 + ζ22

))− 3

8ζ3

1

⎤⎥⎥⎦. (5.32)

Now, repeating the procedure described in the previous section, that is, computing

the indefinite integral∫�Δ−1

ζΨ(2) − 〈Δ−1

ζΨ(2)〉�dt, substituting the general solution of the new

undisturbed system defined by (4.21), and taking d(2) is a null vector, it follows that T (2) is

given by

T (2) =

⎡⎢⎢⎣−3

64ζ3

1 +1

16ζ1ζ

22 +

5

384ζ5

1 −1

192ζ3

1ζ22 +

7

384ζ1ζ

42

− 7

64ζ2

1ζ2 +1

16ζ3

2 −1

384ζ2ζ

41 −

1

96ζ2

1ζ32 −

5

384ζ5

2

⎤⎥⎥⎦. (5.33)


So, the new system of differential equations and the generating function are given, up

to the second-order of the small parameter ε, by

dζ1

dt= ζ2 + ε

1

2ζ1

(1 − 1

4

(ζ2

1 + ζ22

))− ε2 1

8ζ2

(1 − 3

2

(ζ2

1 + ζ22

)+

11

32

(ζ2

1 + ζ22

)2),

dζ2

dt= −ζ1 + ε

1

2ζ2

(1 − 1

4

(ζ2

1 + ζ22

))+ ε2 1

8ζ1

(1 − 3

2

(ζ2

1 + ζ22

)+

11

32

(ζ2

1 + ζ22

)2),

(5.34)

T1 = ε(

1

4ζ2

(1 +

1

8

(ζ2

1 + ζ22

))− 1

8ζ3

2

)+ ε2

(− 3

64ζ3

1 +1

16ζ1ζ

22 +

5

384ζ5

1 −1

192ζ3

1ζ22 +

7

384ζ1ζ

42

),

T2 = ε(

1

4ζ1

(1 +

7

8

(ζ2

1 + ζ22

))− 3

8ζ3

1

)+ ε2

(− 7

64ζ2

1ζ2 +1

16ζ3

2 −1

384ζ2ζ

41 −

1

96ζ2

1ζ32 −

5

384ζ5

2

).

(5.35)

The Lagrange variational equations for the new system of differential equations,

defined by (5.34), are given by

dc

dt=εc

2

(1 − 1

4c2

), (5.36a)

dθ

dt= ε2

(−1

8+

3

16c2 − 11

256c4

). (5.36b)

These differential equations are the well-known averaged equations obtained through

Krylov-Bogoliubov method [5]. Note that (5.28b) and (5.36b) define the phase θ with slightly

different frequencies.

As described in the preceding subsection, the solution of the new system of differential

equations, defined by (5.34), can be obtained by introducing the solution of the above

variational equations into (4.21).The original variables x and x are calculated through (2.6), which provides the

following second-order asymptotic solution,

x = ζ1 + ε1

4ζ2

(1 +

1

8

(ζ2

1 − 3ζ22

))+ ε2

(65

6144ζ5

1 +65

3072ζ3

1ζ22 −

95

6144ζ1ζ

42 −

1

16ζ3

1 +1

16ζ1ζ

22 +

1

32ζ1

),

x = ζ2 + ε1

4ζ1

(1 − 1

8

(5ζ2

1 − 7ζ22

))+ ε2

(− 143

6144ζ5

2 +193

3072ζ3

2ζ21 −

271

6144ζ2ζ

41 +

5

64ζ3

2 −7

64ζ2ζ

21 +

1

32ζ2

).

(5.37)

Finally, note that (5.29) and (5.37) give different second-order asymptotic solution for

van der Pol equation.


5.1.2. Duffing Equation

Consider the well-known Duffing equation:

x + εx3 + x = 0. (5.38)

Introducing the variables z1 = x and z2 = x, this equation can be written in the form:

dz1

dt= z2,

dz2

dt= −z1 − εz3

1. (5.39)

Thus,

Z(1) =[

0

−z31

]. (5.40)

Following the simplified algorithm I and repeating the procedure described in

Section 5.1.1, the first-order terms Z∗(1) and T (1) are obtained as follows. Introducing (4.21)into (5.40), and computing the secular part, one gets

⟨Δ−1ζ Z

(1)⟩=

⎡⎣ 03

8c2

⎤⎦. (5.41)

Thus, from (4.15) and (5.41), it follows that Z∗(1) is given by

Z∗(1) =

⎡⎢⎢⎣−3

8c3 sin(t + θ)

−3

8c3 cos(t + θ)

⎤⎥⎥⎦. (5.42)

In view of (4.21), Z∗(1) can be written explicitly in terms of the new variables ζ1 and ζ2:

Z∗(1) =

⎡⎢⎢⎣3

8ζ2

(ζ2

1 + ζ22

)−3

8ζ1

(ζ2

1 + ζ22

)⎤⎥⎥⎦. (5.43)

Calculating the indefinite integral∫�Δ−1

ζZ(1) − 〈Δ−1

ζZ(1)〉�dt, one finds

∫[Δ−1ζ Z

(1) −⟨Δ−1ζ Z

(1)⟩]dt =

⎡⎢⎢⎣−1

32c3(4 cos 2(t + θ) + cos 4(t + θ))

1

32c2(8 sin 2(t + θ) + sin 4(t + θ))

⎤⎥⎥⎦. (5.44)


Multiplying this result by Δζ, it follows, according to (4.18), that T (1) is given by

T (1) =

⎡⎢⎢⎣−3

16c3 cos(t + θ) +

1

32c3 cos 3(t + θ)

− 3

16c3 sin(t + θ) − 3

32c3 sin 3(t + θ)

⎤⎥⎥⎦ + ΔζD(1). (5.45)

Taking D(1) as a null vector and using (4.21), T (1) can be written explicitly in terms of the new

variables ζ1 and ζ2 as follows:

T (1) =

⎡⎢⎢⎣−5

32ζ3

1 −9

32ζ1ζ

22

15

32ζ2

1ζ2 +3

32ζ3

2

⎤⎥⎥⎦. (5.46)

In the second-order approximation, one finds after lengthy calculations using MAPLE

software:

Ψ(2) =

⎡⎢⎢⎣−69

256ζ4

1ζ2 +27

128ζ2

1ζ32 +

27

256ζ5

2

165

256ζ5

1 +69

128ζ3

1ζ22 −

27

256ζ4

2ζ1

⎤⎥⎥⎦. (5.47)

Repeating the procedure described in the above paragraphs, one finds

⟨Δ−1ζ Ψ(2)

⟩=

⎡⎣ 0

− 51

256c4

⎤⎦,

Z∗(2) =

⎡⎢⎢⎣−51

256ζ2

(ζ2

1 + ζ22

)2

51

256ζ1

(ζ2

1 + ζ22

)2

⎤⎥⎥⎦.(5.48)

Taking D(2) as a null vector, it follows that

T (2) =

⎡⎢⎢⎣19

256ζ5

1 +13

32ζ3

1ζ22 +

65

256ζ1ζ

42

− 95

256ζ2ζ

41 −

13

32ζ2

1ζ32 −

13

256ζ5

2

⎤⎥⎥⎦. (5.49)


The new system of differential equations and the generating function are given, up to

the second-order of the small parameter ε, by

dζ1

dt= ζ2 + ε

3

8ζ2

(ζ2

1 + ζ22

)− ε2 51

256ζ2

(ζ2

1 + ζ22

)2,

dζ2

dt= −ζ1 − ε

3

8ζ1

(ζ2

1 + ζ22

)+ ε2 51

256ζ1

(ζ2

1 + ζ22

)2,

(5.50)

T1 = −ε(

5

32ζ3

1 +9

32ζ1ζ

22

)+ ε2

(19

256ζ5

1 +13

32ζ3

1ζ22 +

65

256ζ1ζ

42

),

T2 = ε(

15

32ζ2

1ζ2 +3

32ζ3

2

)+ ε2

(− 95

256ζ2ζ

41 −

13

32ζ2

1ζ32 −

13

256ζ5

2

).

(5.51)

The Lagrange variational equations for the new system of differential equations,

defined by (5.50), are given by

dc

dt= 0,

dθ

dt= ε

3

8c2 − ε2 51

256c4.

(5.52)

These differential equations are the well-known equations obtained through Krylov-

Bogoliubov method [5].As described in Section 5.1.1, the solution of the new system of differential equations,

defined by (5.50), can be obtained by introducing the solution of the above variational

equations into (4.21).The original variables x and x are calculated through (2.6), which provides the

following second-order asymptotic solution:

x = ζ1 − ε1

32ζ1

(5ζ2

1 + 9ζ22

)+ ε2 1

2048

(227ζ5

1 + 742ζ31ζ

22 + 547ζ1ζ

42

),

x = ζ2 + ε3

32ζ2

(5ζ2

1 + ζ22

)− ε2 1

2048

(77ζ5

2 + 922ζ32ζ

21 + 685ζ2ζ

41

).

(5.53)

These equations are in agreement with the solution obtained through the canonical version

of the Hori method [6].

5.2. Determination of Asymptotic Solutions through Simplified Algorithm II

5.2.1. Van der Pol Equation

For the van der Pol equation, the function f(x, x) is written in terms of the variables z1 = c′

and z2 = θ′ by

f(x, x) = f(z1 cos(t + z2),−z1 sin(t + z2)) = −(z2

1 co s2(t + z2) − 1)z1 sin(t + z2). (5.54)



Z(1) =

⎡⎢⎢⎣1

2z1

(1 − 1

4z2

1 − cos 2(t + z2) +1

4z2

1 cos 4(t + z2))

1

2

(1 − 1

2z2

1

)sin 2(t + z2) −

1

8z2

1 sin 4(t + z2)

⎤⎥⎥⎦. (5.55)

As mentioned before, two different choices of D(m) will be made, and two generating

functions will be determined.

(1) First Asymptotic Solution

Following the simplified algorithm II defined by (4.41) and (4.42), the first-order terms Z∗(1)

and T (1) are calculated as follows.

Taking the secular part of Z(1), with ζ replacing z, one finds

Z∗(1) =

⎡⎣1

2ζ1

(1 − 1

4ζ2

1

)0

⎤⎦, (5.56)

and, integrating the remaining part,

T (1) =

⎡⎢⎢⎣−1

4ζ1 sin 2(t + ζ2) +

1

32ζ3

1 sin 4(t + ζ2)

−1

4

(1 − 1

2ζ2

1

)cos 2(t + ζ2) +

1

32ζ2

1 cos 4(t + ζ2)

⎤⎥⎥⎦ +D(1), (5.57)

with D(1)i = D(1)

i (ζ1), i = 1, 2.

Following the algorithm of the Hori method described in Section 2, the second-order

equation, (2.11), involves the term Ψ(2) given by

Ψ(2) =1

2

[Z(1) + Z∗(1), T (1)

]. (5.58)


After tedious lengthy calculations using MAPLE software, one finds

Ψ(2)1 =

(7

64ζ3

1 −3

128ζ5

1

)sin 2(t + ζ2)

− 1

32ζ3

1 sin 4(t + ζ2) −1

128ζ5

1 sin 6(t + ζ2)

+D(1)1

(1

2− 3

8ζ2

1 −1

4cos 2(t + ζ2) +

3

16ζ2

1 cos 4(t + ζ2))

+D(1)2

(1

2ζ1 sin 2(t + ζ2) −

1

4ζ3

1 sin 4(t + ζ2))

+dD

(1)1

dζ1

(−1

2ζ1 +

1

8ζ3

1 +1

4ζ1 cos 2(t + ζ2) −

1

16ζ3

1 cos 4(t + ζ2)),

Ψ(2)2 = −1

8+

3

16ζ2

1 −11

256ζ4

1 −(

1

32ζ2

1 +1

64ζ4

1

)cos 2(t + ζ2)

+(− 1

32ζ2

1 +1

128ζ4

1

)cos 4(t + ζ2) −

1

128ζ4

1 cos 6(t + ζ2)

+D(1)1

(−1

4ζ1 sin 2(t + ζ2) −

1

8ζ1 sin 4(t + ζ2)

)+D(1)

2

((1

2− 1

4ζ2

1

)cos 2(t + ζ2) −

1

4ζ2

1 cos 4(t + ζ2))

+dD

(1)2

dζ1

(−1

2ζ1 +

1

8ζ3

1 +1

4ζ1 cos 2(t + ζ2) −

1

16ζ3

1 cos 4(t + ζ2)).

(5.59)

In order to obtain the same averaged Lagrange variational equations given by (5.28a)and (5.28b), D(1) must be taken as

D(1) =

⎡⎢⎣ 0

−1

4+

1

16ζ2

1

⎤⎥⎦. (5.60)

Thus, it follows that

Z∗(2) =

⎡⎢⎣ 0

−1

8+

1

8ζ2

1 −7

256ζ4

1

⎤⎥⎦. (5.61)

In view of the choice of D(1), T (1) is then given by

T (1) =

⎡⎢⎢⎣−1

4ζ1 sin 2(t + ζ2) +

1

32ζ3

1 sin 4(t + ζ2)

−1

4+

1

16ζ2

1 −1

4

(1 − 1

2ζ2

1

)cos 2(t + ζ2) +

1

32ζ2

1 cos 4(t + ζ2)

⎤⎥⎥⎦. (5.62)


Repeating the procedure for the third-order approximation, and, taking

D(2) =

⎡⎣− 1

16ζ1 +

15

256ζ3

1 −7

512ζ5

1

0

⎤⎦, (5.63)

one finds

T(2)1 =

1

1536

(−96ζ1 + 90ζ3

1 − 21ζ51

)+(

1

16ζ1 −

9

128ζ3

1 +9

768ζ5

1

)cos 2(t + ζ2)

+(− 1

128ζ3

1 +1

256ζ5

1

)cos 4(t + ζ2) +

1

768ζ5

1 cos 6(t + ζ2),

T(2)2 =

(− 1

16+

3

64ζ2

1 −1

64ζ4

1

)sin 2(t + ζ2) +

(1

128ζ2

1 −1

256ζ4

1

)sin 4(t + ζ2)

− 1

768ζ4

1 sin 6(t + ζ2).

(5.64)

The new system of differential equations is given, up to the second-order of the small

parameter ε, by

dζ1

dt= ε

1

2ζ1

(1 − 1

4ζ2

1

), (5.65a)

dζ2

dt= ε2

(−1

8+

1

8ζ2

1 −7

256ζ4

1

). (5.65b)

These differential equations are exactly the same equations given by (5.28a) and (5.28b).The generating function is obtained from (5.62), (5.64), and it is given, up to the

second-order of the small parameter ε, by

T1 = ε(−1

4ζ1 sin 2(t + ζ2) +

1

32ζ3

1 sin 4(t + ζ2))

+ ε2

(1

1536

(−96ζ1 + 90ζ3

1 − 21ζ51

)+(

1

16ζ1 −

9

128ζ3

1 +9

768ζ5

1

)cos 2(t + ζ2)

+(− 1

128ζ3

1 +1

256ζ5

1

)cos 4(t + ζ2) +

1

768ζ5

1 cos 6(t + ζ2)),

T2 = ε(

1

16ζ2

1 −1

4

(1 − 1

2ζ2

1

)cos 2(t + ζ2) +

1

32ζ2

1 cos 4(t + ζ2))

+ ε2

((− 1

16+

3

64ζ2

1 −1

64ζ4

1

)sin 2(t + ζ2) +

(1

128ζ2

1 −1

256ζ4

1

)× sin 4(t + ζ2) −

1

768ζ4

1 sin 6(t + ζ2)).

(5.66)


The original variables x and x are calculated through (2.7), which provides, up to the

second-order of the small parameter, the following solution:

x = ζ1 cos(t + ζ2) − ε1

32ζ3

1 sin 3(t + ζ2)

+ ε2

{[3

256ζ3

1 −9

2048ζ5

1

]cos(t + ζ2) −

[1

128ζ3

1 +1

1024ζ5

1

]cos 3(t + ζ2)

− 5

3072ζ5

1 cos 5(t + ζ2)},

x = −ζ1 sin(t + ζ2) + ε{[

1

2ζ1 −

1

8ζ3

1

]cos(t + ζ2) −

3

32ζ3

1 cos 3(t + ζ2)}

+ ε2

{[1

8ζ1 −

35

256ζ3

1 +65

2048ζ5

1

]sin(t + ζ2) −

[3

128ζ3

1 −15

1024ζ5

1

]× sin 3(t + ζ2) +

25

3072ζ5

1 sin 5(t + ζ2)},

(5.67)

with ζ1 and ζ2 given by the solution of (5.65a) and (5.65b).Note that (5.29) and (5.67) give the same second-order asymptotic solution for the van

der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations, but they are

related through an equation similar to (4.21).

(2) Second Asymptotic Solution

Now, let us to takeD(1) andD(2) as null vectors. Equations (5.59) simplifies, andZ∗(2) is given

by

Z∗(2) =

⎡⎣ 0

−1

8+

3

16ζ2

1 −11

256ζ4

1

⎤⎦. (5.68)

The functions T (1) and T (2) are then given by

T (1) =

⎡⎢⎢⎣−1

4ζ1 sin 2(t + ζ2) +

1

32ζ3

1 sin 4(t + ζ2)

−1

4

(1 − 1

2ζ2

1

)cos 2(t + ζ2) +

1

32ζ2

1 cos 4(t + ζ2)

⎤⎥⎥⎦,T(2)1 = −

(7

128ζ3

1 −3

256ζ5

1

)cos 2(t + ζ2) +

1

128ζ3

1 cos 4(t + ζ2)

+1

768ζ5

1 cos 6(t + ζ2),

T(2)2 = −

(1

64ζ2

1 +1

128ζ4

1

)sin 2(t + ζ2) +

(− 1

128ζ2

1 +1

512ζ4

1

)sin 4(t + ζ2)

− 1

768ζ4

1 sin 6(t + ζ2).

(5.69)


The new system of differential equations is obtained from (5.56) and (5.68), and it is

given, up to the second-order of the small parameter, by

dζ1

dt= ε

1

2ζ1

(1 − 1

4ζ2

1

),

dζ2

dt= ε2

(−1

8+

3

16ζ2

1 −11

256ζ4

1

).

(5.70)

These differential equations are exactly the same equations given by (5.36a) and (5.36b).The generating function is obtained from (5.69), and it is given, up to the second-order

of the small parameter ε, by

T1 = ε(−1

4ζ1 sin 2(t + ζ2) +

1

32ζ3

1 sin 4(t + ζ2))

+ ε2

(−(

7

128ζ3

1 −3

256ζ5

1

)cos 2(t + ζ2) +

1

128ζ3

1 cos 4(t + ζ2) +1

768ζ5

1 cos 6(t + ζ2)),

T2 = ε(−1

4

(1 − 1

2ζ2

1

)cos 2(t + ζ2) +

1

32ζ2

1 cos 4(t + ζ2))

+ ε2

(−(

1

64ζ2

1 +1

128ζ4

1

)sin 2(t + ζ2) +

(− 1

128ζ2

1 +1

512ζ4

1

)sin 4(t + ζ2)

− 1

768ζ4

1 sin 6(t + ζ2)).

(5.71)

Equations (5.71) are in agreement with the solution obtained by Ahmed and Tapley [7]through a different integration theory for the Hori method.

A second-order asymptotic solution for the original variables x and x is calculated

through (2.7), and it is given by

x = ζ1 cos(t + ζ2) + ε{[

−1

4ζ1 +

1

16ζ3

1

]sin(t + ζ2) −

1

32ζ3

1 sin 3(t + ζ2)}

+ ε2

{[1

32ζ1 −

1

32ζ3

1 +15

2048ζ5

1

]cos(t + ζ2) +

[− 1

32ζ3

1 +5

1024ζ5

1

]× cos 3(t + ζ2) −

5

3072ζ5

1 cos 5(t + ζ2)},

x = −ζ1 sin(t + ζ2) + ε{[

1

4ζ1 −

1

16ζ3

1

]cos(t + ζ2) −

3

32ζ3

1 cos 3(t + ζ2)}

+ ε2

{[− 1

32ζ1 −

1

32ζ3

1 +25

2048ζ5

1

]sin(t + ζ2) +

[3

64ζ3

1 −3

1024ζ5

1

]× sin 3(t + ζ2) +

25

3072ζ5

1 sin 5(t + ζ2)},

(5.72)

with ζ1 and ζ2 given by the solution of (5.70).As before, note that (5.37) and (5.72) give the same second-order asymptotic solution

for the van der Pol equation. Recall that ζ1 and ζ2 have different meaning in these equations,

but they are related through an equation similar to (4.21).


5.2.2. Duffing Equation

For the Duffing equation, the function f(x, x) is written in terms of the variables z1 = c′and

z2 = θ′, by

f(x, x) = f(z1 cos(t + z2),−z1 sin(t + z2)) = −z31 co s3(t + z2). (5.73)


Z(1) =

⎡⎢⎢⎣1

4z3

1 sin 2(t + z2) +1

8z3

1 sin 4(t + z2)

3

8z2

1 +1

2z2

1 cos 2(t + z2) +1

8z2

1 cos 4(t + z2)

⎤⎥⎥⎦. (5.74)

Following the simplified algorithm II and repeating the procedure described in

Section 5.2.1, the first-order terms Z∗(1) and T (1) are obtained as follows. Taking the secular

part of Z∗(1), with ζ replacing z, and, integrating the remaining part, one finds

Z∗(1) =

⎡⎢⎣ 0

3

8ζ2

1

⎤⎥⎦, (5.75)

T (1) =

⎡⎢⎢⎣−1

32ζ3

1(4 cos 2(t + ζ2) + cos 4(t + ζ2))

1

32ζ2

1(8 sin 2(t + ζ2) + sin 4(t + ζ2))

⎤⎥⎥⎦. (5.76)

In the second-order approximation, one finds

Ψ(2) =

⎡⎢⎢⎣1

256ζ5

1(−33 sin 2(t + ζ2) − 12 sin 4(t + ζ2) + 3 sin 6(t + ζ2))

1

256ζ4

1(−51 − 99 cos 2(t + ζ2) − 18 cos 4(t + ζ2) + 3 cos 6(t + ζ2))

⎤⎥⎥⎦. (5.77)

Taking the secular part of Ψ(2), and, integrating the remaining part, one finds

Z∗(2) =

⎡⎣ 0

− 51

256ζ4

1

⎤⎦, (5.78)

T (2) =

⎡⎢⎢⎣1

512ζ5

1(33 cos 2(t + ζ2) + 6 cos 4(t + ζ2) − cos 6(t + ζ2))

1

512ζ4

1(−99 sin 2(t + ζ2) − 9 sin 4(t + ζ2) + sin 6(t + ζ2))

⎤⎥⎥⎦. (5.79)


The new system of differential equations is given, up to the second-order of the small

parameter ε, by

dζ1

dt= 0,

dζ2

dt= ε

3

8ζ2

1 − ε2 51

256ζ4

1.

(5.80)

These differential equations are exactly the same equations given by (5.52).The generating function is obtained from (5.76) and (5.79), and it is given, up to the

second-order of the small parameter ε, by

T1 = −ε 1

32ζ3

1(4 cos 2(t + ζ2) + cos 4(t + ζ2))

+ ε2 1

512ζ5

1(33 cos 2(t + ζ2) + 6 cos 4(t + ζ2) − cos 6(t + ζ2)),

T2 = ε1

32ζ2

1(8 sin 2(t + ζ2) + sin 4(t + ζ2))

+ ε2 1

512ζ4

1(−99 sin 2(t + ζ2) − 9 sin 4(t + ζ2) + sin 6(t + ζ2)).

(5.81)

A second-order asymptotic solution for the original variables x and x is calculated

through (2.7), and it is given by

x = ζ1 cos(t + ζ2) + ε1

32ζ3

1(−6 cos(t + ζ2) + cos 3(t + ζ2))

+ ε2 1

2048ζ5

1(303 cos(t + ζ2) − 78 cos 3(t + ζ2) + 2 cos 5(t + ζ2)),

x = −ζ1 sin(t + ζ2) − ε1

32ζ3

1(6 sin(t + ζ2) + 3 sin 3(t + ζ2))

+ ε2 1

2048ζ5

1(249 sin(t + ζ2) + 162 sin 3(t + ζ2) − 10 sin 5(t + ζ2)).

(5.82)

These equations are in agreement with the solution obtained through the canonical version

of the Hori method [6]. Note that (5.53) and (5.82) give the same second-order asymptotic

solution for the Duffing equation. Recall that ζ1 and ζ2 have different meaning in these

equations, but they are related through an equation similar to (4.21).

6. Conclusions

In this paper, the Hori method for noncanonical systems is applied to theory of nonlinear

oscillations. Two different simplified algorithms are derived from the general algorithm

proposed by Sessin. It has been shown that themth-order terms T(m)j andZ

∗(m)j that define the

near-identity transformation and the new system of differential equations, respectively, are

not uniquely determined, since the algorithms involve at each order arbitrary functions of the

constants of integration of the general solution of the undisturbed system. This arbitrariness

is an intrinsic characteristic of perturbation methods, since some kind of averaging principle


must be applied to determine these functions. The simplified algorithms are then applied in

determining second-order asymptotic solutions of two well-known equations in the theory

of nonlinear oscillations: van der Pol and Duffing equations. For van der Pol equation, the

appropriate use of the arbitrary functions allows the determination of the solution presented

by Hori. This solution defines a new system of differential equations with a different

frequency for the phase in comparison with the solution obtained by Ahmed and Tapley,

who used a different approach for determining the near-identity transformation and the new

system of differential equations for the Hori method, and, with the solution obtained by

Nayfeh through the method of averaging. For the Duffing equation, only one generating

function is determined, and the second simplified algorithm gives the same generating

function obtained through Krylov-Bogoliubov method.

References

[1] S. da Silva Fernandes, “Notes on Hori method for non-canonical systems,” Celestial Mechanics andDynamical Astronomy, vol. 87, no. 3, pp. 307–315, 2003.

[2] W. Sessin, “A general algorithm for the determination of T(n)j and Z

∗(n)j in Hori’s method for non-

canonical systems,” Celestial Mechanics, vol. 31, pp. 109–113, 1983.[3] G. Hori, “Theory of general perturbations with unspecified canonical variables,” Publications of the

Astronomical Society of Japan, vol. 18, pp. 287–295, 1966.[4] G. Hori, “Theory of general perturbations for non-canonical systems,” Publications of the Astronomical

Society of Japan, vol. 23, pp. 567–587, 1971.[5] A. H. Nayfeh, Perturbation Methods, Wiley-VCH, New York, NY, USA, 2004.[6] S. da Silva Fernandes, “Notes on Hori method for canonical systems,” Celestial Mechanics and Dynamical

Astronomy, vol. 85, no. 1, pp. 67–77, 2003.[7] A. H. Ahmed and B. D. Tapley, “Equivalence of the generalized Lie-Hori method and the method of

averaging,” Celestial Mechanics, vol. 33, no. 1, pp. 1–20, 1984.


Research ArticleIMU Fault Detection Based on χ2-CUSUM

Elcio Jeronimo de Oliveira,1 Helio Koiti Kuga,2and Ijar Milagre da Fonseca2

1 Space Systems Division, Institute of Aeronautics and Space (IAE),12.228-904 Sao Jose dos Campos, SP, Brazil

2 Mechanic and Control Division, National Institute for Space Research (INPE),12.227-010 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Elcio Jeronimo de Oliveira, [email protected]

Received 16 December 2011; Accepted 1 February 2012

Academic Editor: Tadashi Yokoyama

Copyright q 2012 Elcio Jeronimo de Oliveira et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The problem of fault detection and isolation (FDI) on inertial measurement units (IMUs) hasreceived great attention in the last years, mainly with growing use of IMU strapdown platformsusing fiber optic gyros (FOG) or micro electro mechanical systems (MEMSs). A way to solvethis problem makes use of sensor redundancy and parity vector (PV) analysis. However, theactual sensor outputs can include some anomalies, as impulsive noise which can be associatedwith the sensors itself or data acquisition process, committing the elementary threshold criteria ascommonly used. Therefore, to overcome this problem, in this work, it is proposed an algorithmbased on median filter (MF) for prefiltering and chi-square cumulative sum (χ2-CUSUM) only forfault detection (FD) applied to an IMU composed by four FOGs.

1. Introduction

The design of a FDI algorithm for applications on IMUs can, in general, be divided in

two types. The first one, named analytical redundancy, takes into account a mathematical

model of the system in which the IMU is used. This method generates a residue vector as a

result of the state observer [1–4] in order to indicate the IMU operational status. The second

one, named sensor redundancy, is based on specific geometrical configurations using extra

sensors and solving the FDI problem with the aid of the parity vector (PV) analysis [5, 6].However, the actual sensor outputs can include some anomalies, as impulsive noise which

can be associated with the sensors itself or with the data acquisition process [7], violating

the elementary threshold criteria as commonly used. This work applies the second type of

algorithm to the FD problem. The use of four sensors (minimal redundancy) constrains the

algorithm to detect the fault not allowing the identification (isolation) of the faulty sensor [8].


(a)

g4

g3g

2

g1

X

Y Z

αα

(b)

Figure 1: Tetrahedral base.

To overcome the anomalies addressed before, in this work is proposed an algorithm based

on MF that performs the prefiltering of FOG outputs and the use of the χ2-CUSUM for FD

problem.

2. Background

2.1. Geometry

The geometrical arrangement used in this work considers four gyros mounted on the faces of

a tetrahedral structure (tetrad), and three accelerometers in a triad configuration internally

fixed in the tetrahedral. The analysis performed here takes into account the gyros only,

and the extension for accelerometers is straightforward. The tetrad configuration and the

reference frame are shown in Figure 1. The mathematical representation of the arrangement

of the gyros is given in terms of direct cosine matrix (DCM), and is obtained from angular

relationship between sensor axes and the analytical triorthogonal axes (analytical triad).Therefore, considering the schema shown in Figure 1, where α = 54.736◦, the DCM is given

by

H =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1√3

0

√6

31√3

√2

2−√

6

61√3

−√

2

2−√

6

61 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (2.1)

The matrix H relates the sensor measurements (gi) with the angular rate in the main axes in

the form of

go = Hω. (2.2)


The estimate of the angular rate components in the main axes can be obtained from (2.2) in

the following manner:

ω = H∗go, (2.3)

H∗ =(HTH

)−1HT , (2.4)

where go = [g1 g2 g3 g4]T is the vector of the gyro outputs, ω = [ωx ωy ωz]

T is the angular

rate vector about main axes, ω is the angular rate estimate vector, and H∗ is the generalized

inverse of H.

Equation (2.4) provides the best state estimation in the least squares sense.

2.2. Parity Vector

The sensor equation considered in (2.2) can be rewritten by adding the faults, biases, and

noises components as follows:

go = Hω + δgo + f + ηs, (2.5)

where δgo is the bias vector whose magnitude is constant, f is the fault vector, and ηs is a

random term vector (Gaussian noise). Applying the singular value decomposition (SVD)on H, it can be obtained the range and null spaces from this matrix [9]. In addition, also is

computed the biases influence on the arrangement. Decomposing H as follows:

UTHV = Λ =(Σ0

), (2.6)

H = UΛVT , (2.7)

where U, Λ, and VT are matrices obtained from SVD of H. The matrix Σ is a diagonal

matrix whose elements are the eigenvalues of H. The superscript (T ) indicates the transpose.

Applying (2.7) into (2.5) and multiplying both sides by UT , it can be obtained the following

relationship:

UT go = ΛVTω +UT(δgo + f + ηs

). (2.8)

Partitioning U as follows:

U =[U1

... U2

], (2.9)


where U1 ∈ R4 × 3 and U2 ∈ R4 × 1, and applying it into (2.8), the resulting equations are

UT1 go = (ΣV)ω +UT

1

(δgo + f + ηs

), (2.10)

UT2 go = UT

2

(δgo + f + ηs

)= p, (2.11)

p = Cgo. (2.12)

Equation (2.10) leads to least squares estimate of ω, what is equivalent to (2.3), and can be

expressed by

ωLS = (ΣV)−1UT1 go. (2.13)

The meaning of (2.11) is that, if sensors faults and biases are zero, the resulting product of

parity vector with sensor measurements is a white noise with zero mean. Otherwise, if the

biases and/or faults values differ from zero, (2.11) is a “weighted” sensor errors summation.

Then, UT2 (or C) is null space of H and p is the parity vector.

2.3. Median Filter

In the image processing field, it is very common to employ filters that preserve edges or

abrupt transitions between distinct parts of the image or remove salt and pepper noise. These

filters generally compare the pixel under observation with its neighbors at certain window

and take a decision based on a statistical or threshold criteria. The simplest filter that meets

these requirements is the median filter (MF). Being the MF not an optimal filter, it preserves

discontinuities as jumps [10] and is also a robust estimator [11], besides being an easy and

fast way to remove outliers from signals [7].

2.3.1. Recursive Median Filter

In the time sequence processing by MF, it is possible to process the samples in the filter

window in two ways. (a) Process the samples in the window to obtain the output at

discrete time (k), shift the window to process the time (k + 1) over the original data without

replacement; this is the nonrecursive processing. (b) Process the samples in the window to

obtain the output at discrete time (k), replace the original sample x(k) by the filter output

y(k), shift the window and process the new samples with replacement; this is the recursivemedian filter. The nonrecursive MF is shown by (2.14) and the recursive MF is shown by (2.15),

y(k) = median{x(k −N), . . . , x(k), . . . , x(k +N)}, (2.14)

y(k) = median{y(k −N), . . . , y(k − 1), x(k), . . . , x(k +N)

}, (2.15)

where x represents the time series and the filter window has 2N + 1 samples.

The recursive form of the MF presents an expressive noise attenuation capacity

comparing with the nonrecursive form, this property leads the signal to a fast convergence

[12].


The appropriate size of the MF can be defined comparing the variance of the signal

filtered with the variance of the same signal filtered by an average filter. In this work it was

chosen the exponentially weighted moving average (EWMA) filter [13], setting the β factor

properly. The EWMA filter is defined as follows:

μn = βμn−1 +(1 − β

)xn, (2.16)

where μ is the estimated mean of the random variable x at time n and β is the EWMA factor.

The absolute value of the difference between variances is defined as

Δvar

(j)= abs

[var(MF(j)

)− var

(FExp(β)

)], (2.17)

where MF(j) is the size j MF output and FExp(β) is the EWMA filter output related to β factor.

2.4. χ2-CUSUM Algorithm

The cumulative sum (CUSUM) algorithm is widely used to detect changes in the mean value

of an independent Gaussian sequence. This algorithm is based on log-likelihood ratio and

defined as follows [14]:

s(y)= ln

pθ1

(y)

pθ0

(y) , (2.18)

where s(y) is the sufficient statistics, pθ(y) is probability density function with conditional

parameter θ.

Before a change, the parameter θ is constant and equals to θ0; after the variation it

assumes the value θ1.

Considering (2.18) and a sampling set of size N, the following decision function can

be established:

Skj =k∑i=j

si,

si = lnpθ1

(yi)

pθ0

(yi) , (2.19)

where the decision is stated as,

d =

{0 if SN1 < λ, H0 is chosen

1 if SN1 ≥ λ, H1 is chosen(2.20)


and the hypothesis test is defined as,

H0 : θ = θ0,

H1 : θ = θ1,(2.21)

where λ is a suitable threshold.

The alarm time (ta) after a change is defined by the following stopping rule:

ta =N · min{K : dK = 1}. (2.22)

In this form, the CUSUM algorithm requires prior knowledge about the parameter θ1 and

cannot be processed recursively. So, for online applications and considering the parameter

θ1 as unknown, it is required a modified version of the CUSUM algorithm. It is presented in

this work a modified version of the CUSUM based upon the weighted likelihood ratio and

the sequential probability ratio test (SPRT) to suit the unknown θ1 and allow for the online

processing as in [14]:

Λkj =∫∞

−∞

pθ1

(yj , . . . , yk

)pθ0

(yj , . . . , yk

)dF(θ1), (2.23)

where the stopping time (or alarm time) ta is defined as

ta = min

{k : max

1≤j≤kln Λk

j ≥ λ}

(2.24)

and dF(θ1) is a weighting function.

Considering the prior knowledge about σ2 and μ0 of a Gaussian sequence with

distribution F(θ1) = F(μ) concentrated on two points μ0−ν, and μ0+ν the weighted likelihood

ratio becomes

Λkj =∫∞

−∞exp

[bSkj −

b2

2

(k − j + 1

)]dF(ν), (2.25)

where b = ν/σ is the signal-to-noise ratio (SNR), ν = μ1 − μ0, and

Skj =1

σ

k∑i=j

(yi − μ0

). (2.26)

Solving (2.25),

Λkj = cosh

(bSkj

)e−(b

2/2)(k−j+1)

= cosh[b(k − j + 1

)χkj

]e−(b

2/2)(k−j+1),(2.27)


where

χkj =1

k − j + 1

∣∣∣Skj ∣∣∣. (2.28)

Equation (2.27) is the probability ratio to noncentral parameter test of a χ2 distribution with

one degree-of-freedom between values 0 and (k − j + 1)b2. So, by using this approach, we

reach χ2-CUSUM algorithm.

As a consequence of the previous derivation, ta is now defined as

ta = min{k : gk ≥ λ

}, (2.29)

where

gk = max1≤j≤k

[ln cosh

(bSkj

)− b2

2

(k − j + 1

)]. (2.30)

By making a slight modification and considering the CUSUM algorithm as a repeated SPRT

[14] with lower and upper thresholds fixed in 0 and λ, respectively, the recursive form of the

χ2-CUSUM algorithm takes the following form:

gk =(Skk−Nk+1

)+,

Skk−Nk+1 = −1

2Nkb

2 + ln cosh(bSkk−Nk+1

),

Skk−Nk+1 = Sk,

Sk = Sk−11{gk−1>0} +yk − μ0

σ,

(2.31)

where Nk =Nk−11{gk−1>0} + 1.

3. Fault Detection Algorithm

3.1. FD Algorithm Structure

The detection of faults in a IMU with redundant sensors can be performed by PV analysis.

Under normal conditions, that is, bias and faults with null values, the PV should present a

normal distribution N(0, σ2). However, in actual systems, impulsive noise may occur. This

situation is described in [7]. As stated above, the MF is used in order to remove those

impulses and to improve the detection performed by χ2-CUSUM algorithm. In Figure 2, it

is shown the FD processing structure. Again, in this form, the algorithm only alarms the

fault occurrence because it is not possible to define which sensor is under fault condition.

After processing the signal from gyros by a data acquisition system, the data are filtered by

a recursive MF (2.15) of size 3 or 5 in order to remove the spikes (or outliers) generated in

this phase. In the sequence, the information (in mV/deg/s) is converted by a scale factor, SF

(in deg/s). The SF block makes a polynomial conversion, whose degree is 7. The PV block


Gyro number 1

Gyro number 2

Gyro number 3

Gyro number 4

MF (1) SF Filtering

PV

MF (2)

Sensor

matrix

Fault ? YN

Alarm

(H∗)

χ2-CUSUM

Fau

lt d

etec

tio

n b

lock

x

zY

GN&C

Figure 2: Gyro processing and fault detection block diagram. MF: median filter (MF(1): size 3//MF(2):size 11); SF: scale factor; GN&C: guidance, navigation, and control; PV: parity vector.

3 5 7 9 11 13 150

1

2

3

MF window size (samples)

×10−4

MF: 3–15 samples

FExp: β = 0.95

ab

s(v

ar(M

F)-

var(F

Ex

p))

Figure 3: Absolute value of the difference between variances of the FM(j) and FExp(0.95).

performs the multiplication of the null space of the actual H matrix (3.1) with the gyro output

vector in deg/s, forming the parity vector. This PV is filtered, again, by another recursive MF

of size 11. This size was obtained using (2.16) and (2.17), setting β = 0.95. The results obtained

by processing (2.17) are shown in Figure 3. It can be seen that difference from the results

obtained for sizes 9 to 15 is negligible. So, after some tests, it was concluded that size 11 is the

best choice. The actual parameters used in the algorithm were obtained from the calibration

of the IMU and are presented in [7]. The values and operations associated to blocks in the

Figure 2 are given as follows.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

SNR (b)

σ = 0.022 ◦/s

μ0 = 0.002 ◦/s

λ = 30

Rati

o[

tota

l(gk>λ)

/

tota

l(n

um

ber

of

sam

ple

s)]

0%

Figure 4: Ratios for b varying from 0.1 to 5.

For the actual sensor matrix (this matrix is obtained from calibration process) (H) and

respective null space matrix (C) and generalized inverse (H∗),

H =

⎛⎜⎜⎝0.57868624 −0.00432436 0.81553879

0.57733766 0.70734792 −0.40784817

0.57695344 −0.70679030 −0.40935582

0.99999805 0.00100995 0.00169158

⎞⎟⎟⎠, (3.1)

C =[−0.40955 −0.40903 −0.40585 0.70729

],

CH = 0,

H∗ =(HTH

)−1HT .

(3.2)

For the scale factor block, the polynomials of degree 7 are,

go1

(◦s

)= 0.0657g7

v − 3.64 × 10−4g6v − 7.799 × 10−2g5

v + 9.911 × 10−4g4v

+ 1.200g3v + 3.085 × 10−4g2

v + 37.432gv − 3.724 × 10−3,

go2

(◦s

)= 0.09452g7

v + 1.083 × 10−5g6v − 0.1965g5

v − 2.041 × 10−3g4v

+ 1.5219g3v + 6.910 × 10−3g2

v + 38.305gv − 4.615 × 10−3,

go3

(◦s

)= 0.0761g7

v − 2.898 × 10−4g6v − 0.13277g5

v − 8.253 × 10−4g4v

+ 1.3711g3v + 2.326 × 10−3g2

v + 38.234gv − 1.1713 × 10−2,

go4

(◦s

)= 0.0814g7

v − 1.0544 × 10−4g6v − 0.18366g5

v − 1.3751 × 10−3g4v

+ 1.4068g3v + 3.9596 × 10−3g2

v + 37.282gv − 4.1806 × 10−3,

(3.3)


0

0.05

0.1

−0.050 0.5 1 1.5 2

Samples ×104

(a)

0

500

1000

−500

−10000 0.5 1 1.5 2

Samples ×104

(b)

0 0.5 1 1.5 2

0

50

100

Samples

−50

×104

(c)

0

20

40

60

80

100

0 0.5 1 1.5 2

Samples ×104

λ = 30gk

(d)

Figure 5: χ2-CUSUM results for 20000 samples of a movement sequence without faults with (SNR) b = 0.1.

(a) PV; (b) Sk ; (c) Sk ; (d) gk .

and the PV block processes the following operation:

p = Cgo, (3.4)

where p is filtered by MF(2) and applied to χ2-CUSUM algorithm (2.31) in the form of yk.

The filtering block in the Figure 2 is an optional filter to comply with the GN&C block

requirements.

3.2. Calibration of the FD Algorithm

The calibration of the FD algorithm was performed offline by using a time series extracted

from the IMU movement on a 2DOF rotating table. The first parameter to be determined is


0

0.02

0.04

0.06

−0.02

−0.04

0 0.5 1 1.5 2

Samples×104

(a)

0

10

20

30

0 0.5 1 1.5 2

Samples×104

(b)

0 0.5 1 1.5 2

0

10

20

30

Samples

−10

×104

(c)

0

10

20

30

0 0.5 1 1.5 2

Samples×104

λ = 30

gk

(d)

Figure 6: χ2-CUSUM results for 20000 samples of a movement sequence without faults with (SNR) b = 3.5.

(a) PV; (b) Sk ; (c) Sk ; (d) gk .

the threshold (λ), obtained by use of the Kullback information [14] as follows:

l ≈ λ

K(ν),

K(ν) =ν2

2σ2=

1

2b2,

(3.5)

λ ≈ 1

2b2l, (3.6)

where l is the average number of samples until the alarm time (ta).Of course, the parameters b and l in (3.6) are unknown. So, to overcome this condition,

it is assumed the SNR (b) equals to 1 and l with properly size. In this work, it was considered

l equals to 60. Thus, the threshold is fixed as

λ =1

212 × 60 = 30. (3.7)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

Samples

−0.1

−0.2

−0.3

(◦/

s)

×104

Parity vector ltered by MF (blue)and nonltered (red)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.05

0.1

0.15Parity vector filtered by MF (blue) and FExp (red)

Samples

−0.050.99 1 1.01

×104

−0.020

0.02

0.04

0.06

(◦/

s)

×104

(b)

Figure 7: PV with step fault for sample 10000. (a) PV nonfiltered (red) and filtered by MF of size 11 (blue);(b) PV filtered by MF of size 11 (blue) and by FExp with β = 0.95 (red); the zoom in (b) indicates thedifference between MF and FExp during the transition time.

Once the threshold value is set and considered the actual values associated to the PV (σand μ0) as given in Table 1, it was performed the following computation the χ2-CUSUM

algorithm:

% = 100 × Total of gk > λ

Total number of samples. (3.8)

The ratio in (3.8) is used to determine the actual value of b that reach 0% of samples crossing

the threshold in normal condition (without fault). In other words, 0% of false alarms. It is

shown in Figure 4 the ratios for b varying from 0.1 to 5, in whose range from 3.5 for SNR

meets the objective. The efficiency of the calibration can be seen in Figures 5 and 6, it is shown

the algorithm outputs for (SNR) b = 0.1 and b = 3.5, respectively. The main information is

presented in Figures 5 and 6(d), where it is illustrated the behavior of the decision function

with respect to the threshold (λ = 30).


0 0.5 1 1.5 2

0

0.05

0.1

0.15

Samples

−0.05

×104

(a)

0 0.5 1 1.5 2

0

1

2

3

Samples×104

×104

−1

(b)

1 1.002 1.004 1.006

0

10

20

30

40

50

Samples

−10

×104

(c)

0

5

10

15

20

25

30

1 1.002 1.004 1.006

Samples×104

gk

λ = 30

ta = 10054

t0 = 10000

(d)

Figure 8: χ2-CUSUM results for 20000 samples of a movement sequence with a step fault at sample 10000

and magnitude 0.15◦/s; (SNR) b = 3.5. (a) PV; (b) Sk ; (c) Sk ; (d) gk .

4. Results

After the calibration phase, it was injected a step bias fault of magnitude 0.15 deg/s into one

of sensors for the sample 10000. This fault generated a step variation in the PV according to

Figures 7 and 8(a) and slopes as a result of the χ2-CUSUM algorithm processing in the Figures

8(b)–8(d). In the Figure 7(a), it is shown the efficiency of the median filter in impulsive

noise reduction process, and in letter (b) is shown a comparison between MF and FExp

in terms of filtering delay. In the decision function curve Figure 8(d), it is illustrated the

difference between actual time occurrence of the fault (t0 = 10000) and the actual time alarm

(ta = 10054), whose delay is 54 samples. In addition, there are another delays that should

be computed which are associated to median filters (MF(1) and MF(2)). The total processing

delay of the filters and algorithm is summarized in Table 2. In this table, it is presented the

processing of the step faults with values 0.10, 0.15, 0.20, and 0.30 deg/s in order to compare

the effect of the fault level on the detection delay. Considering the case of this work, where

the FOGs are sampled at 100 Hz, the delay is less than 0.15 seconds for the fault level higher

than 0.20 deg/s.


Table 1: Tuning parameters for χ2-CUSUM algorithm.

σ(◦/s) μ0(◦/s) b λ

0.022 0.002 0.1 to 5 30

Table 2: Fault detection delay as a function of fault magnitude given in number of samples with respect tothe parameters b = 3.5 and λ = 30.

Step fault 1st MF delay 2nd MF delay Detection Total delay

(◦/s) (MF3) (MF11) delay (ta − t0)0.10 1 5 430 436

0.15 1 5 54 60

0.20 1 5 9 15

0.30 1 5 4 10

5. Conclusions

In this paper, a method based on χ2-CUSUM algorithm combined with median filter was

applied to detect faults in inertial measurement units with minimal redundancy of fiber

optics gyros. The effectiveness of algorithm in case of low level step faults was demonstrated

achieving the requirements of short delay to alarm with high reliability. In addition, the

calibration technique presented here also proved to be efficient.

References

[1] E. Y. Chow and A. S. Willsky, “Analytical redundancy and the design of robust failure detectionsystems,” IEEE Transactions on Automatic Control, vol. AC-29, no. 7, pp. 603–614, 1984.

[2] J. C. Deckert, M. N. Desai, J. J. Deyst, and A. S. Willsky, “F-8 dfbw sensor failure identification usinganalytic redundancy,” IEEE Transactions on Automatic Control, vol. AC-22, no. 5, pp. 795–804, 1977.

[3] X. C. Lou, A. S. Willsky, and G. C. Verghese, “Failure detection with uncertain models,” in Proceedingsof the IEEE American Control Conference, pp. 956–959, 1983.

[4] Z. Han, W. Li, and S. L. Shah, “Fault detection and isolation in the presence of process uncertainties,”Control Engineering Practice, vol. 13, no. 5, pp. 587–599, 2005.

[5] M. A. Sturza, “Skewed axis inertial sensor geometry for optimal performance,” in Proceedings of theDigital Avionics Systems Conference (AIAA ’88), pp. 128–135, San Jose, Calif, USA, 1988.

[6] S. Kim, Y. Kim, and C. Park, “Failure diagnosis of skew-configured aircraft inertial sensors usingwavelet decomposition,” IET Control Theory and Applications, vol. 1, no. 5, pp. 1390–1397, 2007.

[7] E. J. de Oliveira, Fault detection and isolation in inertial measurement units with minimal redundancy offiber optic gyros, Ph.D. thesis, National Institute for Space Research (INPE), Sao Jose dos Campos, SP,Brazil, 2011.

[8] P. G. Savage, Introduction to Strapdown Inertial Navigation Systems, Strapdown Associates, Maple Plain,Minn, USA, 11st edition, 2005.

[9] D. S. Shim and C. K. Yang, “Geometric FDI based on SVD for redundant inertial sensor systems,” inProceedings of the 5th Asian Control Conference, pp. 1094–1100, July 2004.

[10] S. V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction, John Wiley & Sons, New York,NY, USA, 2nd edition, 2000.

[11] P. J. Rousseeuw, “Robust estimation and identifying outliers,” in Handbook of Statistical Methods forEngineers and Scientists, H. M. Wadsworth, Ed., pp. 16.1–16.24, MacGraw-Hill, New York, NY, USA,1st edition, 1990.

[12] G. Qiu, “An improved recursive median filtering scheme for image processing,” IEEE Transactions onImage Processing, vol. 5, no. 4, pp. 646–648, 1996.


[13] V. A. Siris and F. Papagalou, “Application of anomaly detection algorithms for detecting SYN floodingattacks,” Computer Communications, vol. 29, no. 9, pp. 1433–1442, 2006.

[14] M. Basseville and I. V. Nikiforov, Detection of Abrupt Changes, Prentice-Hall, Upper Saddle River, NJ,USA, 1 edition, 1993.


Research ArticleLow-Thrust Orbital Transfers in theTwo-Body Problem

A. A. Sukhanov1 and A. F. B. A. Prado2

1 Space Research Institute (IKI) RAS, Moscow, Russia2 Division of Space Mechanics and Control, National Institute for Space Research (INPE),Av dos Astronautas 1758, Jd. da Granja, Sao Jose dos Campos, SP 12227-010, Brazil

Correspondence should be addressed to A. F. B. A. Prado, [email protected]

Received 6 December 2011; Accepted 6 February 2012


Copyright q 2012 A. A. Sukhanov and A. F. B. A. Prado. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Low-thrust transfers between given orbits within the two-body problem are considered; the thrustis assumed power limited. A simple method for obtaining the transfer trajectories based on thelinearization of the motion near reference orbits is suggested. Required calculation accuracy canbe reached by means of use of a proper number of the reference orbits. The method may be usedin the case of a large number of the orbits around the attracting center; no averaging is necessaryin this case. The suggested method also is applicable to the cases of partly given final orbit and ifthere are constraints on the thrust direction. The method gives an optimal solution to the linearizedproblem which is not optimal for the original nonlinear problem; the difference between theoptimal solutions to the original and linearized problems is estimated using a numerical example.Also examples illustrating the method capacities are given.

1. Introduction

In this paper spiral transfers with power-limited low thrust between two given orbits are

considered. The transfer trajectory may include a large number of orbits around the attracting

center if the transfer is performed in a strong gravity field; this situation takes place, for

example, near Earth. This makes optimization of the transfer more difficult.

There are various mathematical methods for optimization of multirevolution orbital

transfers [1–7], most of them are based on averaging of motion. However, the methods have

some defects: they are complicated [3, 5] or have a limited application, that is, are applicable

only to the circular or neighboring orbits [1, 6, 7] or only to the planar or coaxial orbits [2, 4,

6, 7].This paper suggests a simple mathematical method for calculation of the orbital

transfers within the two-body problem. (A paper describing a generalization of the method


to any force field will be published in the Cosmic Research journal.) The method is based on a

linearization of motion near some reference orbits; in this respect the method is similar to the

Method of Transporting Trajectory (MTT) for solving the two-point boundary value problem

(TPBVP) suggested in [8] and developed in [9–12]. The method suggested here makes it

possible obtaining transfer trajectories of the following types:

(i) transfer between a given position in the initial orbit and an optimally determined

position in the final orbit (this case takes place, for example, if the spacecraft

launches from a given orbit at a given time and phasing of the final orbit is not

necessary);

(ii) transfer between the initial orbit and a given position in the final orbit with

obtaining an optimal position of the launch (this situation takes place, for instance,

in the transfer to a given point of geostationary orbit if the time of launch from the

initial orbit may be selected arbitrarily);

(iii) transfer between two given orbits with obtaining optimal launch and arrival

positions (this is a classical case of the orbital transfer).

The method suggested in this paper does not need any averaging of motion. Required

calculation accuracy is reached by means of use of a proper number of the reference orbits:

the bigger is the number, the shorter are intervals of linearization and the higher is accuracy.

The method does not put any limits on the number of orbits around the attracting center and

on the number of the reference orbits. The suggested method is applicable also to the case

of a partly given final orbit (e.g., only semimajor axis and eccentricity or only orbital energy

may be given) and to the case of constraints imposed on the thrust direction.

However, the method gives an optimal solution to the linearized problem and this

solution is not optimal for the initial nonlinear problem. Note that this nonoptimality is

not caused by the linearization errors, but is a principal sequence of the substitution of the

original problem for the linearized one. A comparison of the optimal solutions to the initial

and linearized problems made for a numerical example is given in the paper. Other numerical

examples illustrate the method capacities.

2. Statement of the Problem

The spacecraft motion is described by the following:

x = f (x, t) + g, (2.1)

where x = {r,v} is the spacecraft state vector, f = f(x, t) = {v, fv}, fv = fv(x, t) is the

acceleration caused by the external forces,

g = {0, α}. (2.2)

In the considered case of the two-body problem,

fv = − μr3r. (2.3)


x(t)ξ(t)

t = 0

yiT t = Tyi0 = x0

yfT = xT

Final orbit

Transfer trajectory

Initial orbitqi

qf

ξT

dddd

Figure 1: Transfer between given orbits.

Let us assume for simplicity that the electric power of the propulsion system is constant. This

case takes place, for example, if a nuclear power source is used or if the transfer is performed

near a planet with nearly circular orbit. Then the performance index for the power-limited

thrust is

J =1

2

∫T0

α2dt, (2.4)

where α = |α|. Minimal value of the functional (2.4) gives minimal propellant consumption

[13].Let qi,qf be 5-dimensional vectors of orbital elements defining the initial and final

orbits. The problem is to find transfer trajectory between the initial and final orbits in given

time T in which minimal value of the performance index (2.4) is reached.

3. Transfer between Given State and Given Orbit

The boundary values for the problem formulated in Section 2 may be written as

x0 = yi0, xT = yfT , (3.1)

where yi,yf are state vectors of the initial and final orbits. The case will be considered in this

section when vector yi0 is given and vector yfT is not given.

3.1. Transfer between Neighboring Orbits

First let us assume that the initial and final orbits are close to each other. Then the equation

of motion (2.1) can be linearized near the initial orbit as follows (Figure 1):

ξ = Fξ + g, (3.2)

where

ξ = ξ(t) = x(t) − yi(t), (3.3)


is state vector of the linearized motion and

F =∂f∂yi

. (3.4)

Vector f and matrix F in (3.2) and (3.4) are calculated in the initial orbit, this is why matrix Fis a function of time even if vector f does not depend on time explicitly. The Hamiltonian for

(3.2) and performance index (2.4) is [14]

H = −α2

2+ ptFξ + Ptvα + pt, (3.5)

where p = {pr ,pv} is a vector of the costate variables corresponding to (3.2), pv is Lawden’s

primer vector, pt is costate variable corresponding to the additional equation t = 1 making

the system autonomous. Vector p satisfies the following costate variational equation:

pt = −(∂H

∂ξ

)t= −ptF, (3.6)

variable pt satisfies the following equation:

pt = −∂H∂t

= −poFξ. (3.7)

Let us consider sixth-order matrix Ψ = Ψ(t) which is a general solution to (3.6) with initial

value Ψ(0) = I. Matrix Ψ is costate transition matrix given by

Ψ = ∂yi0/∂yi. (3.8)

Dividing matrix Ψ into two 6 × 3-dimensional submatrices as follows:

Ψ = [ΨrΨv], (3.9)

the costate variables may be represented as

p = Ψtβ, pv = Ψtvβ, (3.10)

where β is a constant 6-dimensional vector. The Hamiltonian (3.5) reaches its maximum if

α = pv = Ψtvβ. (3.11)

Solution to (3.2) is given by the Cauchy formula

ξ = ξ(t) = ξ0 +∫ t

0

Φ(t, τ)gdτ, (3.12)


where

Φ(t1, t2) =∂yi(t1)∂yi(t2)

(3.13)

is the state transition matrix. Due to (3.1) and (3.3) ξ0 = 0 in (3.12). Using (2.2), (3.9), and

(3.11) and relations

Φ(t, τ) = Φ(t, 0)Φ(0, τ) = Φ(t, 0)Φ−1(τ, 0), Φ = Φ(t, 0) = Ψ−1, (3.14)

the solution (3.12) can be represented as

ξ = ΦSβ, (3.15)

where

S = S(t) =∫ t

0

ΨvΨtvdt (3.16)

is matrix of sixth order (also see [9]). As follows from (3.3), (3.11), and (3.15), in order to find

optimal thrust vector and the state vector of the transfer trajectory, it is sufficient to obtain

vector β. Nonspecified state vector yfT may be found using transversality condition which in

the considered case is

pT = p(T) =

(∂qf∂yfT

)t

σ, (3.17)

where σ is an arbitrary constant 5-dimensional vector. Due to the closeness of the initial and

final orbits, the approximate equality ∂qf/∂yf ≈ U is fulfilled, where

U = U(t) =∂qi∂yi(t)

, (3.18)

is a 5 × 6-dimensional matrix. Then the condition (3.17) takes the form

pT = UtTσ. (3.19)

Equations (3.10) and (3.19) and the last relation of (3.14) give

β = (UTΦT )tσ. (3.20)

The closeness of the initial and final orbits makes it possible to use the following approximate

linear relation:

Δq = qf − qi = UTξT . (3.21)


Equations (3.15), (3.20), and (3.21) give

Δq = Wσ, (3.22)

where

W = UTΦTST (UTΦT )t (3.23)

is 5-dimensional matrix. Using (3.8), (3.9), (3.13), (3.14), (3.16), and (3.18), we obtain

UTΦT =∂qi∂yiT

∂yiT∂yi0

=∂qi∂yi0

= U0, (3.24)

UTΦTST (UTΦT )t =∂qi∂yi0

∫T0

∂yi0∂vi

(∂yi0∂vi

)tdt

(∂qi∂yi0

)t, (3.25)

where vi = vi(t) is velocity vector in the initial orbit. (3.23), (3.25) give

W = U0STUt0 =∫T

0

QQt dt, (3.26)

where

Q = Q(t) = −∂qi∂vi

(3.27)

is a 5× 3-dimensional matrix; as is seen from (3.18) and (3.27), matrix Q is a part of matrix U.

Using (3.20), (3.22), and (3.24), vector β can be found as follows:

β = Ut0W

−1Δq. (3.28)

Now due to (3.3), (3.11), (3.15), and (3.28), the optimal thrust vector and the spacecraft state

vector are

α = ΨtvU

t0W

−1Δq, (3.29)

x(t) = yi(t) +ΦSUt0W

−1Δq. (3.30)

Equations (3.1) and (3.30) give the state vector of entry into the final orbit

yfT = yiT +ΦTSTUt0Δq. (3.31)


Final orbit

Transfer trajectory

Reference orbits

Initial orbit

t0 = 0

t1

t2

tj

tn = Tq1

q2

qj

qn−1

qn = qf

q0 = qi

Figure 2: Transfer between arbitrary orbits.

Substituting (3.29) into (2.4) and taking into account definitions (3.16), (3.26), the minimal

value of the performance index can be found as follows:

J =1

2ΔqtW−1Δq. (3.32)

Equations (3.29), (3.30), and (3.32) give the solution to the problem.

Note that matrices U, W, Φ, Ψ are calculated in [9, 15, 16] in an explicit form which

makes the suggested method analytical.

3.2. Transfer between Arbitrary Orbits

Let us consider transfer between arbitrary orbits and divide the time interval T into n

subintervals defined by instants t0 = t1, . . . , tn−1, tn = T . Also we assume that n−1 intermediate

reference orbits between the initial and final orbits are specified somehow and q1, . . . ,qn−1 are

5-dimentional vectors of elements of the reference orbits (see Figure 2). These vectors may be

given, for instance, in the following way:

qj = qj +j

n

(qf − qi

),(j = 1, . . . , n − 1

). (3.33)

Let us define vectors

Δqj = qj − qj−1,(j = 1, . . . , n

), (3.34)

where q0 = qi,qn = qf (here subscript “0” is the number of the reference orbit). Equations

(3.21) and (3.34) give the following equality:

n∑j=1

Δqj = Δq. (3.35)

Let us divide the transfer trajectory into n arcs corresponding to the time subintervals and

assume that the jth arc begins in the j–1st reference orbit and ends in the jth one. Assuming


number n big enough to make the j–1st and jth reference orbits close to each other for all

j = 1, . . . , n, the results of the Section 3.1 may be applied to each pair of the reference orbits.

Now the problem is to find the reference orbits providing optimality of the transfer. Due to

(3.32) the performance index for the jth transfer arc is

Jj =1

2ΔqtjW

−1j Δqj ,

(j = 1, . . . , n

), (3.36)

where similarly to (3.26), (3.27) the following definitions are used:

Wj =∫ tjtj−1

QjQtjdt, Qj =

∂qj−1

∂vj−1

(j = 1, . . . , n

), (3.37)

where (41) vj is velocity vector of the jth reference orbit. Performance index of the whole

problem is

J =n∑j=1

Jj . (3.38)

In order to find the transfer trajectory that gives minimum value for J , it is sufficient to find

intermediate reference orbits providing a minimum for (3.38). Thus, function (3.38) should

be minimized with respect to vectors Δqj , j = 1, . . . , n taking into account (3.35). Let us

introduce the helping function

L = J − λt⎛⎝ n∑

j=1

Δqj −Δq

⎞⎠, (3.39)

where λ is a Lagrange multiplier. Necessary conditions of a minimum for the functional (3.38)are

(∂L

∂Δqj

)t

= W−1j Δqj − λ = 0

(j = 1, . . . , n

). (3.40)

Thus,

Δqj = Wjλ(j = 1, . . . , n

), (3.41)

and (3.35) and (3.41) give

λ = W−1Δq, (3.42)


where

W =n∑j=1

Wj . (3.43)

Now new values of vectors Δqj can be found from (3.41) and (3.42) as follows:

Δqj = WjW−1Δq(j = 1, . . . , n

). (3.44)

New reference orbits are defined by the new values of the orbital elements given by

qj+1 = qj + Δqj(j = 0, . . . , n − 1

). (3.45)

Assuming optimal locations of the reference orbits found and applying (3.29) and (3.30) to

the jth arc of the transfer trajectory, we obtain optimal thrust vector and the trajectory state

vector in the time interval tj−1 ≤ t ≤ tj(j = 1, . . . , n) as follows:

α(t) = ΨtjvU

tjW

−1Δq, (3.46)

x(t) = yj−1(t) +ΦjSjUtj0W

−1Δq, (3.47)

where

Sj = Sj(t) =∫ ttj−1

ΨjvΨtjvdt, Uj =

∂qj−1

∂yj−1, (3.48)

matrix W is given by (3.43), matrices Φ = Φj(t, 0),Ψjv = Ψjv(t) are calculated in the j − 1st

reference orbit.

Multiplying (3.40) by Δqj and taking into account (3.35), (3.36), and (3.38), we obtain

J =1

2λtΔq. (3.49)

Due to (3.42) and (3.49) minimal value of the performance index in the considered case is

given by (3.32) with matrix W defined by (3.43).

3.3. Calculation Procedure

Let y0j ,y

1j be the state vectors of the jth reference orbit at the beginning and at the end of the

jth time subinterval (i.e., at times tj−1, tj resp.). Then, the solution to the problem considered

here may be obtained by means of the following iterative calculation procedure.


(1) n−1 intermediate reference orbits are specified somehow, for example, using (3.33).The launch position in the initial orbit is specified in the case considered here (i.e.,

state y00 = yi(0) is given) and the respective initial state vector of the transfer

trajectory is x0 = y00.

(2) Vector y0j is calculated for j = 1 using the following equation, that is, similar to

(3.30):

y0j = y1

j−1 +Φ1jS

1jU

tj0W

−1j Δqj , (3.50)

where S1j = Sj(tj), Uj0 = Uj(0) matrix Ψjv = Ψv(tj) is calculated in the j − 1st reference

orbit.

(3) Step 2 is repeated for j = 2, . . . , n−1 and for j = n (3.50) gives state vector yfT in the

final orbit.

(4) New vectors Δqj are calculated using (3.44) and new reference orbits with elements

given by (3.45) are found.

(5) Performance index is calculated using (3.32) and steps 2–4 are repeated until

decrement ΔJ of the performance index gets smaller than a given parameter ε > 0.

As soon as |ΔJ | < ε, the thrust vector α and the state vector x of the transfer

trajectory may be calculated at each time subinterval using (3.46) and (3.47).

The suggested method is approximate, although any desired accuracy can be reached by

means of selecting an appropriate amount n of subintervals. It can be shown that if n −→ ∞then the solution converges to a limit which is an accurate solution to the linearized problem.

4. Other Types of the Transfer

4.1. Transfer between Given Orbit and Given State Vector

Now let us consider the case when the launch position can be selected in the initial orbit

arbitrarily and the position of the entry into the final orbit is given (i.e., vector yi0 is not given

and vector yfT is given in (3.1)). In this case the method described in Section 3 should be

applied in the backward direction with decreasing time, that is, vector y0n−1 of the start from

n − 1st reference orbit can be found for a given state vector yfT of the arrival to the final

orbit, and so forth, until vector yi0 is found. The equations necessary to solve the problem

considered here can be easily derived from the equations given in Section 3.

4.2. Transfer between Two Given Orbits

Let us assume that neither launch position in the initial orbit nor arrival position in the final

orbit are given and should be determined in an optimal way during solving the transfer

problem. This case may be solved using the suggested method in the following way.

A first guess for the launch position should be given somehow. This position defines

the vector yi0 and in the first iteration of the calculation procedure described in Section 3.3 the

final state vector yfT may be found. In the second iteration of the calculation procedure for

this state vector a new value of the vector yi0 may be found as described in Section 4.1, and


so forth, that is, odd iterations of the calculation procedure use the case described in Section 3

and even iterations use the case described in Section 4.1.

5. Partly Given Final Orbit

The suggested method can be also used in the case of partly given elements of the final orbit,

that is, if vector qf has dimension m < 5. For instance, only energy of the final orbit (m = 1)or semimajor axis and eccentricity (m = 2) may be given. In this case the method described

above is applied with m-dimensional vectors qi,qf ,qj(j = 1, . . . , n − 1), m × 6-dimensional

matrix Uj , and m-order matrices Wj , Sj . Nongiven orbital elements are determined using

transversality condition (3.17) and the respective conditions for vectors qj .

6. Constraint Imposed on the Thrust Direction

Here the case when a constraint is imposed on the thrust direction is considered. This

constraint may be caused by specific features of the spacecraft design or of its attitude control

system. Let us assume the constraint given by

Bα = 0, (6.1)

where B = B(x, t) is a matrix of dimension 1×3 (i.e., B is a row) or 2×3 (in this case the thrust

direction is given and the problem is to find optimal thrust value). In this case, as is shown

in [17, 18], the suggested optimization method is also applicable with matrices Wj , Sj and

vector αj from (41), (3.48), and (3.46) replaced by

Wj =∫ tjtj−1

QjPQtjdt, Sj = Sj(t) =

∫ ttj−1

ΨjvPΨtjvdt, αj = PΨt

vUtjW

−1j Δqj , (6.2)

where third-order matrix

P = I − Bt(BBt)−1B, (6.3)

is a projector onto the constraining set given by (6.1). Note that rank of matrix P is less

than 3. This is why matrices Wj given by (6.2) may be singular, while in the case of no

constraint on the thrust direction matrices Wj given by (41) are nonsingular in any interval

of integration [9]. As is shown in [18] nonsingularity of all matrices Wj given by (6.2) is a

sufficient condition of feasibility of the transfer with constraint (6.1).

7. On Optimality of the Method

One of the necessary conditions of optimality of the thrust vector is constancy of the

Hamiltonian in the whole time interval of the transfer [14]. It can be shown that the

Hamiltonian of the linearized problem given by (3.5) is constant in the solution given by the

suggested method, although the Hamiltonian of the original nonlinear problem calculated in


Table 1: Comparison of the solutions to the original and linearized problems.

Transfer durationin periods ofthe initial orbit

Solution to the Solution to the

Difference, %original problem linearized problem

No. ofcomplete orbits

JNo. of

complete orbitsJ

20 9 1, 012 × 10−3 8 1, 053 × 10−3 4,1

40 18 4, 997 × 10−4 16 5, 193 × 10−4 3,9

100 46 1, 991 × 10−4 42 2, 070 × 10−4 4.0

200 93 0, 995 × 10−4 84 1, 034 × 10−4 3.9

the solution of the linear problem is not constant. Thus, the method described in the paper

gives the solution which is not optimal in the original problem.

A comparison of the solutions to the problem of the orbital transfer in the original

formulation and linearized one was performed. Transfer between coplanar circular orbits of

radii ai = 1 and af = 4 with gravitational parameter of the primary body μ = 1 was considered.

Solution to the original problem was provided by Petukhov. He used the mathematical

developed by his method for solving two-point boundary value problem [19]; optimal

solution to the orbital transfer problem was obtained by Petukhov by means of variation

of the arrival point in the final orbit. Results of the comparison are given in Table 1.

As is seen in Table 1, the difference between the values of the performance index for

the optimal solutions to the original and linearized problems is small and practically does not

depend on the transfer duration. Although in more complicated transfers the difference may

be bigger.

8. Illustrative Examples

This section demonstrates potentialities of the suggested method by means of examples of

transfers in the Earth’s sphere of influence. Orbits are given by the orbital elements

q = {rπ , rα, i,Ω, ω}, (8.1)

where rπ , rα are radii of perigee and apogee in thousands of kilometers (Mm), i is the

orbital inclination, Ω is the longitude of the ascending node, ω is the argument of the

perigee. Angular elements are given in degrees. Direction of the thrust vector is given in the

examples by angles between the projection of the thrust vector onto the instantaneous orbital

plane (angle ϕ) and between the thrust vector and orbital plane (angle Ψ). In all examples

given below the optimal positions of departure from the initial orbit and arrival to the final

orbit were obtained (i.e., the classical case of orbital transfer described in Section 4.2 was

considered). The changes ΔV of the velocity by means of the thrust also are given in the

examples. These changes were calculated using numerical integration of the magnitude of

the thrust vector given by (3.46) or (6.2). The number of the time subintervals was taken

n = 5000 in all examples.


x

−80

80

y

x

z

−80 80

−80

80

−80 80

Figure 3: Transfer between two elliptic orbits with high mutual inclination.

8.1. Transfer between Elliptical Orbits with High Mutual Inclination

Transfer between two orbits given by qi = {7, 30, 50, 80,−60}, qf = {40, 80, 80,−80, 70}with T = 400 hour is considered here. The transfer trajectory is shown in Figure 3 in

projections onto the equator plane xy and the polar plane xz. Dashed lines show node lines

of the initial and final orbits. The jet acceleration value divided by g = 9.8066 m/s2 is shown

in Figure 4; Figure 5 shows angles ϕ, Ψ. Performance index and total ΔV for the transfer are

J = 44.42 m2s−3, ΔV = 10.05 km/s.


0 100 200 300 400

Time of flight (hour)

0

0.0005

0.001

0.0015

α(g)

Figure 4: Jet acceleration for transfer between two elliptic orbits with a high mutual inclination.

0 100 200 300 400

0

0

30

ψ(d

eg)

ϕ(d

eg)

−90

−30


90

Figure 5: Angles of the thrust direction transfer between two elliptic orbits with a high mutual inclination.

8.2. Transfer to an Orbit with Given Perigee and Apogee Radii

Transfer in time T = 400 hrs to a partly given orbit, namely, to an orbit with given only perigee

and apogee radii, is considered here. Optimal transfer is planar in this case. Only perigee and

apogee radii of the initial orbit are specified, because the other initial orbital elements are not

important and may be taken equal to zero. Elements of the initial and final orbits are taken as

follows: qi = {7, 20, 0, 0, 0}, qf = {40, 80, 0, 0, 0}. Transfer orbit is shown in Figure 6; as

is seen in this figure optimal is coaxial collocation of the final orbit with respect to the initial

orbit. Figure 7 shows respective propulsion acceleration value divided by g, and angle ϕ is


x

y

−60

60

60−100

Figure 6: Transfer trajectory to an orbit with given perigee and apogee radii.

0

0.0001

0.0002

0.0003

0.0004

0.0005

0 100 200 300 400


α(g)

Figure 7: Jet acceleration for transfer to orbit with given perigee and apogee distances.

shown in Figure 8 (Ψ = 0 in the planar transfer). Performance index and total ΔV are J = 3.01

m2s−3, ΔV = 2.82 km/s.

8.3. Transfer to an Orbit with Given Energy

A transfer to an orbit with a given energy, namely, to a hyperbolic orbit given only by C3

= 1 km2/s2, is considered here. Initial orbit with elements qi = {7, 20, 0, 0, 0} is taken, the

transfer duration is T = 1000 hrs. The transfer trajectory (which is obviously planar in this

case) and the jet acceleration are shown in Figures 7 and 8. The dot at the end of the spiral

trajectory in Figure 7 marks the entry into the hyperbolic orbit. Corresponding performance

index and total ΔV are J = 4.85 m2 s−3, ΔV = 4.93 km/s. The thrust direction is tangential in

this case; this confirms nonoptimality of the solution to the linearized problem of the orbital

transfer, because in the original nonlinear problem the optimal thrust is not tangential (see,

e.g., [20]).


0 100 200 300 400


0

10

ϕ(d

eg)

−10

Figure 8: Angle ϕ for transfer to orbit with given perigee and apogee distances.

x

y

1500

500

−1500

−500

Figure 9: Transfer trajectory to the hyperbolic orbit with given energy.

8.4. Constrained Thrust Direction

A transfer between the orbits given by the elements qi = {7, 20, 0, 0, 0}, qf ={50, 100, 60, 60, 60}, in a time T = 400 hours, is considered here. The transfer trajectory is

shown in Figure 9. The performance index and total ΔV are J = 10.83 m2s−3, ΔV = 5.39 km/s.

Now let us consider the following constraint on the thrust direction: the thrust is

always orthogonal to the spacecraft position vector, that is, B = rt in (6.1). The projector (6.3)in this case is P = I−rrt/r2. The transfer trajectory for the constrained thrust direction visually

does not differ from the one for the unconstrained direction shown in Figure 9. Performance

index and total ΔV in the case of the constrained thrust direction are J = 11.44 m2s −3, ΔV =5.52 km/s.

Acceleration value versus time for the unconstrained and the constrained thrust

direction is shown in Figure 10, the transfer trajectories are shown in Figure 11, the Jet

acceleration for the transfers are shown in Figure 12 and the angles of the thrust direction

for the transfers are shown in Figure 13.

9. Conclusion

The mathematical method for calculation of low-thrust orbital transfers presented in this

paper has two essential disadvantages, as follows.

(1) The method is applicable to the power-limited thrust, while the existing thrusters

are close to ones with constant or given exhaust velocity.

(2) The original nonlinear problem is replaced in the method by a linearized problem

solution to which it is not optimal for the original problem.


200 400 600 800 10000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0


α(g)

Figure 10: Jet acceleration for transfer to hyperbolic orbit with given energy.

−80 80

−80

80

x

y

x

z

−80 80

80

(a)

−80 80

−80

80

x

y

x

z

−80 80

80

(b)

Figure 11: Transfer trajectories without (a) and with (b) constraint on the thrust direction.


0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

α(g)

0 100 200 300 400


(a)

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

α(g)

0 100 200 300 400


(b)

Figure 12: Jet acceleration for transfer without (a) and with (b) constraint on the thrust direction.

These disadvantages are compensated by simplicity of the method and its analytical form at

each iteration, and also by the wide applicability of the method: it works well in the case of a

big difference between the initial and final orbits, for a very high number of orbits around the

attracting center; also it is applicable for different transfer types (such as point-to-orbit, orbit-

to-point, and orbit-to-orbit transfers), in the cases of partly given final orbit and of a constraint

imposed on the thrust direction. Despite the fact that the method is based on linearization of

motion, any necessary accuracy of calculations may be reached by means of augmentation of

the number n of the reference orbits.

The suggested method may be used at early phases of the mission design when a high

optimization accuracy is not needed and at the same time massive calculations are necessary

for selection of a best mission scheme.


0 100 200 300 400


0

90

0

90

180

270ψ(d

eg)

ϕ(d

eg)

−90

−90

(a)

0 100 200 300 400


0

90

0

90

180

270

ψ(d

eg)

ϕ(d

eg)

−90

−90

(b)

Figure 13: Angles of the thrust direction for transfer without (a) and with (b) constraint on the thrustdirection.

Nomenclature

r, v: Position and velocity vectors

t: Current time

t = 0, t = T : Initial and final instants of the transfer

I: Unit matrix

α: Jet acceleration vector (thrust vector)α: |α|μ: Gravitational parameter of the attracting center

ϕ: Angle between the projection of the thrust vector onto the orbital

plane and the velocity vector

ψ: Angle between the thrust vector and the orbital plane

Subscripts “0” and “T”: Values of the parameters at the time instants 0 and T (if another

meaning of the “0” subscript is not stipulated), superscript “t”

denotes transposition.

Acknowledgment

The authors are grateful to the Brazilian Sao Paulo Research Foundation (FAPESP) for fi-

nancial support of this study.


References

[1] F. W. Gobetz, “Optimal variable-thrust transfer of a power-limited rocket between neighboringcircular orbits,” AIAA Journal, vol. 2, no. 2, pp. 339–343, 1964.

[2] T. N. Edelbaum, “Optimum power-limited orbit transfer in strong gravity fields,” AIAA Journal, vol.3, pp. 921–925, 1965.

[3] J. P. Marec and N. X. Vinh, “Optimal low-thrust, limited power transfers between arbitrary ellipticalorbits,” Acta Astronautica, vol. 4, no. 5-6, pp. 511–540, 1977.

[4] C. M. Haissig, K. D. Mease, and N. X. Vinh, “Minimum-fuel, power-limited transfers betweencoplanar elliptical orbits,” Acta Astronautica, vol. 29, no. 1, pp. 1–15, 1993.

[5] B. N. Kiforenko, “Optimal low-thrust orbital transfers in a central gravity field,” International AppliedMechanics, vol. 41, no. 11, pp. 1211–1238, 2005.

[6] S. Da Silva Fernandes and W. A. Golfetto, “Numerical computation of optimal low-thrust limited-power trajectories—transfers between coplanar circular orbits,” Journal of the Brazilian Society ofMechanical Sciences and Engineering, vol. 27, no. 2, pp. 178–185, 2005.

[7] S. Da Silva Fernandes and W. A. Golfetto, “Numerical and analytical study of optimal low-thrustlimited-power transfers between close circular coplanar orbits,” Mathematical Problems in Engineering,vol. 2007, Article ID 59372, 23 pages, 2007.

[8] V. V. Beletsky and V. A. Egorov, “Interplanetary flights with constant output engines,” Cosmic Research,vol. 2, no. 3, pp. 303–330, 1964.

[9] A. A. Sukhanov, “Optimization of flights with low thrust,” Cosmic Research, vol. 37, no. 2, p. 191, 1999.[10] A. A. Sukhanov, “Optimization of low-thrust interplanetary transfers,” Cosmic Research, vol. 38, no. 6,

pp. 584–587, 2000.[11] A. A. Sukhanov and A. F. B. D. A. Prado, “A modification of the method of transporting trajectory,”

Cosmic Research, vol. 42, no. 1, pp. 103–108, 2004.[12] A. A. Sukhanov and A. F. B. D. A. Prado, “Optimization of low-thrust transfers in the three body

problem,” Cosmic Research, vol. 46, no. 5, pp. 413–424, 2008.[13] J. H. Irving, “Low thrust flight; variable exhaust velocity in gravitational fields,” in Space Technology,

H. S. Seifert, Ed., chapter 10, John Wiley and Sons, New York, NY, USA, 1959.[14] L. S. Pontryagin and R. V. Gamkrelidze, TheMathematical Theory of Optimal Processes, Gordon & Breach

Science Publishers, 1986.[15] B. Bakhshiyan and A. A. Sukhanov, “First and second isochronous derivatives in the two-body

problem,” Cosmic Research, vol. 16, no. 4, p. 391, 1978.[16] A. A. Sukhanov, “Isochronous derivatives in the two-body problem,” Cosmic Research, vol. 28, no. 2,

pp. 264–266, 1990.[17] A. A. Sukhanov and A. F. B. De A Prado, “Optimization of transfers under constraints on the thrust

direction: I,” Cosmic Research, vol. 45, no. 5, pp. 417–423, 2007.[18] A. A. Sukhanov and A. F. B. D. A. Prado, “Optimization of transfers under constraints on the thrust

direction: II,” Cosmic Research, vol. 46, no. 1, pp. 49–59, 2008.[19] V. G. Petukhov, “Optimization of interplanetary trajectories for spacecraft with ideally regulated

engines using the continuation method,” Cosmic Research, vol. 46, no. 3, pp. 219–232, 2008.[20] R. A. Jacobson and W. F. Powers, “Asymptotic solution to the problem of optimal low-thrust energy

increase,” AIAA Journal, vol. 10, no. 12, pp. 1679–1680, 1972.


Research ArticleFour-Impulsive Rendezvous Maneuversfor Spacecrafts in Circular Orbits UsingGenetic Algorithms

Denilson Paulo Souza dos Santos,1 Antonio Fernando Bertachinide Almeida Prado,1 and Guido Colasurdo2

1 Division of Space Mechanics and Control INPE, CP 515, 12227-310 Sao Jose dos Campos SP, Brazil2 Department of Aerospace and Mechanical Engineering, Universita degli Studi di Roma “La Sapienza”,Piazzale Aldo Moro 5, 00185 Roma, Italy

Correspondence should be addressed to

Denilson Paulo Souza dos Santos, [email protected]

Received 1 December 2011; Accepted 27 January 2012

Academic Editor: Maria Zanardi

Copyright q 2012 Denilson Paulo Souza dos Santos et al. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Spacecraft maneuvers is a very important topic in aerospace engineering activities today. In a moregeneric way, a spacecraft maneuver has the objective of transferring a spacecraft from one orbit toanother, taking into account some restrictions. In the present paper, the problem of rendezvousis considered. In this type of problem, it is necessary to transfer a spacecraft from one orbit toanother, but with the extra constraint of meeting another spacecraft when reaching the final orbit.In particular, the present paper aims to analyze rendezvous maneuvers between two coplanarcircular orbits, seeking to perform this transfer with lowest possible fuel consumption, assumingthat this problem is time-free and using four burns during the process. The assumption of fourburns is used to represent a constraint posed by a real mission. Then, a genetic algorithm is usedto solve the problem. After that, a study is made for a maneuver that will make a spacecraftto encounter a planet, in order to make a close approach that will change its energy. Severalsimulations are presented.

1. Introduction

This paper aims to analyze optimal rendezvous maneuvers between two spacecrafts that are

initially in circular coplanar orbits around the Earth. The main goal is to perform this transfer

having the fuel consumption as a penalty function, so the minimization of this quantity is

searched during the process of finding the solutions. The approach used here is to assume

that the problem is time-free, which means that the time of the transfer is not important. The

control assumed to perform this task is an engine that can deliver four burns. This assumption


is used to represent a common constraint posed by real missions. In the present paper, we are

considering a generic problem, not a specific mission, but this type of constraint appears very

often in space activities.

Then, a genetic algorithm is used in order to solve the problem. This type of

approach represents a new alternative to solve this problem and can be used for comparisons

with results obtained by standard procedures available in the literature, as shown in [1–

28]. Preliminary studies showed that, in some situations, this algorithm can be faster in

convergence and more accurate, while in some others, it is slower and presents less accuracy.

A detailed comparison still has to be made to evaluate under which circumstances this

algorithm can be more efficient. In any case, several kinds of missions can use the benefits

of the techniques based on the genetic algorithm showed in this work. The main types are

transference with free time (to change the orbit of the space vehicle without restrictions in the

time required by the execution of the maneuver), “rendezvous” (when one desires that the

space vehicle stands alongside another spacecraft), “flyby” (a mission to intercept another

body, however without the objective to remain next to it), “swing-by” (a close approach

to a celestial body to gain or lose energy), and so forth. But, in the present paper, only the

rendezvous maneuver is considered.

2. Description of the Problem

The problem of orbital maneuvers has been studied in several published papers. Some of

them are shown in [1–28]. The different approaches to solve this problem can be appreciated

in those references. Some authors assumed that a low magnitude force is applied to the

spacecraft during a finite time. This is the so-called continuous thrust approach. References

[7, 10, 17, 28] have some details on this topic. As an alternative approach, the idea of an

impulsive maneuver is also studied. In this situation, a high magnitude force is applied

during a time that can be considered negligible. References [3, 5, 8, 27] used this important

approach. More recently, two more ideas appeared in the literature to perform orbital

maneuvers. The first one is the use of a close approach with a celestial body to change the

orbit of a spacecraft. It is the swing-by maneuver. References that used this approach are

[2, 13]. The second recent approach is the gravitational capture, where the force generated

by the perturbation of a third body [14] can be used to decrease the fuel expenditure of a

space maneuver. References [11, 12] have some details of this idea. Some publications cover

all those topics in more details, like [6, 9, 15]. Studies more related to the research shown in

the present paper are the ones considering the Lambert’s problem ([1, 16]), the rendezvous

maneuver ([20–26]), and genetic algorithms itself ([18, 19]).In the present research, in order to solve the transfer proposed here, the Lambert’s

problem is used, in the way described below. The Lambert’s problem can be formulated

as follows: “Find an unperturbed orbit, under the mathematical model given by a law that

works with the inverse square of the distance (Newtonian formulation), that connects two

given points P1 and P2, with the transfer time (Δt) specified.” In the literature, several

researchers have solved this problem by using distinct formulations. Reference [1] shows

several of them. In this way, the parameters of the transfer orbit can be defined by

(1) ν1 is the true anomaly of the departure point P1 on the initial orbit. ν1 ∈ [0, 2π],

(2) Δν is the angular length of the transfer. Δν ∈ [0, 2π],


F2

F1

c

ν

P1P2

Figure 1: Instantaneous scenario of the problem.

(3) at is the semimajor axis of the transfer orbit. Note that, for each pair of departure

and arrival points, a minimum value amin exists for at. Two transfer orbits can be

found for the same value of at, depending on the sense of the transfer.

The parameter at is usually replaced by a different parameter y. The advantage of this

substitution is that the new variable has values between 0 and 1. The relationship is shown

below [19]

at =amin

4y(1 − y

) . (2.1)

The parameters ν1 and Δν determine the position of the points P1 and P2 that can be

related to the radius vectors �r1 and �r2. Any permitted value of the parameter y determines

univocally one transfer orbit. These parameters are, from the point of view of the genetic

optimizer, the genes of the members of the population.

The genetic algorithm searches for the best solution among a number of possible

solutions, represented by vectors in the solution space. To find a solution is to look for some

extreme value (minimum or maximum) in the solution space. The fitness of each individual

is represented by the total velocity impulse ΔV required to perform the orbital transfer. The

total impulse is given by the sum of the single impulses ΔVi provided in each thrust point in

order to pass from an orbital arc to the following one. It corresponds to the velocity difference

at the relevant thrust point.

The positions of the thrust points and the parameters of the transfer orbit are obtained

using as input the three genes, that is, the parameters previously chosen. The velocities at

the thrust points, before and after firing the engine, are easily computed, and it provides the

total velocity impulse, which is the measurement of the individual fitness. The evolutionary

process will select individuals with the genes corresponding to the optimal maneuver.

Figure 1 shows an instantaneous scenario of the problem.

Note that ν1 is the true anomaly of the point P1 on the initial orbit; ν2 is the true

anomaly of the point P2 on the final orbit; Δν is the angular length of the transfer; the

orientation of the transfer orbit is defined by the angle ω between its axis and the axis of

the initial orbit; c is the distance between P1 and P2 (2.4); Fi are the focus of the ellipse.


Random population

of chromosomes

Fitness of each individual

Select next

generation

Crossover

Mutation

Epidemic

Population

Displayresults

Figure 2: The genetic algorithm.

2.1. The Genetic Algorithm

The procedure starts with a random population of up to 800 individuals. The initial

population is generated randomly, and consider its characteristics distance and angles

according to the constrains of each variable. The vectors �x are assembled according to the

allowed boundary condition. Then, the fitness of each individual is verified, following the

criteria of the objective function, which is to minimize the fuel consumption (measured by

the ΔV ) found by solving the Lambert’s problem. So, the best individuals are selected to go

to the next generation, parents, and children. The procedure of crossover is then applied, as

well as a mutation to insert diversity in the population (Figure 2).The random variables used for the implementation of the algorithm are �x =

(Δθ1, Δθ2, R1, R2, y1, y3). Those symbols have the meaning that θi = νi − ω is the true

anomaly of the Pi points that determine the transfer orbit, as shown in Figure 3;Ri determines

the radius vector (position) in each thrust; the y (2.7) are the angles between (F1, P1, P2) (see

Figure 3 again).Eventually, there are epidemics, with the goal of inserting diversity and reducing the

elitism. After that, a new population is created, and the procedure is started again, finishing

after n attempts. The block diagram of the genetic algorithm (Figure 2) shows the procedures

followed to solve the problem. More details of the genetic algorithm can be obtained in [18,

19].

2.2. Selecting the Next Generation and Performingthe Crossover and the Epidemic Process

The selection of the new generation is made after the analysis of each individual by

measuring its objective function (Fitness). The ones with better values for this measurement

are selected to undergo a process of crossing or reproduction (crossover), where parents are

selected, and the children of this intersection are raised (Figure 2). When the population is too


uniform, measured by the values of their objective functions, part of the population suffers

an epidemic process, where many individuals are killed and replaced by others using again a

random process, to insert diversity in the population and to prevent premature convergence

to local optimal values. The crossover starts by separating the chromosomes of the parents

in two parts. After this separation, the first part of the parent 1 is combined with the second

part of the parent 2, and the first part of the parent 2 is combined with the second part of the

parent 1. In this way, a second generation is created. See [18, 19] for more details.

2.3. Chromosome

The chromosome representation is vital for a genetic algorithm (GA), because it is the

way that we translate the information from the problem to a format that can be handled

by the computer. This representation is completely arbitrary, so it varies according to the

choice made by each developer, without any kind of obligation to adopt any representation

available in the literature. This is a very important point to emphasize. The vast majority

of researchers use the binary representation for this problem because it is the simplest

one. In fact, many people, when they imagine a GA, quickly make an association with

binary chromosomes (used to facilitate the crossing). However, other formulations using real

chromosomes, modifying the way of performing crossover, get satisfactory results [18]. In

this paper, each gene is chosen to be a real number between 0 and 1, being generated in a

binary form and then converted in a real number. The value of the corresponding parameter

is Xi = Xmini + ui(Xmax

i −Xmini ), where Xmin

i and Xmaxi are the minimum and maximum values

of those variables, which means that they are the boundary conditions. The main reason to

use the binary approach is to validate this usual approach, in GA problems, in the particular

type of problem considered here. References [18, 19], that studied this same problem using

GA, used different approaches, so the validation of the binary approach was considered

important.

2.4. Objective Function

Most of the selection techniques used in this procedure require comparisons of the fitness to

decide which solutions should be propagated to the next generation. Normally, the fitness

has a direct relation with the value of the objective function, according to the rule that

better values of the objective function generate higher values of the fitness parameter. When

the genetic algorithm calls the objective function, it transfers an array of parameters that

specify the selected solution. This selection parameter must not be changed in any way

by the objective function. Genetic algorithms are based on biological evolution, and they

are able to identify and explore environmental factors to converge to optimal solutions, or

approximately optimal global levels. Then, the fitness of each individual can be computed by

using the five data that define the problem (a1, e1, a2, e2,Δω, the first one being unit because

of the normalization of the variables) and the three genes (ν1,Δν, y) that characterize the

individual. Then, we can obtain several important parameters [15]. The true anomaly of the

arrival point is given by

νi = νi−1 + Δν. (2.2)


P1P2

F1

F2

c

c

1

c2

y2

y1

r2 r1ν

Figure 3: Geometry of the problem and the angles involved in the problem.

The radii of the departure and arrival point are given by

r1 =a1

(1 − e2

1

)1 + e1 cos ν1

,

r2 =a2

(1 − e2

2

)1 + e2 cos ν2

.

(2.3)

The distance between P1 and P2 is

c =√r2

1 + r22 − 2r1r2 cosΔν. (2.4)

The semimajor axis of the transfer orbit is

amin =r1 + r2 + c

4. (2.5)

The distances c1 and c2 of P1 and P2 from the vacant focus F2 can be specified by the equations

ci = 2a − ri. (2.6)

Figure 3 shows a description of several important variables.

The angles can be calculated by (y = y1 + y2),

y = arcos

(r2

1 − r22 + c2

2r1c

),

y1 = arcos

(c2

1 − c22 + c

2

2c1c

).

(2.7)

The eccentricity of the transfer orbit is given by

et =

√c2

1 + r21 − 2c1r1 cos γ2

2at. (2.8)


The true anomaly θ1 of the P1 on the transfer orbit is

r1 =at(1 − e2

t

)1 + et cos θ1

,

θ1 = arcos

(at(1 − e2

t

)− r1

r1et

).

(2.9)

The argument of perigee for the transfer orbit is

ω = ν1 − θ1, (2.10)

which is the angle between the perigees of the transfer and the initial orbits.

Now that the geometry of the maneuver has been shown, it is possible to calculate

the radial and the tangential components of the spacecraft velocity before and after both

impulses, what permits the computation of the total ΔV , which has been assumed to be the

measurement of the individual fitness.

2.5. Normalization

Nondimensional variables are used in the procedure. They are shown below

r =r

a1,

v =v√μ/a1

.(2.11)

The distance and velocity units for the normalized variables are the semimajor axis of

the initial orbit and the velocity on a circular orbit with the same energy as the initial one. So,

the reference time is Δt =√a3

1/μ.

3. Numerical Solutions

Several maneuvers were simulated with the procedure developed here, using the genetic

algorithm. Then, the equivalent Hohmann maneuvers were calculated to provide a level of

comparison. The idea is not to find a transfer that has a smaller total ΔV , when compared

to the Hohmann transfers, but to try to minimize the difference in costs, assuming that the

engine of the spacecraft has a limitation that does not allow two impulsive maneuvers to

be performed. In theory, for the cases simulated here, the two impulses maneuvers always

have a lower consumption. So, the idea is to find the best maneuver that has four impulses,

in order to compare with other works [18, 19]. The number of impulses is a parameter that

can be modified in the input data of the algorithm to be useful for other applications. The

results shown in the present paper always consider a rendezvous maneuver between two

spacecrafts, where the radius of the orbit of the first spacecraft is ro = 1, and several values

were used for the radius of the spacecraft that is in the final orbit (see Table 1). The genetic


Table 1: Rendezvous between coplanar circular orbits showing the values of the ΔV for each burn and thetotal expenditure (ΔVT ). The results for the Hohmann are included for comparison.

n◦Simulation Cost using the genetic algorithm Hohmann transfer ΔV T −ΔVHT(ro = 1) ΔV1 ΔV2 ΔV3 ΔV4 ΔV T =

∑4i=1 ΔV i ΔVHT =

∑2i=1 ΔV i

1 rf = 1.2 0.104455 0.118925 0.159234 0.154917 0.537531 0.087000 0.450500

2 rf = 1.5 0.044987 0.118996 0.165893 0.189594 0.519469 0.181600 0.337900

3 rf = 1.6 0.028621 0.120939 0.163349 0.204023 0.516932 0.206600 0.310300

4 rf = 1.8 0.000000 0.131481 0.169234 0.225721 0.526435 0.249300 0.277100

5 rf = 1.9 0.000000 0.127047 0.178111 0.233180 0.538338 0.267700 0.270600

6 rf = 2 0.008136 0.117267 0.185660 0.242748 0.553811 0.284500 0.269300

7 rf = 2.5 0.134621 0.000000 0.222416 0.298130 0.655167 0.349600 0.205300

8 rf = 3 0.171286 0.000000 0.242839 0.369209 0.783334 0.393800 0.271100

9 rf = 5 0.172864 0.310322 0.000000 0.423150 0.906336 0.480000 0.426300

10 rf = 10 0.004673 0.000000 0.372418 0.311566 0.688657 0.499300 0.189400

V

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.12

1

0.8

0.6

0.4

0.2

0

Comparison GA versus Hohmann

VGA

Hohmann

Radius of target orbit (rf)

Figure 4: Comparison between the ΔVGA and ΔVHohmann.

algorithm provided satisfactory solutions, when compared with the solutions of the literature

[18], as shown in Table 1. The population is composed by 800 individuals, and up to 400

generations of individuals were used.

The results indicated that the maneuvers using the GA with 4 impulses do not provide

savings over the Hohmann transfer for all cases simulated (see Figure 4 and Table 1), as

expected and explained before, but it minimizes the difference in costs for the assumed four

impulsive maneuvers. Figure 4 shows all the details for this comparison.

Figures 5, 6, 7, 8, and 9, as well as Table 1, show a series of maneuvers. In general, an

impulse is applied in the initial orbit (ΔV1), generating the first elliptical transfer arc, and

then, according to the procedure, the second impulse is applied (ΔV2), leading to another

elliptical transfer orbit. The third point of burn will happen (ΔV3) to put the spacecraft in

the last transfer arc, and, finally, the last impulse (ΔV4) is applied to locate the vehicle in

the desired orbit. The total consumption is the sum of all the intermediate impulses, and

it is named ΔVT (Table 1). This total consumption serves as an index of measurement and

comparison between the methods. In other words, the information of the extra cost is due to


2

1.5

1

0.5

0

−0.5

−1

−2.5−2

−2 −1.5

−1.5

−1 −0.5 0 0.5 1 1.5 2 2.5

V1 = 0.008094V2 = 0.117318V3 = 0.185638

V4 = 0.24276

VT = 0.553811

R(θ)(u.n.)

R(θ)(u

.n.)

V4

V3

V2

V1

Four-burn orbit transfers—genetic algorithm

Figure 5: Four-burn orbit transfers for simulation 6.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.10 25 50 75 100 125 150 175 200 225 250 275

θ1

θ2

R1

R2

Y1

Y2

Best fitness

Number of objective function evaluations

Figure 6: The variables of the problem using the method of genetic algorithm and the best fitness forsimulation 6.

the fact that a two-impulse maneuver is not possible and a detailed vision of the best four-

impulse strategy is generated by the GA.

The variables of the problem are �x = (Δθ1,Δθ2, R1, R2, y1, y3) (see Figures 5 and 7). In

each new generation of the population, the individuals are approaching the values suggested

by the algorithm, converging to a solution of the problem. The best fitness values of the

parameters show the convergence to the optimal value. Table 2 shows a detailed view of

the maneuver, explaining all the intermediate Keplerian orbits obtained.


Table 2: The Keplerian elements of the intermediate orbits for the case where rf = 2.

Orbit a E w Vc

0 1 0 0 1

1 1.067591 0.247661 4.720904 0.967826586

2 1.553574 0.460257 5.401124 0.802294893

3 1.553574 0.460257 5.401124 0.802294893

4 2 0 0.707106781

3

2

1

0

−1

−3−3

−2

−2 −1 0 1 2 3

V1 = 0.171286V2 = 0V3 = 0.242839

V4 = 0.369209VT = 0.783334

R(θ)(u.n.)

R(θ)(u

.n.)

V4

V3

V2

V1



Simulation 8 and Figure 7 show some new results that confirm that the use of

the procedure with four impulses provides results with higher consumption than the bi-

impulsive maneuver (Table 1), but that minimizes the four-impulsive burn technique.

This study can also be applied to find orbital maneuvers that search for the minimum

fuel consumption for a spacecraft that leaves one celestial body and goes back to this same

body (Figures 9 and 10). This question is of great importance for missions whose objective

is to shift the position of the satellite in a given orbit, without changing the other orbital

elements. Prado and Broucke [1] also studied this problem using the Lambert method, under

different circumstances.

3.1. The Swing-By Maneuvers

The next step is to use the algorithm developed here to study a maneuver that will make a

spacecraft to encounter a planet, in order to make a close approach that will change its energy.

This problem can be seen as a rendezvous problem, where the second spacecraft, the one to be

reached, is a planet and not a space vehicle. Using this approach, a transfer maneuver using


0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.10 25 50 75 100 125 150 175 200 225 250

θ1

θ2

R1

R2

Y1

Y2

Best fitness

Number of objective function evaluations

Best individual

Figure 8: The variables of the problem using the method of genetic algorithm and the best fitness forsimulation 8.

0.6

0.8

1

0.4

0.2

0

−0.6

−0.4

−0.2

−1.5

−1

−0.8

−1 −0.5 0 10.5 1.5

V1 = 0.104455V2 = 0.118925V3 = 0.159234

V4 = 0.154917VT = 0.537531

R(θ)(u.n.)

R(θ)(u

.n.)

V4

V3

V2V1



an impulsive engine with four burns is followed by a gravity-assisted maneuver to send the

spacecraft further in the solar system. This technique will reduce the cost of an interplanetary

mission. This is a standard procedure in orbital maneuvers, and a more detailed description

is available in references [2, 9]. In this case, the system consists of three bodies:

(1) the bodyM1, with finite mass, situated in the center of mass of the Cartesian system

of reference;


0.6

0.8

1

0.4

0.2

0

−0.6

−0.4

−0.2

−1

−0.8

−1 −0.5 0 10.5

V1 = 0.130534V2 = 0.143611V3 = 0.143608

V4 = 0.130537VT = 0.54829

R(θ)(u.n.)

R(θ)(u

.n.)

V4

V3

V2

V1


Figure 10: Transfer maneuver from one orbit back to the same orbit.

(2) M2, a smaller body, that can be a planet or a satellite of M1, in a Keplerian orbit

around M1;

(3) a body M3, a space vehicle with infinitesimal mass, traveling in a Keplerian orbit

around M1, when it passes close to M2.

This close approach changes the orbit of M3 and, by the hypothesis assumed for

the problem, it is considered that the orbits of M1 and M2 do not change. Using the

“patched conics” approximation, the equations that quantify those changes are available in

the literature [9].The standard maneuver can be identified by the following three parameters

(Figure 11):

(i) | �V∞|, the magnitude of the velocity of the spacecraft with respect to M2 when

approaching the celestial body;

(ii) rp, the distance between the spacecraft and the celestial body during the closest

approach;

(iii) ΨA, the angle the approach.

Having those variables, it is possible to obtain δ, half of the total deflection angle, by

using the equation [2]

δ = arsin

(1

1 +(rpV 2

∞/μ2

)). (3.1)

Note that V2 is the velocity of the celestial body with respect to the main body and

VP is the velocity of the smaller mass when passing by the periapsis. A complete description


−→V −∞

−→V +∞

δ

δ

δ

δVP

rP

−→V2

ψAM1 M2 x

Figure 11: The parameters of the swing-by maneuver.

of this maneuver and the derivation of the equations can be found in Prado [9]. The final

equations are reproduced below

ΔE = −2V2V∞ sin(δ) sin(ΨA),

ΔC =−2V2V∞ sin(δ) sin(ΨA)

ω2,

(3.2)

where ω2 is the angular velocity of the motion of the primaries, ΔE is the variation of energy,

ΔC is the variation of the angular momentum, and ΔV is the variation of the magnitude of

the velocity due to the swing-by. For the ΔV , we have the equation

ΔV =∣∣∣Δ �V∣∣∣ = 2

∣∣∣ �V∞∣∣∣ sin(δ) = 2V∞ sin(δ). (3.3)

The gravity-assisted maneuver (swing-by) can provide a considerable change of the

velocity and energy of the spacecraft, reducing the costs of the mission. During this approach,

the spacecraft will be transferred to another orbit of interest of the mission. The dynamics

used to solve this problem is the traditional model given by the “Patched Conics,” so it

is assumed that all three bodies involved are points of mass and do not suffer external

disturbances. The variations given by the swing-by, in terms of velocity variation (ΔV ) and

energy variation (ΔE), can now be obtained.

Figure 12 shows the maneuver obtained by the genetic algorithm. The spacecraft

comes from an initial orbit with radius ro = 1 u.a., which represents the position of the Earth’s

heliocentric system, in astronomical units. It means that the spacecraft starts from the Earth.

Then, it performs a maneuver with 4 impulses, using three elliptic intermediate transfer

orbits, and finally it arrives in an orbit with rf = 5.202803 u.a. (Jupiter). At this moment,

it realizes a maneuver of Swing-by with the planet Jupiter. Note that the gain in velocity

was ΔVSB = 1.104368 and the gain in energy was ΔE = 2.017347. During this approach, the

space vehicle place itself in another orbit of the interest of the mission. In this mission, the

participation of the GA is to find the best procedure to make the spacecrafts reach the planet


5

4

3

2

1

0

−1

−2

−3

−4

−5

−6 −4 −2 0 2 4 6

V3 V2

V1

V4

R(θ)(u.n.)

R(θ)(u

.n.)

V1 = 0.256974V2 = 0.281067V3 = 0

V4 = 0.394731VT = 0.932772

Four-burn orbit transfers—Earth-Jupiter-genetic algorithm.

Swing-by Jupiter

ψ = 259.4993

E = 2.017347

VSB = 1.104368

Figure 12: Simulation of a maneuvers leaving the Earth (ro = 1) and reaching the orbit of Jupiter (rf =5.202803), performing a swing-by maneuver on the planet to gain velocity and energy.

Jupiter. From this point, standard procedures of interplanetary trajectories can complete the

mission.

4. Conclusion

Based on the analysis of the results obtained, the genetic algorithm implemented here

shows that this technique brings good results for the proposed four impulsive rendezvous

maneuvers, when compared with the ones obtained by the traditional impulsive methods. It

means that it can be used in real cases, specially when a bi-impulsive transfer is not possible

due to the limitations of the engine of the spacecfraft. The procedure is also effective in

maneuvering the spacecraft from one body back to the same body, that is, making it leaving

and returning to the same orbit.

The results indicate that the maneuver using the genetics algorithm with four impulses

does not provide better fuel consumption in any case simulated, since the bi-impulsive

maneuver is better in this situation, but the method proves to be efficient in minimizing

the four impulsive maneuvers. It is necessary to take into account that, in many cases, the

limitations of the propellers of the spacecraft require that the maneuver has to be performed

using several impulses, passing through intermediate orbits to reach the target.

Then, we studied a maneuver where the goal is to send a spacecraft to encounter the

planet Jupiter to make a swing-by maneuver. The algorithm worked well in finding a good

solution for this problem.

In general, the proposed technique can be used when a rendezvous maneuver is

required between two given circular orbits for a spacecraft that has an engine that requires

the application of four impulses.


In the future, it is possible to apply this technique in three dimensions, in maneuvers

that requires more impulses, and also in maneuvers to avoid collisions between a spacecraft

and asteroids.

Acknowledgments

This work was accomplished with the support of Sao Paulo State Science Foundation

(FAPESP) under Contract 2009/16517-7 and National Institute for Space Research, INPE,

Brazil.

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[1] A. F. B. A. Prado and R. A. Broucke, “Study of henon’s orbit transfer problem using the lambertalgorithm,” AIAA Journal of Guidance, Control, and Dynamics, vol. 17, no. 5, pp. 1075–1081, 1993.

[2] D. P. S. Santos, A. F. B. A. P. Prado, and E. M. Rocco, “The use of consecutive collision orbits to obtainswing-by maneuvers,” in Proceedings of the 56th International Astronautical Congress, Fukuoka, Japan,October 2005.

[3] F. W. Gobetz and J. R. Doll, “A survey of impulsive trajectories,” AIAA Journal, vol. 7, pp. 801–834,1969.

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[5] W. Hohmann, Die Erreichbarkeit Der Himmelskorper, Oldenbourg, Munique, 1925.[6] J. P. Marec, Optimal Space Trajectories, Elsevier, New York, NY, USA, 1979.[7] D. P. S. Santos, L. Casalino, G. Colasurdo, and A. F. B. A. P. Prado, “Optimal trajectories using gravity

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[23] J. E. Prussing, “Optimal two-and three-impulse fixed-time rendezvous in the vicinity of a circularorbit,” AIAA Journal, vol. 8, no. 7, pp. 1221–1228, 1970.

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[27] E. M. Rocco, A. F. B. A. Prado, M. L. O. Souza, and J. E. Baldo, “Optimal bi-impulsive non-coplanarmaneuvers using hyperbolic orbital transfer with time constraint,” Journal of Aerospace Engineering,Sciences and Applications, vol. 1, no. 2, pp. 43–51, 2008.

[28] D. P. S. Santos, A. F. B. A. Prado, L. Casalino, and G. Colasurdo, “Optimal Trajectories towards near-earth-objects using solar electric propulsion (SEP) and gravity assisted maneuver,” Journal of AerospaceEngineering, Sciences and Applications, vol. 1, no. 2, pp. 51–64, 2008.


Research ArticleSemianalytic Integration of High-Altitude Orbitsunder Lunisolar Effects

Martin Lara,1 Juan F. San Juan,2 and Luis M. Lopez3

1 C/Columnas de Hercules 1, ES-11100 San Fernando, Spain2 Departamento de Matematicas y Computacion, Universidad de La Rioja, 26004 Logrono, Spain3 Departamento de Ingenierıa Mecanica, Universidad de La Rioja, 26004 Logrono, Spain

Correspondence should be addressed to Martin Lara, [email protected]


Academic Editor: Josep Masdemont

Copyright q 2012 Martin Lara et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The long-term effect of lunisolar perturbations on high-altitude orbits is studied after a doubleaveraging procedure that removes both the mean anomaly of the satellite and that of the moon.Lunisolar effects acting on high-altitude orbits are comparable in magnitude to the Earth’soblateness perturbation. Hence, their accurate modeling does not allow for the usual truncationof the expansion of the third-body disturbing function up to the second degree. Using canonicalperturbation theory, the averaging is carried out up to the order where second-order terms inthe Earth oblateness coefficient are apparent. This truncation order forces to take into accountup to the fifth degree in the expansion of the lunar disturbing function. The small values of themoon’s orbital eccentricity and inclination with respect to the ecliptic allow for some simplification.Nevertheless, as far as the averaging is carried out in closed form of the satellite’s orbit eccentricity,it is not restricted to low-eccentricity orbits.

1. Introduction

In an increasingly saturated space about the Earth, aerospace engineers confront the

mathematical problem of accurately predicting the position of Earth’s artificial satellites. This

is required not only for the correct operation of satellites but also for preserving the integrity

of space assets. Thus, operational satellites are threatened by the not so remote possibility of a

collision with a defunct satellite [1] but most probably by the impact with other uncontrolled

man-made space objects as spent rocket stages or collision fragments—all of them commonly

called space debris.

Precise predictions require the integration of complete force models including both

gravitational and nongravitational effects, like a high-degree and order geopotential,


ephemerides-based lunisolar perturbations, drag, solar radiation pressure including eclipses,

and so forth (see [2, 3], for instance). The most accurate integration is expected from numeri-

cal methods, although precision ephemeris can also be obtained by means of semianalytical

integration [4]. In fact, both approaches, numerical and semi-analytical, do not need to enter

a competition. Thus, for instance, while semi-analytical methods may be efficient in keeping a

running catalog of hundreds of thousands of space objects within a reasonable accuracy, if the

probability of impact with an operational satellite is detected to surpass a certain threshold,

then a more accurate numerical integration will help control engineers in deciding whether

a collision avoidance maneuver is required or, on the contrary, the integrity of the satellite is

not in risk [5].

In a semi-analytical approach the highest frequencies of the motion, which normally

have small amplitudes, are filtered analytically via averaging procedures. This filtering

allows the numerical integration of the averaged system to proceed with very long step sizes.

Then, the short-period terms are recovered analytically, if desired, at any step of the numerical

integration [6–8].

Averaging techniques are also used for exploring questions affecting stability, such

as those derived from tesseral resonances or third-body perturbations, in a reduced-phase

space [9]. In this respect, much attention has been recently paid to the long-term evolution of

classical GNSS constellations, either for operational or disposal orbits [10].

While the noncentralities of the Earth gravitational potential play a key role in the

motion of low altitude satellites, third-body perturbations have also a decisive influence

in the long-term evolution of medium- and high-altitude Earth orbits. The third-body

disturbing function is commonly given by a series expansion in Legendre polynomials. Often,

the series is truncated to the first term in the expansion [11], but this early truncation is not

always accurate enough to represent the real dynamics [4, 12, 13]. Nevertheless, recursions in

the literature allow to extend the Legendre polynomial expansion to any desired order either

in classical or nonsingular elements [14–16].

Because of the physical characteristics of the Earth gravitational potential, where the

second-order zonal coefficient (J2) clearly dominates over all other harmonics, second-order

effects of J2 may be important and must be taken into account when the effect of higher-

order harmonics is studied. Correspondingly, the truncation in the expansion of third-body

perturbations must include terms of magnitude comparable to J2-squared. Because the third-

body disturbing function is expanded in the ratio semi-major axis of the satellite’s orbit

to semi-major axis of the third-body’s orbit, the degree at which the expansion must be

truncated depends on the altitude of the satellites to be studied.

In this paper we study the effect of lunisolar perturbations on high-altitude orbits

about a noncentral Earth, which is assumed to be oblate although without equatorial symme-

try. More specifically, we are interested in the semi-analytical integration of satellites on alti-

tudes of classical global navigation satellite system (GNSS) constellations such as GPS, Glo-

nass, or Galileo. Note, however, that the assumption of an axisymmetric geopotential pre-

vents to tackle the tesseral resonance problem that commonly suffer this kind of orbits. With

respect to previous research [17], where we approached the general case of third-body pertur-

bations via double averaging, we release here the common simplification of assuming the

third-body in circular orbit. Also, we focus on the case of Earth’s artificial satellites deal-

ing with more real lunisolar perturbations. We do that because recent results [18] seem to con-

tradict the claim that taking the third-body in circular orbit does not produce any noticeable

degradation in the long-term propagation of real Earth orbits [19]. Besides, for the actual


values of the orbits of the sun and moon, neglecting terms in the eccentricity is not

consistent with a higher-order expansion of the lunisolar disturbing function. Hence, both the

eccentricity of the orbits of the sun and moon are taken into account in the present work. For

the latter, the moon, the orbit is assumed to remain with constant inclination with respect to

the ecliptic plane, over which the longitude of the ascending node moves with linear motion.

The argument of the perigee of the moon is also assumed to evolve linearly, while we take

the apparent orbit of the sun to be purely Keplerian.

We use canonical perturbation theory by Lie transforms [20–22]. The order of each

term of the disturbing function is determined by a virtual small parameter that is taken

proportional to the ratio of the satellite’s orbit semimajor axis to the moon’s orbit semi-major

axis. Then, we check that the magnitude of second order terms due to the Earth oblateness is

comparable to that of the fifth degree in the expansion of the moon’s third-body potential, and

hence we truncate the moon’s third-body potential up to the fifth degree. On the contrary, the

usual truncation up to the second degree is enough for modeling the sun disturbing function

assumed that we limit the theory to the order of J2-squared terms. The Hamiltonian model

also takes into account the asymmetries in the Earth potential caused by the J3 term, whose

influence in the dynamics of the lower eccentricity orbits is clearly noted.

The initial Hamiltonian is of three degrees of freedom and time dependent. Note, how-

ever, that the time appears in different scales related to the mean motion of the moon, that of

the sun, and the rate of variation of the moon’s argument of the perigee and longitude of the

ascending node. In our averaging procedure, we eliminate the mean anomaly of the satellite

and that of the moon to obtain a two-degrees-of-freedom Hamiltonian, which still depends

on time albeit only through very slowly varying quantities. As far as we do not find reso-

nances between the mean motions of the satellite and the moon, the averaging can be done in

two steps: the mean anomaly of the satellite is removed first by means of a classical Delaunay

normalization [23]; then, a similar transformation is used to eliminate the mean anomaly

of the moon. The splitting of the averaging has the advantage of simplifying computations.

Specifically, the generating function of each transformation is obtained from the solution of

simple quadratures.

The averaging is carried out not only in closed form of the satellite’s orbit eccentricity,

thus making the simplified Hamiltonian useful for studying the long-term evolution of any

orbit, but also in closed form of the eccentricity of the moon’s orbit. Nevertheless, because of

the small values of the eccentricity of the orbits of both perturbing bodies as well as the small

inclination of the orbit of the moon with respect to the ecliptic, further simplifications can be

done by neglecting terms of order higher than the truncation order of the theory.

Numerical experiments using the doubly averaged Hamiltonian demonstrate that the

inclusion in the model of the orbital eccentricities of the sun and moon does not cause

qualitative differences with respect to the circular orbit approximation. Besides, the re-

covery of the short-period effects by means of the analytical transformation equations of the

averaging provides a quite reasonable precision in the long-term integrations.

2. Model

We study the motion of an Earth artificial satellite of negligible mass whose Keplerian motion

is slightly distorted by the noncentralities of the geopotential due to the Earth’s oblateness

and latitudinal asymmetry, as represented by the harmonic coefficient J2 and J3, respectively,


and under the point-mass attraction of the sun and moon. Thus, the motion of the satellite is

derived from the potential

V = −μr+μ

r

α2

r2J2

(3

2

z2

r2− 1

2

)+μ

r

α3

r3J3z

r

(5z2

2r2− 3

2

)+ V� + V�, (2.1)

where μ is the Earth’s gravitational parameter, α its equatorial radius, Ji is the zonal harmonic

coefficient of degree i, r is distance from the origin, and z is the satellite’s coordinate in the

direction of the symmetry axis of the Earth. In the mass-point approximation, the third-body

disturbing potential V�, � ∈ (�,�), is

V� = −μ�r�

(r�

‖r − r�‖− r · r�

r2�

), (2.2)

where μ� is the third-body’s gravitational parameter, r and r� are the radius vector of the

satellite and of the third-body, respectively, of corresponding modulus r and r�. If the

disturbing body is far away from the perturbed body, (2.2) can be expanded in power series

of the ratio r/r�:

V� = −βn2�a

3�

r�

∑j≥2

(r

r�

)jPj(cosψ

), (2.3)

where β = m�/(m�+m), n� is the mean motion of the third-body in its orbit of semi-major axis

a�, Pj are Legendre polynomials, and ψ is the angle encompassed by r and r�. The absence

of the Keplerian term −μ�/r� as a summand in (2.3) has no effect in the restricted problem

approximation, in which the mass of the satellite has negligible effects on the third-body’s

orbit.

In our model we assume an Earth-centered frame with the origin defined by the

intersection of the Earth’s equator and the ecliptic; we both of which consider fixed planes.

Whereas we take the sun’s apparent orbit about the Earth to be purely Keplerian, we assume

that the moon moves with Keplerian motion on a precessing ellipse that evolves with constant

rate of the argument of perigee. Besides, the moon’s orbital plane is assumed to have constant

inclination with respect to the ecliptic; yet it is also assumed to be precessing over the ecliptic

with linear variation of the longitude of the ascending node.

Calling a, e, i, ω, Ω, and M to semimajor axis, eccentricity, inclination, argument of

the perigee, longitude of the ascending node, and mean anomaly, respectively, we use the

approximate values

a� = 150 × 106 km, e� = 0.017, i� = ε = 23.44 deg, (2.4)


for the sun, where ε notes the obliquity of the ecliptic, we assume a fictitious epoch in which

ω� = Ω� = 0, and take n� = 2π/sidereal year. For the moon,

a� = 384 400 km, e� = 0.055, i� = J = 5.1 deg = 0.089, (2.5)

and take Ω� =N =N0 + nNt, ω� = w = w0 + nwt, with

nN = − 2π

18.61 year, nw =

2π

8.85 year− nN, n� =

2π

27.32 day− nw − nN. (2.6)

We recall that J and N are referred to the ecliptic. Besides, β = 1 for the sun and β = 1/28.8245

for the moon.

Remark that this simple model is not, of course, valid for computing precise ephemeris

of the moon, who may be clearly out of the predicted position because of the amplitude of

periodic oscillations in N and J , and also in e� and i�, comper [24]. Nevertheless, it will be

enough for our purposes of investigating the qualitative features that lunisolar perturbations

introduce in the long-term behavior of Earth’s artificial satellites.

3. Perturbation Approach

By using canonical variables we can study the problem in Hamiltonian form. Besides, in order

to apply perturbation theory, we arrange the Hamiltonian as a power series in a small para-

meter:

H =∑m≥0

δm

m!Hm,0. (3.1)

We are interested in high-altitude orbits in the range of 20 000 to 35 000 km, the so-

called upper mean earth orbit (MEO) region. In order to take account of all relevant lunisolar

perturbations, we take a virtual small parameter proportional to the ratio semimajor axis of

the satellite semimajor axis of the moon. For the altitudes of interest, this ratio is δ ∼ O(10−1).Then, we find that the Earth’s oblateness coefficient is a fourth-order quantity, and the J3 effect

appears at the eighth order. With respect to the moon the consecutive terms of the Legendre

polynomials expansion of the moon disturbing function appear at consecutive orders of the

Hamiltonian, starting by the fifth order. We include up to the fifth degree, whose effect may

be comparable to second-order effects of J2. For the sun, it is enough to take the first term in

the Legendre polynomials expansion. Then, the zero-order term is the Keplerian

H0,0 = −1

2n2a2. (3.2)


Besides, Hi,0 = 0 for i < 4, and

H4,0

4!H0=α2

a2J2a3

r3

(1 − 3

z2

r2

),

H5,0

5!H0= β

n2�n2

a3�r3� M�

2 +n2�n2

a3�r3�M�

2 ,

H6,0

6!H0= β

n2�n2

a4�r4� M�

3 ,

H7,0

7!H0= β

n2�n2

a5�r5� M�

4 ,

H8,0

8!H0= β

n2�n2

a6�r6� M�

5 +α3

a3J3a4

r4

z

r

(3 − 5

z2

r2

),

(3.3)

where

M�i = 2

ai−2

ai−2�

ri

aiPi(cosψ�

), (3.4)

with

cosψ� =xx� + yy� + zz�

rr�, cosψ� =

xx� + yy� + zz�rr� ,

r =aη2

1 + e cos f,

(3.5)

where f is the true anomaly and η =√

1 − e2 is the usual eccentricity function. The equations

of motion are obtained from the Hamilton equations:

d(, g, h

)dt

=∂H

∂(L,G,H),

d(L,G,H)dt

= − ∂H∂(, g, h

) . (3.6)

Recall that the Cartesian coordinates of the satellite are expressed in orbital elements

by means of simple rotations. Thus,

⎛⎝x

y

z

⎞⎠ = R3(−Ω)R1(−i)R3(−θ)

⎛⎝r

0

0

⎞⎠, (3.7)

where θ = f + ω is the argument of latitude and R1 and R3 are the usual rotation matrices

about the x- and z-axes, respectively.

Since the Hamiltonian must be expressed in canonical variables, we assume hereafter

that the orbital elements of the satellite are always expressed as function of the Delaunay


variables (, g, h, L, G, H), given by the known relations = M, g = ω, h = Ω, L = √μ a,

G = L η, and H = G cos i. Nevertheless, we use the orbital elements notation because of its

immediate physical insight.

Delaunay’s variables are singular for zero eccentricity and/or zero inclination. In

spite of that, the Lie transforms theory is naturally computed in Delaunay variables. Once

the generating function of the averaging is computed, it can be applied to any function of

the Delaunay’s variables. Specifically nonsingular variables are assumed to be functions of

Delaunay variables to obtain the transformation equations of the averaging in nonsingular

variables [25].We note that the sun and moon coordinates are assumed to be known functions of

time. Therefore, (3.1) is a time-dependent Hamiltonian of three degrees of freedom. In our

perturbation approach, we avoid dealing with time by moving to an “extended” phase space

(see, for instance, [26]). Since the time-dependent variables evolve in quite different time

scales, we find convenient to introduce four new pairs of canonical conjugate variables: (l,L)for the mean anomaly of the sun and its conjugate momentum (λ,Λ) the same for the moon,

(w,W) for the moon argument of perigee and its conjugate momentum, and (N,Γ) for the

moon longitude of the ascending node and its conjugate momentum. The specific values of

the introduced canonical momenta are irrelevant for our purposes, and we find convenient

to make the ordering

H1,0 = Λn�, H2,0 = 2!Ln�, H3,0 = 3!(Wnw + ΓnN). (3.8)

Once we have set the Hamiltonian order, we will apply perturbation theory by the

Lie transforms in order to filter the short-period effects in the potential equation (2.1). More

precisely, we base our computations on Deprit’s algorithm, which is specifically designed for

automatic computation by machine [21, 27].

3.1. The Delaunay Normalization

We compute the canonical transformation T : (, g, h, L,G,H) → (′, g ′, h′, L′, G′,H ′) that

removes the mean anomaly of the satellite from the Hamiltonian up to the eighth order.

Note that the mean anomaly does not appear explicitly but through its dependence on the

true anomaly. This fact introduces some subtleties in the averaging procedure, but the usual

differential relations between true, mean, and eccentric anomalies allow to carry out the

normalization on closed form of the eccentricity.

After normalization, we get the new Hamiltonian

(H′ ≡ T : H

)=∑m≥0

δm

m!H0,m, (3.9)


where the H0,m terms are expressed in the prime variables. We find H0,m = Hm,0 for m < 4,

and

H0,4

4!H0,0=α2

a2J2

1

η3

(1 − 3

2s2

),

H0,5

5!H0,0= β

n2�n2

a3�r3�⟨M�

2

⟩+n2�n2

a3�r3�

⟨M�

2

⟩,

H0,6

6!H0,0= β

n2�n2

a4�r4�⟨M�

3

⟩,

H0,7

7!H0,0= β

n2�n2

a5�r5�⟨M�

4

⟩,

H0,8

8!H0,0= β

n2�n2

a6�r6�⟨M�

5

⟩+ J3

α3

a3

3e

4η5

(4 − 5s2

)s sinω +

α4

a4J2

2

3

32η7

×{+

1

2

[8(

5 + 2η − η2)− 8(

10 + 6η − η2)s2 +(

35 + 36η + 5η2)s4]

−1 − η1 + η

[2(

15 + 30η + 7η2)− 5(

7 + 14η + 3η2)s2]s2 cos 2ω

},

(3.10)

where s and c stand for sine and cosine of the inclination, respectively;

〈M�m〉 =

am−2

am−2�

m/2∑j=0

Em,2j+k

m∑i=−m

pm,2j+k,i(t�m,i cosφ + d�m,i sinφ

),

φ =(2j + k

)ω + iΩ, k = m mod 2, � ∈ (�,�),

(3.11)

where m/2 is an integer division. The eccentricity coefficients E, the inclination ones p, and

the third-body direction coefficients t� and d� are given in Tables 1, 2, and 3.

Note that except for J2-squared terms the new Hamiltonian is obtained by the simple

average over the mean anomaly of all the terms in (3.1). Besides, we introduced Kozai’s

arbitrary constant [28] in the solution of the fourth-order generating function to keep the

prime variables as close as possible to the average value of corresponding osculating ones.

The semimajor axis of the satellite remains constant after averaging, as well as its

canonical partner, the Delaunay action L′. Then, the time evolution of the mean anomaly

decouples from the two-degrees-of-freedom, time-dependent system

d(g ′, h′

)dt

=∂H′

∂(G′,H ′),

d(G′,H ′)dt

= − ∂H′

∂(g ′, h′

) . (3.12)

The numerical integration of this system can be done with longer step sizes than the

original one because of the filtering of short periodic effects via averaging. At each step of

the numerical integration, the osculating elements can be recovered analytically using the

transformation equations computed also by the Lie transforms method.


Table 1: Direction coefficients tj,k and dj,k and eccentricity coefficients Ej,k . The unit vector (x, y, z) definesthe direction of the center of mass of the third body.

j k tj,k dj,k Ej,|k|

2

0 1 − 3z2 0 2 + 3e2

±1 yz −kxz±2 x2 − y2 kxy e2

3

0 0 z(3 − 5z2)±1 x(1 − 5z2) ky(1 − 5z2) (4 + 3e2)e±2 4xyz −k(x2 − y2)z±3 3x(x2 − 3y2) ky(3x2 − y2) e3

4

0 3 − 30z2 + 35z4 0 8 + 40e2 + 15e4

±1 yz(3 − 7z2) −kxz(3 − 7z2)±2 (x2 − y2)(1 − 7z2) kxy(1 − 7z2) (2 + e2)e2

±3 3y(3x2 − y2)z −kx(x2 − 3y2)z±4 x4 − 6x2y2 + y4 kxy(x2 − y2) e4

5

0 0 z(15 − 70z2 + 63z4)±1 x(1 − 14z2 + 21z4) ky(1 − 14z2 + 21z4) (8 + 68e2 + 29e4)e±2 4xyz(1 − 3z2) −k(x2 − y2)z(1 − 3z2)±3 3x(x2 − 3y2)(1 − 9z2) ky(3x2 − y2)(1 − 9z2) (8 + 7e2)e3

±4 16xy(x2 − y2)z −k(x4 − 6x2y2 + y4)z±5 5x(x4 − 10x2y2 + 5y4) ky(5x4 − 10x2y2 + y4) e5

Table 2: Inclination coefficients p2,j,k and p3,j,k ; for brevity, we use the notation γ ≡ (1 ± c).

k p2,0,k p2,2,k p3,1,k p3,3,k

0 (1/8)(2 − 3s2) (15/8)s2 (15/64)s(1 − 5c2) −(175/64)s3

±1 (3/4)cs ∓(15/4)sγ (15/256)γ(1 ± 10c − 15c2) −(525/256)s2γ

±2 (3/16)s2 (15/16)γ2 ±(75/256)sγ(1 ∓ 3c) ±(525/256)sγ2

±2 −(25/256)s2γ −(175/768)γ3

Table 3: Inclination coefficients p4,j,k and p5,j,k ; for brevity, we use the notation γ ≡ (1 ± c).

j k p4,j,k p5,j+1,k

0

0 (3/2048)(3 − 30c2 + 35c4) −(15/4096)s(1 − 14c2 + 21c4)±1 −(15/512)sc(3 − 7c2) −(15/8192)γ(1 ± 28c − 42c2 ∓ 84c3 + 105c4)±2 −(15/1024)s2(1 − 7c2) ∓(105/4096)sγ(1 ∓ 3c − 9c2 ± 15c3)±3 (35/512)s3c (35/16384)s2γ(1 ± 6c − 15c2)±4 (105/4096)s4 ±(315/32768)s3γ(1 ∓ 5c)±5 −(63/16384)s4γ

2

0 −(105/512)s2(1 − 7c2) (245/8192)s3(1 − 9c2)±1 ±(105/256)sγ(1 ± 7c − 14c2) (735/16384)s2γ(1 ± 6c − 15c2)±2 (105/256)γ2(1 ∓ 7c + 7c2) ±(735/8192)sγ2(1 ∓ 12c + 15c2)±3 ∓(245/256)sγ2(1 ∓ 2c) −(245/98304)γ3(13 ∓ 54c + 45c2)±4 (735/1024)s2γ2 ±(2205/65536)sγ3(3 ∓ 5c)±5 −(441/32768)s2γ3

4

0 (2205/2048)s4 −(6615/8192)s5

±1 ∓(2205/512)s3γ −(33075/16384)s4γ

±2 (2205/1024)s2γ2 ±(33075/8192)s3γ2

±3 ∓(735/512)sγ3 −(11025/32768)s2γ3

±4 (2205/4096)γ4 ±(33075/65536)sγ4

±5 −(1323/32768)γ5


3.2. Elimination of the Mean Anomaly of the Moon

A new canonical transformation T′ : (′, g ′, h′, L′, G′,H ′) → (′′, g ′′, h′′, L′′, G′′,H ′′) removes

the mean anomaly from the Hamiltonian. The algorithm starts now by setting the new Hami-

ltonian

K =∑m≥0

δm

m!Km,0, (3.13)

where Km,0 = H0,m. Then, we express the direction of the moon (x�, y�, z�) in terms of the

moon’s orbital elements by the composition of rotations:

⎛⎝x�y�z�

⎞⎠ = R1(−ε)R3(−N)R1(−J)R3

(−θ�)

⎛⎝r�0

0

⎞⎠, (3.14)

where θ� = f� + ω�. The components of the moon direction vector are then replaced

in coefficients t�, d� of Table 1. Finally, application of the Lie triangle provides the new

Hamiltonian

(K′ ≡ T′ : K

)=∑m≥0

δm

m!K0,m, (3.15)

where the K0,m terms are expressed in the double prime variables, as well as the generating

function of the transformation.

The new averaging is similar to the preceding Delaunay normalization in the sense

that we base on the differential relations between the true and mean anomalies to perform the

averaging. As before, up to the computed order in the perturbation theory, there is no coupl-

ing between the different Hamiltonian terms, which, therefore, reduce to their averaging over

the mean anomaly of the moon λ. Thus, K0,m = Km,0 for m < 5, and

K0,5

5!K0,0= β

n2�n2

⟨a3�r3�⟨M�

2

⟩

⟩λ

+n2�n2

a3�r3�

⟨M�

2

⟩,

K0,6

6!K0,0= β

n2�n2

⟨a4�r4�⟨M�

3

⟩

⟩λ

,

K0,7

7!H0,0= β

n2�n2

⟨a5�r5�⟨M�

4

⟩

⟩λ

,

K0,8

8!K0,0= β

n2�n2

⟨a6�r6�⟨M�

5

⟩

⟩λ

+ J3α3

a3

3e

4η5

(4 − 5s2

)s sinω +

α4

a4J2

2

3

32η7

×{+

1

2

[8(

5 + 2η − η2)− 8(

10 + 6η − η2)s2 +(

35 + 36η + 5η2)s4]

−1 − η1 + η

[2(

15 + 30η + 7η2)− 5(

7 + 14η + 3η2)s2]s2 cos 2ω

},

(3.16)


Table 4: Averaged moon direction coefficients tj,k . Here, c ≡ cos J , s ≡ sin J , and e ≡ e�.

j k tj,k

2

0

(1 − 3

2e2

)[1

4(2 − 3σ2)(2 − 3s2) − 3σκcs cosN +

3

4σ2s2 cos 2N

]

±1

(1 − 3

2e2

)[1

4σκ(2 − 3s2) +

1

2(1 − 2σ2)cs cosN − 1

4σκs2 cos 2N

]

±2

(1 − 3

2e2

)[1

4σ2(2 − 3s2) + σκcs cosN +

1

4(2 − σ2)s2 cos 2N

]

3

0 0

±1 e

[1

4(4 − 5σ2) cos(N +ω�) − 5

2σκs cos(2N +ω�)

]±2 e[σκ cos(N +ω�) + (1 − 2σ2)s cos(2N +ω�)]

±3 e

[9

4σ2 cos(N +ω�) +

9

2σκs cos(2N +ω�)

]

4

03

8(8 − 40σ2 + 35σ4) − 15

2(4 − 7σ2)σκs cosN

±13

8(4 − 7σ2)σκ +

3

8(4 − 29σ2 + 28σ4)s cosN

±23

8(6 − 7σ2)σ2 +

3

2(3 − 7σ2)σκs cosN

±39

8σ3κ +

9

8(3 − 4σ2)σ2s cosN

±43

8σ4 +

3

2σ3κs cosN

where, from (3.11),

⟨am+1�rm+1�

⟨M�

m

⟩

⟩λ

=am−2

am−2�m/2∑j=0

Em,2j+k

m∑i=−m

pm,2j+k,i(tm,i cosφ + dm,i sinφ

),

φ =(2j + k

)ω + iΩ, k = m mod 2,

(3.17)

with

tm,i =

⟨(a�r�)m+1

t�m,i⟩

λ

, dm,i =

⟨(a�r�)m+1

d�m,i

⟩λ

. (3.18)

To get some further simplification, we note that e�∼ i�∼ O(δ). Thus, in our eighth-

order theory, we neglect higher-order terms factored by em�sinnJ δk such that m + n + k > 8.

Under these simplifying assumptions, the coefficients tm,i and dm,i are presented in Tables 4

and 5, respectively.


Table 5: Averaged moon direction coefficients dj,k . Here, c = cos J , s = sin J , and e ≡ e�.

j k dj,k

2

0 0

±1 k

(1 − 3

2e2

)(1

2κcs sinN − 1

4σs2 sin 2N

)

±2 k

(1 − 3

2e2

)(1

2σcs sinN +

1

4κs2 sin 2N

)

3

0 e

[3

2(2 − 5σ2)κs sinω� +

3

4(4 − 5σ2)σ sin(N +ω�) − 15

4σ2κs sin(2N +ω�)

]

±1 ke

[−3

2(4 − 5σ2)σs sinω� +

1

4(4 − 15σ2)κ sin(N +ω�) − 5

4(2 − 3σ2)σs sin(2N +ω�)

]

±2 ke

[3

2σ2κs sinω� − 1

4(2 − 3σ2)σ sin(N +ω�) − 1

4(2 − 3σ2)κs sin(2N +ω�)

]

±3 ke

[−3

2σ3s sinω� +

3

4σ2κ sin(N +ω�) +

3

4(2 − σ2)σs sin(2N +ω�)

]

4

0 0

±1 k3

8(4 − 7σ2)κs sinN

±2 k3

8(6 − 7σ2)σs sinN

±3 k9

8σ2κs sinN

±4 k3

8σ3s sinN

As in the preceding averaging, adequate arbitrary constants have been introduced in

the computation of the generating function to guarantee its average to zero.

After the double averaging, the Hamiltonian only depends on long-period terms

related to the sun’s apparent motion and very long-period terms related to the precession of

the nodes and recession of the line of apsides of the moon’s orbit. The numerical integration

of corresponding Hamilton equations

d(g ′′, h′′

)dt

=∂K′

∂(G′′,H ′′),

d(G′′,H ′′)dt

= − ∂K′

∂(g ′′, h′′

) (3.19)

is, now, very much faster and efficient.

4. Numerical Experiments

For the numerical tests we used a higher-order Runge-Kutta method. Specifically, the

numerical integration was performed with the Dormand and Prince implementation of the

Runge-Kutta method coded in FORTRAN 77 by Hairer et al. [29].


0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

−300−100

100300

−3000−100010003000

a(m

)a(m

)a(m

)a(m

)a(m

)a(m

)60

20

−20

2

−2

0

2

−2

0.4

0

−0.4

Figure 1: Semianalytical theory versus numerical integration. From top to bottom, errors in semimajor axisof the secular terms and after computing the fourth, fifth, sixth, seventh, and eighth order transformationequations, respectively. Abscissas are days.

Table 6: Initial conditions in the single- (top) and double-averaged phase space (bottom) after differenttruncation orders n of the semi-analytical theory. Distances are km, and angles are deg.

n 28560 − a 0.2 − e 56 − i 0 −ω 72 −Ω 0 − 4 3.06217 0.0000723 0.0014902 0.0000000 0.0000000 0.0000000

5 3.06923 0.0000722 0.0014713 −0.0016703 0.0000132 0.0016800

6 3.06073 0.0000720 0.0014808 −0.0017529 0.0000081 0.0018807

7 3.06077 0.0000720 0.0014820 −0.0017477 0.0000075 0.0018817

8 3.06085 0.0000721 0.0014819 −0.0017510 0.0000042 0.0018783

4 3.06073 0.0000453 −0.0030387 −0.0149534 −0.0027445 0.0167622

5 3.06923 0.0000431 −0.0026903 −0.0153934 −0.0028968 0.0172767

6 3.06073 0.0000432 −0.0026846 −0.0153427 −0.0029047 0.0173255

7 3.06077 0.0000432 −0.0026744 −0.0153287 −0.0028995 0.0173220

8 3.06085 0.0000433 −0.0026745 −0.0153319 −0.0029027 0.0173187

To illustrate the significance of recovering the short-period effects up to higher orders,

we first show in Figure 1 a sequence of the errors obtained in the semimajor axis after one

month semi-analytical propagation. For this example, we take the initial osculating elements

a = 28 560 km, e = 0.2, i = 56 deg., ω = 0, Ω = 72 deg., and = 0; besides we assumed

that both the mean anomalies of the moon and the sun are zero at the origin of time.

Corresponding elements in the single- and double-averaged phase space depend on the

truncation order of the perturbation theory and are presented in the top and bottom parts

of Table 6, respectively.


e

100806040200

0.05

0.1

0.15

0.2

Years

(a)

100806040200

Years

58

57

56

55

54

53

i(d

eg)

(b)

Figure 2: Sample long-term propagations for ω = 0 and Ω = 0 full line, 90 dashed, 180 dotted, and 270 deg.dash-dotted. (a) Eccentricity variation. (b) Inclination variations.

In reference to Figure 1, the top plot shows a direct comparison between the

numerical integration of the Hamilton equations of the original, nonaveraged problem—

whose disturbing potential is given in (2.1)—and that of the long-term Hamiltonian after the

double averaging up to the eight order of the small parameter. Then, from top to bottom, we

show the errors obtained at each step of the integration when recovering short-period terms

by computing the transformation equations of the averaging up to the fourth, fifth, sixth,

seventh, and eighth orders. For the latter, the amplitude of the periodic errors is reduced to

several centimeter; note that, in addition to short-period errors related to the orbital period of

the satellite, we can appreciate a two-week modulation related to the moon’s mean motion.

Remark that the amplitude of the periodic errors roughly divides by ten with each order of

the transformation equations, which is consistent with the assumed magnitude of the virtual

small parameter δ ∼ 0.1.

The short-period effects can be ignored in the study of the long-term orbital behavior,

where the simple propagation of the double-averaged equations appears very fast and

efficient. Sample propagations are shown in figures below, which show the important effects

that have the initial right ascension of the ascending node and argument of perigee of the

satellite’s orbit in the long term and specifically in the eccentricity and inclination evolution

of the satellite’s orbit. Thus, Figure 2 shows the notably different evolution of the satellite’s

eccentricity and inclination for different initial nodes; for the other initial conditions we have

taken the same as in the preceding short-term propagations but now assuming directly that

they are mean elements. In fact, the more relevant parameter is the difference between the

node of the satellite’s orbit and that of the moon’s orbit, as illustrated in Figure 3 where the

initial longitude of the ascending node of the moon’s orbit over the ecliptic has been taken as

N = 180 instead of N = 0 deg.

Figure 4 shows that the effect of the initial argument of the perigee of the satellite’s

orbit is also important in the long term, eccentricity evolution. The effect is almost negligible

in the orbital inclination in the long-term and it is not presented.

Finally, we must mention that further tests demonstrated that there are no important

qualitative differences in the long term when using lower-order truncations of the theory,

resulting in faster numerical integration of the mean elements. So the fifth-order truncation

should be the preferred long-term Hamiltonian. Besides, we also checked for a variety of

orbits that making e� = e� = 0 does not either introduce qualitative differences in the long-

term, in agreement with [19].


100806040200

e

0.05

0

0.1

0.15

0.2

0.25

0.3

Years

(a)

100806040200

Years

58

59

60

57

56

55

54

53

i(d

eg)

(b)

Figure 3: The same as Figure 2 but now N = 180 instead of N = 0 deg.

100806040200

Years

e

0.12

0.1

0.08

0.06

0.04

0.02

Figure 4: Sample long-term propagations for Ω = 0 and ω = 0 full line, 30 dashed, 60 dotted, and 90 deg.dash-dotted. Eccentricity variation.

5. Conclusions

Modeling lunisolar perturbations on high-altitude Earth orbits requires to retain high degrees

in the Legendre polynomials expansion of the third-body disturbing function of the moon. In

consequence the eccentricity of the third-body’s orbit cannot be neglected in the case of either

the moon or the sun.

The long-term behavior of high-altitude Earth orbits is approached in a semianalytical

way via averaging procedures, in which we take advantage of the different scales in which

appears the time to do the averaging in the extended phase space. In addition, up to second-

order terms in the Earth’s oblateness coefficient, the averaging has been computed in closed

form of the eccentricity, and, therefore, the semianalytical integration can be applied to any

orbit.

Sample numerical propagations of test cases show that the more relevant parameter on

the long-term behavior is the difference between the right ascension of the ascending node of

the satellite’s orbit and the longitude of the ascending node of the moon’s orbit over the eclip-

tic, having an apparent effect manifested by the almost-secular growing of the orbit eccentric-

ity and also by very-long-period oscillations of the inclination with an amplitude of several

degrees. Also the initial argument of the perigee of the satellite’s orbit has notable effects in

the satellite’s orbit, but only in what respects to the long-term evolution of the eccentricity.


Acknowledgments

Part of this research has been supported by the Government of Spain (Projects AYA 2009-

11896, AYA 2010-18796, and Grant Gobierno de La Rioja Fomenta 2010/16).

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Research ArticleError Modeling and Analysis forInSAR Spatial Baseline Determination ofSatellite Formation Flying

Jia Tu, Defeng Gu, Yi Wu, and Dongyun Yi

Department of Mathematics and Systems Science, College of Science,National University of Defense Technology, Changsha 410073, China

Correspondence should be addressed to Jia Tu, tu jia [email protected]

Received 30 September 2011; Revised 9 December 2011; Accepted 12 December 2011


Copyright q 2012 Jia Tu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Spatial baseline determination is a key technology for interferometric synthetic aperture radar(InSAR) missions. Based on the intersatellite baseline measurement using dual-frequency GPS,errors induced by InSAR spatial baseline measurement are studied in detail. The classificationsand characters of errors are analyzed, and models for errors are set up. The simulations of singlefactor and total error sources are selected to evaluate the impacts of errors on spatial baselinemeasurement. Single factor simulations are used to analyze the impact of the error of a single type,while total error sources simulations are used to analyze the impacts of error sources inducedby GPS measurement, baseline transformation, and the entire spatial baseline measurement,respectively. Simulation results show that errors related to GPS measurement are the main errorsources for the spatial baseline determination, and carrier phase noise of GPS observation andfixing error of GPS receiver antenna are main factors of errors related to GPS measurement. Inaddition, according to the error values listed in this paper, 1 mm level InSAR spatial baselinedetermination should be realized.

1. Introduction

Close formation flying satellites equipped with synthetic aperture radar (SAR) antenna

could provide advanced science opportunities, such as generating highly accurate digital

elevation models (DEMs) from Interferometric SAR (InSAR) [1, 2]. Compared to a single

SAR satellite system, the performance of two SAR satellites flying in close formation can be

greatly enhanced. Nowadays, close satellite formation flying has become the focus of space

technology and geodetic surveying.

In order to realize the advanced space mission goal of InSAR mission, the high-

precision determination of inter-satellite interferometric baseline [3] is a fundamental issue.


Take the TanDEM-X mission for instance. TanDEM-X mission is the first bistatic single-pass

SAR satellite formation, which is formed by adding a second TanDEM-X, almost identical

spacecraft, to TerraSAR-X and flying the two satellites in a closely controlled formation. The

primary mission goal is the derivation of a high-precision global DEM according to high-

resolution terrain information (HRTI) level 3 accuracy [4–6]. The generation of accurate

InSAR-derived DEMs requires a precise knowledge of the interferometric baseline with an

accuracy of 1 mm (1D, RMS) [7]. Therefore high-precision determination of inter-satellite

interferometric baseline is a prerequisite for InSAR mission.

The interferometric baseline is defined as the separation between two SAR antennas

that receive echoes of the same ground area [8]. Based on this definition, the interferometric

baseline can be denoted as the resultant vector of temporal baseline and spatial baseline, that

is,

S2(t2) − S1(t1) = S2(t2) − S1(t2) + S1(t2) − S1(t1), (1.1)

where t1, t2 are epochs that two SAR antennas receive echoes of the same ground area, S1(t),S2(t) represent the positions of SAR antenna phase centers of satellite 1 and satellite 2 at

epoch t in International Terrestrial Reference Frame (ITRF), respectively, S2(t2) − S1(t2) is

the spatial baseline, S1(t2) − S1(t1) is the temporal baseline which is the velocity integral of

satellite 1. For close formation flying (1 km-2 km) with single-pass bistatic acquisitions, the

deviation of epochs that two SAR antennas receive echoes of the same ground area is typically

on the millisecond level. When the velocity is determined on the mm/s level, its influence in

the temporal baseline can be neglected. Therefore, the accuracy of interferometric baseline

is mainly determined by the accuracy of spatial baseline. Note that only spatial baseline is

considered in this paper.

The spaceborne dual-frequency GPS measurement scheme [9–11] is widely used

for inter-satellite baseline determination currently. This scheme for spatial baseline deter-

mination consists of two steps. Firstly, the relative position of two formation satellites is

determined by dual-frequency GPS measurement, and then spatial baseline is transformed

from inter-satellite relative position. The relative position here is the vector that links the

mass centers of two formation satellites.

In our research, impacts of the errors introduced by spatial baseline measurement are

analyzed. This paper starts with a description of spatial baseline measurement using dual-

frequency GPS. The baseline transformation from the relative position to spatial baseline

is given. In a second step, errors are classified into two groups: errors related to GPS

measurement and errors related to baseline transformation. The error characters are studied,

and the impact of each error on spatial baseline determination is analyzed from theoretical

aspect. Then the impacts of each error and total errors on spatial baseline determination are

analyzed by single factor simulations and total error sources simulations. At last, conclusions

are shown.

2. Generation of Spatial Baseline

In preparation for latter description some coordinate systems are introduced at first, which

are illustrated in Figure 1. Coordinate systems employed in this paper contain Conventional

Inertial Reference Frame (CIRF), ITRF, satellite body coordinate system, and satellite orbit


ZCIRF

ZITRF

XCIRF

Vernal equinox

Equator

Greenwich meridian

tG

OS

OE

XITRF

YITRF

XBody

YBody

YOrbit

XOrbit

ZBody

ZOrbit

Satellite orbit

YCIRF

Figure 1: Definitions of coordinate systems employed in this paper. CIRF, ITRF, satellite body coordinatesystem, and satellite orbit coordinate system are denoted as OE-XCIRFYCIRFZCIRF, OE-XITRFYITRFZITRF, OS-XBodyYBodyZBody, and OS-XOrbitYOrbitZOrbit, respectively. OE is the geocenter, and OS is the mass center ofsatellite.

Spatial baseline

Relative positionSatellite 1 Satellite 2

G1 G2

O1 O2

S1S2

Figure 2: Geometric relation for spatial baseline determination. G1 and G2 are GPS receiver antenna phasecenters, O1 and O2 are mass centers, and S1 and S2 are SAR antenna phase centers.

coordinate system. CIRF used here is J2000.0 inertial system and ITRF is ITRF2000 system.

The definitions of these coordinate systems can be found in [12].As the spatial baseline is determined by spaceborne dual-frequency GPS measurement

scheme, the entire process of spatial baseline determination consists of relative positioning

and baseline transformation. Figure 2 is the geometric relation for spatial baseline determina-

tion.

Relative positioning is the determination of O1O2 by dual-frequency GPS observation

data. As the real position of signal reception is the phase center Gi (i = 1, 2) of GPS receiver

antenna, GPS observation data has to be revised to the mass center Oi (i = 1, 2) of satellite

using the phase center data of GPS receiver antenna during relative positioning.

From Figure 2, baseline transformation can be described as follows:

S1S2 = O1O2 +M1 · S1O1 −M2 · S2O2, (2.1)

where S1S2 is the spatial baseline in ITRF, O1O2 is the relative position of two satellites

in ITRF, SiOi (i = 1, 2) is a vector that links SAR antenna phase center to mass center of

satellite in body coordinate system of Satellite i, Mi (i = 1, 2) is a transformation matrix of

Satellite i from satellite body coordinate system to ITRF. The flow chart of spatial baseline

determination is shown in Figure 3.


GPS observation

data of satellite 1

GPS observation

data of satellite 2

GPS relative positioning

Position and velocity of

satellite 1 in ITRF

SAR antenna phase center

position of satellite 2 in

ITRF

SAR antenna phase center position of satellite 1 in

ITRF

Baseline

transformation

Baseline

transformation

Subtracting at same

epochSpatial baseline

Position and velocity of

satellite 2 in ITRF

Figure 3: Flow chat of spatial baseline determination.

3. Errors of Spatial Baseline Measurement

According to the generation of spatial baseline in Section 2, the errors of spatial baseline

measurement can be classified into two groups: errors related to GPS measurement, which

are introduced by relative positioning using dual-frequency GPS measurement, and errors

related to baseline transformation, which are generated by the transformation from relative

position to spatial baseline.

3.1. Errors Related to GPS Measurement

The relative positions of two satellites are determined by the reduced dynamic carrier

phase differential GPS approach. In this approach, the absolute orbits of one reference

satellite (Satellite 1) are fixed, which are determined by the zero-difference reduced dynamic

batch least squares approach based on GPS measurements of single satellite. Only the

relative positions are estimated by reduced dynamic batch least-squares approach based

on differential GPS measurements. The integer double difference ambiguities for relative

positioning are resolved by estimating wide-lane and narrow-lane combinations [13]. The

well-known Least-Squares Ambiguity Decorrelation Adjustment (LAMBDA) method [14,

15] is implemented for the integer estimate.

By differenced GPS observation, common errors can be eliminated or reduced.

International GNSS Service (IGS) final GPS ephemeris product (orbit product and clock

product) [16] is often adopted for orbit determination based on GPS observation. The

accuracy of GPS final orbit product is presently on the order of 2.5 cm. For 2 km separation of

satellite formation, the impact of GPS ephemeris error on single-difference GPS observation

is about 0.0025 mm [17], which can be neglected. The impact of GPS clock error can be

well cancelled out by differential GPS observation. Due to the close separation (1 km-

2 km) and similar materials, configuration, and in-flight environment of formation satellites,

near-field multipath, thermal distortions of satellites, and other external perturbations can

also be effectively reduced by differential GPS observation. In addition, the influence of

differential ionospheric path delay is mainly from the first order, which can be eliminated by

constructing ionosphere free differential GPS observation. Therefore, the errors related to GPS

measurement that have to be considered consist of noise of GPS carrier phase measurement,

ground calibration error of GPS receiver antenna phase center, error of satellite attitude

measurement, and fixing error of GPS receiver antenna.


3.1.1. Noise of GPS Carrier Phase Measurement

The quality of GPS carrier phase observation data used is of utmost importance for relative

positioning. The noise of GPS carrier phase measurement belongs to random error, which

cannot directly be eliminated by GPS differential observation. Take the BlackJack receiver

and its commercial Integrated GPS and Occultation Receiver (IGOR) version, for example,

which are widely used for geodetic grade space missions and exhibit a representative noise

level of 1 mm for L1 and L2 carrier phase measurements [18]. The reduced dynamic relative

positioning approach makes use of dynamical models of the spacecraft motion to constrain

the resulting relative position estimates, which allows an averaging of measurements from

different epochs. The influence of GPS carrier phase noise can be effectively reduced by

reduced dynamic relative positioning approach.

3.1.2. Ground Calibration Error of GPS Receiver Antenna Phase Center

The phase center location accuracy of the GPS receiver antenna will directly affect the veracity

of GPS observation modeling. GPS receiver antenna phase center is the instantaneous location

of the GPS receiver antenna where the GPS signal is actually received. It depends on intensity,

frequency, azimuth, and elevation of GPS receiving signal.

The phase center locations can be described by the mechanical antenna reference point

(ARP), a phase center offset (PCO) vector, and phase center variations (PCVs). The PCO

vector describes the difference between the mean center of the wave front and the ARP.

PCVs represent direction-dependent distortions of the wave front, which can be modeled

as a consistent function that depends on azimuth and elevation of the observation from

the position indicated by the PCO vector. The position of GPS receiver antenna phase

center can be measured by ground calibration, such as using an anechoic chamber and

using field calibration techniques [18, 19]. Take the SEN67-1575-14+CRG antenna system

for instance. It is a dual-frequency GPS receiver antenna and has been used for TanDEM-

X mission. Its phase center has been measured by automated absolute field calibration [20].The mean value of calibration result is shown in Figure 4 that the pattern of PCVs has obvious

character of systematic deviation. The maximum value for the mean PCVs on ionosphere-

free combination can reach to 1.5 cm. In addition, there also exist random errors in the same

direction of different receptions. The random errors are similar to the noise of GPS carrier

phase measurement and can also be smoothed by reduced dynamic relative positioning

approach.

As there is a slim difference between the line of sight (LOS) vectors of two satellites

during close formation flying, the common systematic errors of GPS receiver antenna phase

center and near-field multipath can be eliminated by differential GPS observation. Therefore,

the same type of GPS receiver antenna has to be selected for both formation satellites in order

to reduce the impact of these errors.

3.1.3. Error of Satellite Attitude Measurement

Satellite attitude data are obtained from star camera observations and provided as

quaternion. The error of satellite attitude measurement consists of a slowly varying bias and

a random error. Its impact on GPS relative positioning appears on the correction for GPS

observation data of single satellite, that is, the reference point of GPS observation data has to


X (azimuth = 0)

Y(a

zim

uth=

90)

−12 −6 0 6 12

Antenna phase center (mm)

030

6090

Figure 4: Ground calibrated mean PCVs result of SEN67-1575-14+CRG antenna on ionosphere-freecombination.

be corrected from GPS receiver antenna phase center to the mass center of satellite by satellite

attitude data and GPS receiver antenna phase center data. Take Satellite 1 for instance. The

correction in direction of LOS vector ej1 (in CIRF) is given by

δOffset,1 = −(ej1)T

·MBody CIRF ·O1G1,

MBody CIRF = MOrbit CIRF ·MBody Orbit,(3.1)

where O1G1 is GPS receiver antenna phase center location in body coordinate system of

Satellite 1, MBody CIRF is the transformation matrix from body coordinate system of Satellite

1 to CIRF and can be obtained by attitude quaternion data, MBody Orbit is the transformation

matrix from body coordinate system to orbit coordinate system of Satellite 1, and MOrbit CIRF

is the transformation matrix from orbit coordinate system of Satellite 1 to CIRF.

Assuming that the Euler angles are ϕ, θ, and ψ respectively, we can get

MBody Orbit = RX

(ϕ)· RY (θ) · RZ

(ψ), (3.2)

where RX(ϕ), RY (θ), RZ(ψ) are rotation matrices around roll axis, pitch axis, and yaw axis,

respectively.

Assuming that the errors of Euler angle measurements are εϕ, εθ, and εψ , respectively,

and the corresponding error matrix of MBody CIRF is εM, the relation between εM and εϕ, εθ, εψcan be expressed as

εM = MOrbit CIRF ·(∂RX

∂ϕRYRZ · εϕ + RX

∂RY

∂θRZ · εθ + RXRY

∂RZ

∂ψ· εψ). (3.3)


Furthermore, the impact of Euler angle errors on MBody CIRF · O1G1 in (3.1) can be

obtained as

εMBody CIRF·O1G1= MOrbit CIRF

·(∂RX

∂ϕRYRZ ·O1G1 · εϕ + RX

∂RY

∂θRZ ·O1G1 · εθ + RXRY

∂RZ

∂ψ·O1G1 · εψ

).

(3.4)

εMBody CIRF·O1G1is a three-dimensional random vector and its magnitude can be described

as the mean value of space radius, that is

σ2MBody CIRF·O1G1

= E(∣∣∣εMBody CIRF·O1G1

∣∣∣2) = E(εTMBody CIRF·O1G1

· εMBody CIRF·O1G1

), (3.5)

where |·| denotes the magnitude of a vector, E(·) denotes the expectation of a random variable.

Assuming Euler angle errors of different axes are independent, we can get

σ2MBody CIRF·O1G1

=∣∣∣∣MOrbit CIRF ·

∂RX

∂ϕRYRZ ·O1G1

∣∣∣∣2 · (Var(εϕ) + (E(εϕ))2)

+∣∣∣∣MOrbit CIRF · RX

∂RY

∂θRZ ·O1G1

∣∣∣∣2 · (Var(εθ) + (E(εθ))2)

+∣∣∣∣MOrbit CIRF · RXRY

∂RZ

∂ψ·O1G1

∣∣∣∣2 · (Var(εψ) + (E(εψ))2),

(3.6)

where Var (·) denotes the variation of a random variable.

As RX(ϕ), RY (θ), RZ(ψ), and MOrbit CIRF are orthogonal matrices, for any v ∈ R3, we

can get

|RX · v|2 = |v|2,∣∣∣∣∂RX

∂ϕ· v∣∣∣∣2 ≤ |v|2

|RY · v|2 = |v|2,∣∣∣∣∂RY

∂θ· v∣∣∣∣2 ≤ |v|2

|RZ · v|2 = |v|2,∣∣∣∣∂RZ

∂ψ· v∣∣∣∣2 ≤ |v|2

|MOrbit CIRF · v|2 = |v|2.

(3.7)

Taking (3.7) into (3.6), we can get

σ2MBody CIRF·O1G1

≤ |O1G1|2 ·[Var(εϕ)+(E(εϕ))2 +Var(εθ) + (E(εθ))2 +Var

(εψ)+(E(εψ))2].

(3.8)


Hence,

σ2δOffset,1

= E(|εδOffset

|2)= E

(∣∣∣∣(ej1)T· εMBody CIRF·O1G1

∣∣∣∣2)

≤ E

(∣∣∣εMBody CIRF·O1G1

∣∣∣2). (3.9)

For differential GPS observation, the impact of attitude determination error on two

satellites can be given as follows

σΔδOffset≤√σ2δOffset,1

+ σ2δOffset,2

. (3.10)

According to the TanDEM-X missions, the attitude determination accuracy has a

slowly varying bias of ±0.005◦ in the yaw, pitch, and roll components plus a 0.003◦ sigma

random error [21]. From (3.8), (3.9), and (3.10), we can get

σΔδOffset≤

√√√√3 ·(|O1G1|2 + |O2G2|2

)·[(

0.005

180π

)2

+(

0.003

180π

)2]. (3.11)

Take the GPS receiver antenna ARP location of TanDEM-X mission for instance, that

is,

|O1G1| = |O2G2| = 1.8976 m, (3.12)

we can get

σΔδOffset≤ 0.47 mm. (3.13)

3.1.4. Fixing Error of GPS Receiver Antenna

The fixing error of GPS receiver antenna is caused by the inaccuracy of the fixed position

of antenna onboard the satellite. This error is a random error for multiple repeated satellite

missions. But for a single launch, it is considered to be a fixed bias vector in satellite body

coordinate system during satellite flying.

The fixing errors of GPS receiver antenna in body coordinate system of two satellites

are assumed as follows:

ΔE1 =(Δx1;Δy1;Δz1

),

ΔE2 =(Δx2;Δy2;Δz2

).

(3.14)


For a mutually observed GPS satellite j, the LOS vectors of two formation satellites

are assumed to be ej1 and ej2. The impact of fixing errors of GPS receiver antenna for both

formation satellites on GPS observation data can be denoted as

Δδj1 =(ej1)T

·MBody CIRF,1 ·ΔE1,

Δδj

2 =(ej

2

)T·MBody CIRF,2 ·ΔE2.

(3.15)

The impact of fixing error of GPS receiver antenna on differential GPS observation is

Δδj12 =(ej2)T

·MBody CIRF,2 ·ΔE2 −(ej1)T

·MBody CIRF,1 ·ΔE1. (3.16)

Due to the close separation of two satellites, we can assume

ej1 ≈ ej2. (3.17)

From (3.16) and (3.17), we can get

Δδj

12 ≈(ej2)T

·(MBody CIRF,2 ·ΔE2 −MBody CIRF,1 ·ΔE1

)=(ej2)T

·MBody CIRF,2 · (ΔE2 −ΔE1) +(ej2)T

·(MBody CIRF,2 −MBody CIRF,1

)·ΔE1.

(3.18)

As the magnitudes of ΔE1 and ΔE2 are small (generally less than 0.5 mm) and the

difference between MBody CIRF,1 and MBody CIRF,2 is insignificant; therefore, the impact of (ej2)T·

(MBody CIRF,2 − MBody CIRF,1) · ΔE1 in (3.18) can be neglected and the main influence is from

(ej2)T·MBody CIRF,2 ·(ΔE2−ΔE1). If the magnitude of GPS receiver antenna fixing error is 0.5 mm

for each formation satellite, the maximum 3-dimensional impact on relative positioning can

reach to 1 mm.

In addition, we can also draw a conclusion from the aforementioned analysis that the

GPS receiver antenna bias caused by thermal distortions of satellites can be cancelled out by

differential GPS observation.

3.2. Errors Related to Baseline Transformation

From (2.1), errors related to baseline transformation consist of two parts: one part is

introduced by transformation matrices M1 and M2, which is mainly caused by the satellite

attitude measurement error; the other part is introduced by S1O1 and S2O2, which is caused

by the inconsistency of two SAR antenna phase centers.


3.2.1. Error of Satellite Attitude Measurement

Take M1 for instance,

M1 = MCIRF ITRF ·MBody CIRF, (3.19)

where MCIRF ITRF is a transformation matrix from CIRF to ITRF, MBody ITRF has been defined

in (3.2).Note that the transformation from CIRF to ITRF is in accordance with IERS 1996

conventions [22] and this transformation error can be neglected. The errors of M1 and M2 are

also introduced by satellite attitude measurement errors. Similar to the analysis of satellite

attitude measurement error related to GPS measurement, from (3.8), we can obtain

σ2M1·S1O1

≤ |S1O1|2 ·[Var(εϕ)+(E(εϕ))2 +Var(εθ) + (E(εθ))2 +Var

(εψ)+(E(εψ))2]. (3.20)

Hence, the impact of attitude determination errors on baseline transformation is given

as follows:

σAtt ≤√σ2M1·S1O1

+ σ2M2·S2O2

. (3.21)

Take the attitude determination accuracy of TanDEM-X mission for instance and select

the magnitudes of S1O1 and S2O2 as follows

|S1O1| = |S2O2| = 2 m, (3.22)

we can get

σAtt ≤ 0.50 mm. (3.23)

3.2.2. Consistency Error of SAR Antenna Phase Center

Unlike GPS receiver antenna, active phased array antenna is selected for SAR antenna.

The phase center of the SAR antenna describes the variation of the phase curve within the

coverage region against a defined origin, here the origin of the antenna coordinate system

[18]. For two formation satellites of InSAR mission, the same type of SAR antenna should be

selected. As the identical processes of the scheme designing, manufacturing, and testing are

selected for SAR antennas of the same type, theoretically the consistency in configuration

and electric performance of SAR antennas should be well achieved. But factually there

exist the errors during manufacturing, fixing, and deploying of SAR antenna, therefore, the

consistency error of the SAR antenna phase center corresponding to the same beam occurs. It

is mainly caused by two factors:

(1) The inconsistency between receiver channels, which is introduced by manufac-

turing process, such as the instrument difference, machining art level, module

assembling level and the work temperature difference, et al.


Table 1: Orbit elements of formation satellites.

Parameters of satellites Satellite 1 Satellite 2

Semimajor axis 6886478 m 6886478 m

Inclination 97.4438◦ 97.4438◦

Eccentricity 0.00117 0.001073

Argument of perigee 90◦ 90◦

Right ascension of ascending node 0◦ 0.01171◦

True anomaly 269.99677◦ 270.00622◦

(2) The inconsistency between the locations of apertures, which is mainly caused

by the fixing flatness difference, relative dislocation difference, the deployment

inconsistency of SAR antennas and the configuration distortions caused by different

thermal circumstances, and others.

According to current ability of engineering, the phase inconsistency between T/R

modules at X-band can be constrained to 15◦ (3σ) and the inconsistency between the

locations of apertures can be constrained to λ/10 (3σ) [23] that equals to 36◦ (3σ) of

phase inconsistency. Assuming that the number of T/R modules of an SAR antenna is

N, the synthetic phase consistency error can be constrained to

√((15◦)2 + (36◦)2)/N =

39◦/√N (3σ). Hence, the consistency error of two SAR antenna phase center locations can

be constrained to (39◦/

√N)

360◦·√

2 · λ = 0.153 · λ√N

(3σ). (3.24)

Take the TanDEM-X mission, for example. Setting N = 384, λ = 0.032 m, the

consistency error of SAR antenna phase center location can be constrained to 0.25 mm (3σ).

4. Simulations for InSAR Spatial Baseline Determination

4.1. Simulation Settings

The HELIX satellite formation is selected for the simulations and the orbit elements of two

satellites are shown in Table 1. The spaceborne SAR is assumed to work at X-band with a

wavelength of 0.032 m and consist of 384 T/R modules.

The entire simulation consists of GPS measurement simulation and baseline transfor-

mation simulation. The flow chart of GPS observation data simulation is shown in Figure 5.

The International Reference Ionosphere 2007 (IRI2007) model is used to simulate ionospheric

delay, Allan variation is used to simulate the clock offset of GPS receiver, and the ARP data,

PCO data [18] and PCVs data of GPS receiver antenna system SEN67-1575-14+CRG are

used to simulate the GPS receiver antenna phase center locations. The PCVs data contains

the mean values and RMS values corresponding to frequency, azimuth, and elevation of

received signal. The attitude data of formation satellite is generated as follows: at first, a

transformation matrix from CIRF to satellite orbit coordinate system is obtained from orbit

data of a formation satellite in CIRF; second, assuming the real Euler angles are 0◦, that

is, satellite orbit coordinate system and satellite body coordinate system are the same, the


Orbit elements of formation satellite,

orbit dynamical parameters

CODE final orbit and clock product for

GPS satellites

Orbit integral Lagrange interpolation

Visibility analysis of GPS satellites

Standard positions and velocities of

formation satellite

Positions, velocities and

clock offsets of GPS satellites

GPS observation data file

The real distances between visible GPS satellites and formation satellite

Mea

sure

men

t n

ois

es o

f c

od

e an

d p

hase

Att

itu

de

data

of

form

ati

on

sate

llit

e

Ph

ase

cen

ter

data

of

GP

S r

ecei

ver

an

ten

na

Rel

ati

vis

tic

eec

t

Ion

osp

her

ic d

elay

GP

S r

ecei

ver

clo

cko

sets

of

form

ati

on

sate

llit

e

Figure 5: Flow chart of GPS observation data simulation.

simulating data of Euler angles are generated by attitude measurement error model list in

Table 2; third, the transformation matrix from satellite orbit coordinate system to satellite

body coordinate system can be obtained by the simulating data of Euler angles; at last, the

attitude quaternion is obtained by the transformation matrix from CIRF to satellite body

coordinate system.

Baseline transformation simulation is the process that the spatial baseline in ITRF is

obtained by mass center data of formation satellites in ITRF, attitude simulation data, and

SAR antenna phase center simulation locations in satellite body coordinate system. The

real SAR antenna phase center simulation location in satellite body coordinate system is

(1.2278 m, 1.5876 m, 0.0223 m). The error accuracies and models in the simulations are shown

in Table 2.

4.2. Simulations of Errors Related to GPS Measurement

Each error related to GPS measurement is analyzed by single factor simulation, which is

intended to obtain its impact on relative positioning based on dual-frequency GPS. The

impact of each error is drawn by the comparison residuals between the relative position


Table 2: Error accuracies and modeling descriptions in simulations.

Error type Error accuracy Modeling description

GPS code measurementnoise

0.5 m (1σ) Gaussian white noise model with mean value of0 m and standard deviation of 0.5 m

GPS carrier phasemeasurement noise

0.002 m (1σ) AR(2) model with mean value of 0 m and standarddeviation of 0.002 m

Ground calibration errorof GPS receiver antennaphase center

—ARP data, PCO data, and PCVs data (mean valuedata and RMS data) of SEN67-1575-14+CRG

Attitude measurementerror

Fixed bias of 0.005◦

in the yaw, pitch,and roll components

plus a 0.003◦ (1σ)random error

Gaussian white noise model with mean value of0.005◦ and standard deviation of 0.003◦ in the yaw,pitch, and roll components

Fixing error of GPSreceiver antenna

0.5 mm (3σ)A fixed vector with direction randomly drawn inunit ball and magnitude of 0.5 mm in each satellitebody coordinate system

Consistency error of SARantenna phase center

0.25 mm (3σ)A fixed vector with direction randomly drawn inunit ball and magnitude of 0.25 mm in bodycoordinate system of Satellite 1

solutions determined by GPS observation data and relative positions obtained by standard

orbits of formation satellites. The relative position solutions are implemented in the separate

software tools as part of the NUDT Orbit Determination Software 1.0. The GPS observation

data processing consists of GPS observation data preprocessing [24], reduced dynamic

precise orbit determination for single satellite [25], GPS observation data editing [17, 24], and

reduced dynamic precise relative positioning. The RMS values of KBR comparison residuals

of GRACE relative position solution are about 1-2 mm implemented by this software.

4.2.1. Simulations for GPS Carrier Phase Measurement Noise

The noises of GPS carrier phase (L1 and L2) measurements are separately simulated by

second-order autoregressive model (AR(2)) as follows

ej

L(ti) = ej

L(ti−1) − 0.67 · ejL(ti−2) + εj

L(ti), (4.1)

where εj

L(ti) is the noise of carrier phase L measurement for GPS satellite j at epoch ti, εj

L(ti)is the Gaussian white noise.

From the following formula

E(ej

L(ti))= 0 m; σ

(ej

L(ti))= 0.002 m, (4.2)

where σ(·) denotes the standard deviation of a random variable, we can get

E(εj

L(ti))= 0 m; σ

(εj

L(ti))= 0.0012 m. (4.3)

One instance of carrier phase noise simulation is shown in Figure 6.


0 2000 4000 6000 8000−8

−6

−4

−2

0

2

4

6

8

×10−3

Number of epochs

Carr

ier

ph

ase

no

ise

(m)

Mean value = 0 m; standard deviation = 0.002 m

Figure 6: One instance of carrier phase measurement noise simulation.

50 groups of 24 h GPS observation data (interval of 30 s) for two formation satellites are

simulated by only adding the noises of GPS carrier phase (L1 and L2) measurements. By the

processing of relative positioning, the mean RMS values of comparison residuals of relative

position solutions in ITRF (Figure 7) are 0.340 mm of x-axis, 0.333 mm of y-axis, 0.288 mm of

z-axis, and 0.560 mm of 3 dimensions. It is shown by simulation results that the GPS carrier

phase noise can be well smoothed by reduced dynamic relative positioning approach.

4.2.2. Simulations for Ground Calibration Error of GPS Receiver Antenna Phase Center

Ground calibration error of GPS receiver antenna phase center is mainly caused by PCVs.

The PCVs values are described by the mean value and RMS value corresponding to the

direction of received signal. The PCV value corresponding to the direction of received signal

is simulated by Gaussian white noise with mean value and RMS value obtained from ground

calibration result of GPS receiver antenna system SEN67-1575-14+CRG.

The GPS observation data are simulated only considered ground calibration error of

GPS receiver antenna phase center. By the precise orbit determination for single satellite, the

mean RMS values of comparison residuals of orbit solutions in ITRF are 4.018 mm of x-axis,

4.154 mm of y-axis, 2.427 mm of z-axis, and 6.269 mm of 3 dimensions. The impacts of PCVs

on single satellite orbit solutions are mainly made by the mean value part of PCVs, while the

impacts of RMS part in ITRF are only 0.119 mm of x-axis, 0.094 mm of y-axis, 0.116 mm of

z-axis, and 0.191 mm of 3 dimensions, and the RMS part of PCVs can nearly be smoothed.

By the processing of relative positioning, the mean RMS values of comparison residuals of

relative position solutions in ITRF are 0.067 mm of x-axis, 0.070 mm of y-axis, 0.056 mm of z-

axis, and 0.112 mm of 3-dimensions. As the nearly equal models of ground calibration errors

of GPS receiver antenna phase centers for two formation satellites are selected and the LOS

vectors are nearly the same for close satellite formation, the impacts of mean value part of

PCVs can nearly be cancelled out by differential GPS observation and impacts of RMS part

can be well smoothed by the constraints of orbit dynamical models. It is shown by the results


0 10 20 30 40 500

x (

mm

)

Average RMS value = 0.34 mm

Number of simulations

0.5

(a)y

(m

m)

0


0 10 20 30 40 50


0.5

(b)

z (

mm

)

00 10 20 30 40 50


Average RMS value = 0.288 mm0.5

(c)

Figure 7: Simulation results of GPS carrier phase measurement noise for relative positioning.

of single satellite orbit solutions and relative position solutions that the characters of GPS

receiver antenna phase centers onboard two formation satellites must have great consistency.

4.2.3. Simulations of Satellite Attitude Measurement Error for GPS Relative Positioning

The Euler angle errors are simulated by Gaussian white noise with 0.005◦ of mean value and

0.003◦ of standard deviation, and 50 groups of 24 h GPS observation data for two formation

satellites are simulated by only adding the attitude measurement errors. The mean RMS

values of comparison residuals of relative position solutions in ITRF (Figure 8) are 0.069 mm

of x-axis, 0.075 mm of y-axis, 0.081 mm of z-axis, and 0.128 mm of 3 dimensions. The 3

dimensional maximum of comparison residuals in these 50 simulations is 0.219 mm, which is

less than 0.47 mm and is well consistent with aforementioned analysis in Section 3.1.3.

4.2.4. Simulations for Fixing Error of GPS Receiver Antenna

The fixing error of GPS receiver antenna belongs to systematic error and it is a fixed

bias vector in satellite body coordinate system. At first, four representatively “extreme”

circumstances of fixing errors of GPS receiver antennas onboard two formation satellites

are simulated. The so-called “extreme” circumstance is that the directions of two fixed bias

vectors are opposite. Four representatively “extreme” circumstances of fixing errors of GPS

receiver antennas here are directions along X-axis, Y -axis, Z-axis, and diagonal of X-axis,


x (

mm

)

00 10 20 30 40 50



0.05

0.1

(a)y

(m

m)

00 10 20 30 40 50


Average RMS value = 0.075 mm0.2

0.1

(b)

z (

mm

)

00 10 20 30 40 50



0.2

0.1

(c)

Figure 8: Simulation results of satellite attitude measurement error for relative positioning.

Table 3: Relative positioning results for four representatively “extreme” circumstances of fixing errors ofGPS receiver antenna.

x/mm y/mm z/mm 3 dimension/mm

X-axis 0.512 0.499 0.701 1.002

Y-axis 0.677 0.694 0.133 0.979

Z-axis 0.072 0.083 0.081 0.136

Diagonal 0.502 0.503 0.429 0.830

Y -axis, Z-axis in satellite body coordinate system, respectively. All the magnitudes of fixed

bias vectors are selected 0.5 mm. 24 h GPS observation data for two formation satellites are

simulated by only considering the four representatively “extreme” circumstances of fixing

errors of GPS receiver antennas. The results of relative positioning are shown in Table 3.

From Table 3, it is shown that the fixing errors of GPS receiver antenna along X-axis

and Y -axis will mainly be absorbed by relative position solutions and the impact can reach

to 1 mm, but the error along Z-axis can be smoothed by the constraints of orbit dynamical

models.

In practice, the occurrence of “extreme” circumstances is extremely low and they are

just analyzed as the ultimate circumstances. For multiple repeated satellite missions, the

fixing error of GPS receiver antenna is a random error. So this error can be simulated as a

fixed vector with direction randomly drawn from unit ball and magnitude of 0.5 mm in each

satellite body coordinate system. 50 groups of 24 h GPS observation data for two formation

satellites are simulated by only adding the simulations of fixing error of GPS receiver


x (

mm

)

00 10 20 30 40 50



0.5

1

(a)y

(m

m)

00 10 20 30 40 50



0.5

1

(b)

z (

mm

)

00 10 20 30 40 50



0.5

1

(c)

Figure 9: Simulation results of fixing error of GPS receiver antenna for relative positioning.

antenna. The mean RMS values of comparison residuals of relative position solutions in ITRF

(Figure 9) are 0.295 mm of x-axis, 0.294 mm of y-axis, 0.249 mm of z-axis, and 0.495 mm of 3

dimensions.

From aforementioned simulations of each error related to GPS measurement, it is

shown that the impacts of GPS carrier phase measurement noise and fixing error of GPS

receiver antenna on GPS relative positioning are much bigger than other errors related to GPS

measurement and these two errors are the main factors of errors related to GPS measurement.

4.3. Simulations of Errors Related to Baseline Transformation

In this section, the impact of each error on baseline transformation is obtained by single factor

simulation. Each impact is given by the comparison between the spatial baseline solutions

obtained with and without errors.

4.3.1. Simulations of Satellite Attitude Measurement Error for Baseline Transformation

The satellite attitude simulation data used here are the same as Section 4.2.3. By baseline

transformation with attitude simulation data, the mean RMS values of comparison residuals

of spatial baseline solutions in ITRF (Figure 10) are 0.115 mm of x-axis, 0.115 mm of y-

axis, 0.133 mm of z-axis, and 0.210 mm of 3 dimensions. The 3 dimensional maximum of


x (

mm

)

00 10 20 30 40 50



0.1

0.2

(a)

00 10 20 30 40 50



0.1

0.2

y (

mm

)

(b)

00 10 20 30 40 50



0.1

0.2

z (

mm

)

(c)

Figure 10: Simulation results of satellite attitude measurement error for baseline transformation.

comparison residuals in these 50 simulations is 0.213 mm, which is less than 0.50 mm and is

consistent with aforementioned analysis in Section 3.2.1.

4.3.2. Simulations for Consistency Error of SAR Antenna Phase Center

It is shown by the analysis in Section 3.2.2 that the accuracy of consistency error of SAR

antenna phase center is better than 0.25 mm (3σ) in current simulation circumstances. This

error is only added to the SAR antenna phase center of Satellite 1 and can be simulated as

a fixed vector with direction randomly drawn from unit ball and magnitude of 0.25 mm in

body coordinate system of satellite 1. By 50 groups of simulations, the mean RMS values of

comparison residuals of spatial baseline solutions in ITRF (Figure 11) are 0.142 mm of x-axis,

0.142 mm of y-axis, and 0.153 mm of z-axis.

4.4. Simulations of Total Error Sources

In this section, all the errors are added to the flow of spatial baseline determination

simulations according to the error models listed in Table 2. By 50 groups of total error sources

simulations, the mean RMS values of comparison residuals of spatial baseline solutions in

ITRF (Figure 12) are 0.500 mm of x-axis, 0.500 mm of y-axis, 0.452 mm of z-axis, and 0.845 mm

of 3 dimensions.


x (

mm

)

00 10 20 30 40 50



0.2

0.1

(a)

00 10 20 30 40 50



0.2

0.1

y (

mm

)

(b)

00 10 20 30 40 50



0.2

0.1

z (

mm

)

(c)

Figure 11: Simulation results of consistency error of SAR antenna phase center for baseline transformation.

0

1

x (

mm

)

0 10 20 30 40 50



0.5

(a)

0

1

0 10 20 30 40 50



0.5

y (

mm

)

(b)

0

1

0 10 20 30 40 50



0.5

z (

mm

)

(c)

Figure 12: Simulation results of total error sources for spatial baseline determination.


x (

mm

)

00 10 20 30 40 50



0.5

1

(a)

00 10 20 30 40 50



0.5

1

y (

mm

)

(b)

00 10 20 30 40 50



0.5

1

z (

mm

)

(c)

Figure 13: Simulation results of total errors related to GPS measurement for relative positioning.

In addition, the impact of total errors related to GPS measurement on GPS relative

positioning in ITRF (Figure 13) is 0.454 mm of x-axis, 0.452 mm of y-axis, 0.388 mm of z-

axis, and 0.755 mm of 3 dimensions, and the impact of total errors related to baseline

transformation in ITRF (Figure 14) is 0.185 mm of x-axis, 0.185 mm of y-axis, 0.206 mm of

z-axis, and 0.334 mm of 3 dimensions.

It is shown by the simulations of total error sources that errors related to GPS

measurement are the main error sources for the spatial baseline determination and 1 mm

level InSAR spatial baseline determination can be realized according to current simulation

conditions.

5. Conclusions

In this paper, the errors introduced by spatial baseline measurement for InSAR mission are

deeply studied. The impacts of errors on spatial baseline determination are analyzed by single

factor simulations and total error sources simulations. The main conclusions are drawn as

follows.

(1) The spatial baseline measurement errors can be classified into two groups: errors

related to GPS measurement and errors related to baseline transformation. By

simulations, the three-dimensional impacts of these errors on spatial baseline

determination in ITRF are 0.755 mm and 0.334 mm, respectively. It is shown that


x (

mm

)

00 10 20 30 40 50



0.2

0.4

(a)

00 10 20 30 40 50



0.2

0.4

y (

mm

)

(b)

00 10 20 30 40 50



0.2

0.4

z (

mm

)

(c)

Figure 14: Simulation results of total errors related to baseline transformation for baseline transformation.

the errors related to GPS measurement are the main influence on spatial baseline

determination.

(2) By the results of single factor simulations, the three dimensional impacts of GPS

carrier phase measurement noise and the fixing error of GPS receiver antenna on

GPS relative positioning in ITRF are 0.560 mm and 0.495 mm, respectively. These

two errors are the main factors of errors related to GPS measurement.

(3) It is shown by total error sources simulations that the impact of all the errors on

spatial baseline determination in ITRF is 0.500 mm of x-axis, 0.500 mm of y-axis,

0.452 mm of z-axis, and 0.845 mm of 3 dimensions. Therefore, 1 mm level InSAR

spatial baseline determination can be realized.

Acknowledgments

The mean antenna phase center description for the Sensor Systems SEN67157514 antenna has

been contributed by the German Space Operations Center (GSOC), Deutsches Zentrum fur

Luft- und Raumfahrt (DLR), Wessling, to enable the simulation of antenna phase center data

of dual-frequency GPS receiver. Precise GPS ephemerides for use within this study have been

obtained from the Center for Orbit Determination in Europe at the Astronomical Institute of

the University of Bern (AIUB). The authors extend special thanks to the support of the above

institutions. This paper is supported by the National Natural Science Foundation of China


(Grant no. 61002033 and no. 60902089) and Open Research Fund of State Key Laboratory of

Astronautic Dynamics of China (Grant no. 2011ADL-DW0103).

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[21] J. H. Gonzalez, M. Bachmann, G. Krieger, and H. Fiedler, “Development of the TanDEM-X calibrationconcept: analysis of systematic errors,” IEEE Transactions on Geoscience and Remote Sensing, vol. 48, no.2, pp. 716–726, 2010.

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[23] W. T. Wang and Y. H. Qi, “A new technique to compensate for error in SAR antenna power pattern,”Chinese Space Science and Technology, vol. 17, no. 3, pp. 65–70, 1997.

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Research ArticleAnalysis of Attitude Determination Methods UsingGPS Carrier Phase Measurements

Leandro Baroni1 and Helio Koiti Kuga2

1 Institute of Science, Engineering and Technology, Federal University of Jequitinhonha andMucuri Valleys (UFVJM), Rua do Cruzeiro, 1, 39803-371 Teofilo Otoni, MG, Brazil

2 Space Mechanics and Control Division (DMC), National Institute for Space Research (INPE),12227-010 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Helio Koiti Kuga, [email protected]



Copyright q 2012 L. Baroni and H. K. Kuga. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

If three or more GPS antennas are mounted properly on a platform and differences of GPSsignals measurements are collected simultaneously, the baselines vectors between antennas canbe determined and the platform orientation defined by these vectors can be calculated. Thus, theprerequisite for attitude determination technique based on GPS is to calculate baselines betweenantennas to millimeter level of accuracy. For accurate attitude solutions to be attained, carrierphase double differences are used as main type of measurements. The use of carrier phasemeasurements leads to the problem of precise determination of the ambiguous integer numberof cycles in the initial carrier phase (integer ambiguity). In this work two algorithms (LSAST andLAMBDA) were implemented and tested for ambiguity resolution allowing accurate real-timeattitude determination using measurements given by GPS receivers in coupled form. Platformorientation was obtained using quaternions formulation, and the results showed that LSASTmethod performance is similar to LAMBDA as far as the number of epochs which are necessaryto resolve ambiguities is concerned, but with processing time significantly higher. The final resultaccuracy was similar for both methods, better than 0.1◦ to 0.2◦, when baselines are considered indecoupled form.

1. Introduction

Global Navigation Satellite Systems (GNSSs) are satellite-based radionavigation systems,

providing to worldwide users precise position and timing. System satellites transmit radio-

frequency signals containing information required for the user equipment to compute its

navigation solution (position, velocity, and time). GNSS can also be used to determine

attitude of a platform, in which three or more antennas are needed to calculate attitude

parameters [1].


If three or more GNSS antennas properly mounted on a platform and differences of

GNSS signals measurements are collected simultaneously, baselines vectors formed between

antennas can be determined, and orientation of the platform defined by these vectors can be

calculated. Thus, the prerequisite for the attitude determination technique based on GNSS

systems is to calculate the baselines between the antennas.

Accurate attitude solutions can be obtained using carrier phase double difference

observables as the main type of measurements, including all independent combinations

of antenna positions. Baselines between antennas must be determined in millimeter level

of accuracy. Typically, the distance between the antennas is a few meters or less, and all

spatially correlated errors between the antennas are almost eliminated in differencing (single

and double) process, including orbital, ionospheric, and tropospheric errors. Therefore, main

error sources affecting attitude determination are the multipath, receiver internal noise, and

antenna phase center variation [2].The use of carrier phase measurements leads to the problem of determining precisely

the ambiguous initial carrier phase integer number of cycles (integer ambiguity). To increase

confidence and accelerate the process by limiting the search space, three types of restrictions

can be established from prior knowledge of the antenna fixed geometry: (i) length of the

baseline, (ii) angle between baselines, and (iii) knowledge of the geometry of the antennas

as a network in which double difference ambiguities must satisfy a closed loop condition [2].Recently, several studies have focused on increasing the success rate of resolving ambiguity

process using the restrictions, as in [3, 4]. References [5, 6] show phase measurements

usage and performance of solving integer ambiguity methods in positioning applications.

[7] Reference develops a procedure to determine satellite attitude in three axes with GPS

associated with a gyro, and resolving ambiguities with the method described in [8]. Thus,

the resolving ambiguity process is an important step to determine an accurate baseline vector,

resulting in an accurate attitude determination.

A common attitude representation is done by Euler angles. This parameterization

has difficult computation in general, because of the use of trigonometric functions

and the appearance of a singularity in the motion modeling. Another way to attitude

parameterization is using quaternions. This parameterization has some advantages over

Euler angles; it is computationally efficient, there is no singularity, and it does not depend

on trigonometric functions [9].So, in this work an implementation and analysis of algorithms for integer ambiguity

resolution allowing accurate attitude determination in real time, using measurements

provided by GPS receivers, will be tested. Algorithm tests, using quaternions for attitude

representation, will be implemented with real data, collected at INPE and described in [7]. In

this experiment, three antennas were fixed on a structure with known baseline lengths (1 m)and angle between baselines of 90◦. The tests were executed with the methods LAMBDA and

LSAST for ambiguity resolution.

least-squares ambiguity decorrelation adjustment (LAMBDA) method is a procedure

for integer ambiguity estimation in carrier phase measurements. This method executes the

integer ambiguity estimation through a Z transform, in which ambiguities are decorrelated

before the integer values search process. Then, a minimization problem is approached as

a discrete search inside an ellipsoidal region defined by decorrelated ambiguities, which is

smaller than original ones. As a result, integer least-squares estimates for the ambiguities are

obtained. This method was introduced in [10, 11]. References [12, 13] show computational

implementation aspects and ambiguity search space reducing performance.


Least-Squares Ambiguity Solution Technique (LSAST) method, also known as LSAST

method, was proposed in [14]. This method involves a modified sequential least-squares

technique, in which ambiguity parameters are divided into two groups: primary ambiguities

(typically three double difference ambiguities), and the secondary ambiguities. Only the

primary ambiguities are fully searched, in ±5 cycles around the corresponding float

ambiguity, after rounded to the nearest integer. For each set of the primary ambiguities, there

is a unique set of secondary ambiguities. Therefore, the search dimension is smaller and the

computation time is significantly shorter than the full search approach. The choice of primary

group measurements is based on GDOP value. geometric dilution of precision (GDOP) is a

quantity which measures the influence of satellite geometry on positioning errors. Satellites

with low GDOP will lead to a search with less-potential solutions. However, GDOP cannot

be very low, in order to avoid the position uncertainty including more than one solution for

secondary group measurements. The procedure is to choose primary group of satellites which

have a reasonable GDOP.

2. Attitude Representation

Euler angles are a means of representing the spatial orientation of any coordinate system

as a composition of rotations from a frame of reference. These angles uniquely determine

the orientation of a rigid body in three-dimensional space. There are several conventions for

defining the Euler angles, depending on the choice of axes and the order in which rotations

about these axes are performed. A matrix expression can be found for any frame given its

Euler angles, performing three rotations in sequence. Here, Euler angles are denoted as θ is

pitch, φ is roll, and ψ is yaw. The resulting rotation matrix is given in convention

R(θ, ψ, φ

)= Rx

(φ)Ry(θ)Rz

(ψ). (2.1)

Expanding the rotation matrix,

R =

⎡⎣ cos θ cosψ cos θ sinψ − sin θ

sinφ sin θ cosψ − cosφ sinψ sinφ sin θ sinψ + cosφ cosψ cos θ sinφ

cosφ sin θ cosψ + sinφ sinψ cosφ sin θ sinψ − sinφ cosψ cos θ cosφ

⎤⎦. (2.2)

Quaternion q is determined in function of rotation angle ϕ and rotation axis n and is

composed by a scalar term q4 and a vector term �q [15]

q =[q1 q2 q3 q4

]T=[�qT q4

]T, (2.3)

with

�q =[q1 q2 q3

]T = sin(ϕ

2

)n, q4 = cos

(ϕ2

). (2.4)

Rotation matrix is given in terms of quaternions as

R(q) =(∣∣q4

∣∣2 − ∣∣�q∣∣2)I3 + 2�q�qT + 2q4��q�, (2.5)


where I3 is an third-order identity matrix, and

��q� =

⎡⎣ 0 q3 −q2

−q3 0 q1

q2 −q1 0

⎤⎦. (2.6)

Or explicitly,

R(q) =

⎡⎣q21 − q2

2 − q23 + q

24 2

(q1q2 + q3q4

)2(q1q3 − q2q4

)2(q1q2 − q3q4

)−q2

1 + q22 − q2

3 + q24 2

(q2q3 + q1q4

)2(q1q3 + q2q4

)2(q2q3 − q1q4

)−q2

1 − q22 + q

23 + q

24

⎤⎦. (2.7)

Euler angles can be calculated from rotation matrix as

⎡⎣φθψ

⎤⎦ =

⎡⎣atan2[2(q2q3 + q1q4

), −q2

1 − q22 + q

23 + q

24

]asin[2(q2q4 − q1q3

)]atan2

[2(q1q2 + q3q4

), q2

1 − q22 − q2

3 + q24

]⎤⎦. (2.8)

3. Ambiguity Resolution

When distance between receivers is short (until 10 km), ionospheric and tropospheric

residuals are small compared to multipath and internal receiver noise errors. Thus, for a short

baseline, carrier phase double-difference measurements (φijub) are [16]

φij

ub=(1ib − 1j

b

)· xub + λNij

ub+ εij

ub,φ, (3.1)

where 1ib

is unit vector pointing from base to satellite i, xub is baseline vector, Nij

ubis the

integer ambiguity, and ε is a vector representing unmodeled errors. If m satellites were in

view simultaneously, then there are (m− 1) double difference measurements. In matrix form,

taking M as master satellite, (3.1) becomes

⎡⎢⎢⎣φ1Mub...

φ(m−1)Mub

⎤⎥⎥⎦ =

⎡⎢⎢⎣11b− 1M

b...

1(m−1)b

− 1Mb

⎤⎥⎥⎦xub + λ⎡⎢⎢⎣

N1Mub...

N(m−1)Mub

⎤⎥⎥⎦ + ε, (3.2)

or

y = Hxub + λN + ε. (3.3)

Equation (3.3) is the measurement model considered for ambiguity resolution. Results were

obtained for ambiguity resolution from the same estimation process using a Kalman filter,

and processing code and carrier phase measurements. Ambiguity resolution is done epoch to

epoch, applying LAMBDA and LSAST methods on real-valued (float) ambiguities given by

the Kalman filter.


Antenna 3

Antenna 4

Antenna 2

North

East

Figure 1: Antenna configuration in data acquisition.

4. Results

Data used in this study were originally collected for use in [7] on March 30th, 2005. Three

GPS receivers were used, and their antennas were fixed on a frame allowing the knowledge

of the baseline lengths (1 m) and the angle between them (90◦). The sampling rate was

1 Hz, collected only in L1 carrier frequency. Figure 1 illustrates that the antenna mounting

configuration and equipments used are listed below:

(i) 3 AllStar CMC GPS receivers (Canadian Marconi Space Company) and

(ii) 3 AllStar CMC model AT 575-70 GPS antennas.

GPS measurements were processed to obtain a precise baseline length. Ambiguity

resolution was made using LAMBDA and LSAST methods. Each baseline was independently

determined to form a frame, whose attitude is calculated referred to the east-north-up

reference system. Rotation matrix, obtained by quaternions, is transformed to Euler angles

using (2.8), in order to give a geometric view of orientation. All algorithms were implemented

in Matlab language, using a computer with 4 Gb RAM, Intel Core i3 processor, and Windows

7 operating system. Three data sets were tested.

Data Set 1

In this test, same 8 satellites were kept in view (SV04, SV08, SV13, SV19, SV23, SV27, SV28 e

SV31), resulting in 7 double difference measurements. SV13 was chosen as master satellite,

because of its high elevation. Ambiguity resolution methods use a Kalman filter to obtain

real-valued (float) ambiguities. Kalman filter needs some epochs to converge to an ambiguity

solution, and thus to a given baseline. Data were free of cycle slips.

In this way, LAMBDA method presents a solution with correct values of ambiguities

after 107 epochs, while LSAST method takes 157 epochs to reach the correct values. Graphs

in Figure 2 show Euler angles values after the correct solution is reached.

Table 1 shows the statistics for the angle values in each method after ambiguities

are correctly solved. These results show that baseline orientation could be determined with

accuracy better than 0.1◦ in both methods although LSAST method has taken longer to deliver

the correct solution.

LSAST method sweeps a fixed number of cycles in satellite primary set, leading to

a slower search. This method had a mean processing time of 140 ± 32 ms for resolving 7

ambiguities. Mean processing time of LAMBDA method was 11 ± 1 ms after ambiguities are


100 125 150 175

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4A

ng

le(◦)

Time (s)

(a) LAMBDA

100 125 150 175

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4

An

gle(◦)

Time (s)

(b) LSAST

Figure 2: Euler angles for data set 1.

100 150 200 250 300 350

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4

An

gle(◦)

Time (s)

(a) LAMBDA

100 150 200 250 300 350

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4A

ng

le(◦)

Time (s)

(b) LSAST


solved due to small search space of candidate ambiguities. Considering whole set, processing

time was 18±25 ms because search space was larger. Processing time is shown on Figure 5(a).

Data Set 2

In this set, 7 satellites were visible (SV08, SV13, SV19, SV23, SV27, SV28, and SV31), resulting

in 6 double difference measurements. SV13 was taken as master satellite. Data were free of

cycle slips.

Graphs in Figure 3 show platform roll, pitch, and yaw angles. For this set, both

LAMBDA and LSAST methods converged to correct ambiguity values after 141 epochs.

Table 2 shows mean and standard deviation values for Euler angles after a stable solution

is reached. The accuracy is very similar in both methods, once number of epochs necessary

for getting ambiguity solution is the same.


300 450 600 750 900 1050 1200

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4A

ng

le(◦)

Time (s)

(a) LAMBDA

Yaw

Pitch

Roll

−8

−6

−4

−2

0

2

4

An

gle(◦)

Time (s)

300 450 600 750 900 1050 1200

(b) LSAST


Time (s)

Pro

cess

ing

tim

e(s)

LAMBDA

Hatch

0

0.05

0.1

0.15

0.2

100 125 150 175

(a) Data set 1

100 150 200 250 300 3500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (s)

Pro

cess

ing

tim

e(s)

LAMBDA

Hatch

(b) Data set 2

Time (s)

300 450 600 750 900 1050 12000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Pro

cess

ing

tim

e(s)

0.4

LAMBDA

Hatch

(c) Data set 3

Figure 5: Processing time for all data sets.


Table 1: Mean and standard-deviation (SD) for data set 1.

LAMBDA LSAST

Angle [◦] Mean SD Mean SD

Roll 0.136 0.132 0.097 0.014

Pitch 0.482 0.060 0.520 0.028

Yaw −6.488 0.045 −6.478 0.001

Table 2: Mean and standard-deviation (SD) for data set 2.

LAMBDA LSAST


Roll 1.839 0.190 1.839 0.190

Pitch −0.820 0.079 −0.820 0.079

Yaw −5.608 0.056 −5.608 0.056

Table 3: Mean and standard deviation (SD) for data set 3.

LAMBDA LSAST


Roll 1.049 0.306 1.049 0.306

Pitch 0.557 0.116 0.557 0.116

Yaw −5.612 0.178 −5.612 0.178

In this test, mean processing time was 11 ± 17 ms for LAMBDA method in whole set,

and 8 ± 1 ms after reaching correct solution. For LSAST, processing time was 154 ± 55 ms to

solve 6 ambiguities (Figure 5(b)).

Data Set 3

This data set had the same visible satellites as Data Set 2. Both LAMBDA and LSAST

methods present solutions for ambiguities after 320 epochs. Graphs in Figure 4 show the

Euler angles obtained by both ambiguities resolution methods. Table 3 shows mean and

standard deviation values for roll, pitch, and yaw angles.

The average processing time for ambiguity resolution step, whenever a successful

solution is obtained, is 7 ± 2 ms, and whole set processing time was 9 ± 11 ms with LAMBDA

method, and 137 ± 39 ms with LSAST method. Processing time in LSAST method increases

along the time due to changing satellite geometry (Figure 5(c)).

5. Conclusions

Due to short baseline (1 m), both methods have a tendency to solve ambiguities for the same

number of epochs, as filter converges. Both LAMBDA and LSAST methods had a similar

performance in number of epochs needed to give the correct solution; however, processing

time of LSAST method was significantly longer. This is due to the fact that LSAST method

performs a systematic search throughout the primary set of measurements, while LAMBDA

optimizes the search space as a whole. Processing time for LAMBDA method is search space


size dependant [13]. The smaller search space, the shorter processing time. This can be

verified by mean time to process the whole set compared to partial solution.

The accuracy of final result is also similar for both methods, better than 0.1◦ to 0.2◦,

once they find the same ambiguity set. The small variations are due to difference in number

of epochs to obtain the solution. Data Set 2 and Data Set 3 showed same mean values and

standard deviation for Euler angles. This occurs because the only difference in algorithms is

resolution ambiguity routine. Once ambiguities were resolved to the same values, statistics

must be equal. LAMBDA method also has an extensive series of studies documented in the

literature.

Attitude estimation using quaternions led to the same values when using a rotation

matrix based on Euler angles, as results showed in [17]. The main difference is processing

time, slightly longer with quaternions. This is probably due to conversion of results to Euler

angles, which uses trigonometric functions. However, quaternions can be useful to avoid

singularity situations.

Using dual frequency measurements is not necessary for the mitigation of ionospheric

errors since the baselines are short but can be useful for a faster ambiguity resolution, for

example, using widelane technique. If compared with the use of GPS alone, measurements

from two different GNSS systems can certainly improve the attitude determining accuracy

as result of increased number of measurements and improved satellite geometry. This can be

accomplished with the use of GPS/GLONASS and future GPS/Galileo receivers.

References

[1] B.W. Parkinson and J.J. Spilker Jr., Global Positioning System: Theory and Applications, vol. 1-2, 1996.[2] A. El-Mowafy and A. Mohamed, “Attitude determination from GNSS using adaptive Kalman

filtering,” Journal of Navigation, vol. 58, no. 1, pp. 135–148, 2005.[3] J. Pinchin, “Enhanced integer bootstrapping for single frequency GPS attitude determination,” in

Proceedings of the 21st International Technical Meeting of the Satellite Division of the Institute of Navigation(ION GNSS ’08), pp. 1524–1532, Savannah, Ga, USA, September 2008.

[4] G. Zheng and D. Gebre-Egziabher, “Enhancing ambiguity resolution performance using attitudedetermination constraints,” in Proceedings of the 22nd International Technical Meeting of the SatelliteDivision of the Institute of Navigation (ION GNSS ’09), pp. 3598–3610, Savannah, Ga, USA, September2009.

[5] L. Baroni, H. K. Kuga, and K. O’Keefe, “Analysis of three ambiguity resolution methods for real timestatic and kinematic positioning of a GPS receiver,” in Proceedings of the 22nd International TechnicalMeeting of the Satellite Division of the Institute of Navigation (ION GNSS ’09), pp. 2392–2400, Savannah,GA, USA, September 2009.

[6] L. Baroni., Algoritmos de navegacao em tempo real para um sistema GPS de posicionamento relativo de precisao(Real time navigation algorithms for a precise GPS relative positioning system), Ph.D. thesis, INPE, Sao Josedos Campos, Brazil, 2009, (INPE-16598-TDI/1584).

[7] A. C. Louro, Determinacao autonoma de atitude de satelites utilizando GPS (Autonomous satellite attitudedetermination using GPS), Ph.D. thesis, INPE, Sao Jose dos Campos, Brazil, 2006, (INPE-14091-TDI/1074).

[8] R. V. F. Lopes and H. K. Kuga, “Determinacao de atitude em 3 eixos por interferometria GPS (Three-axis attitude determination by GPS interferometry),” in Plataforma Integrada Sensores Inerciais/GPS(Inertial sensor/GPS integrated platform), O. S. C. Durao, V. R. Schad, H. K. Kuga, R. V. F. Lopes, H. C.Carvalho, and M. Esper, Eds., INPE, Sao Jose dos Campos, Brazil, 2002, (INPE-9293-PRP/234).

[9] R. V. Garcia, Filtro nao linear de Kalman sigma-ponto com algoritmo unscented aplicado a estimativa dinamicada atitude de satelites artificiais (Sigma point nonlinear Kalman filter with unscented algorithm applied toattitude dynamics estimation of artificial satellites), Ph.D. thesis, INPE, Sao Jose dos Campos, Brazil, 2011,(10.16.14.32-TDI).

[10] P. J. G. Teunissen, “Least-squares estimation of the integer GPS ambiguities,” in Proceedings of the IAGGeneral Meeting (IAG ’93), Beijing, China, 1993.


[11] P. J. G. Teunissen, “New method for fast carrier phase ambiguity estimation,” in Proceedings of theIEEE Position Location and Navigation Symposium, pp. 562–573, Las Vegas, Mev, USA, April 1994.

[12] P. J. de Jonge and C. C. J. M. Tiberius, “The LAMBDA method for integer ambiguity estimation:implementation aspects,” Tech. Rep., Delft University of Technology, 1996.

[13] P. J. de Jonge, C. C. J. M. Tiberius, and P. J. G. Teunissen, “Computational aspects of the LAMBDAmethod for GPS ambiguity resolution,” in Proceedings of the 9th International Technical Meeting of theSatellite Division of the Institute of Navigation (ION GPS ’96), pp. 935–944, Kansas City, Mo, USA, 1996.

[14] R. R. Hatch, “Instantaneous ambiguity resolution,” in Proceedings of the KIS ’90, pp. 299–308, Springer,Banff, Canada, 1990.

[15] R. V. Garcia, H. K. Kuga, and M. C. Zanardi, “Unscented Kalman filter for spacecraft attitudeestimation using quaternions and Euler angles,” in Proceedings of the 22 International Symposium onSpace Flight Dynamics, Sao Jose dos Campos, Brazil, 2011.

[16] P. Misra and P. Enge, Global Positioning System: Signals, Measurements and Performance, Ganga-JamunaPress, Lincoln, Mass, USA, 2001.

[17] L. Baroni and H. K. Kuga, “Integer ambiguity resolution in attitude determination using GPSmeasurements,” in Proceedings of the 8th International Conference on Mathematical Problems inEngineering, Aerospace and Sciences (ICNPAA ’10), Sao Jose dos Campos, Brazil, 2010.


Research ArticleNumerical Analysis of Constrained, Time-OptimalSatellite Reorientation

Robert G. Melton

Department of Aerospace Engineering, Pennsylvania State University, 229 Hammond Building.,University Park, PA 16802, USA

Correspondence should be addressed to Robert G. Melton, [email protected]

Received 11 July 2011; Accepted 11 October 2011


Copyright q 2012 Robert G. Melton. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Previous work on time-optimal satellite slewing maneuvers, with one satellite axis (sensor axis)required to obey multiple path constraints (exclusion from keep-out cones centered on high-intensity astronomical sources) reveals complex motions with no part of the trajectory touchingthe constraint boundaries (boundary points) or lying along a finite arc of the constraint boundary(boundary arcs). This paper examines four cases in which the sensor axis is either forced to followa boundary arc, or has initial and final directions that lie on the constraint boundary. Numericalsolutions, generated via a Legendre pseudospectral method, show that the forced boundary arcsare suboptimal. Precession created by the control torques, moving the sensor axis away fromthe constraint boundary, results in faster slewing maneuvers. A two-stage process is proposedfor generating optimal solutions in less time, an important consideration for eventual onboardimplementation.

1. Introduction

The problem of reorienting a spacecraft in minimum time, often through large angles (so-

called slew maneuvers) and subject to various constraints, can take a number of forms. For

example, the axis normal to the solar panels may be required to lie always within some

specified minimum angular distance from the sun-line. In cases where the control authority is

low and the slew requires a relatively long time, certain faces of the vehicle may benefit from

being kept as far as possible from the sun-line to avoid excessive solar heating. For many

scientific missions, observational instruments must be kept beyond a specified minimum

angular distance from high-intensity light sources to prevent damage.

Before addressing the time-optimal, constrained problem, it is useful to review what is

known about the unconstrained problem. In a seminal paper, Bilimoria and Wie [1] consider


time-optimal slews for a rigid spacecraft whose mass distribution is spherically symmetric,

and which has equal and independently limited control-torque authority for all three axes.

Despite the symmetry of the system, the intuitively obvious time-optimal solution is not a

λ-rotation about the eigenaxis (the appendix contains a discussion of λ-rotations and the

eigenaxis of a direction cosine matrix). Indeed, the fallacy here is that one easily confuses the

minimum-angle of rotation problem (i.e., the angle about the eigenaxis) with the minimum-

time problem, ignoring the constraints imposed by Euler’s equations of rigid-body motion.

Bilimoria and Wie find that the time-optimal solution includes precessional motion to achieve

a lower time (approximately 10% less) than that obtained with an eigenaxis maneuver.

Further, they determine that the control history is bang-bang, with a switching structure that

changes depending upon the magnitude of the angular maneuver (referring here to the final

orientation in terms of a fictitious λ-rotation, with associated angle θ): for values of θ less

than 72 degrees, the control history is found to contain seven switches between directions of

the control torque components; larger values of θ require only five switches.

Several subsequent papers have revisited the unconstrained problem, including

such modifications as axisymmetric mass distribution and only two-axis control [2],asymmetric mass distribution [3], small reorientation angles [4], and combined time and

fuel optimization [5]. Recently, Bai and Junkins [6] have reconsidered the original problem

(spherically symmetric mass distribution, three equal control torques) and find that at least

two locally optimum solutions exist for reorientations of less than 72 deg. (one of which

requires only six switches) if the controls are independently limited. Further, they prove

that if the total control vector is constrained to have a maximum magnitude (i.e., with

the orthogonal control components not independent), then the time-optimal solution is the

eigenaxis maneuver.

Hablani [7] and Mengali and Quarta [8] consider constrained maneuvers, but focus

upon generating feasible solutions without attempting to find optimal solutions. Melton [9]considers time-optimal, constrained slewing maneuvers for cases involving multiple path

constraints. That work uses the Swift spacecraft [10] as an example of a vehicle that must

be rapidly reoriented to align two telescopes at a desired astronomical target, namely, a

gamma-ray burst. The satellite’s burst alert telescope (wide field of view) first detects the

gamma-ray burst and the spacecraft then must reorient to allow the X-ray and UV/optical

instruments to capture the rapidly fading afterglow of the event. To prevent damage to these

instruments, the slewing motion is constrained to prevent the telescopes’ common axis from

entering established “keep-out” zones, defined as cones with central axes pointing to the Sun,

Earth, and Moon, with specified half-angles. A somewhat surprising result is that all of the

cases studied yield trajectories of the sensor axis that neither travel along the boundary of

the keep-out cone nor even touch it (so-called boundary arcs and boundary points). Figure 1

shows an example of this behavior, in which the sensor axis traverses a narrow gap (0.1 deg.)between the Sun and Moon cones, but does not intersect either cone.

This paper presents a preliminary study of boundary arcs and boundary points in

this same problem. A full analytical solution is not possible; however, some insight to the

problem can be gained by examining instances where the sensor axis is constrained to follow

the constraint boundary, and those where the initial and final sensor axis directions lie exactly

on the boundary. The paper also addresses the practical challenge of implementing onboard

optimal control for this type of maneuver and proposes a means for reducing computation

time for the reference trajectory.


Initial position

Final position

EarthMoon

Sensor axis path

Sun

−1.5

−1

−1

−1

−0.5

−0.5

−0.5

0

00

0.5

0.50.5

1

1 1

1.5

1.5

Figure 1: Trajectory of sensor axis between Sun (yellow) and Moon (gray) cones.

2. Problem Statement

The problem is formulated as a Mayer optimal control problem, with performance index

J = tf , (2.1)

where tf is the final time to be minimized. Euler’s equations of rigid-body motion describe

the system dynamics

ω1 =[M1 −ω2ω3(I3 − I2)]

I1

ω2 =[M2 −ω3ω1(I1 − I3)]

I2

ω3 =[M3 −ω1ω2(I2 − I1)]

I3.

(2.2)

Note that the control torques are assumed to be independently bounded, and no assumption

is made about the type of control actuator being used. Euler’s equations must be augmented

with kinematic relationships in order to determine the orientation of the body over time.

In this work, a formulation that uses Euler parameters is employed, with the relationship

between the Euler parameters εi and the angular velocity components ωj given by

⎡⎢⎢⎢⎢⎢⎣ε1

ε2

ε3

ε4

⎤⎥⎥⎥⎥⎥⎦ =1

2

⎡⎢⎢⎢⎢⎢⎣ε4 −ε3 ε2 ε1

ε3 ε4 −ε1 ε2

−ε2 ε1 ε4 ε3

−ε1 −ε2 −ε3 ε4

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣ω1

ω2

ω3

0

⎤⎥⎥⎥⎥⎥⎦. (2.3)


The appendix discusses the relationship between the Euler parameters, λ-rotations,

and direction cosine matrices. The problem to be considered is a rest-to-rest maneuver, with

initial conditions

ω1(0) = ω2(0) = ω3(0) = 0

ε1(0) = ε2(0) = ε3(0) = 0, ε4(0) = 1,(2.4a)

and two possible sets of final conditions

ω1

(tf)= ω2

(tf)= ω3

(tf)= 0

ε1

(tf)= ε1f , ε2

(tf)= ε2f , ε3

(tf)= ε3f , ε4

(tf)= ε4f ,

(2.4b)

for which the final orientation at the final time is completely specified, or

ω1

(tf)= ω2

(tf)= ω3

(tf)= 0

f1 (ε1, ε2, ε3, ε4) = 0

...

fn(ε1, ε2, ε3, ε4) = 0,

(2.4c)

for which only some aspect of the final orientation is specified (this is discussed further in

Section 2.1).For the unconstrained optimal control problem formulated using (2.1)–(2.3), the

Hamiltonian is

H = λω1ω1 + λω2

ω2 + λω3ω3 + λε1

ε1 + λε2ε2 + λε3

ε3 + λε4ε4. (2.5)

For spacecraft of the type being modeled here (Swift, or other astronomical missions), one

or more sensors are fixed to the spacecraft bus; these sensors all have the same central axis

for their fields of view and this axis is designated here with the unit vector σ and referred

to as the sensor axis. This axis must be kept at least a minimum angular distance αA from

each of several high-intensity astronomical sources. Denoting the directions to these sources

as σA, where the subscript A can be S (Sun), E (Earth), or M (Moon), the so-called keep-outconstraints are then written as follows:

αA ≤ cos−1(σ · σA). (2.6)


Without loss of generality, σ is assumed to lie along the body-fixed x-axis and its orientation

with respect to the inertial frame is then determined from (A.6), with A as the inertial frame

and B as the body-fixed frame

σ1 = 1 − 2(ε2

1 + ε23

)σ2 = 2(ε1ε2 + ε3ε4)

σ3 = 2(ε3ε1 − ε2ε4).

(2.7)

It is further assumed that the reorientation maneuver can occur quickly enough that the

spacecraft’s orbital position remains essentially unchanged, and that therefore, the inertial

directions to the high-intensity sources also remain constant during the slew maneuver.

Analytically, the path constraint given by (2.6) can be adjoined to the Hamiltonian by

creating the following constraint function:

C = cos−1(σ · σA) − αA, (2.8)

then substituting for σ using (2.7). The result must then be differentiated twice with respect

to time (and using (2.2) and (2.3)) in order to get a form in which the control torques

appear [11]. Finally, a new Hamiltonian H is formed by adjoining C to H with Lagrange

multiplier μ,

H = H + μC, (2.9)

with the conditions

μ ≥ 0 if C = 0

μ = 0 if C > 0.(2.10)

Further, the so-called tangency conditions [11] must also be applied (i.e., for any part of

the trajectory on the boundary, not only C, but also its first and second time derivatives

must be zero). The resulting form is analytically intractable for assessing whether necessary

conditions are met along a boundary arc or at a boundary point; however, by using a direct

method (Legendre pseudospectral), it is possible to obtain numerical solutions that meet the

necessary conditions of optimality. Fleming and Ross [12] show that a pseudospectral method

is effective for solving the unconstrained minimum-time reorientation problem.

2.1. Semifree Final Orientation

For some missions, the reorientation strategy may be altered if the final orientation is not

completely specified. An example would be the need to reorient the sensor axis to a desired

target direction in minimum time, with no constraints on the orientation of the other body-

fixed axes at the final time. In practice, some subsequent rotation about the sensor axis

might be required to optimize some other parameter (e.g., maximizing illumination of solar

panels, or minimizing solar heating of sensitive components), but the principal reorientation

maneuver could be achieved faster. The corresponding optimal control problem is the same


Sensor axis path along

the constraint boundary

λ = σA

σi

σf

Figure 2: Constrained rotation about the keep-out cone axis.

as before, but with the final conditions on the Euler parameter values given in (2.4c) specified

by

σ1

(tf)= σ1,f = 1 − 2

[ε2

1

(tf)+ ε2

3

(tf)]

σ2

(tf)= σ2,f = 2

[ε1

(tf)ε2

(tf)+ ε3

(tf)ε4

(tf)]

σ3

(tf)= σ3,f = 2

[ε3

(tf)ε1

(tf)− ε2

(tf)ε4

(tf)].

(2.11)

2.2. Constrained Rotation Axis

Consider now the suboptimal approach of a simple λ-rotation that carries the sensor axis σ

from initial to final orientation along the constraint boundary, with the other body-fixed axes

undergoing the same λ-rotation This amounts to a λ-rotation about the keep-out cone axis σA(Figure 2). Such a constrained rotation is a simple one-dimensional problem whose solution

is bang-bang, with the maximum torque being applied along the direction of λ. Because the

control torque components are each limited to a maximum magnitude Mmax, the maximum

torque along λ has magnitude

∣∣∣ ⇀Mλ

∣∣∣ = ∣∣∣∣∣Mmaxλ

λmax

∣∣∣∣∣, (2.12)

where λmax = max(λ1, λ2, λ3). Equation (2.12) thus maximizes⇀

Mλ while obeying the limits on

the individual control torques M1, M2, and M3. Assuming that a feasible rotation axis λ can

be identified, then the rotation angle θ can be determined from (A.5), and the rotation time is

given by

tf,λ = 2

√Iλθ

|Mλ|, (2.13)


where Iλ is the moment of inertia about the λ-axis. This provides a useful benchmark to which

the optimal solution can be compared. As a practical matter, such a forced-axis rotation could

be an acceptable suboptimal solution if the optimal path required too much computing time

or resources to calculate.

3. Results

Four cases are considered: two of these (cases BA1 and BA2) constrain the sensor axis to

move along the constraint boundary; the initial and final sensor axis directions also lie on

the boundary. These cases serve as proxies for boundary arcs, at least in that they provide

some indication of the slewing time and other qualitative properties of the motion. The other

spacecraft axes are unconstrained at the final time. The other two cases (BP1 and BP2) have the

sensor axis beginning and terminating on the constraint boundary (and of course, prohibited

from entering the constraint cone); the other spacecraft axes are unconstrained at the final

time.

The numerical results are generated using a Legendre pseudospectral method,

implemented in the software package DIDO [13]. In all cases, the discrete solution for the

control history produced by the pseudospectral method has been subsequently employed

to numerically propagate the dynamics from the given initial conditions; the results give

solutions that match the discrete solutions to within an absolute error of 10−16 (corresponding

units) at each node. These cases all require significant computation time (as much as 72 hours

on a computer with an Intel Core 2 2.0 GHz processor, with the number of pseudospectral

nodes in the range of 100–250.

Note that the Hamiltonian and costate values are not evaluated directly during the

pseudospectral solution process, but rather are reconstructed via the covector mapping

principle [14] after the problem solution is found. In all cases here, the Hamiltonian is found

to be reasonably constant given the relatively small number of nodes and level of accuracy

specified for the nonlinear programming aspect of the calculations.

A system of nondimensional units is employed, partly to provide somewhat more

general results, but chiefly because this system of units provides the kind of scaling needed

for the pseudospectral method to perform well. In physical units, the angular velocities,

moments of inertia, and control torques about the spacecraft’s principal axes are denoted

as ωi, Ii, and Mi, respectively; the corresponding nondimensional quantities are defined as

ξ =

√√√√ Imax

Mmax

ωi = ξωi

Ii =Ii

Imax

Mi =Mi

Mmax

,

(3.1)

where ξ is the time unit, and Imax and Mmax are the maximum values of the principal inertias

and control torques, respectively. In all of the cases presented, the spacecraft is assumed to


have a spherically symmetric mass distribution (i.e., all three principal moments of inertia

are equal) and three-axis control capability, with equal and independently limited control

authority about all three axes.

3.1. Case BA1

This is a forced boundary arc trajectory of the sensor axis σ. The constraint cone has half-angle

of 45 deg. (corresponding approximately to the Sun keep-out cone for the Swift spacecraft)and the sensor axis must traverse an arc of −90 deg. about the cone’s central axis. This solution

uses 151 nodes in the pseudospectral method (even a modest increase in the number of nodes

(to 171) resulted in nonconvergence with the available computing resources).A notable feature of the motion (Figure 3) is its qualitative similarity to the

unconstrained time-optimal solution (shown in Figure 4). Referring to the angular velocity

components, it is evident that the motion includes precession, with portions of the trajectory

including fully saturated control torques. Some finite time intervals have intermediate

torques, necessitated by the boundary arc constraint. This solution yields a final time of tf =1.9480 (nondimensional units). For comparison, if the rotation axis is constrained to lie along

the constraint cone’s axis, the final time would be tf,λ = 2.1078.

Figure 5 depicts the Hamiltonian and costate histories. It is evident that the

Hamiltonian is fairly constant, giving some confidence that the optimal solution has been

determined.

3.2. Case BA2

For this forced boundary arc, the constraint cone has half-angle αA = 23 deg. (corresponding

to the Moon cone for Swift), and the sensor axis must traverse an angle of −70 deg. about

the cone’s central axis. The final orientation of the sensor axis σf is calculated via (A.1), with⇀a = σi = [1, 0, 0],

⇀

b = σf , θ = −70 deg., and λ = [cos(αA), sin(αA), 0].For this case, the solution uses 100 nodes and has a final time of tf = 1.3020. For

comparison, if the rotation axis is constrained to lie along the constraint cone’s axis, the

final time would be tf,λ = 2.0966. Figure 6 depicts the dynamic response and control

torques; Figure 7 shows the Hamiltonian and costate histories. The motion and controls are

qualitatively similar to those in case BA1. It should be noted that in both BA1 and BA2, the

path of the sensor axis is verified to lie within 10−16 radians of the constraint boundary.

3.3. Case BP1

This case is identical to case BA1, except that only the initial and final directions of the sensor

axis are constrained to lie on the constraint boundary, corresponding to two forced boundary

points. Two of the control torques (M1 and M2) exhibit bang-bang behavior whereas M3

shows some chatter (even with 250 nodes employed), as seen in Figure 8. The somewhat

larger variation in the Hamiltonian (Figure 9) occurs near the initial time; however, this

happens at a finite time t = 0.025 (corresponding to 22 pseudospectral nodes after t = 0) after

the sensor axis has moved away from the constraint boundary (Figure 10). Such behavior

is therefore not the theoretically expected discontinuity in the Hamiltonian and costates at


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

Time

ωi

ω1

ω2

ω3

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time

−1

0

1

2

ε i

ε1

ε2

ε3

ε4

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time

−1

0

1

Mi

M1

M2

M3

(c)

Figure 3: Dynamic response and controls for the case BA1.

a point where the trajectory leaves the constraint boundary [11]; indeed, such discontinuities

may not be observable in numerical solutions where the equality condition in (2.10) is

unlikely to occur. The numerical solutions may be improved by using a Bellman chain, based

upon a sequence of embedded optimal solutions each of which can use a relatively small

number of nodes [15].Nevertheless, the trend is clear: precession created by the control torques, works to

reduce the final time to 1.9258, approximately 1% faster than the solution in BA1.

3.4. Case BP2

This case is identical to case BA2, except that only the initial and final directions of the sensor

axis are constrained to lie on the constraint boundary. Unlike case BP1, the control torques

display more intermediate behavior (Figure 11); solution accuracy is not as good here since

convergence could not be obtained for more than 100 nodes in the 72 hours of computer

time available. This is also evident in the slight variation in the Hamiltonian (Figure 12). As

with case BP1, the only points where the sensor axis contacts the constraint boundary (see

Figure 13) are at the initial and final times. The final time achieved is tf = 1.2967.


0 0.5 1 1.5 2 2.5 3 3.5−1

012

Angular velocity

Time

ωi

ω1

ω2

ω3

(a)

0 0.5 1 1.5 2 2.5 3 3.5

Time

−1012

Euler parameters

ε i

ε1

ε2

ε3

ε4

(b)

0 0.5 1 1.5 2 2.5 3 3.5

Time

−1

0

1Control torques

Mi

M1

M2

M3

(c)

Figure 4: Dynamic response and controls [9] for the motion shown in Figure 1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1.04

−1.02

−1

−0.98

−0.96

Time

Ham

ilto

nia

n

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1500

−1000

−500

0

500

Time

Co

state

s

λω1

λω2

λω3

λε1

λε2

λε3

λε4

(b)Figure 5: Hamiltonian and costates for the case BA1.


0 0.2 0.4 0.6 0.8 1 1.2 1.4−1

−0.5

0

0.5

Time

ωi

ω1

ω2

ω3

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

−1

0

1

2

ε i

ε1

ε2

ε3

ε4

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

−1

0

1

Mi

M1

M2

M3

(c)

Figure 6: Dynamic response and controls for the case BA2.

4. Practical Consideration

Regardless of whether boundary points or boundary arcs exist as part of an optimal trajectory,

the numerical determination of the solution via a pseudospectral method frequently requires

considerable computation time; indeed, the actual slewing maneuver of the spacecraft can

be accomplished in seconds or minutes (depending upon the control authority) while

the pseudospectral solver requires from 20 minutes to 72 hours to compute the solution.

Experience shows that providing even a rudimentary estimate of the states and controls as

an initial guess for the pseudospectral solver can reduce the computation time significantly.

A two-stage process is proposed, wherein a random-process algorithm, such as a particle

swarm optimizer (PSO) which can rapidly explore the solution space, provides the initial

guess to the pseudospectral algorithm. The literature abounds with hybrid methods used in

related control problems. For example, Ahmed et al. [16] successfully apply just a PSO to the

problem of tuning a satellite’s attitude controller while Sentinella and Casalino [17] examine

a hybrid evolutionary algorithm that comprises differential evolution, genetic algorithms,

and a PSO applied to the problem of spacecraft trajectory optimization.


0 0.2 0.4 0.6 0.8 1 1.2 1.4−1.04

−1.02

−1

−0.98

−0.96

Ham

ilto

nia

nTime

(a)

−300

−200

−100

0

100

Co

state

s

λω1

λω2

λω3

λε1

λε2

λε3

λε4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

(b)

Figure 7: Hamiltonian and costates for the case BA2.

Initial efforts that employ a two-stage process for optimal slewing maneuvers show

promising results. A PSO produces a low-quality approximation of the states, controls,

and node times for the solution, followed by the pseudospectral solver, which takes the

approximate solution as its initial starting point. An efficient method for generating the first-

stage solution is to represent each control torque component Mi as a sum of N Chebyshev

polynomials of the form [18]

Mi(t) ≈[N−1∑k=0

ckTk(t)

]− 1

2c0, (4.1)

where Tk is the Chebyshev polynomial of degree k, and the coefficients ck are determined by

the PSO using an explicit integration of the equations of motion ((2.2), (2.3)). Implementation

of (4.1) makes use of the Clenshaw recursion relation [18] for rapid evaluation of the

Chebyshev polynomials.

As an example, the control torque for a one-dimensional slewing maneuver (with no

keep-out cones) is shown in Figure 14(a). The PSO runs for only 50 iterations (requiring 64

seconds of computation time), producing a crude approximation to the bang-bang control

solution that is known to be optimal. That approximate solution is then employed by the

DIDO pseudospectral solver to calculate the optimal solution (Figure 14(b)), requiring 12

seconds. Using this two-stage method, the total computation time (PSO plus pseudospectral

solver) requires approximately half the time needed using only the pseudospectral method


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

0

1

Time

ωi

ω1

ω2

ω3

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time

−1

0

1

2

ε i

ε1

ε2

ε3

ε4

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time

−1

0

1

Mi

M1

M2

M3

(c)

Figure 8: Dynamic response and controls for the case BP1.

(148 seconds) with no initial guess for the solution. Future study should examine the utility

of the two-stage method for the full three-dimensional, constrained problem.

5. Conclusion

This preliminary study indicates that, although boundary arcs and boundary points may exist

in time-optimal spacecraft slewing maneuvers with path constraints, they are at best part of

a suboptimal solution. The numerical calculations (completed via a Legendre pseudospectral

method) show that even if the initial and final states are boundary points, the solution moves

away from the constraint boundary, resulting in a lower final time than if the motion is forced

to move along the boundary (a forced boundary arc). The necessary conditions lead to an

unwieldy set of relations, making it impossible to determine analytically if boundary points

or boundary arcs are excluded. Further examination of the problem using a Bellman chain

to improve the numerical accuracy may provide additional insight. A two-stage method for

generating optimal solutions in less time than that required by the pseudospectral method

alone shows some promise, but further work is needed to determine its utility for the three-

dimensional constrained problem.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1.08

−1.06

−1.04

−1.02

−1

−0.98

Time

Ham

ilto

nia

n(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time

−3000

−2000

−1000

0

1000

2000

Co

state

s

λω1

λω2

λω3

λε1

λε2

λε3

λε4

(b)

Figure 9: Hamiltonian and costates for the case BP1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Dis

tan

ce f

rom

“k

eep

-ou

t” c

on

e(r

ad)

Time

Figure 10: Distance from the sensor axis to the constraint boundary (case BP1).

Appendix

λ-Rotations and the Eigenaxis

Consider a dextral orthonormal basis set aj , fixed in reference frame A. A copy of this set,

denoted bi, is rotated in a right-handed sense, through an angle θ, about a unit vector λ,

which has fixed orientation with respect to A. The new orientation is denoted as reference

frame B and the rotation axis λ will have the same orientation with respect to B as it has to A.


0 0.2 0.4 0.6 0.8 1 1.2 1.4−1

−0.5

0

0.5

Time

ωi

ω1

ω2

ω3

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

−1

0

1

2

ε i

ε1

ε2

ε3

ε4

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

−1

0

1

Mi

M1

M2

M3

(c)

Figure 11: Dynamic response and controls for the case BP2.

This representation of the orientation of B with respect to A is called a λ-rotation [19]. For

two vectors⇀a and

⇀

b , fixed in frames A and B, respectively, if⇀

b =⇀a initially, and B undergoes

a λ-rotation, then the subsequent relationship between⇀a and

⇀

b is given by

⇀

b =⇀a cos θ− ⇀

a × λ sin θ + a · λ λ(1 − cos θ), (A.1)

a result named after Olinde Rodrigues (Goldstein [20] claims that the relationship was known

before Rodrigues, and that the vector form used here was first published by Gibbs [21]). This

form is useful in describing rotations of a sensor axis about an axis fixed in both the inertial

and spacecraft frames.

If one uses a direction cosine matrix CAB to express the orientation of B with respect

to A, then the constituent basis vectors are related by⎡⎢⎢⎢⎣b1

b2

b3

⎤⎥⎥⎥⎦ = CAB

⎡⎢⎢⎣a1

a2

a3

⎤⎥⎥⎦, (A.2)


0 0.2 0.4 0.6 0.8 1 1.2 1.4−1.1

−1.05

−1

−0.95

−0.9

TimeH

am

ilto

nia

n(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time

−15000

−10000

−5000

0

5000

Co

state

s

λω1

λω2

λω3

λε1

λε2

λε3

λε4

(b)

Figure 12: Hamiltonian and costates for the case BP2.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.002

0.004

0.006

0.008

0.01

0.012

Dis

tan

ce f

rom

“k

eep

-ou

t” c

on

e(r

ad)

Time

Figure 13: Distance from the sensor axis to the constraint boundary (case BP2).

and the elements of CAB are given by

Cij = bi · aj . (A.3)

Every direction cosine matrix has one unity-valued eigenvalue with corresponding

eigenvector (in matrix form) e, that is,

CABe = e, (A.4)


0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.51D slew maneuver

Time

Co

ntr

ol

torq

ue

PSO values

Chebyshev approx

(a)

0 0.5 1 1.5 2 2.5 3

−1

−0.6

−0.2

0.2

0.6

1

Time

Co

ntr

ol

torq

ue

Dido result—200 nodes, accurate mode

(b)

Figure 14: (a) Approximate solution for 1D slewing maneuver, generated by 50 iterations of a particleswarm optimizer. (b) Optimal control solution for the same maneuver, generated by DIDO (running inaccurate mode). Total required cpu time for solution: 148 sec using only DIDO, with no initial guess; 76 secfor the combined PSO-DIDO solution.

and the elements of e clearly constitute the components of the associated λ-rotation vector.

The vector e is commonly referred to as the eigenaxis for the rotation that generates B from A.

By taking dot products of both sides of (A.1) with⇀a and then with

⇀

b , and using

appropriate vector identities, it can be shown that

cos θ =

⇀a ·

⇀

b −(⇀a · λ)2

1 −(⇀a · λ)2

, sin θ =−1 +

(⇀b · ⇀a

)cos θ +

(⇀a · λ)2(1 − cos θ)

⇀

b ·(⇀a × λ

) , (A.5)

which permits the calculation of θ for given⇀a,

⇀

b , and λ. Another useful relationship gives the

direction cosine matrix CAB in terms of the Euler parameters

CAB =

⎡⎢⎢⎣1 − 2

(ε2

2 + ε23

)2(ε1ε2 + ε3ε4) 2(ε1ε3 − ε2ε4)

2(ε2ε1 − ε3ε4) 1 − 2(ε2

1 + ε23

)2(ε2ε3 + ε1ε4)

2(ε3ε1 + ε2ε4) 2(ε3ε2 − ε1ε4) 1 − 2(ε2

1 + ε22

)⎤⎥⎥⎦. (A.6)

Nomenclature

C: Constraint function

H: Hamiltonian without the path constraint

H: Hamiltonian with the path constraint

Ii: Principal moment of inertia (nondimensional)Ii: Principal moment of inertia (in physical units)J : Performance index


Mi: Control torque (nondimensional)Mi: Control torque (in physical units)tf : Final time

αA: Half-angle of the keep-out cone for object A

εi: Euler parameter

λ: Rotation axis

λx: Costate corresponding to state x

μ: Lagrange multiplier associated with constraint function C

θ: Rotation angle about the λ-axis

ξ: Time unit

σ: Direction of the sensor axis

σA: Direction of the central axis of the keep-out cone for object A

ωi: Angular velocity component (nondimensional)ωi: Angular velocity component (in physical units).

Acknowledgment

This paper has been presented at the 6th International Workshop and Advanced School

“Spaceflight Dynamics and Control,” Covilha, Portugal, March 28–30, 2011, http://www

.aerospace.ubi.pt/workshop2011/.

References

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[2] H. Shen and P. Tsiotras, “Time-optimal control of axisymmetric rigid spacecraft using two controls,”Journal of Guidance, Control, and Dynamics, vol. 22, no. 5, pp. 682–694, 1999.

[3] R. J. Proulx and I. M. Ross, “Time-optimal reorientation of asymmetric rigid bodies,” in Proceedingsof Advances in the Astronautical Sciences AAS/AIAA Astodynamics Conference, vol. 109, pp. 1207–1227,August 2002.

[4] S. L. Scrivener and R. C. Thompson, “Time-optimal reorientation of a rigid spacecraft usingcollocation and nonlinear programming,” in Proceedings of Advances in the Astronautical SciencesAAS/AIAA Astodynamics Conference, vol. 85, pp. 1905–1924, August 1993.

[5] S. W. Liu and T. Singh, “Fuel/time optimal control of spacecraft maneuvers,” Journal of Guidance,Control, and Dynamics, vol. 20, no. 2, pp. 394–397, 1997.

[6] X. Bai and J. L. Junkins, “New results for time-optimal three-axis reorientation of a rigid spacecraft,”Journal of Guidance, Control, and Dynamics, vol. 32, no. 4, pp. 1071–1076, 2009.

[7] H. B. Hablani, “Attitude commands avoiding bright objects and maintaining communication withground station,” Journal of Guidance, Control, and Dynamics, vol. 22, no. 6, pp. 759–767, 1999.

[8] G. Mengali and A. A. Quarta, “Spacecraft control with constrained fast reorientation and accuratepointing,” Aeronautical Journal, vol. 108, no. 1080, pp. 85–91, 2004.

[9] R. G. Melton, “Constrained time-optimal slewing maneuvers for rigid spacecraft,” in Proceedings ofthe Advances in the Astronautical Sciences AAS/AIAA Astrodynamics Specialist Conference, vol. 135, pp.107–126, Univelt, San Diego, Calif, USA, 2010.

[10] Swift, http://heasarc.nasa.gov/docs/swift/swiftsc.html.[11] A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, Hemisphere

Publishing, New York, NY, USA, 1975.[12] A. Fleming and I. M. Ross, “Minimum-time maneuvering of CMG-driven spacecraft,” in Proceedings

of the Advances in the Astronautical Sciences AAS/AIAA Astrodynamics Specialist Conference, vol. 129, pp.1623–1644, 2007.


[13] I. M. Ross, “A beginner’s guide to DIDO (ver 7.3): a MATLAB application package for solving optimalcontrol problems,” Tech. Rep. TR-711, Elissar LLC, Monterey, Calif, USA, 2007.

[14] Q. Gong, I. M. Ross, W. Kang, and F. Fahroo, “Connections between the covector mapping theoremand convergence of pseudospectral methods for optimal control,” Computational Optimization andApplications, vol. 41, no. 3, pp. 307–335, 2008.

[15] I. Michael Ross, Q. Gong, and P. Sekhavat, “Low-thrust, high-accuracy trajectory optimization,”Journal of Guidance, Control, and Dynamics, vol. 30, no. 4, pp. 921–933, 2007.

[16] R. Ahmed, H. Chaal, and D. W. Gu, “Spacecraft controller tuning using particle swarm optimization,”in Proceedings of the International Joint Conference 2009 (ICCAS-SICE ’09), pp. 73–78, August 2009.

[17] M. R. Sentinella and L. Casalino, “Cooperative evolutionary algorithm for space trajectoryoptimization,” Celestial Mechanics and Dynamical Astronomy, vol. 105, no. 1, pp. 211–227, 2009.

[18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, The Art ofScientific Computing, Cambridge University Press, 2nd edition, 2002.

[19] T. R. Kane, P. W. Likins, and D. A. Levinson, Spacecraft Dynamics, McGraw-Hill, 1983.[20] H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd edition, 1981.[21] J. W. Gibbs., Vector Analysis, edited by E. B. Wilson, Scribner, 1901; Yale University Press, London, UK,

1931.


Research ArticleLow-Thrust Out-of-Plane Orbital Station-KeepingManeuvers for Satellites

Vivian M. Gomes and Antonio F. B. A. Prado

DMC, National Institute for Space Research (INPE), Avenida dos Astronautas 1758,12227-010 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Vivian M. Gomes, [email protected]

Received 30 October 2011; Revised 30 November 2011; Accepted 8 January 2012


Copyright q 2012 V. M. Gomes and A. F. B. A. Prado. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

This paper considers the problem of out of plane orbital maneuvers for station keeping of satellites.The main idea is to consider that a satellite is in an orbit around the Earth and that it has its orbitis disturbed by one or more forces. Then, it is necessary to perform a small amplitude orbitalcorrection to return the satellite to its original orbit, to keep it performing its mission. A low thrustpropulsion is used to complete this task. It is important to search for solutions that minimize thefuel consumption to increase the lifetime of the satellite. To solve this problem a hybrid optimalcontrol approach is used. The accuracy of the satisfaction of the constraints is considered, inorder to try to decrease the fuel expenditure by taking advantage of this freedom. This type ofproblem presents numerical difficulties and it is necessary to adjust parameters, as well as detailsof the algorithm, to get convergence. In this versions of the algorithm that works well for planarmaneuvers are usually not adequate for the out of plane orbital corrections. In order to illustratethe method, some numerical results are presented.

1. Introduction

The present paper studies the orbital maneuvers required by a spacecraft that needs to

perform an orbital correction maneuver in order to remain with its orbital elements inside

a region where it can achieve the goals of the mission. It is assumed that the orbit of the

satellite is given, as well as a nominal orbit for this satellite that allows it to be useful

for the planned activities. Then, it is necessary to maneuver this satellite from its current

position to the nominal specified orbit. Emphasis will be given in out-of-plane maneuvers,

because this situation was not much explored in previous studies, like in [1, 2], and it has a

high cost in terms of fuel consumption. Another contribution of the present paper is that it

studies the problem of station-keeping maneuvers. In terms of theoretical definition, orbital


transfers and station keeping are quite similar. But, when facing practical problems, from

the engineering point of view, they are different. This fact can be seen in the literature that

presents many methods for station-keeping and orbital transfers, considering them as two

different problems. For station-keeping, it is necessary much lower values for the parameters

that control the convergence of the method. It means that convergence is much more difficult

in the present case. To solve this problem, it was necessary to divide the convergence of the

method in several steps. First a larger number for the tolerance was used and then, using the

solution obtained in this step as first guess, we searched for new values of the parameters

α1 and β1, in order to make the steps in the direction of the convergence smaller than in

the previous case. This technique is repeated until the goal defined for the convergence

is reached. So, the present paper shows that this method is also valid for the important

application of station keeping maneuvers.

The control available to perform this maneuver is the application of a low thrust,

and the objective is to perform this maneuver with minimum fuel consumption. An optimal

approach will be used. There is no time restriction involved here, and the spacecraft can leave

from any point in the initial orbit and arrive at any point in the final orbit. The stochastic

version of the projection of the gradient method is used. This version allows us to include

the fact that the constraints do not need to be exactly satisfied (see [1, 2]). This is done

to realistically treat the numerical inaccuracies and/or flexibilities in terms of tolerance in

mission requirements, leading to situations where the final state is constrained to lie inside a

given region, instead of having an exact value.

2. Review of Orbital Maneuvers

A very important result in this field was obtained by Hohmann [3]. He solved the problem

of minimum ΔV transfers between two circular coplanar orbits. The Hohmann transfer

would later be generalized to the elliptic case (transfer between two coaxial elliptic orbits)by Marchal [4].

After that, can be found in the literature the problem of a two-impulse transfer, where

the magnitude of the two impulses are fixed, like in Jin and Melton [5] and Jezewski and

Mittleman [6].Then, the three-impulse concept was introduced in the literature by Hoelker and Silber

[7] and Shternfeld [8]. They showed that a bielliptical transfer between two circular orbits

has a ΔV lower than the one required by the Hohmann transfer, for some combinations of

the initial and final orbits. Continuing those studies, Ting [9] showed that the use of more

than three impulses does not lower the ΔV , considering only situations where impulsive

maneuvers are used.

Some other researchers worked on methods where the number of impulses is a

parameter to be optimized, and not a value fixed in advance. It is the case of the papers

written by Jezewski and Rozendaal [10] and Eckel [11]. Most of the research done in this

situation is based on the “Primer-Vector” theory, developed by Lawden and showed in

[12, 13].Later, two more techniques were introduced in the orbital maneuvers field, using the

concepts of swing-by and gravitational capture. Those techniques are based on the use of the

gravitational forces of a third body to increase or decrease the energy of the spacecraft, so

reducing the fuel consumption of the maneuver. References [14–20] describe this problem

and show some applications of those techniques in more details.


3. Definition of the Problem

The objective of the problem considered here is to change the orbit of a spacecraft. The

amplitude of this orbital maneuver has to be compatible with the requirement that the

spacecraft needs to return to its nominal orbit that was affected by some perturbations

[21, 22], but this is done in a small amplitude framework. Note that the third-body

perturbation, when the perturbed body is not in the same plane of the perturbing body, can

change the inclination of the perturbed body, as shown in [21]. Then, an initial and a final

orbit around the Earth are completely specified. The problem is to find how to transfer the

spacecraft between those two orbits and make this transfer in a form that minimizes the fuel

consumption. Time restrictions will not be considered in the present investigation, and the

spacecraft can leave and arrive at any point in the given initial and final orbits. The maneuver

is made by an engine that can deliver a thrust with constant magnitude and variable direction.

Although the algorithm is generic regarding the orbital elements changed, some adaption is

made in order to search for faster convergence in the case of a maneuver that changes the

inclination of the orbit.

The spacecraft is assumed to be travelling in a Keplerian orbit without perturbation

forces during the maneuver. This orbit is controlled by the thrusts, so the orbit is no longer

keplerian when the engines are turned on. In a situation like this, there are two types of

motion:

(i) a Keplerian orbit, assuming that the Earth is a point of mass and that it dominates

the motion of the spacecraft. This motion occurs when the engine is not being

applied to the satellite;

(ii) the motion governed by the two forces: the gravity of the Earth and the force

delivered by the thrusts. This motion occurs when the engine is working.

The thrusts have the following characteristics:

(i) fixed magnitude, so intermediate values are not allowed and the thrusts can deliver

the full power or are not working;

(ii) constant ejection velocity that has to occur to be consistent with the assumption that

the velocity of the gases ejected from the thrusts is constant;

(iii) free angular motion: the direction of the thrusts is not constant during the transfer.

This direction is specified by the angles u1 and u2, called pitch (the angle between

the direction of the thrust and the perpendicular to the line Earth-spacecraft) and

yaw (the angle with respect to the orbital plane);

4. Optimal Control Formulation

This approach is based on optimal control theory. First-order necessary conditions for a local

minimum are used to obtain the adjoint equations the Pontryagin’s maximum principle,

which allow us to obtain the control angles at each instant of time (in fact, we search for

those control angles as a function of the range angle that is an independent variable that

replaces the time and is equivalent to the true anomaly). These assumptions lead us to

a Two-Point Boundary Value Problem (TPBVP), where the difficulty is to find the initial

values of the Lagrange multipliers. The approach used in the present research is the hybrid

approach of guessing a set of values, numerically integrating all the differential equations,


and then searching for a new set of values, based on a nonlinear programming algorithm.

With this approach, the problem is reduced to a parametric optimization, where the Lagrange

multipliers are variables to be optimized. This idea follows the same principles used by Biggs

in [23, 24].The method also showed by Biggs [24], where the “adjoint-control” transformation

is performed and, instead of the initial values of the Lagrange multipliers, one guesses the

control angles and their rates at the beginning of the “burning arc,” is used here. Then, it is

easier to find a good initial guess, and the convergence is faster. This hybrid approach has the

advantage that, since the Lagrange multipliers remain constant during the “ballistic arcs,” it

is necessary to guess values of the control angles and their rates only for the first “burning

arc.” This transformation reduces very much the number of variables to be optimized and, as

a consequence, the time of convergence.

The next paragraphs show how to write this problem using an optimal control

approach.

Objective Function

Let Mf be the final mass of the vehicle. It has to be maximized with respect to the control

u(·);Subject to

x = f(x, u, s

),

Ce(x, u, s

)= Ee,

Cd(x, u, s

)≤ Ed,

h(x(tf),(tf))

= Eh, t0, x(t0) given,

(4.1)

where x is the state vector, f(·) is the right-hand side of the equations of motion, in the same

way that was used by Biggs [23, 24] and by Prado and Rios-Neto [25] and Bryson and Ho

[26] s is the independent variable that replaces the time (s0 ≤ s ≤ sf), Ce(·) and Cd(·) are the

algebraic dynamic constraints on state and control that have dimensions me and md, h(·) are

the boundary constraints of dimension mh, and Ee, Ed, Eh are error vectors satisfying

|Eei| ≤ EeTi , i = 1, 2, 3, . . . , me,

|Edi| ≤ EdTi , i = 1, 2, 3, . . . , md,

|Ehi| ≤ EhTi , i = 1, 2, 3, . . . , mh,

(4.2)

where the fixed given values EeTi , EdTi , and EhTi characterize the region around zero within

which errors are considered tolerable.


To avoid singularities or close approaches to them, we use the following variables:

X1 =

[a(1 − e2

)μ

]1/2

,

X2 = e cos(ω − φ

),

X3 = e sin(ω − φ

),

X4 =Fuel Consumed

m0

,

X5 = t,

X6 = cos

(i

2

)cos

(Ω + φ

2

),

X7 = sin

(i

2

)cos

(Ω − φ

2

),

X8 = sin

(i

2

)sin

(Ω − φ

2

),

X9 = cos

(i

2

)sin

(Ω + φ

2

),

(4.3)

where: a = semi-major axis, e = eeccentricity, i = inclination, Ω = argument of the ascending

node, ω = argument of perigee, f = true anomaly, s = range angle, φ = f + ω − s,μ =gravitational constantl, m0 = initial mass of the spacecraft.

In those variables, the equations of motion are the following ones:

dX1

ds= f1 = SiX1F1,

dX2

ds= f2 = Si{[(Ga + 1) cos(s) +X2]F1 + GaF2 sin(s)},

dX3

ds= f3 = Si{[(Ga + 1) sin(s) +X3]F1 − GaF2 cos(s)},

dX4

ds= f4 =

SiGaF(1 −X4)X1W

,

dX5

ds= f5 =

SiGa(1 −X4)m0

X1,

dX6

ds= f6 = − SiF3

[X7 cos(s) +X8 sin(s)]2

,

dX7

ds= f7 = SiF3

[X6 cos(s) −X9 sin(s)]2

,

dX8

ds= f8 = SiF3

[X9 cos(s) +X6 sin(s)]2

,

dX9

ds= f9 = SiF3

[X7 sin(s) −X8 cos(s)]2

,

(4.4)


where

Ga = 1 +X2 cos(s) +X3 sen(s),

Si =μX4

1[Ga3m0(1 −X4)

] . (4.5)

F, F1, F2, F3 are the forces generated by the thrust and they are given by

�F = �F1 + �F2 + �F3,∣∣∣�F∣∣∣ = F,F1 = F cos(α) cos

(β),

F2 = F sen(α) cos(β),

F3 = F sen(β),

(4.6)

where α is the angle between the direction of the thrust and the line that is perpendicular to

the radius vector and β is the angle between the direction of the thrust and the orbital plane.

The equations for the Lagrange multipliers are given as follows:

dp1

ds= −

4∑9

j=1 pjfj + p1f1 − p4f4 − p5f5

X1,

dp2

ds=

cos(s)Ga

⎡⎣39∑j=1

pjfj − p4f4 − p5f5

⎤⎦− Si p2F1 − Si cos2(s)

(p2F1 − p3F2

)− Si cos(s)sen(s)

(p2F2 + p3F1

),

dp3

ds=

sen(s)Ga

⎡⎣39∑j=1

pjfj − p4f4 − p5f5

⎤⎦− Si p3F1 − Si cos(s) sen(s)

(p2F1 − p3F2

)− Si sen2(s)

(p2F2 + p3F3

),

dp4

ds= −

⎡⎣∑9j=1 pjfj − p4f4 − p5f5

m0(1 −X4)

⎤⎦,dp5

ds= 0,

dp6

ds=

−SiF3

[p7 cos(s) + p8 sen(s)

]2

,

dp7

ds=

SiF3

[p6 cos(s) − p9 sen(s)

]2

,

dp8

ds=

SiF3

[p6 sen(s) + p9 cos(s)

]2

,

dp9

ds=

−SiF3

[p8 cos(s) − p7 sen(s)

]2

.

(4.7)


The control that is applied to the spacecraft also needs a substitution of variables, with

the objective of solving numerical problems. In this situation we use the following set of

variables:

u1 = s0,

u2 =(sf − s0

)cos(β0

)cos(α0),

u3 =(sf − s0

)cos(β0

)sin(α0),

u4 =(sf − s0

)sin(β0

),

u5 = α′,

u6 = β′.

(4.8)

The first-order necessary conditions of the optimal problem, which are the conditions

that allow us to obtain the optimal control, can be obtained, at every instant of time, by

extremizing the Hamiltonian of the system. The equations are given as follows:

sin(α) =q2

S′ ,

sin(B) =q3

S′′ ,

cos(α) =q1

S′ ,

cos(B) =S′

S′′ ,

(4.9)

where

S′ = ±[q2

1 + q22

]1/2,

S′′ = ±[q2

1 + q22 + q

23

]1/2,

q1 = p1X1 + p2[X2 + (Ga + 1) cos(s)] + p3[X3 + (Ga + 1) sin(s)],

q2 = p2Ga sin(s) − p3Ga cos(s),

q3 = − p6[X7 cos(s) +X8 sin(s)]

2+ p7[X6 cos(s) −X9 sin(s)]

+p8[X6 sin(s) +X9 cos(s)] + p9[X7 sin(s) −X8 cos(s)].

(4.10)

It is possible to include constraints in this type of problem. They can be represented by the

following equations:

S(·) ≥ 0,

(a − a∗)|a0 − a∗|

= 0,

[a(1 + e) − a∗(1 + e∗)]|a0(1 + e0) − a∗(1 + e∗)| = 0,


(i − i∗)|i0 − i∗|

= 0,

(Ω −Ω∗)|Ω0 −Ω∗| = 0,

(ω −ω∗)|ω0 −ω∗| = 0.

(4.11)

In those constraints, the first equation represents a generic inequality constraint and the other

five represent the important constraint of having an orbit specified.

5. Numerical Method

To solve this problem, the stochastic version of the projection of the gradient method (Rios-

Neto and Pinto [1]) can be used. It is showed in the next paragraphs.

We start by assuming an initial value for p, the vector of parameters p that represents

the variables that we are searching. It may come from an initial guess or from a value that

belongs to an immediately previous iteration. Then, a first-order direct search approach is

adopted in a typical iteration to obtain an approximate solution for the increment Δp:

Minimize: J(p + Δp

)(5.1)

Subject to: Ce(p + Δp

)= α1Ce

(p)+ Ee, (5.2)

Cd(p + Δp

)= β1Cd

(p)+ Ed, (5.3)

where J(p) is the objective function, Ce(p) is the equality constraints, Cd(p) is the active

inequality constraints at p, and 0 ≤ α1 < 1, 0 ≤ β1 < 1 are parameters that have to be chosen

close enough to one to make the increments Δp be of the first order of magnitude.

Linearizing the left-hand sides of (5.2) and (5.3) and using a stochastic interpretation

for the errors Ee and Ed, we have

(α1 − 1)Ce(p)=

⎛⎜⎝d[Ce(p)]

dp

⎞⎟⎠Δp + Ee, (5.4)

(β1 − 1

)Cd(p)=

⎛⎜⎝d[Cd(p)]

dp

⎞⎟⎠ Δp + Ed, (5.5)


where Ed and Ee are zero mean uniformly distributed errors, given by

E[EeEeT

]= diag[ei, i = 1, 2, . . . , me],

E[EdEdT

]= diag[di, i = 1, 2, . . . , md],

(5.6)

where E[·] indicate the expected value of its argument.

The condition shown in (5.1) is approximated by

−g · ∇ JT(p)= Δp + n, (5.7)

where g ≥ 0 needs to be adjusted to guarantee that Δp is small enough to permit the use of

the linearized representation of J(p+Δp), and n is taken as a zero mean uniformly distributed

random vector, modeling the a priori searching error in the direction of the gradient ∇J(p),with

E[nnT]= P (5.8)

as its diagonal covariance matrix. The values of the variances in P are chosen such as to

characterize an “adequate order of magnitude” for the dispersion of n. The diagonal form

adopted is to model the assumption that it does not impose any a priori correlation between

the errors in the gradient components.

The simultaneous consideration of conditions of (5.4), (5.5), and (5.7) characterizes a

problem of parameter estimation, which can be written as

U = U + n, (5.9)

Y =MU + V , (5.10)

where UΔ − g · ∇JT (p) is the “a priori information”; UΔΔp; Y Δ [(α1 − 1)CeT (p) : (β1 −1)CdT (p)] is the observation vector; MTΔ[(d(Ce(p))/dp)T : (d(Cd(p))/dp)T ]; V T = [EeT :

EdT ].Adopting a criterion of linear, minimum variance estimation, the optimal search

increment can be determined using the classical Gauss-Markov estimator, which in Kalman

form gives

U = U +K(Y −MU

), (5.11)

P = P −KMP, (5.12)

K = P MT(MPMT + R

)−1, (5.13)


where P is defined as before, RΔE[VV T ] = diag [Rk, k = 1, 2, . . . , me + md], and P has the

meaning of being the covariance matrix of the errors in the components estimates of U, that

is,

P = E[(U − U

)(U − U

)T]. (5.14)

To build a numerical algorithm using the proposed procedure, the following types of

iterations are considered:

(i) initial phase of acquisition of constraints: when starting from a feasible point

that satisfies the inequality constraints, the search is done to capture the equality

constraints, including those inequality constraints that eventually became active

along this phase;

(ii) search of the minimum: when from a point that satisfies the constraints in the limits

of the tolerable errors V in (5.10), the search is done to take the objective function

(5.1) to get closer to the minimum; this search is conducted by relaxing the order of

magnitude of the error bounds around the constraints;

(iii) restoration of the constraints: when from a point that resulted from a type (ii)iteration, the search is done to restore constraints satisfaction, within the limits

imposed by the error V in (5.10).

Rios-Neto and Pinto [1] suggest how to choose good values for the numerical

parameters that must be different for each type of iteration.

6. Simulations and Numerical Tests

To verify if the algorithm proposed is useful for this particular type of problem, three

maneuvers of station keeping were simulated, all of them having modifications in the

inclination. These results were compared with the ones obtained by the deterministic version,

without flexibility in the constraint’s satisfaction. Similar problems can be found in [27–29].The first maneuver will occur with the data given in Table 1. The thrust level is 2.0 N. Table 2

shows the errors allowed in the final Keplerian elements of the orbit. The goal is to combine

a modification in semimajor axis, eccentricity, and inclination in a single maneuver.

Several numbers of “burning arcs” were used for the same maneuver, in order to

obtain some information about the importance of this parameter.

The consumptions found are showed in Table 3, as well as comparisons with

deterministic methods that we also simulated for comparison.

The second maneuver will occur with the data given in Table 4. Now, only the

inclination is changed. The thrust level is again 2.0 N. Table 5 shows the errors allowed in

the final Keplerian elements of the orbit.

The choice of the number of “burning arcs” was done for several different values, in

the same way that was made in the first maneuver.

The consumptions found are showed in Table 6, as well as comparisons with

deterministic methods.

The third maneuver occurs with the data given in Table 7. Now, inclination and

eccentricity are changed. The thrust level is again 2.0 N. Table 8 shows the errors allowed

in the final Keplerian elements of the orbit.


Table 1: Data for Maneuver 1.

Orbits Initial Final

Semimajor axis 7060.00 7090.00


Inclination (degrees) 1.00 0.50

Ascending node (degrees) 0.00 Free

Argument of perigee (degrees) 0.00 Free

Mean anomaly (degrees) 0.00 Free

Table 2: Errors allowed for final keplerian elements for Maneuver 1.

Semimajor axis 1.0 Km

Eccentricity 0.005

Inclination 0.001 deg

Table 3: Fuel expenditure comparisons (kg) for Maneuver 1.

Approach Stochastic Deterministic

2 Arcs 0.301 0.303

3 Arcs 0.300 0.301

4 Arcs 0.298 0.299

5 Arcs 0.297 0.298

6 Arcs 0.296 0.297

7 Arcs 0.295 0.296

8 Arcs 0.294 0.296

Table 4: Data for transfer Maneuver 2.








The number of “burning arcs” was again varied from two to eight, in the same way

that was made in the other maneuvers. The consumptions found are showed in Table 9, as

well as comparisons with deterministic methods.

Next, a new maneuver is considered to emphasize the effects of the errors allowed

for the final Keplerian elements. This maneuver considers the data shown in Table 1 for the

initial and final orbits, but increases the tolerable errors to the values shown in Table 10.

Several numbers of “burning arcs” were also used for this maneuver, to be compatible

with the previous studies. The consumptions found are showed in Table 11, as well as

the usual comparisons with the deterministic methods. Note that the difference in savings

increased, when larger values for the errors are allowed.




Eccentricity 0.005




2 Arcs 0.130 0.132

3 Arcs 0.128 0.130

4 Arcs 0.127 0.129

5 Arcs 0.127 0.128

6 Arcs 0.127 0.128

7 Arcs 0.127 0.128

8 Arcs 0.127 0.128

Table 7: Data for transfer Maneuver 3.










Eccentricity 0.005




2 Arcs 0.170 0.172

3 Arcs 0.167 0.169

4 Arcs 0.165 0.167

5 Arcs 0.163 0.166

6 Arcs 0.162 0.160

7 Arcs 0.162 0.160

8 Arcs 0.162 0.160



Eccentricity 0.05





2 Arcs 0.295 0.303

3 Arcs 0.294 0.301

4 Arcs 0.292 0.299

5 Arcs 0.291 0.298

6 Arcs 0.290 0.297

7 Arcs 0.290 0.296

8 Arcs 0.289 0.296

7. Conclusions

Optimal control was used to build an algorithm to search for solutions for the problem of

minimum fuel consumption to make orbital maneuvers for a satellite that needs to return to

its nominal orbit after deviations caused by perturbations forces.

In this research, emphasis was given in the problem considering station keeping with

out-of-plane maneuvers. The adjustments made in the algorithm, as well as in the parameters

used, allow us to get convergence in most of the cases for this version of the problem.

This problem took into account the accuracy in the constraint’s satisfaction by using

the nonlinear programming algorithm proposed by Rios Neto and Pinto [1].The results showed that it is possible to reduce the costs by exploring tolerable errors

shown in Tables 2, 5, 8 and 10 in the constraint’s satisfaction. The amount saved can be

important in many cases.

It is also clear that increasing of the number of propulsion arcs decreases the fuel

costs. It can be seen from Tables 3, 6, 9 and 11, because the fuel consumption is smaller

for larger numbers of arcs. The reason is that increasing this number causes an increase in

the degrees of freedom available for the optimization technique. Since the maneuvers have

small amplitudes, this increase in the burning arcs has a limit in the savings, so the fuel

consumption reaches a constant value after a certain value.

Acknowledgments

The authors are grateful to Fundacao de Amparo a Pesquisa do Estado de Sao Paulo,

(FAPESP), Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), and

Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior (CAPES) for supporting this

research.

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360–382, 1953.[13] H. B. Griffiths, “The fundamental group of two spaces with a common point,” Quarterly Journal of

Mathematics, vol. 5, no. 1, pp. 175–190, 1954.[14] V. M. Gomes and A. F. B. A. P. Prado, “Effects of the variation of the periapsis velocity in a swing-

by maneuver of a cloud of particles,” in Proceedings of the Recent Advances in Applied and TheoreticalMechanics, pp. 106–108, Tenerife, Spain, 2009.

[15] V. M. Gomes and A. F. B. A. Prado, “A numerical study of the dispersion of a cloud of particles,” inProceedings of the 20th International Congress of Mechanical Engineering (COBEM ’09), 2009.

[16] V. M. Gomes and A. F. B. D. A. Prado, “The use of gravitational capture for space travel,” in Proceedingsof the 60th International Astronautical Congress (IAC ’09), vol. 2, pp. 1066–1071, Daejeon, South Korea,2009.

[17] V. M. Gomes and A. F. B. De Almeida Prado, “A study of the close approach between a planet and acloud of particles,” in Proceedings of the Recent Advances in Signal Processing, Robotics and Automation,vol. 109, pp. 1941–1958, Cambridge, UK, 2002.

[18] A. F. B. De Almeida Prado, “Numerical and analytical study of the gravitational capture in thebicircular problem,” Advances in Space Research, vol. 36, no. 3, pp. 578–584, 2005.

[19] A. F. B. D. A. Prado, “Numerical study and analytic estimation of forces acting in ballistic gravitationalcapture,” Journal of Guidance, Control, and Dynamics, vol. 25, no. 2, pp. 368–375, 2002.

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[21] A. F. B. De Almeida Prado, “Third-body perturbation in orbits around natural satellites,” Journal ofGuidance, Control, and Dynamics, vol. 26, no. 1, pp. 33–40, 2003.

[22] A. P. M. Chiaradia, H. K. Kuga, and A. F. B. A. Prado, “Single frequency GPS measurements in real-time artificial satellite orbit determination,” Acta Astronautica, vol. 53, no. 2, pp. 123–133, 2003.

[23] M. C. B. Biggs, The Optimization of Spacecraft Orbital Manoeuvres. Part I: Linearly Varying Thrust Angles,The Hatfield Polytechnic, Numerical Optimization Centre, 1978.

[24] M. C. B. Biggs, The Optimisation of Spacecraft Orbital Manoeuvres. Part II: Using Pontryagin’s MaximunPrinciple, The Hatfield Polytechnic, Numerical Optimisation Centre, 1979.

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[26] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Wiley, New York, NY, USA, 1975.[27] V. M. Gomes, A. F. B. A. P. Prado, and H. K. Kuga, “Orbital maneuvers using low thrust,”

in Proceedings of the Recent Advances in Signal Processing, Robotics and Automation, pp. 120–125,Cambridge, UK, 2009.

[28] A. F. B. A. Prado, V. M. Gomes, and I. M. Fonseca, “Low thrust orbital maneuvers to insert a satellite ina constellation,” in Proceedings of the 4th International Workshop on Satellite Constellations and FormationFlying, Instituto Nacional de Pesquisas Espaciais, Sao Jose dos Campos, Brazil, February 2005.



Research ArticleAn Adaptive Remeshing Procedure for ProximityManeuvering Spacecraft Formations

Laura Garcia-Taberner1 and Josep J. Masdemont2

1 Departament d’Informatica i Matematica Aplicada, Universitat de Girona, C/Lluıs Santalo s/n,17071 Girona, Spain

2 IEEC and Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647,08028 Barcelona, Spain

Correspondence should be addressed to Laura Garcia-Taberner, [email protected]



Copyright q 2012 L. Garcia-Taberner and J. J. Masdemont. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a methodology to optimally obtain reconfigurations of spacecraft formations. Itis based on the discretization of the time interval in subintervals (called the mesh) and theobtainment of local solutions on them as a result of a variational method. Applied to a librationpoint orbit scenario, in this work we focus on how to find optimal meshes using an adaptiveremeshing procedure and on the determination of the parameter that governs it.

1. Introduction

Formation flying concepts have been growing in interest during the last years. In different

scenarios, formations or clusters of small satellites can perform like a virtual larger telescope,

obtaining equivalent information, but with a reduced cost. Mission concepts like the NASA

Terrestrial Planet Finder [1], the ESA Darwin [2], the NASA MAXIM [3], or the ESA XEUS [4]are just few examples that remark the importance of this new technology for space telescopes.

Nevertheless, formation flying still demands many new technologies to be successfully

implemented. Usually the spacecraft must be located and maintained within a very narrow

range of relative distances, and severe constraints like this, increasing the already high

complexity of the mission design (see for instance the works of Farrar et al. [5] and references

therein). Also there are many other issues that need to be addressed as well. For instance, a

main one is collision avoidance when maneuvering or reconfiguring the formation. Typically,

from one observation to the next one, the formation needs to change the pointing goal and

eventually change its pattern. To this end, some representative techniques considered are


the computation of proximity maneuvers by means of artificial potential functions in the

works of Badawy and McInnes [6], rotation techniques introduced by Wang et al. [7, 8], or the

FEFF methodology based on a finite element implementation introduced by Garcia-Taberner

and Masdemont in [9].In this paper we consider the FEFF methodology (Finite Elements for Formation

Flight) for the reconfiguration and proximity maneuvering of a formation about a libration

point. With this methodology, individual trajectories of the spacecraft inside the formation

are computed by means of solving a direct optimal control problem which is formulated

in terms of the discretization given by a finite element procedure. We briefly summarize

this procedure in Section 2. Then in Section 3 we focus on an adaptive remeshing strategy

that produces optimal discretizations for the trajectories, in the sense that the error in the

trajectories obtained is kept below a given threshold with the coarsest possible mesh. Finally,

the paper ends with some numerical examples and conclusions.

2. The FEFF Methodology

In this section we present a brief summary of the basics of the FEFF methodology that can be

found fully developed in [9, 10]. This methodology was made with the purpose of system-

atically computing reconfigurations of spacecraft formations located in libration point orbits.

However, it could also be generalized for formations about the Earth or for in free space.

As it is well known, the vicinity of the Lagrangian points L1 and L2 is a very convenient

place for space observatories (L1 for the Sun and L2 for deep space). In this paper we consider

a formation of spacecraft located in a halo orbit about L2. We assume that the formation

is made of N spacecraft flying with a particular pattern and our objective is to perform a

reconfiguration in a fixed time T . We also assume that the spacecraft are in a small formation.

This is, the distance between them is only of the order of few hundreds of meters, both in the

initial and the final configurations. The objective of the FEFF methodology is to find a suitable

trajectory for each of the spacecraft that delivers it to the goal position, with minimum fuel

consumption and avoiding collisions with other spacecraft.

Since the formations are small with respect to the amplitude of the halo orbit, we

consider the linearized equations of motion about the nonlinear orbit. In [11] we have

studied the impact of the nonlinear part, concluding that, for orbits with a diameter of a few

hundreds of meters (i.e., the usual length for a formation), these linearized equations model

the dynamics of the formation in a very good way.

Then, according to these hypotheses, associated to each spacecraft in the formation,

we have an equation of the form

X(t) = A(t)X(t), (2.1)

where A(t) is a 6 × 6 matrix and X refers to the state of the spacecraft (see The appendix).Usually the origin of the reference frame for the X coordinates is the nominal point on the

base halo orbit at time t being the orientation of the coordinate axis parallel to the ones of the

RTBP model.


In order to perform the reconfigurations, we consider a control function for each of the

spacecraft and also we include the boundary conditions corresponding to the initial and final

states:

Xi(t) = A(t)Xi(t) +Ui(t),

Xi(0) = X0i ,

Xi(T) = XTi .

(2.2)

Here X0i and XTi stand for the initial and final states of the ith spacecraft inside the formation,

and U1, . . . ,UN , are the controls we are searching with the aim of being optimal in terms of

fuel consumption.

The basis of the FEFF methodology is to use the finite element method in time to

discretize the spacecraft trajectories and to obtain the controls. Essentially, for each spacecraft,

the time interval [0, T] which we consider for the reconfiguration is split in Mi subintervals

of the domain called elements. For a given trajectory, its mesh can have elements of different

lengths and of course different satellites can have associated different meshes. This will

depend on the nature of the trajectories of the reconfiguration and the path they follow. Using

this mesh we impose that controls are maneuvers applied at the points where two elements

join (the nodes). The finite element methodology is used to formulate the problem and to

obtain, by means of this discretization, a relation between the states of each spacecraft at the

nodes of the elements and the Δv maneuvers applied. We note that in our discretization we

use elements with two nodes located at the ends of each element (consecutive elements share

the connecting node). These elements are called linear elements and their associated trun-

cation error is according to the linear approximation taken about the nonlinear orbit when

considering (2.1). Usually, in the finite element methodology, the solution inside each element

is approximated by a polynomial and, for linear elements, this polynomial is of degree

one. Finite elements in time and greater orders have also been considered in more general

formulations of optimal control problems. An interesting presentation can be found in [12].By means of the procedure FEFF, we reduce the reconfiguration problem to an optimal

control problem with constrains. The functional to be minimized is related to the fuel

consumption and is taken as the sum of the norm of the maneuvers:

J1 =N∑i=1

Mi∑k=0

ρi,k‖Δvi,k‖, (2.3)

where || ∗ || denotes the Euclidean norm, vi,k is the maneuver applied at the kth node of

trajectory, and i and ρi,k are weighting parameters that can be used, for instance, to penalize

fuel consumptions of selected spacecraft with the purpose of balancing fuel resources. For

clarity, in (2.3) we consider that ρi,k multiplies the modulus of the maneuver, but in a similar

way we can impose a weight on each of its components.

An important issue in the formulation of the procedure is collision avoidance between

satellites. It enters in the optimization problem as constraints. We consider that each

spacecraft is surrounded by a security sphere that cannot intersect during the reconfiguration

process. Again, the discretization of the time interval made by the finite element methodology

provides an efficient implementation to check these constraints.


3. Adaptive Remeshing Applied to Reconfigurations

The objective of this paper is how to systematically obtain good meshes for the

reconfiguration problem. We note that, for a given mesh, the FEFF methodology computes

the trajectory of the spacecraft in such a way that J1 in (2.3) is minimized. But at the end, the

trajectories of the satellites have been obtained after a discretization process and the error of

the discretized approximated trajectories with respect to the exact solutions of the problem

is not obvious. If we take a small number of elements, we can have a poor model that is not

accurate enough. On the other side, as it is well known in the finite element methodology, the

approximated trajectories converge towards the true solution when the diameter of the mesh

(the length of the longest time interval) tends to zero.

Of course when we increase the number of elements in the mesh, it also increases

the complexity of the computations, the required CPU time, and the representation of the

solution someway. Also we could end up with ill conditioned problems when the number

of elements is very high due to the presence of very small maneuvers. To overcome these

difficulties, adaptive remeshing is a technique that allows us to work in the other way round.

Fixing an acceptable level of accuracy, and by means of an iterative procedure, one can find

“the coarsest mesh” providing approximate trajectories with the required accuracy.

Another issue we have to deal is related to the minimization of the functional (2.3).Its derivatives are ill conditioned when one or more delta-v are near zero. Since the objective

is to find these maneuvers as small as possible, one may expect problems if we perform the

computations in a naive way.

We address these two facts in a two-step methodology. First we find an initial

approximation of the solution minimizing an alternative functional. In a natural way we have

chosen the functional:

J2 =N∑i=1

Mi∑k=0

ρi,k‖Δvi,k‖2, (3.1)

because it is also directly related to fuel consumption and it does not have any ill-conditioning

problems when computing derivatives near zero.

Using this target functional, there are no problems in finding a solution; moreover, the

errors associated to a coarse mesh are not critical for the second step. Let us call FEFF-DV2

the procedure that provides this approximated solution. Starting with FEFF-DV2, we usually

consider a mesh with a small number of elements (generally from 5 to 10) and we take all

of them of the same length. We note that since FEFF-DV2 is an optimization problem, we

need an initial seed. For this purpose we consider each spacecraft alone, this is, we solve N

independent problems, where we compute the trajectory which minimizes J2 without taking

into account possible collision risks.

Using the same discretization as in FEFF formulation, these initial seeds can be found

semianalytically by means of solving a linear system [10, The proof]. The solution is unique

considering that the elements are all of the same length. Moreover, if the obtained trajectories

do not have collision risks (the exclusion spheres do not intersect), they are already the output

of FEFF-DV2 (i.e., the approximate solution for the given mesh that minimizes J2).In the second step of the procedure, we consider an adaptive remeshing strategy with

two purposes: to control the error due to the finite element methodology and to suppress

all the nodes that can give ill-conditioning problems in the minimization of J1. Let us call

FEFF-DV1 the formulation that uses the FEFF methodology to optimize the functional J1 once


Fail

Generate new mesh

Adapted solution

Initial coarse mesh

Generate new mesh

Estimate error

Estimate error

FEFF-DV

FEFF-DV2

Pass(error below the threshold)

Figure 1: Schema of the procedure of adaptive remeshing.

the nodes that could give ill-conditioned problems have been removed. Then the general

idea of the adaptive methodology follows the scheme displayed in Figure 1. Once we have

the approximation given by FEFF-DV2, we start the iterative second step which involves

FEFF-DV1 and an estimation of the error that is used to generate a new mesh. This iterative

procedure is repeated till the estimated error is below the given threshold requirement.

Finally let us comment more in detail how the adaptive remeshing works. The general

idea is that, given a threshold value e, we want to find a mesh that provides an approximate

solution with error (understood as the difference between the solution of the problem and its

approximation inside of an element) less than e in some norm.

Adaptive remeshing methods penalize the elements where the error is considered

big, dividing them into smaller elements. On the other hand, if the estimation of the error

is small in an element, this element is made bigger in the next iteration. Since, as we will

see, our estimation of the error is basically related to the value of the delta-v maneuvers to

be implemented; roughly speaking, this method tends to increase the length of the elements

which have associated small delta-v and tends to decrease the length of the elements which

have associated big delta-v’s. As a consequence, it is also suitable to avoid the ill-conditioned

problems that FEFF-DV1 might have.

Essentially, to decide whether the current mesh is good enough or not, we base on a

criterion which compares the modulus of the estimated error, ||e||, on the mesh with the total

gradient of the solution (related in our problem with velocities). For this purpose we compute

the following integral by means of adding the results obtained in each element. We compute

‖u‖ =∫T

0

v2dt, (3.2)


where in each element we numerically propagate the initial condition at the starting point by

means of the dynamical equations, in order to obtain the velocity function v2 inside the kth

element. Then a Simpson quadrature is employed to compute the integral.

To get an estimation of the error inside a given element, we consider the former v2

velocities inside the element (vk2) and the velocities vk1 obtained taking the derivative of the

approximate solution given by the finite element method as well inside this element (a linear

function in our case). An estimator of the error inside the element is computed by means of

ek =∥∥∥vk1 − vk2

∥∥∥ =

(∫ tk+1

tk

(vk1 − vk2

)·(vk1 − vk2

)dt

)1/2

, (3.3)

where tk and tk+1 are the ends of the kth element. From these values we have an estimation

of the error of the mesh: ||e|| =√e2

1 + e22 + · · · e2

M, and we accept the mesh when

‖e‖ ≤ ν‖u‖, (3.4)

where ν is the acceptability criteria, the threshold parameter control of the adaptive

remeshing procedure that will be discussed and tuned in the examples of Section 4.

In order to compute a new mesh when it is not accepted, we use the Li and Bettess

remeshing strategy (see [13]). This strategy is based on the idea that the error distribution on

an optimal mesh is uniform

‖ek‖ =ν‖u‖√M

, (3.5)

where ν is again the acceptability criteria, ek is the computed error on element k, M is

the number of elements of the mesh, and the hat distinguishes the parameters of the new

mesh to be generated. The strategy consists on finding the new length of the elements using

the number of elements of the new mesh, M. According to Li and Bettess, if d denotes

the dimension of the problem and m the maximum degree of the polynomials used in the

interpolation for the approximate solutions inside an element k, then the number of elements

that should have the new mesh is

M = (ν‖u‖)−d/m(

M∑k=1

‖ek‖d/(m+d/2)

)(m+d/2)/m

. (3.6)

Since we work with linear elements in dimension one, we have m = 1 and d = 1, and the

recommended number of elements of our new mesh is

M = (ν‖u‖)−1

(M∑k=1

‖ek‖2/3

)3/2

. (3.7)


Once we have the estimation of the number of new elements, we can find the length of them

by means of

hk =

(ν‖u‖√M‖ek‖

)1/m+d/2

hk, (3.8)

that, in our case, turns out to be

hk =

(ν‖u‖√M‖ek‖

)3/2

hk. (3.9)

4. Simulations with Adaptive Remeshing

As it has been mentioned, in the following simulations we have located the formation in

the vicinity L2 in the Sun-Earth system. In particular we choose a halo orbit of 120000 km of

z-amplitude.

We have considered two limiting cases. The first one involves no collision risk. It is

known that in this case the optimal solution for each spacecraft is a bang-bang control and

in this section we see that our methodology converges towards it. We note that this is the

most critical case for our procedure, since the optimal maneuver is not a continuous function

but it consists of two impulsive delta-v: one at departure and another one at the arrival

position. The remaining nodal delta-v must be zero, and consequently this is a case where

the computation of derivatives for J1 is very ill conditioned.

The second limiting case corresponds to reconfigurations with collision risks. In this

case the simple bang-bang controls would cause collision between the spacecraft, and the

FEFF methodology solves the problem tending to low thrust solutions when the diameter of

the mesh tends to zero. This is, the methodology can cope with both impulsive and smooth

function controls selecting the optimal one for each case or part of the trajectory.

With the purpose of calibrating the acceptability parameter ν, in this section we present

some simulations with reconfigurations in different situations involving and not involving

collision risks.

4.1. A Bang-Bang Simulation Considering a Single Spacecraft

When the reconfiguration maneuver is not affected by collision risk for one or more spacecraft

of the formation, these satellites follow independent trajectories (i.e., the collision avoidance

constraints in fact will not be active). So we can consider a formation just consisting of a single

spacecraft to exemplify the procedure.

Let us consider in this example a shift of a single spacecraft. The reference frame for

(2.2) is aligned with respect to the RTBP one but with origin on the nominal point of the base

halo orbit (which has been taken of 120000 km in the z-amplitude). When t = 0, this point

on the halo orbit corresponds to the “upper” position, this is, when it crosses the RTBP plane

Y = 0 with Z > 0. The initial condition for this example is taken 100 meters far from the base

nominal halo orbit in the X direction, and the goal is to transfer it to a symmetrical position

with respect to the halo orbit in 8 hours. This is to 100 meters in the opposite X direction


0

0.1

0.2

0 4 8

Time (hours)

v(c

m/

s)

Figure 2: Delta-v obtained in the minimization of the J2 functional (3.1) in a case of no collision risk.

performing a total shift of 200 meters during the transfer maneuver. For this particular case

we know that the optimal solution is a bang-bang control with maneuvers of 0.69 cm/s at

departure and arrival.

Our procedure starts with FEFF-DV2 minimizing the J2 functional (3.1) and obtaining

a trajectory with the delta-v profile of Figure 2. In fact, as we have discussed previously, since

there are no collision risks, this optimal solution corresponds to the initial seed we provide

for FEFF-DV2. Moreover, if we do not take into account the magnitude of the maneuvers for

this particular example, the delta-v profile displayed in Figure 2 is the usual one we find in

similar situations. As expected, since it does not correspond to the optimal solution of the J1

functional (2.3), it is not a bang-bang control.

Using this solution as the initial seed, we start the second part of the procedure corres-

ponding to the iterative part in the schema of Figure 1. It involves FEFF-DV, the estimation of

the error for the current mesh and the generation of the new one. In Figure 3 we can see the

delta-v profile that we obtain after the iterations one and three; this last one is already very

close to the bang-bang control (the method converges after 5 iterations).Of course in a real situation, and specially for small thrusters, the maximum value of

delta-v may be constrained. In Figure 4 we show the delta-v profile for the same simulation

but constraining its maximum value. Values of 0.4 cm/s and 0.3 cm/s have been chosen. We

see how the methodology splits the optimal impulsive bang-bang control of about 0.69 cm/s

in longer arcs at the end points of the trajectory.

4.1.1. Considerations and Calibration of the ν Parameter

Let us consider now the impact of the parameter ν in the performance of the procedure. We

note that this parameter does not only appear in the acceptability criteria (3.4), but it is also

used to obtain the new mesh in (3.9).Intuitively one can expect that if we take a small value of ν, we could end up with a

mesh with a big number of nodes, which turns into an optimization problem with a very large

number of variables. If we take into account that to a mesh of 100 elements we have associated

an optimization problem of 594 variables, we could end up with unsolvable problems in


0

0.3

0.6

0 4 8

Time (hours)

v(c

m/

s)

(a)

0

0.3

0.6

0 4 8

Time (hours)

v(c

m/

s)

(b)

Figure 3: Delta-v profile obtained with the minimization of the J1 functional (2.3) in a case without collisionrisk. (a) we have the result after the first iteration and (b) we show the result after three iterations.

0

0.3

0.6

0 4 8

Time (hours)

v(c

m/

s)

(a)

0

0.3

0.6

0 4 8

Time (hours)

v(c

m/

s)

(b)

Figure 4: Delta-v profile obtained with the minimization of the J1 functional (2.3) for a case withoutcollision risk and constraining the maximum delta-v allowed. (a) we have imposed a maximum delta-v of 0.4 cm/s and (b) a maximum value of 0.3 cm/s.

practice. On the other way round, if we use a big ν, we could end up accepting some meshes

with big errors. In Table 1, we have a summary of the results obtained for different values of

the parameter ν, the number of iterations needed to reach the bang-bang solution, and the

number of elements after the first iteration of the methodology.

We note that when ν is very small, convergence can fail. The case with ν equal to

0.0001 makes the optimal procedure awkward. When ν is 0.001, the final number of elements

is greater than 1 (that we know is our final target number) although the maneuvers associated

to the central nodes are very small. Moreover, when ν is big, there is no convergence: the final

mesh contains more elements than expected because it passes the acceptability criteria before

converging to the bang-bang control. For this bang-bang case, we can conclude that the best

values for ν are inside the range [0.04, 0.06].


Table 1: Number of iterations necessary to obtain the bang-bang solution of the first example dependingon the parameter ν. We have indicated by “fail” the cases where the procedure does not converge and inM(1) the number of elements in the first iteration.

ν 0.0001 0.001 0.002 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07

M(1) 3008 301 149 61 31 15 11 7 6 4 4

Iter Fail Fail 25 16 14 10 6 4 4 2 Fail

(a)

0

0

80

80

0

80

12

34

5

−80

−80

−80

Z(m

)

Y (m)X (m)

(b)

Figure 5: Example of reconfiguration with collision risk: the switch of two pairs of spacecraft of the TPFformation. (a) shows the schema of the reconfiguration and (b) is a solution of the reconfiguration obtainedwith the FEFF methodology.

4.2. A Simulation Using the TPF Formation

For this case we consider a configuration based on the Terrestrial Planet Finder (TPF) model

(see [1]). We assume that the satellites are initially contained in the local plane Z = 0, with

the interferometry baseline aligned on the X axis. The length of the baseline is 150 meters.

We simulate the swap between two pairs of satellites in the baseline: each inner satellite

changes its location with the outer satellite which is closest to it in position (inner satellites are

maneuvered to attain outer positions and vice versa as shown in Figure 5). Again we consider

8 hours for the reconfiguration maneuver. The process of swapping positions is affected of

collision risk for any radius of the sphere of influence, and simple bang-bang controls are no

longer valid. We have considered a sphere of radius 10 meters, and the FEFF methodology

obtains solutions of the form displayed in Figure 5. We note that the convergence of the

methodology does not depend on the radius of the sphere. The reconfiguration cost increases

with the radius. The final number of elements also increases with the radius.

A discussion for the parameter ν similar to the one in the previous example is also

valid here: using a small ν, we can end up with a mesh with too many elements. For example,

taking ν = 0.0005, in the first iteration we have around 1000 elements and we do not have only

the problem of having very small elements. The optimization problem has 29970 variables

something that it is not desirable at all. Also, if we take a big ν, we end up with a mesh with

very few elements and a big truncation error.

In Table 2 we display a summary of the results obtained for different values of the

parameter ν including the number of iterations till the methodology converges (Iter), and the

number of elements in the first and last iterations, M(1) and M(F). Due to symmetry reasons,

the number of elements in each spacecraft trajectory is the same. We note that now the best

values are inside the range [0.005, 0.05] and that the value ν = 0.05 is appropriated for the two


Table 2: Number of iterations and elements involved in the swapping example of TPF depending on theparameter ν. M(1) refers to the number of elements in the first iteration and M(F) to the final ones.

ν 0.0001 0.001 0.002 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07

M(1) 3504 350 175 70 34 18 12 9 6 6 5

Iter Fail 10 8 8 7 6 4 3 3 3 2

M(F) 232 202 171 89 45 33 27 15 9 7

Figure 6: Example of reconfiguration with mixed bang-bang and collision risk: the swap of a pair ofspacecraft and a shift.

cases studied. Since the shift case and the swapping case are someway the building blocks

of the reconfiguration maneuver, we could suggest that ν = 0.05 is a good value for the

computation of reconfiguration maneuvers by means of adaptive remeshing.

4.3. Mixed Case Simulation: 3 Spacecraft

We consider in this section a case that would demand both a bang-bang and low thrust arc in

the reconfiguration maneuver. The formation consists of 3 spacecraft which are in the same

plane as shown in Figure 6. The reconfiguration is the swap of two spacecraft and the shift of

the third one. If we perform the transfers sequentially in time, the maneuver decouples into

two independent problems like the ones considered in the previous examples (bang-bang

plus swap). However, we are going to consider all the transfers in parallel in the same time

interval, this is, with a collision risk of the three satellites in the center of the formation.

Again, we have a similar discussion about the parameter ν and the simulation results

are summarized in Table 3. In this case, the values of ν that are good for our purposes are

inside the range [0.002, 0.05].

4.4. Summary of Considerations about the Value of ν

In the previous sections we have seen that a desirable value for the adaptive remeshing

control parameter ν should be in the range [0.005, 0.05]. This range already gives us an idea


Table 3: Number of iterations needed to obtain the mixed solution depending on ν.

ν 0.0001 0.001 0.002 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07

M(1) 3305 331 165 66 33 17 11 8 7 6 5

Iter Fail Fail 36 27 21 16 12 6 5 Fail Fail

Table 4: Mean value of the number of iterations as a function of ν for the 25 test bench reconfigurationsconsidered in the simulation.

ν 0.005 0.01 0.02 0.03 0.04 0.05 0.055 0.06

Iter. 10.2 8.4 7.1 4.2 3.7 3.2 4.3 5.2

of the value of ν that we should choose when using adaptive remeshing for the computation

of reconfiguration maneuvers by means of FEFF.

We have applied the reconfiguration procedure to a test bench of 25 reconfigurations

which include swaps between spacecraft located at opposite vertices of polygons (6 cases),swaps in the TPF formation (9 cases), and parallel shifts (10 cases). Different sizes and

number of spacecraft, from 3 to 10, have been considered. Ten of the reconfigurations would

be converging to a bang-bang solution while the other 15 would converge to low-thrust arcs

when the diameter of the mesh tends to zero. We have considered our methodology taking

different values of ν, and we have computed the mean of the number of iterations of the

adaptive process necessary to converge. The obtained results are summarized in Table 4 and

point again to the value of ν = 0.05 as a convenient parameter for this kind of proximity

maneuver computations.

5. Conclusions

This paper presents an adaptive remeshing strategy applied to a methodology to find

trajectories for reconfigurations of spacecraft formations. The procedure adapts the mesh

in a systematic and optimal way, and a suitable value for the parameter controlling the

procedure has been found. Moreover, the methodology is robust in all the ranges of possible

reconfiguration cases: from the ones that should result in a bang-bang control to the ones that

should be performed with low thrust arcs.

Appendix

Let us consider the usual restricted three-body problem (RTBP) in synodic coordinates, where

the unit of mass and length is taken such as the sum of the masses of the primaries and the

distance between the primaries is equal to 1, and the unit of time is taken such as the period

of the primaries is equal to 2π . In this synodic coordinate system, the origin is located on the

center of mass and the x axis is defined by the line of the two primaries, from the smallest

primary to the larger one. The z axis is normal to the rotation plane, in the direction of the

angular momentum of the primaries, and the y axis is chosen orthogonal to the previous ones

in order to have a positively orientated coordinate system.


Using this reference frame, the equations of motion of the RTBP are

x − 2y =∂Ω∂x

, y + 2x =∂Ω∂y

, z =∂Ω∂z

, (A.1)

where Ω(x, y, z) = (x2 + y2)/2 + (1 − μ)/r1 + μ/r2 + (1 − μ)μ/2, μ is the mass of the small

primary, and r1 and r2 are the distances from the spacecraft to the big and small primaries

respectively.

Writing (A.1) as a system of first order equations, x = f(x), we have that x =(x1, x2, x3, x4, x5, x6) is the state vector (x, y, z, x, y, z), and f is given by

f1(x) = x,

f2(y) = y,

f3(z) = z,

f4(x) = 2y +∂Ω∂x

,

f5(x) = −2x +∂Ω∂y

,

f6(x) =∂Ω∂z

.

(A.2)

The reference system we consider in this paper for (2.1) is parallel to the one of the

RTBP but with origin in a halo orbit; that is, it is a time-dependent translation of the synodic

RTBP one given by

X(t) = x(t) − xh(t), (A.3)

where xh(t) is the current state on the chosen halo orbit.

Linearizing then x = f(x) about the halo orbit, we have X + xh = f(xh) +Df(xh)X, and

since xh(t) is a solution of x = f(x), we obtain X = Df(xh)X, which defines A(t) = Df(xh(t))in (2.1).

Acknowledgments

This research has been supported by the MICINN-FEDER Grant, MTM2009-06973, and the

CUR-DIUE Grant 2009SGR859.

References

[1] The TPF Science Working Group, The Terrestrial Planet Finder: A NASAOrigins Program to Search for Ha-bitable Planets, JPL Publication, 1999.

[2] The Science and Technology Team of Darwin and Alcatel Study Team, Darwin. The Infrared Space Interf-erometer. Concept and Feasibility Study Report, ESA-SCI 12, European Space Agency, 2000.


[3] K. C. Gendreau, N. White, S. Owens, W. Cash, A. Shipley, and M. Joy, “The MAXIM X-ray inter-ferometry mission concept study,” in Proceedings of the 36th Liege International Astrophysics Colloquium,pp. 11–16, Liege, Belgium, July, 2001.

[4] A. N. Parmar, G. Hasinger, M. Arnaud, X. Barcons, D. Barret, and A. Blanchard, “XEUS—the X-rayevolving universe spectroscopy mission,” in X-Ray and Gamma-Ray Elescopes and Instruments for Astro-nomy, pp. 304–313, 2003.

[5] M. Farrar, M. Thein, and D. C. Folta, “A comparative analysis of control techniques for formation fly-ing spacecraft in an earth/moon-sun L2-centered lissajous orbit,” AIAA Paper 2008-7358, 2008.

[6] A. Badawy and C. R. McInnes, “On-orbit assembly using superquadric potential fields,” Journal ofGuidance, Control, and Dynamics, vol. 31, no. 1, pp. 30–43, 2008.

[7] P. K. C. Wang and F. Y. Hadaegh, “Minimum-fuel formation reconfiguration of multiple free-flyingspacecraft,” Journal of the Astronautical Sciences, vol. 47, no. 1-2, pp. 77–102, 1999.

[8] C. Y. Xia, P. K. C. Wang, and F. Y. Hadaegh, “Optimal formation reconfiguration of multiple spacecraftwith docking and undocking capability,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 3, pp.694–702, 2007.

[9] L. Garcia-Taberner and J. J. Masdemont, “Maneuvering spacecraft formations using a dynamicallyadapted finite element methodology,” Journal of Guidance, Control, and Dynamics, vol. 32, no. 5, pp.1585–1597, 2009.

[10] L. Garcia-Taberner, Proximity maneuvering of libration point orbit formations using adapted finite elementmethods, Ph.D. dissertation, Universitat Politecnica de Catalunya, Barcelona, Spain, 2010.

[11] L. Garcia-Taberner and J. J. Masdemont, “FEFF methodology for spacecraft formations reconfigura-tion in the vicinity of libration points,” Acta Astronautica, vol. 67, no. 7-8, pp. 810–817, 2010.

[12] M. Vasile, “Finite elements in time: a direct transcription method for optimal control problems,” AIAAPaper 2010-8275, 2010.

[13] L.-Y. Li, P. Bettess, J. W. Bull, T. Bond, and I. Applegarth, “Theoretical formulations for adaptive finiteelement computations,” Communications in Numerical Methods in Engineering, vol. 11, no. 10, pp. 857–868, 1995.


Research ArticleClosed Relative Trajectories for Formation Flyingwith Single-Input Control

Anna Guerman,1 Michael Ovchinnikov,2 Georgi Smirnov,3and Sergey Trofimov4

1 Centre for Aerospace Science and Technologies, Department of Electromechanical Engineering, Universityof Beira Interior, Calcada Fonte do Lameiro, 6201-001 Covilha, Portugal

2 Orientation and Motion Control Department, Keldysh Institute of Applied Mathematics, Miusskaya pl. 4,Moscow 125047, Russia

3 Centre of Physics, Department of Mathematics and Applications, School of Sciences, University of Minho,Campus de Gualtar, 4710-057 Braga, Portugal

4 Department of Control and AppliedMathematics, Moscow Institute of Physics and Technology, Institutskijper. 9, Dolgoprudny, Moscow 141700, Russia

Correspondence should be addressed to Georgi Smirnov, [email protected]

Received 13 July 2011; Revised 11 November 2011; Accepted 15 November 2011


Copyright q 2012 Anna Guerman et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study the problem of formation shape control under the constraints on the thrust direction.Formations composed of small satellites are usually subject to serious limitations for powerconsumption, mass, and volume of the attitude and orbit control system (AOCS). If the purposeof the formation flying mission does not require precise tracking of a given relative trajectory,AOCS of satellites may be substantially simplified; however, the capacity of AOCS to ensure abounded or even periodic relative motion has to be studied first. We consider a formation of twosatellites; the deputy one is equipped with a passive attitude control system that provides one-axis stabilization and a propulsion system that consists of one or two thrusters oriented alongthe stabilized axis. The relative motion of the satellites is modeled by the Schweighart-Sedwicklinear equations taking into account the effect of J2 perturbations. We prove that both in the caseof passive magnetic attitude stabilization and spin stabilization for all initial relative positions andvelocities of satellites there exists a control guaranteeing their periodic relative motion.

1. Introduction

Nowadays, design of formation flying missions is one of the main directions of modern

space system development. Many studies have been carried out, and a number of books on

dynamics of such distributed systems have been published (see, e.g., [1, 2]).


One of the main problems to be solved in design of a formation flying mission is that of

maintenance of the required spatial configuration of satellites. The straightforward approach

is to correct orbits using one or several thrusters. Usually no constraints are imposed on the

thrust direction. However, for a nano- or picosatellite formation subject to severe restrictions

on mass, volume, and energy resources, the number of thrusters is limited, and the available

control systems rarely provide three-axis orientation. Therefore, the thrust direction cannot

be arbitrary changed.

Consider a two-satellite formation the aim of which is to perform measurements or

observations, at several points of the orbit. Suppose that the deputy satellite is equipped

with a propulsion system with its thrust axis fixed in the body of satellite. The thrust can

be directed in both ways or in only one, depending on the propulsion system employed.

(As the simplest example of such a system, one can suggest a cold gas thruster.) A number

of simple and lightweight attitude control systems are available that can stabilize motion

of the thrust axis. Thus one can formulate the problem of orbital control assuming the

thrust axis orientation to be known at any moment in time. In control theory, the above

control is referred to as single-input control. The principal question is whether the above

AOCSs suffice to provide the required formation shape at least at some points of the

orbit.

The cases of successively implemented single-input control are known since the

early days of space exploration. One of them occurred by accident as a result of hull

depressurization during one of the first Veneras, Soviet Venus probe missions. The spacecraft

was spinning in a sun-oriented mode, and so the average jet force of the leaking air happened

to be directed towards the Sun, resulting unexpectedly in the proper orbital correction

[3].Development of modern miniature satellites, such as Cubesats, motivates research

on single-input control to simplify satellite control system. Applications of the single-input

control concept to the problem of formation maintenance have been considered for several

missions. For example, the microsatellite Magion-2 launched in 1989 was equipped with a

passive one-axis magnetic attitude control system and a propulsion system with a thrust

vector along the oriented axis. The aim was to keep it at 10 km distance from the chief satellite

[4]; however, due to the thruster failure, formation maintenance was not possible.

Much research is focused on compensation of the relative drift of satellites caused by

the J2 harmonic of the Earth’s gravitational potential. In [5] this problem is studied assuming

the deputy satellite to be equipped with a passive magnetic attitude control system and two

thrusters installed along the axis of the magnet. The use of solar radiation pressure to solve

this problem is studied in [6] (see also [7]).Another approach to the decoupling of the attitude and orbital control in formation

is presented in [8]. The authors interpret a formation as a quasirigid body. It is shown that

control of such a formation can be effectively separated into a control torque that maintains

the attitude and control forces that maintain the rigidity of a formation. The respective control

strategy is based on the Lyapunov controller synthesis [9].In this paper, we analyze the general problem of compensation of J2 perturbations

for the deputy satellite in two-satellite formation. The chief satellite is assumed to move

passively. We study the Schweighart-Sedwick linear equations [10], that is, the modification

of the Hill-Clohessy-Wiltshire equations of relative motion. This modification well describes

the effect of J2 perturbations and has been successfully used to study many problems of

relative dynamics, such as formation keeping and rendezvous (see, e.g., [11–14]).


We consider two different types of single-input control:

(1) bilateral control oriented along a vector fixed in the inertial space (the case of spin

stabilization);

(2) bilateral and unilateral control oriented along the vector of local geomagnetic field

(the case of passive magnetic stabilization).

We prove that for any initial conditions there exists a control that provides a periodic

relative motion of chief and deputy satellites with a period T between 1 and 2 orbital periods.

This means that the maximum distance between satellites does not become very large.

Though the shape of relative trajectory is not controlled, the existence of bounded short-

period relative motion suffices to perform the required measurements in many nano- and

picosatellite formation missions.

Throughout this paper, the set of real numbers is denoted by R and theN-dimensional

space of vectors with components in R by RN . We denote by 〈a, b〉 the usual scalar product

in RN and by ‖ · ‖ the Euclidean norm. The transposition of a matrix A is denoted by AT .

2. Existence and Stabilization of Closed Trajectories fora Single-Input Control System

Consider a linear single-input control system

η(t) = Aη(t) + a(t) +w(t)b(t), η(t) ∈ Rn, w(t) ∈ R, (2.1)

where A is a (N ×N)-matrix, a : R → RN and b : R → RN are given continuous functions,

and w(t) is a control. The control w(t) may be subjected to the constraint

w(t) ≥ 0. (2.2)

The set of admissible controls w(·) is denoted by W and consists of locally integrable

functions. The general solution to (2.1) is given by the Cauchy formula

η(t) = etAη0 +∫ t

0

e(t−s)A(a(s) +w(s)b(s))ds. (2.3)

We say that system (2.1) has a T -closed trajectory η(·) satisfying η(0) = η0, if and only if there

exists an admissible control wη0(·) such that

η0 = eTAη0 +∫T

0

e(T−t)A(a(t) +wη0

(t)b(t))dt. (2.4)

Put

KT =

{∫T0

e(T−t)Aw(t)b(t)dt | w(·) ∈ W}. (2.5)


If w(t) ∈ R, then KT is a subspace. In the case w(t) ≥ 0, the set KT is a convex cone. If

KT = RN , then for any η(0) = η0 there exists an admissible control wη0(·) satisfying (2.4).

Moreover, for any initial point η1 and any terminal point η2 there exists an admissible control

wη1,η2such that

η2 = eTAη1 +∫T

0

e(T−t)A(a(t) +wη1,η2

(t)b(t))dt, (2.6)

that is, the system is controllable.

The established controllability allows one to correct closed trajectories, that is, if there

is a deviation in the initial condition of the closed trajectory, it can be compensated for by an

appropriate choice of control.

To verify the controllability condition KT = RN in the case of unconstrained control,

we use the following direct consequence of the Pontryagin Maximum Principle.

Theorem 2.1. Assume that there is no nontrivial solution of the equation

p(t) = −ATp(t) (2.7)

satisfying

〈p(t), b(t)〉 = 0, t ∈ [0, T], (2.8)

then the equalityKT = RN holds.

In the case of controls subject to constraint (2.2), the situation is more involved.

Later on we consider only the τ-periodic functions b(·). This assumption is satisfied in all

applications considered here and significantly simplifies the study.

The following two propositions are well known to the specialists in the control theory.

However, to make the presentation self-contained, we include their short proofs in the

Appendix.

First of all, note that the periodicity condition b(t + τ) = b(t) implies that the cones

KMτ , M = 1, 2, . . ., form a monotonously increasing sequence.

Lemma 2.2. Assume that b(t) is τ-periodic. LetM be a positive integer. Then the inclusion KMτ ⊂K(M+1)τ holds.

The next theorem is also a consequence of the Pontryagin Maximum Principle and

contains sufficient conditions of controllability for system (2.1) when the control satisfies

condition (2.2).

Theorem 2.3. Assume that there is no nontrivial solution to the differential equation

p(t) = −ATp(t) (2.9)


satisfying

〈p(t), b(t)〉 ≥ 0, t ≥ 0. (2.10)

Then the equalityK∞ =⋃MKMτ = RN holds.

Since the sequence of convex cones KMτ is monotonous, the equality⋃M>0 KMτ = RN

implies the existence of a positive integer M such that

KMτ = RN. (2.11)

Indeed, let points ξk, k = 1, . . . ,N+1, be the vertices of a simplex Ξ containing the origin as an

interior point. Then any point ξ ∈ RN can be represented as ξ =∑

k λkξk with λk ≥ 0. For any

k = 1, . . . ,N + 1, there exist a positive integer Mk and an admissible control uk(·) satisfying

ξk =∫Mkτ

0

e(Mτ−t)Ab(t)uk(t)dt. (2.12)

So from Lemma 2.2 we see that any vertex ξk can be represented in the form

ξk =∫Mτ

0

e(Mτ−t)Ab(t)wk(t)dt, (2.13)

where M = max{Mk | k = 1, . . . ,N + 1} and wk(·) is an admissible control. This implies the

equality

ξ =∫Mτ

0

e(Mτ−t)Ab(t)∑k

λkwk(t)dt, (2.14)

arriving at (2.11).Let η0 ∈ RN . Condition (2.11) leads to the existence of a control w0(·) such that

η0 − eMτAη0 −∫Mτ

0

e(Mτ−t)Aa(t)dt =∫Mτ

0

e(Mτ−t)Ab(t)w0(t)dt. (2.15)

Therefore, the control w0(·) corresponds to a closed trajectory of (2.1) satisfying η(0) = η0.

The above results permit one also to compensate for the errors caused by the model or

measurements not requiring considerable computational efforts. Under condition (2.11) it is

easy to develop an algorithm that reaches the point η0 even if the initial point η′0 is different

from η0. Note that this algorithm does not require solving the integral equation (2.6).


Consider a simplex Σ containing η0 in its interior. Let {η1, . . . , ηN+1} be the vertices of Σ.

Condition (2.11) implies the existence of admissible controls wk(·), k = 1, . . . ,N+1, satisfying

the equalities

η0 − eMτAηk −∫Mτ

0

e(Mτ−t)Aa(t)dt =∫Mτ

0

e(Mτ−t)Ab(t)wk(t)dt, k = 1, . . . ,N + 1. (2.16)

If η′0 ∈ Σ, there exist nonnegative numbers λk, k = 1, . . . ,N + 1, such that

η′0 =N+1∑k=1

λkηk,N+1∑k=1

λk = 1, (2.17)

and so the control

w(t) =N+1∑k=1

λkwk(t) (2.18)

drives system (2.1) to the point η0. Thus, if the controls wk(·), k = 1, . . . ,N + 1, are known, it

suffices to find nonnegative numbers λk, k = 1, . . . ,N + 1, satisfying (2.17) in order to reach

the point η0 from η′0.

3. Equations of Relative Motion with Single-Input Control

To take into account the influence of the J2-harmonic on relative motion of two satellites with

close near-circular orbits, the following modification of the Hill-Clohessy-Wiltshire equations

has been introduced by Schweighart and Sedwick [10]:

x + 2ncz = w(t)ex(t),

y + q2y = 2lq cos(qt + φ

)+w(t)ey(t),

z − 2ncx −(

5c2 − 2)n2z = w(t)ez(t).

(3.1)

The linearization is done with respect to the circular reference orbit with the mean motion n.

Here x, y, and z are coordinates in the respective orbital reference frame Oxyz. The axes are

chosen in the following way: Oz indicates the radial direction outwards from the Earth, Ox is

directed along the velocity of the pointO, and y is normal to the orbital plane. The coefficients

c, q, l, and φ are properly defined constants (see the appendix, Proof of Lemma 3.1).The direction of the control acceleration w(t) is defined by the vector function

e(t) =(ex(t), ey(t), ez(t)

)T. (3.2)


Using the notations

η =(x, y, z, x, y, z

)T,

a(t) =(0, 0, 0, 0, 2lq cos

(qt + φ

), 0)T,

b(t) =(0, 0, 0, ex(t), ey(t), ez(t)

)T,

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 0 −2nc

0 −q2 0 0 0 0

0 0(5c2 − 2

)n2 2nc 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠,

(3.3)

we obtain a system of type (2.1). System (2.7) that describes the evolution of the vector p =(p1, p2, p3, p4, p5, p6)

T takes the form

p1 = 0,

p2 = q2p5,

p3 = −(

5c2 − 2)n2p6,

p4 = −p1 − 2ncp6,

p5 = −p2,

p6 = −p3 + 2ncp4.

(3.4)

Its general solution is given by

p1(t) = p01,

p2(t) = A2 cos(qt + φ2

),

p3(t) = p03 −A6

(5c2 − 2

)n

√2 − c2

sin(√

2 − c2nt + φ6

)+ 2nc

5c2 − 2

2 − c2p0

1t,

p4(t) =p0

3

2nc−A6

2c√2 − c2

sin(√

2 − c2nt + φ6

)+

5c2 − 2

2 − c2p0

1t,

p5(t) = −A2

qsin(qt + φ2

),

p6(t) = A6 cos(√

2 − c2nt + φ6

)−

2cp01

(2 − c2)n,

(3.5)

where p01, p0

3, A2, A6, φ2, and φ6 are constants. Conditions (2.8) and (2.9) are equivalent to the

conditions

p4(t)ex(t) + p5(t)ey(t) + p6(t)ez(t) = 0, t ∈ [0, T], (3.6)

p4(t)ex(t) + p5(t)ey(t) + p6(t)ez(t) ≥ 0, t ∈ [0, T], (3.7)


respectively. According to Theorem 2.3, to prove the controllability of the Schweighart-

Sedwick system with single-input control it suffices to show that there are no nontrivial

functions (p1(t), . . . , p6(t)) satisfying (3.6) or (3.7).Denote the radius and the inclination of the reference circular orbit by rref and iref,

respectively. Assume that the chief satellite moves passively in an orbit with inclination i1.

The orbit inclination of the deputy satellite is denoted by i2. Set

ω0 = nc, ω1 = q, ω2 =(√

2 − c2 − c)n, ω3 =

(√2 − c2 + c

)n. (3.8)

The following lemma proved in the Appendix is crucial for the analysis of the Schweighart-

Sedwick system controllability.

Lemma 3.1. If 2iref /= arccos(−1/3), then ωj /= 0, j = 0, 1, 2, 3, and ω2 < ω0 < ω1 < ω3.

Below we assume that the main condition of this lemma is satisfied and consider two

systems with single-input control relevant for practical applications.

4. Bilateral Control Oriented along the Geomagnetic Field

Consider first a formation with the deputy satellite equipped with a passive magnetic attitude

control system (PMACS) and has two thrusters installed along its axis of orientation (i.e., axis

of permanent magnet included in PMACS) in opposite directions. We also assume that at any

moment in time this axis coincides with the direction of geomagnetic field described by the

direct dipole model:

ex(t) =cos θ(t) sin i2√

1 + 3 sin2θ(t)sin2i2

,

ey(t) =cos i2√

1 + 3 sin2θ(t)sin2i2

,

ez(t) =−2 sin θ(t) sin i2√1 + 3 sin2θ(t)sin2i2

.

(4.1)

The argument of latitude is given by θ(t) = nct.Under some nonrestrictive conditions the system is controllable in any time interval

[0, T]; for example, one can take T = 2π/(nc).

Theorem 4.1. Let T > 0. If sin 2i2 /= 0, then there exists a T -closed trajectory of system (2.1).Moreover, an error in the initial conditions can be compensated for.

The proof of this theorem can be found in the appendix.


5. Bilateral Control Oriented along a Fixed Vector in the Inertial space

Spin-stabilized satellites represent another interesting possibility for orbit correction by

single-input control. Once rapidly rotated about an axis, the spacecraft keeps spinning around

this direction in the inertial space in the absence of perturbing torques.

Assume that the deputy satellite possesses a spherically symmetrical mass distribu-

tion, is spin stabilized, and has two thrusters oriented in opposite directions along its spin

axis fixed in the inertial space. Suppose that λ is the angle between this axis and the vector

pointing to the vernal equinox direction, and ε is the inclination of the plane containing

these vectors with respect to the Earth’s equator. Then in the Earth-centered inertial reference

frame the spin axis direction has the components (cosλ, sinλ cos ε, sinλ sin ε)T . In the Oxyz

reference frame the expressions are

ex(t) = σz sin θ(t) − σx cos θ(t),

ey = −σy,ez(t) = −σx sin θ(t) − σz cos θ(t).

(5.1)

Here the vector σ = (σx, σy, σz)T defines the direction of spin axis in the ascending node of

the orbit via the inclination i2 and the right ascension Ω2:

σx = cosΩ2 cos i2 sinλ cos ε − sinΩ2 cos i2 cosλ + sin i2 sinλ sin ε,

σy = − cosΩ2 sin i2 sinλ cos ε + sinΩ2 sin i2 cosλ + cos i2 sinλ sin ε,

σz = cosΩ2 cosλ + sinΩ2 sinλ cos ε.

(5.2)

We set θ(t) = nct. As in the case of the satellite oriented along the local geomagnetic field,

under some nonrestrictive conditions the system is controllable in any time interval [0, T],for example, for T = 2π/(nc).

Theorem 5.1. Let T > 0. If σ2x + σ

2z /= 0 and σy /= 0, then there exist a T -closed trajectory of system

(2.1). Moreover, an error in the initial conditions can be compensated for.

See the appendix for the proof.

6. Unilateral Control Oriented along the Geomagnetic Field

Now assume that the control w(t) has to satisfy the nonnegativity condition (2.2). Set τ =2π/(nc). In this case we have the following result.

Theorem 6.1. If sin 2i2 /= 0, there is a positive integer M > 0 and a Mτ-closed trajectory of system(2.1). Moreover, an error in the initial conditions can be compensated for.

The theorem is proved in the Appendix.

Note that a similar result can be proved for the case of the satellite oriented along a

fixed vector in the inertial space. However, this result is of quite limited practical importance.

Indeed, while in the case of magnetic orientation M = 2 (see the numerical example in the

next section), in the case of the satellite oriented along a fixed vector in the inertial space, the


100

300 400 500 600

0

50

200

−150

−100

−50

150

−200

0 100

m

m

m

0100

200300

−100

x

z

y

Figure 1: Closed trajectory of linearized system (LT): bilateral control with magnetic ACS.

value of M is very large, and so is the distance between the chief and deputy satellites. In this

case the linearized equations cease to describe adequately the system dynamics and so the

generated periodic trajectories are of merely academic interest.

7. Numerical Results

The aim of the following numerical simulations is to verify the analytical results listed above

and to compare the trajectories of initial and linearized systems in the presence of the control.

On solving the integral equation (2.4) numerically, we substitute the obtained control into

the Gauss variational equations for the deputy satellite and propagate them in time. For

the passively flying chief satellite the propagation can be done directly. Then, subtracting

one motion from another, we convert the result to the Oxyz reference frame. Only the J2

perturbing effect is taken into account. Indeed, for time intervals of several orbital periods

the influence of atmospheric drag and solar radiation pressure on relative motion of identical

satellites in close orbits is negligible (3 to 4 orders smaller than J2 perturbations).Integral equation (2.4) has many solutions; we use the minimal one in the sense of L2-

norm. This criterion can be interpreted as that of minimal energy consumption for low-thrust

constant-power engines (see, e.g., [15]).Below, we compare the trajectories of the linearized Schweighart-Sedwick system

(LT) and the trajectories obtained by integration of the nonlinear equations of motion (NT).Figure 1 shows a T -closed LT with the numerically obtained bilateral single-input control

oriented along the geomagnetic field. This trajectory has a length T = τ = 2π(nc)−1 and

corresponds to the following initial conditions: x0 = 70.71 m; y0 = 70.71 m; z0 = 35.36 m;

x0 = 76.25 mm/s; y0 = 76.32 mm/s; z0 = −38.07 mm/s. The radius of the circular reference

orbit is rref = 7000 km; the inclination of the chief satellite i1 is the same as the reference

inclination iref = 35 deg. The projections of LT on xy and xz planes are demonstrated in

Figures 2 and 3. As we see, the shape of trajectories is rather complex. Modelling errors of LT

and the corresponding control are shown in Figures 4 and 5, respectively.

Figure 4 shows that the difference between the LT and NT obtained with the same

control is not significant. The difference appears because the in-plane drift is not completely

eliminated. It is caused by the errors of linearization in the Schweighart-Sedwick model.

In the case of free flight, these errors can be compensated for by a proper choice of initial

conditions, which should be done numerically (see [10]). We obtain a similar situation with


0 100 200 300 400 500 600−100

−50

0

50

100

150

200

250

m

m

x

y

Figure 2: Projection on xy plane.

0 100 200 300 400 500 600−200

−150

−100

−50

0

50

100

150

m

m

x

z

Figure 3: Projection on xz plane.

0 1000 2000 3000 4000 5000 6000−4

−2

0

2

4

6

8

10

12

Time s

Err

or

m

x

z

y

Figure 4: Coordinate-wise modelling error.


0 1000 2000 3000 4000 5000 6000−1. 5

−1

−0. 5

0

0. 5

1

1. 5

×10−3

Time (s)

Co

ntr

ol(m

/s2)

Figure 5: Control.

Table 1: Results of simplex experiment.

Number of vertex 1 2 3 4 5 6 7

x0, m 70.71 −141.42 70.71 70.71 70.71 70.71 70.71

y0, m 70.71 70.71 −141.42 70.71 70.71 70.71 70.71

z0, m 35.36 35.36 35.36 −70.71 35.36 35.36 35.36

x0, mm/s 76.25 76.25 76.25 76.25 −152.51 76.25 76.25

y0, mm/s 76.32 76.32 76.32 76.32 76.32 −152.64 76.32

z0, mm/s −38.07 −38.07 −38.07 −38.07 −38.07 −38.07 76.15

T/τ 2 2 2 2 2 2 2

‖w‖1, m/s 0.16 0.16 0.14 0.18 0.08 0.15 0.16

‖w‖22, 10−4 m2/s3 6.83 6.84 5.35 7.93 1.64 5.59 6.84

the controlled flight, but now the control has to be corrected; for example, it can be used as

the first approximation in an iteration procedure (such as Newton’s method) applied to the

Gauss system. A Newton-type method suitable to solve control problems with nonnegativity

constraints can be found in [16]. The control problem for nonlinear system is to be described

in a future paper.

Now proceed with the case of unilateral single-input control oriented along the

geomagnetic field. The construction described in Section 2 is fulfilled numerically. We show

that it is possible to construct a 2τ-closed trajectory for all vertices of a simplex containing the

origin in its interior. Therefore a 2τ-closed trajectory exists for any initial point. The results of

the “simplex” experiment are summarized in Table 1. The last two rows of this table contain

the values of L1-norm

‖w‖1 =∫2τ

0

|w(t)|dt (7.1)

and of squared L2-norm

‖w‖22 =∫2τ

0

|w(t)|2dt, (7.2)


−1500 −1000 −500 0 500 1000 1500−200

0200

400

mm

−500

−400

−300

−200

−1000

100

200

300

400

m

x

z

y

Figure 6: Closed LT: unilateral control with magnetic ACS.

−1500 −1000 −500 0 500 1000 1500−200

−150

−100

−50

0

50

100

150

200

250

300

m

m

x

y

Figure 7: Projection on xy plane.

of the controls w(t) providing closed trajectories for the simplex vertices 1, . . . , 7. 2τ-closed

LT along with their projections on xy and xz planes, the coordinate-wise errors, and the

corresponding nonnegative LT-control are shown in Figures 6, 7, 8, 9, 10. Since the time

interval becomes twice as long, the error of linearization results in larger modelling errors

due to considerable along-track drift.

For the bilateral control oriented along a fixed vector in the inertial space, the results

are qualitatively similar to the case of bilateral magnetic control (see Figures 11, 12, 13). We

use the same initial conditions, and the fixed vector is defined by the following angles: λ =45 deg, ε = 23.45 deg. This choice of ε may correspond to stabilization of the spacecraft axis

in the Sun direction.

8. Conclusions

We consider the problem of formation maintenance under constraints on the thrust vector

directions. The formation consists of two satellites; the deputy satellite is equipped with one

or two thrusters oriented along a given axis. We assume that the orientation of this axis is

kept by an available passive ACS and is known at any instant of time. A possibility to obtain a

periodic relative motion of the chief and deputy satellites is demonstrated for several types of


−1500 −1000 −500 0 500 1000 1500−500

−400

−300

−200

−100

0

100

200

300

400

m

m

x

z

Figure 8: Projection on xz plane.

0 2000 4000 6000 8000 10000 12000−30

−20

−10

0

10

20

30

40

Time s

Err

or

m

x

y

z

Figure 9: Coordinate-wise modeling error.

0 2000 4000 6000 8000 10000 12000−1

0

1

2

3

4

5

6

7

8×10−4

Time (s)

Co

ntr

ol(m

/s2)

Figure 10: Control.


0200

400600

800

−200−100

0100

200

mm

m−200

−150

−100

−500

50

100

150

200

x

z

y

Figure 11: Closed LT: bilateral control along axis fixed in absolute space.

0 1000 2000 3000 4000 5000 6000−8

−6

−4

−2

0

2

4

6

×10−4

Time (s)

Co

ntr

ol(m

/s2)

Figure 12: Control.

0 1000 2000 3000 4000 5000 6000−6

−4

−2

0

2

4

6

8

Time s

x

y

z

Err

or

m

Figure 13: Coordinate-wise modelling error.


single-input control. In each case sufficient controllability conditions are deduced. In general,

these conditions can be formulated as follows: the vector of control direction should have

nonzero components both in the orbital plane and along the normal to the orbit. For the

unilateral control oriented along the geomagnetic field, the existence of a closed trajectory

of relative motion with double period is established for arbitrary initial conditions. A single-

input control numerically obtained for the system of Schweighart-Sedwick equations suffices

to guarantee almost closed trajectories. We also prove that the inaccuracy caused by the errors

of the Schweighart-Sedwick model can be corrected.

Appendix

Proof of Lemma 2.2. Let z ∈ KMτ . Then an admissible control u(·) exists such that

z =∫Mτ

0

e(Mτ−t)Ab(t)u(t)dt. (A.1)

Set

w(s) =

{0, s ∈ [0, τ),u(s − τ), s ∈ [τ, (M + 1)τ].

(A.2)

Then we have

z =∫Mτ

−τe(Mτ−t)Ab(t)w(τ + t)dt

=∫Mτ+τ

0

e(Mτ+τ−s)Ab(s − τ)w(s)ds

=∫ (M+1)τ

0

e((M+1)τ−s)Ab(s)w(s)ds.

(A.3)

Thus we get KMτ ⊂ K(M+1)τ .

Proof of Theorem 2.3. Suppose that K∞ /=RN . From Lemma 2.2 we see that K∞ is a convex

cone. So there exists a vector p∞ /= 0 satisfying 〈x, p∞〉 ≥ 0, for all x ∈ K∞. Therefore, we have

〈x, p∞〉 ≥ 0, for all x ∈ KMτ and any positive integer M. Consider the functions

pM(t) =exp(AT (Mτ − t)

)p∞∥∥exp

(ATMτ

)p∞∥∥ , M = 1, 2, . . . . (A.4)

From the Pontryagin maximum principle we have 〈pM(t), b(t)〉 = 0, t ∈ [0,Mτ], if w(t) ∈ R,

and 〈pM(t), b(t)〉 ≥ 0, t ∈ [0,Mτ], if w(t) ≥ 0. Consider the sequence pM(0). Without loss of

generality it converges to a vector p0 satisfying ‖p0‖ = 1. Thus we have

limM→∞

pM(t) = p0(t) = exp(−ATt

)p0, t ∈ [0,Mτ]. (A.5)


Obviously the solution p0(·) to (2.7) is nontrivial and satisfies (2.8) if w(t) ∈ R, and (2.9) if

w(t) ≥ 0, a contradiction.

Proof of Lemma 3.1. By definition,

q = nc +3nJ2R

2⊕

2r2ref

(cos2i2 −

(cos i1 − cos i2)(cot i1 sin i2 cosΔΩ0 − cos i2)

sin2ΔΩ0 + (cot i1 sin i2 − cos i2 cosΔΩ0)2

),

ΔΩ0 =y0

rref sin iref, y0 = y(0), c =

√1 + s, s =

3J2R2⊕

8r2ref

(1 + 3 cos 2iref),

(A.6)

where R⊕ is the Earth’s radius, J2 ≈ 10−3 is the second zonal harmonic. Since the orbits of

satellites are close, the difference i2 − i1 is small. Taking into account sin 2i1 /= 0 and sin 2i2 /= 0,

we have

q ≈ nc + 3nJ2R2⊕

2r2ref

cos2i2. (A.7)

Since 2iref /= arccos(−1/3), one can see that c /= 1. At the same time |c − 1| � 1. Thus, all the

frequencies ωj, j = 0, 1, 2, 3, are nonzero and pairwise different: ω2 < ω0 < ω1 < ω3.

Proof of Theorem 4.1. Indeed, in this case condition (3.6) reads

κ(t) = gt cosω0t +3∑k=0

(gk cosωkt + hk sinωkt

)≡ 0. (A.8)

From Lemma 3.1 we see that all the frequencies are pairwise different, and therefore the

coefficients of the quasipolynomial κ(t) equal zero. Since

g =5c2 − 2

2 − c2p0

1 sin i2,

g0 =p0

3

2ncsin i2,

h0 =4cp0

1

(2 − c2)nsin i2,

g1 = −A2

qcosφ2 cos i2,

h1 = −A2

qsinφ2 cos i2,


g2 = A6

(1 − c√

2 − c2

)sin i2 sinφ6,

h2 = A6

(1 − c√

2 − c2

)sin i2 cosφ6,

g3 = −A6

(1 +

c√2 − c2

)sin i2 sinφ6,

h3 = −A6

(1 +

c√2 − c2

)sin i2 cosφ6

(A.9)

from the condition sin 2i2 /= 0, we obtain p(t) ≡ 0. Hence we have KT = RN , T > 0. This implies

the existence of a T -closed trajectory for any initial point.

Proof of Theorem 5.1. The proof is almost identical to that of the previous theorem. Indeed,

condition (3.6) of Theorem 2.1 takes the form

κ(t) = gt cosω0t + ht sinω0t +3∑k=0


)≡ 0. (A.10)

As in the previous proof, one can apply Lemma 3.1 and see that all the frequencies are

pairwise different. Consequently the coefficients of the quasipolynomial κ(t) equal zero. Since

g = −5c2 − 2

2 − c2p0

1σx,

h =5c2 − 2

2 − c2p0

1σz,

g0 = −p0

3

2ncσx +

2cp01

(2 − c2)nσz,

h0 =p0

3

2ncσz +

2cp01

(2 − c2)nσx,

g1 =A2

qσy sinφ2,

h1 =A2

qσy cosφ2,

g2 = −A6

(1

2+

c√2 − c2

)(σz cosφ6 − σx sinφ6

),

h2 = A6

(1

2+

c√2 − c2

)(σx cosφ6 + σz sinφ6

),

g3 = −A6

(1

2− c√

2 − c2

)(σz cosφ6 + σx sinφ6

),

h3 = A6

(1

2− c√

2 − c2

)(σz sinφ6 − σx cosφ6

),

(A.11)


from the conditions σ2x +σ

2z /= 0, σy /= 0, we obtain p(t) ≡ 0. Thus we have KT = RN , T > 0. This

implies the existence of a T -closed trajectory for any initial point.

Proof of Theorem 6.1. Assume that⋃MKMτ /=RN . Then, by Theorem 2.3, there exists a

nontrivial solution to (2.7) satisfying

p4(t)ex(t) + p5(t)ey(t) + p6(t)ez(t) ≥ 0, t ∈ [0,∞). (A.12)

Condition (A.12) takes the form

κ(t) = gt cosω0t +3∑k=0


)≥ 0, t ∈ [0,∞), (A.13)

where the coefficients are those defined in the proof of Theorem 4.1. Now we show that the

coefficients of the quasipolynomial κ(t) are equal to zero. Indeed, we have

0 ≤ κ(t)t

= g cosω0t +O(

1

t

), t −→ ∞. (A.14)

So g = 0, and we obtain

κ(t) =3∑k=0


)≥ 0. (A.15)

From Lemma 3.1 we have ω2 < ω0 < ω1 < ω3 and ωj /= 0, j = 0, . . . , 3. Multiplying κ(t) by

1 ± cosωjt, j = 0, 1, 2, 3, we get

0 ≤ limτ→∞

1

T

∫T0

κ(t)(1 ± cosωjt

)dt = ±

gj

2. (A.16)

Multiplying κ(t) by 1 ± sinωjt, j = 0, 1, 2, 3, we obtain

0 ≤ limτ→∞

1

T

∫T0

κ(t)(1 ± sinωjt

)dt = ±

hj

2. (A.17)

Therefore all of the coefficients of κ(t) are equal to zero. As in the proof of Theorem 4.1, we

have p(t) ≡ 0, a contradiction.

Acknowledgments

The authors are grateful to Arun Misra for the bibliographical support and to the

reviewers for their comments and valuable suggestions. This research is supported by the

Portuguese Foundation for Science and Technologies (FCT), the Portuguese Operational

Programme for Competitiveness Factors (COMPETE), the Portuguese Strategic Reference


Framework (QREN), the European Regional Development Fund (FEDER), and by the

Russian Foundation for Basic Research (RFBR), Grant 09-01-00431.

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pp. 1219–1230, 2012.


Research ArticleModified Chebyshev-Picard Iteration Methods forStation-Keeping of Translunar Halo Orbits

Xiaoli Bai and John L. Junkins

Department of Aerospace Engineering, Texas A&M University, TAMU 3141, College Station,TX 77843-3141, USA

Correspondence should be addressed to Xiaoli Bai, [email protected]

Received 15 July 2011; Revised 7 November 2011; Accepted 14 November 2011


Copyright q 2012 X. Bai and J. L. Junkins. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The halo orbits around the Earth-Moon L2 libration point provide a great candidate orbit for a lunarcommunication satellite, where the satellite remains above the horizon on the far side of the Moonbeing visible from the Earth at all times. Such orbits are generally unstable, and station-keepingstrategies are required to control the satellite to remain close to the reference orbit. A recentlydeveloped Modified Chebyshev-Picard Iteration method is used to compute corrective maneuversat discrete time intervals for station-keeping of halo orbit satellite, and several key parametersaffecting the mission performance are analyzed through numerical simulations. Compared withpreviously published results, the presented method provides a computationally efficient station-keeping approach which has a simple control structure that does not require weight turning and,most importantly, does not need state transition matrix or gradient information computation. Theperformance of the presented approach is shown to be comparable with published methods.

1. Introduction

For the spatial circular restricted three-body problem (CR3BP), where the two large bodies

move in planar circular orbits about their center of mass and a third body of negligible

mass moves under only the 1/r2 gravitational influence of the large bodies, there exist five

stationary points in the rotating reference frame. These points, called the Lagrangian points or

libration points, have unique roles for many scientific missions because they lie in the plane of

the primary bodies’ motion and have fixed positions with respect to these two bodies, similar

to the geostationary orbits although a little more complicated. Among the five points, three

of them are collinear with the other two bodies, and the motion near these points is unstable;

the other two libration points with the primaries form equilateral triangles in the plane of

motion of the two large bodies and the motions near these two points are neutrally stable.


There has been a lot of interest in finding and controlling the periodic orbits near the

collinear libration points. Farquhar originally proposed using the orbits near the L2 point of

the Earth-Moon system for communication with the far side of the Moon [1]. By appropriate

station-keeping and with a modest cost, the satellite is visible from the Earth all the time.

However, such orbits are not generally periodic. Later, Farquhar and Kamel found that if

the amplitude of the in-plane motion is large enough so that the nonlinear effects become

significant, purely periodic three dimensional orbits, or halo orbits, exist that are permanently

visible from the Earth [2]. The interest in the Earth-Moon halo orbits later shifted to the Earth-

Sun system following the end of the Apollo missions and the launch of International Sun-

Earth Explore-3 (ISEE3) which is the first libration-point satellite [3].The techniques for station-keeping of halo orbits can be classified as (1) using

continuous control, or (2) generating impulses at discrete time intervals. Breakwell et al.

studied station-keeping for translunar communication station using a continuous feedback

control to minimize a cost function, which is a weighted combination of position deviation

from the nominal orbit and the control acceleration based on a linearization about the

reference orbit [4]. Xin et al. developed a suboptimal closed-form continuous feedback

controller by approximately solving a Hamiltonian-Jacobian-Bellman equation [5]. Kulkarni

et al. applied the H∞ approach for station-keeping of the halo orbit around the L1 of the

Earth-Sun system [6]. Cielaszyk and Wie modeled the nonlinearities of the halo orbit problem

as persistent disturbances and applied disturbance accommodation with a linear quadratic

regulator for the control [7]. Although using continuous control derived from optimal control

theory usually generates orbits closer to their reference orbits than using impulses, discrete

maneuvers have the advantages of less complexity and risk and higher precision orbit

estimation can be obtained during coasting arcs. Dunham and Roberts described the station-

keeping strategies for three Earth-Sun libration point missions, all of which used discrete

controls [8]. Two station-keeping techniques for the Earth-Moon libration point orbits were

studied by Gomez et al. [9]. One is what they called the target point strategy where a cost

function, which is a weighted summation of the position and velocity deviations from several

future points and the fuel cost, is minimized to find corrective maneuvers. Another approach

they studied is based on Floquet methods which combine invariant manifold theory and

Floquet modes to only eliminate the unstable component of the error. Howell and Pernicka

also studied using the target point strategy for the Earth-Sun system in a later paper [10]. The

optimal spacing time between impulsively applied controls has been studied by Renault and

Scheeres [11], and the optimal time to update control law with continuous control has been

recently studied by Gustafson and Scheeres [12].A newly developed numerical computation approach, Modified Chebyshev-Picard

Iteration (MCPI) method, is used in this paper for station-keeping to maintain a halo

orbit in the Earth-Moon system. This orbit provides an excellent parking orbit for a lunar

communication satellite. Details about MCPI methods can be found in the papers by Bai

and Junkins [13, 14] and Bai’s dissertation [15]. Fusing Chebyshev polynomials with the

classical Picard iteration method, MCPI methods iteratively refine an orthogonal function

approximation of the entire state trajectory. MCPI methods can solve both initial value

problems (IVPs) and two-point boundary value problems (BVPs) by constraining the coeffi-

cients of the Chebyshev polynomials. A unique characteristic of MCPI is that the Chebyshev

coefficients are constrained linearly without approximation on each Picard iteration. As a

consequence, shooting techniques to impose boundary conditions are not required. Although

perhaps the most striking feature about MCPI methods is their naturally parallel structure

because computation of the integrand along each path iteration can be rigorously distributed


over many parallel cores with negligible cross communication needed, MCPI methods are

computationally efficient even prior to parallelization according to the results reported by

Bai and Junkins, [13, 14] where MCPI methods have been compared with several classical

methods in solving IVPs and BVPs. In several examples, it has been demonstrated that both

greater efficiency and accuracy can be obtained, compared to the pseudospectral method, for

example. The MCPI method can be applied to either the impulsive or the continuous control

case; in this paper, we will provide only the case of impulsive control.

This paper is structured as follows. We first present the numerical approach to generate

the reference halo orbit. We use the third order analytical formulations derived by Richardson

[16] to provide a starting guess for the halo orbit, and then use a differential-correction

approach to numerically find the accurate initial conditions to generate the periodic halo

orbit. The reference orbit is integrated using MCPI method, and the Chebyshev coefficients

that precisely represent the nominal orbit are saved for future reference. We then use MCPI

methods to generate impulses at fixed discrete time intervals for corrective maneuvers.

Several important criteria such as fuel cost and deviations from the reference orbit are

analyzed through simulations and we compare our results with previous published results.

Conclusions and future directions are presented at the end.

2. The Reference Orbit

2.1. Equations of Motion

As is well known, a good nominal reference orbit reduces the fuel cost for unnecessary

maneuvers, so the dynamical model used to generate the reference orbit ideally should

include the gravitational perturbation forces from the Sun and other planets as well as

solar radiation and other forces. Farquhar and Kamel presented fairly detailed equations of

motion where the effects of the Sun’s gravitation force, solar pressure, and the Moon’s orbital

eccentricity were considered [2]. Gomez et al. [9] used JPL DE403 to include the gravitational

influences of the Sun, Moon, and all nine planets. To illustrate our novel approach, we

have chosen to only consider the CR3BP idealization of the Earth-Moon system. Because

the station-keeping technique we present later is a numerical approach to solve for corrective

maneuvers and is essentially an open loop control, using a more detailed dynamical model

will not change the structure of the algorithm but may moderately affect the performance of

the presented technique. For real mission operations, detailed dynamical models should be

used to generate the reference orbit as well as the corrective maneuvers.

Most of the development of the equations of motion can be found in the book by

Schaub and Junkins [17]. One difference here is that we use the L2 Lagrangian point as the

origin. Figure 1 shows how the rotating reference frame E : {er , eθ, e3} is defined. m1, m2,

and m represent the Earth, Moon, and spacecraft, respectively. Note the vectors denoting the

location of the Earth, the Moon, and the center of mass are defined such that r1 = r1er , which

results in r1, r2, and rCM all being negative quantities since the vectors extend from the origin

along the negative er axis. The two central bodies rotate in circles about their center of mass

at a constant angular velocity, defined by

ω2 =G(m1 +m2)

r312

, (2.1)


m

m1

m2

r

ri

r2

r1r12

ξ1

ξ2

L2

er

eθ

rCM

Figure 1: System Diagram.

where r12 is the distance between m1 and m2. Thus, the E frame in use rotates at a constant

velocity ω with respect to the inertial frame, as given by the following equation:

ω = ωe3. (2.2)

The inertial position vector of the spacecraft is defined as follows:

ri = (rx − rCM)er + ryeθ + rze3, (2.3)

where rx, ry, and rz are the components of the r vector. The inertial derivative of this equation

can be taken using the transport theorem [17] and (2.2) to produce the inertial velocity vector:

ri =(rxer + ryeθ + rze3

)+ω ×

((rx − rCM)er + ryeθ + rze3

)=(rx − ryω

)er +(ry + (rx − rCM)ω

)eθ + rze3.

(2.4)

Finally, the above equation can be differentiated once more to produce the inertial accelera-

tion vector:

ri =(rx − 2ryω − (rx − rCM)ω2

)er +(ry + 2rxω − ryω2

)eθ + rze3. (2.5)

The massless force due to gravity from the two massive bodies is defined in E frame compo-

nents as

Fx = −G(m1

ξ31

(rx − r1) +m2

ξ32

(rx − r2)

),

Fy = −G(m1

ξ31

+m2

ξ32

)ry,

Fz = −G(m1

ξ31

+m2

ξ32

)rz,

(2.6)


with ξi defined as follows:

ξi =√(rx − ri)2 + r2

y + r2z. (2.7)

When combined with (2.5), the equations of motion can be written as

rx − 2ryω − (rx − rCM)ω2 +G

(m1

ξ31

(rx − r1) +m2

ξ32

(rx − r2)

)= 0,

ry + 2rxω − ryω2 +G

(m1

ξ31

+m2

ξ32

)ry = 0,

rz +G

(m1

ξ31

+m2

ξ32

)rz = 0.

(2.8)

The equations of motion are nondimensionalized by first using a new time variable, τ , de-

fined as

τ = ωt, (2.9)

and then using the distance between the Earth and L2 point (|r2|), which leads to the new

distance variables as

x =rx

|r2|, y =

ry

|r2|, z =

rz

|r2|, x1 =

r1

|r2|, x2 =

r2

|r2|,

xCM =rCM

|r2|, ρi =

ξi

|r2|, x12 =

r12

|r2|,

(2.10)

and lastly, the masses are eliminated in favor of a nondimensionalized mass ratio μ, defined

as

μ =m2

m1 +m2. (2.11)

Now, rearrange (2.8) and divide by the distance r2 and also by the factor ω2, yielding:

◦◦x= 2

◦y + (x − xCM) − G

ω2

(m1

ξ31

(x − x1) +m2

ξ32

(x − x2)

),

◦◦y= −2

◦x + y − G

ω2

(m1

ξ31

+m2

ξ32

)y,

◦◦z= − G

ω2

(m1

ξ31

+m2

ξ32

)z.

(2.12)


Replacing m2 by μ(m1 +m2) and m1 by (1 − μ)(m1 +m2) using (2.11), and introducing ρi, the

equations of motion become

◦◦x= 2

◦y + x − xCM − G(m1 +m2)

ω2r32

(1 − μρ3

1

(x − x1) +μ

ρ32

(x − x2)

),

◦◦y= −2

◦x + y − G(m1 +m2)

ω2r32

(1 − μρ3

1

+μ

ρ32

)y,

◦◦z= −G(m1 +m2)

ω2r32

(1 − μρ3

1

+μ

ρ32

)z.

(2.13)

Combining (2.13) with (2.1) yields the final, nondimensionalized equations of motion:

◦◦x= 2

◦y + x − xCM − x3

12

(1 − μρ3

1

(x − x1) +μ

ρ32

(x − x2)

),

◦◦y= −2

◦x + y − x3

12

(1 − μρ3

1

+μ

ρ32

)y,

◦◦z= −x3

12

(1 − μρ3

1

+μ

ρ32

)z.

(2.14)

Using (2.15), the nondimensionalized distance x, which is the distance from the center of

mass to the L2 point, is calculated:

x − 1 − μ(μ + x

)2− μ(

x − 1 + μ)2

= 0. (2.15)

The other values, r1, r2, and rCM, are found using the center of mass equation:

m1(r1 − rCM) +m2(r2 − rCM) = (m1 +m2)r12. (2.16)

Table 1 provides the values of the three parameters used in this model.

2.2. Finding Initial Conditions

A third-order closed-form solution for the equations of motion is given in Richardson’s paper

[16]. We use Ay = 45000 km which is defined as the amplitude of the linearized motion along

the y direction in Richardson’s paper [16]. This is the same magnitude used in the paper by

Breakwell et al. [4], at which position a satellite is visible from the Earth all the time. The

approximate initial conditions obtained from the third order formula quickly diverge when

propagated using the full equations of motion due to the inherent instability of the system.

A differential corrections method [18] was used to find the initial conditions that lead to


Table 1: Parameters used.

Parameter Value

Earth mass 5.976 × 1024 kg

Moon mass 7.3477 × 1022 kg

r12 0.3844 × 106 km

Table 2: Approximate initial conditions.

Variable Nondimensional value

x(t0) −0.390895010335809

z(t0) 0.353556629315019

vy(t0) 1.554577497503360

Tp 3.336429964438981

a bounded periodic orbit. In order to find a halo orbit one important property of halo orbits

was exploited: halo orbits display symmetry about the x-z plane. This means that at the point

where the orbit crosses that plane (y = 0), the x and z components of velocity are both zero.

As such, choosing a point on the x- z plane as the initial conditions reduces the variable

set from three position components (x, y, z), three velocity components (vx, vy, vz), and the

orbit period (T) to two position components (x, z), one velocity component (vy), and the

orbit period (T). Additionally, after a single orbit, or after half an orbit, y, vx, and vy will all

be zero again. Table 2 gives the initial conditions we found, where the physical time of the

period is Tp ≈ 14.5 days. Note that the solution accuracy in the numerical correction process

to find these initial conditions has been set as 10−15.

2.3. Using MCPI Method to Save the Reference Orbit

A nice feature about MCPI methods is that the achieved solutions are represented to high

precision in the form of orthogonal Chebyshev polynomials, thus if the coefficients of the

states are saved, the state values at arbitrary time can be obtained straightforward (by

numerically computing the orthogonal polynomials and then a simple inner product matrix

multiplication). We integrate the reference orbit using MCPI method for one orbit and save

the coefficients of the position and velocity for future station-keeping reference. An important

parameter to choose here is the order of the polynomials to use. Generally, higher order

approximation leads to higher accuracy solutions but requires larger storage space to save

all the coefficients. We use closure error (after one period) of the reference orbit, defined in

(2.17), as the accuracy criterion to find the optimal order to use:

Ep = S(Tp)− S(T0), (2.17)

where S(Tp) is the state values (position or velocity) after one complete orbit and S(T0)is the state values (position or velocity) at the initial time. Figure 2(a) shows how the

nondimensional closure errors change with the order, and Figure 2(b) shows the dimensional

results. Both figures demonstrate the spectral accuracy that the MCPI method achieved,


102

100

10−2

10−4

10−6

10−8

10−10

10−12

10−14

10 20 30 40 50 60 70 80

Position

Velocity

Polynomial order: N

Nondimensional closure error versus order

S(T

p)−S(T

0)

(a) Nondimensional

10 20 30 40 50 60 70 80

Polynomial order: N

Closure error versus order

Position (km)

Velocity(km/s)

105

100

10−5

10−10

10−15

S(T

p)−S(T

0)

(b) Dimensional

Figure 2: Closure error versus polynomial order.

where the closure errors of both position and velocity decrease exponentially as the order

increases. Thirteen significant digits can be achieved with N ≥ 60, but this is higher

precision than required for Earth-Moon halo orbits. In the simulations we present in the

next section, we use polynomials of order 30, which still leads to a position accuracy about

0.1 km and velocity accuracy about 1 mm/s as proved by Figure 2(b). As we will discuss

in the next section, this obtained accuracy is one magnitude smaller than the range of typical

measurement noise. Of course this order can be tuned according to the accuracy requirement,

and we simply illustrate the methodology. The reference orbit in three dimension is as well

as the projection in the three orthogonal reference planes is shown in Figure 3.

3. Simulation Studies

We use MCPI method for station-keeping of the halo orbit generated in the last section.

The matrix-vector form of MCPI methods for solving this special type of two-point BVPs,

where the positions at the two boundaries are constrained, is shown in Figure 4. The matrices

shown in the formulas,Cx andCBα , are constant (once the order of approximation polynomials

are specified), and Θxif is an (N + 1) × 1 vector depending on the order and the two-point

boundary conditions. Detailed derivations of the approach can be found in the paper by Bai

and Junkins [14]. Procedures to use the formula and definitions of the matrices and vectors

are briefly discussed in the appendix of the present paper for completeness.

For the current problem, at the time when a maneuver is possible, the current position

X(ti) = (xi, yi, zi) and velocity V(ti) = (vx, vy, vz) are measured. The reference position at the

target time X(tf) = (xf , yf , zf) is computed from the coefficients of the reference orbit through

a matrix multiplication and the time can be arbitrary. Then MCPI method is used to solve this

two-point BVP using the formula in Figure 4, and the desired velocity Vr(ti) = (vxr , vyr , vzr )at the maneuver time is obtained. The corrective maneuver is computed from

ΔV (ti) =(Δvx(ti),Δvy(ti),Δvz(ti)

)= Vr(ti) −V(ti). (3.1)


−10000−20000

−4

−2

0

0

2

4

×104

×104

x (km)

y (km)

z(k

m)

2

1

0

−1

−2

−3

(a) 3D

×104

×104x (km)

y(k

m)

−4

−3

−2

−1

0

1

2

3

4

−6 −4 −2 0 2 4

(b) x-y plane

x (km)

×104

×104

z(k

m)

2

1

0

−1

−2

−3

−4 −3 −2 −1 0 1 2

(c) x-z plane

×104

×104

z(k

m)

2

1

0

−1

−2

−3

−4

y (km)

−4 −3 −2 −1 0 1 2 3 4

(d) y-z plane

Figure 3: Reference Orbit.

As a numerical approach to solve BVPs, MCPI method can be easily adjusted if the dynamical

model used for the real mission is more complicated than the model used in the preliminary

design. MCPI method does not require neither gradient information nor the computation of

state transition matrices, and a prior knowledge can be wisely used to provide a good initial

guess to start the interaction, all of which make the method computational very attractive.

The position and velocity of the satellite when it is inserted to the halo orbit will be

away from the ideal initial conditions of the reference orbit. The measurement of the position

and velocity from Earth will not be perfect. And the implementation of the impulses will

also have “execution errors”. We include these three kinds of errors in the simulation. All the

errors are assumed to be Gaussian distribution with a mean of zero. The standard deviations

of these errors are chosen to be consistent with those used by Gomez et al. [9] and are listed

as follows

(i) tracking errors: σx = σy = σz = 1 km; σvx = σvy = σvz = 1 cm/s,

(ii) insertion errors: σx = σy = σz = 1 km; σvx = σvy = σvz = 1 cm/s,

(iii) maneuver errors: σΔvx = 0.05; σΔvy = 0.05; σΔvz = 0.02.


Constant matrix initialization

Force evaluation

YesNo

Correction calculation

State update

Stopping criterion check

Exit

Starting guess xi(τ)

−→Xold =

−→Xnew

eold = enew

enew < δ?

eold < δ?and

−→Xold xi τ0 , x

i τ1 , xi τ2 , . . ., x

i τNT

−→g g τ0, xi τ0 , g τ1, x

i τ1 , g τ2, xi τ2 , . . ., g τN, x

i τNT

−→Xnew ω2

2CxCBα f Cx xif

−→Xnew xi 1 τ0 , x

i 1 τ1 , xi 1 τ2 , . . ., x

i 1 τNT

enew−→Xnew − −→

Xold max

Figure 4: Vector-matrix form of MCPI method for solving second order bvps.

Table 3: Baseline case results.

Variable Averaged Results

Max ΔV 27.894 cm/s

Min ΔV 1.178 cm/s

Total ΔV 1606.5 cm/s

Mean deviation distance from the reference 3.98 km

Max deviation distance from the reference 17.59 km

Maneuver numbers 182

Iterations of MCPI 3.05

We look at station-keeping of the halo orbit for 26 revolutions, which corresponds to 377 days,

a little more than one year. 100 Monte Carlo simulations are run and the results in Table 3 are

the averaged numbers from the 100 trials. Here we have used 1/ 7Tp as the spacing between

two impulsive maneuvers which corresponds to a physical time about 2 days. Later, we will

study how this parameter affects the results. Significantly, it might be surprising to notice

that MCPI methods only took about three iterations to converge to such accurate solutions.

This is because we have used the coefficients of the reference orbit to find the corresponding

state values and then use these values to start the MCPI iteration. Thus MCPI method starts

from an ideal trajectory then takes only three Picard iterations to converge to the neighboring

solutions that restore the halo orbit to within the tolerance adopted.

Considering most of the thrusters used to generate impulses will have some limit on

the minimum Δv it can provide; we added one parameter Δvmin = 2 cm/s as a threshold

for the control: if the solved impulses are less than the threshold, no correction maneuver

will be made. Table 4 shows the results for this setting. As expected, the deviations from the

reference become larger and the maximum Δv become larger. The good thing is that both


Table 4: Δvmin = 2 cm/s.

Variable Averaged results

Max ΔV 29.692 cm/s

Min ΔV 0 cm/s

Total ΔV 1523.5 cm/s: about 1476.5 cm/s per year





the overall Δv and maneuver numbers are reduced, which in part is because that the small

maneuvers that might be in the same magnitude of the measurement noise are avoided.

We also increased this threshold to Δvmin = 3 cm/s. Unfortunately and not surprisingly,

the deviations from the reference become larger and more Δv is required. We notice when

we set Δvmin = 2 cm/s, we have a similar parameter setting as those studied by Gomez

et al. [9], for the case where they set the tracking and minimum maneuver intervals as two

days. Compared with their results, our total impulses (1476.5 cm/s) are larger than theirs

(1111.8 cm/s), and we have more impulsive maneuvers (168) than theirs (80); however, our

maximum tracking deviations (17.38 km) are much smaller than their results (60.9 km) and

our maximum impulse (29.692 cm/s) is also much less than theirs (73.554 cm/s). The reason

we have more fuel required than theirs can result from several reasons. First, our reference

halo orbit is different from their orbit. Second, they have used two targeting points and have

minimized a cost function which is a weighted summation of both position deviations and

fuel cost. As they commented in the paper, the weights have been tuned, which very likely

emphasized a large penalty on the fuel cost. Instead, we have not optimized anything in our

approach. This situation is similar as the one reported by Renault and Scheeres [11], where

the LQR control is more fuel efficient than targeting the stable manifold of the equilibrium

point. Additionally, as shown in Figure 4, using the MCPI method only involves evaluating

the differential equations on the discrete nodes and doing vector-matrix multiplication and

summation. Of special interest for the current case, only three iterations are required for

the method to converge. However, the approach by Gomez et al. [9] requires weight tuning

beforehand and also a complicated state transition matrix derivation and then computation

during the mission when a more realistic dynamical model is used. Thus, we believe our

method is computationally very attractive, due to the absence of required tuning, and

extremely simple and easy to implement. Figure 5(a) shows the history of the magnitude

of the impulses from one Monte Carlo simulation. Due to measurement noise and simulated

control errors, the computed solution of the two-point BVP will of course contain errors that

propagate into an error at the frequent maneuver time. These errors along 26 revolutions

(377 days) are shown in Figure 5(a). Note the numbers shown in Tables from 3 to 5 are

the averaged numbers from 100 Monte Carlo simulations. For the 168 maneuvers along

the same orbit of Figures 5(a) and 5(b), Figure 5(c) shows the number of Picard iterations

required to achieve convergence. We conclude, 92.2% of the cases converge in three iterations

and 100% converged in four or fewer iterations, a remarkable algorithm! Another important

parameter is the time interval to solve for and then execute the maneuver. Usually there exists

a minimum maneuver time because of the necessity for orbit determination process and the

mission operation capability. On the other hand, if the maneuver intervals become too large,

the deviations from the reference can become too large to be easily corrected. We have studied


V from one Monte Carlo simulation

V(k

m/

s)

20 40 60 80 100 120 140 160 180

10−3

Impulse time (1/7Tp)

(a) Impulse history

Deviation

Mean deviation

20 40 60 80 100 120 140 160 180

1

2

3

4

5

6

7

8

9

10

11

Distance from the reference from one Monte Carlo simulation

R(k

m)


(b) Deviation

Iterations

Mean iterations

Iter

ati

on

s

20 6040 80 100 120 140 160 180

3.1

3

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

Number of iterations from one Monte Carlo simulation


(c) Iteration numbers

Figure 5: Results from One Monte Carlo Simulation.

the performance while changing the interval times from a half day to a little more than three

and a half days. The results below show how the position and velocity deviations, number

of maneuvers, total ΔV , and maximum ΔV change with respect to the interval time, see

Figures 6(a) to 6(d). As expected, Figures 6(a) and 6(b) show that the deviations increase

and the number of maneuvers decrease as time interval between corrections becomes larger.

It is interesting to see that there exists a broad window (of one and half to three days) for

time intervals between corrections, that gives near-minimum fuel cost, as evident in Figure

6(c). This is similar as a situation reported by Renault and Scheeres [11], where the optimal

spacing for both LQR approach and targeting equisetum method exists for the problems they

studied. From Figure 6(d), the fact that the maximum impulse is minimum for 1 to 1.5 days

implies that separation is significant.


Table 5: Δvmin = 3 cm/s.

Variable Averaged results

Max ΔV 33.86 cm/s

Min ΔV 0 cm/s

Total ΔV 1574.1 cm/s





4. Approaches to Reduce the Fuel Cost

In this section, we study how to reduce the fuel cost using the presented method

while maintaining its advantages such as no requirement for any gradient information,

computational efficiency, and the final results in an orthogonal polynomial form.

The above-method results in exact satisfaction of terminal boundary conditions on

each subinterval if there are no control errors and measurement errors. As a result, the posi-

tion errors are much smaller than those in the literature [9], whereas the control required is

about 24% higher. To gain some qualitative insight to the tradeoff between allowing increased

terminal errors and the associated cost, the current method can be used in a heuristic fashion.

We reasoned that applying less than 100% of the control impulse computed would result in

increased terminal errors and smaller control, and in fact, we found that using 80% of the

computed impulse in the current problem results in a control cost only 4% higher than those

in reference [9], while our terminal errors in fact still remain significantly smaller than those

in the references. We emphasize that the relationships between the amount of the control to

apply and the performance (such as the total fuel cost, the maximum position deviation,

and the mean position deviation) are not linear. In Figure 7, we show how the total fuel

cost, the maximum deviation, and the mean deviation change with respect to the introduced

coefficient. The simulation uses the same conditions we studied for the results reported in

Table 4. Notice all these three performance variables have been averaged from the 100 Monte

Carlo simulations. Especially, we found, by applying 80% of the fuel required for solving the

exact two-point boundary value problem, the total fuel cost is 1160 cm/s, the mean position

deviation from the reference orbit is 4.84 km, and the maximum position deviation is 22.2 km.

Thus with a fuel cost only 4% larger than the reported results (1111.8 cm/s) [9], our position

deviations from the reference orbit are much smaller than the reported results, for which the

mean and maximum position deviations are 61 km and 74 km, respectively.

Another approach to reduce the fuel cost is to adopt a hybrid propulsion system

which uses low thrust during the flight and impulses at some specified points. The hybrid

propulsion concept has been studied by Bai et al. for an Earth to Apophis mission design

[19]. Since the presented methods can also solve optimal control problems for continuous

system [14], utilizing this hybrid propulsion system is straightforward. We could also reduce

the fuel cost by varying the time interval used for the station-keeping instead of using the

current constant time interval. However, this will increase the computation requirements,

since all the matrices used in the current approach are constant and are precomputed, which

have to be recomputed if the time interval changes. We will also explore to efficiently utilize

the dynamic information of some specified orbits to reduce the fuel cost in the future study.


20

40

60

80

100

120

140

160

180

Maneuver spacing (days)

Dev

iati

on(k

m)

Averaged deviation versus spacing

0.5 1 1.5 2 2.5 3 3.5

Mean deviation

Max deviation

(a) Deviations.


0.5 1 1.5 2 2.5 3 3.5 4100

200

300

400

500

600

700

800

Averaged number of maneuver versus spacing

Nu

mb

er o

f m

an

euv

er

(b) Number of maneuvers


0.5 1 1.5 2 2.5 3 3.5 41000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

V(c

m/

s)

Averaged total V versus spacing

(c) Total impulses


0.5 1 1.5 2 2.5 3 3.5 4

V(c

m/

s)

20

40

60

80

100

120

140

160

180Averaged max V versus spacing

(d) Max impulses

Figure 6: Effects of the spacing of maneuvers (averaged from 100 Monte Carlo simulation).

5. Conclusions

Modified Chebyshev-Picard Iteration method is used for station-keeping of a halo orbit

around the L2 libration point of the Earth-Moon system. Compared with other station-

keeping techniques which also generate impulses at discrete times, the presented method

is much simpler, since it does not need to compute state transition matrix or gradient

information, which can be complicated for the full dynamical model and might be impossible

to implement for the on-board system. Previous studies [13, 14] have shown that the MCPI

algorithm works well for highly perturbed orbits; it is anticipated that little change will be

required in the present algorithm upon including solar and other non-CR3BP effects. There is

also no tuning process (e.g., for the weight matrices) which is required for other approaches.

Although the fuel cost from this approach, for the examples presented herein, might be a

little higher than other optimal control results, the simple structure of the method as well as

its salient computation efficiency makes this technique very attractive. The method can be

used for station-keeping other orbits with the only change being the use of the appropriate


0.65 0.7 0.75 0.8 0.85 0.9 0.95 11150

1200

1250

1300

1350

1400

1450

1500

1550

1600

1650

Coefficient of control

V(c

m/

s)

Averaged total V versus coe cient of control

(a) Averaged total impulses

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1


15

20

25

30

35

40

45

50

55

60

Dev

iati

on

(k

m)

Averaged max deviation versus coe cient of control

(b) Averaged maximum deviation

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1


Dev

iati

on

(k

m)

4

4.5

5

5.5

6

6.5

7

7.5

8

Averaged mean deviation versus coe cient of control

(c) Averaged mean deviation

Figure 7: Performance versus coefficient of control.

equations of motion. We believe this method can take the same role as the classical Battin’s

form of the Lambert algorithm for solving orbit transfer problems, with the differences that

this method is a numerical approach and can be used for rather general nonlinear systems,

including perturbations. We are investigating extending the current approach which solves

two-point boundary value problems to solve optimal control problems which include several

target points as well as hybrid propulsion mode with both impulsive and continuous thrust.

Appendix

MCPI Method for Solving Second Order BVPS

Assume the second order dynamic system is described as

d2x(t)dt2

= f(t, x(t)), t ∈[t0, tf

], (A.1)


and the boundary conditions are x(t = t0) = x0 and x(t = tf) = xf . The first step of MCPI

methods is to transform the generic independent variable t to a new variable τ , which is

defined on the valid range, the closed interval [−1, 1], of Chebyshev polynomials

t = ω1 +ω2τ, ω1 =tf + t0

2, ω2 =

tf − t02

. (A.2)

Introducing this time transformation of (A.2) into (A.1), it is rewritten as

d2x(τ)dτ2

= ω22f(ω1 +ω2τ, x(τ)) ≡ g(τ, x(τ)). (A.3)

The position update equation is

�Xk+1 = ω22CxC

Bα�f + CxΘxif , (A.4)

where �X = [x(τ0), x(τ1), x(τ2), . . . , x(τN)]T is the vector representing the position trajectory

evaluated at all the (N + 1) Chebyshev-Gauss-Lobatto (CGL) nodes, which are computed

through

τj = cos

(jπ

N

), j = 0, 1, 2, . . . ,N. (A.5)

The components of Cx ≡ TW and CBα = SBTV are defined as below, and �f =

[f(τ0), f(τ1), f(τ2), . . . , f(τN)]T . Θxif is defined by the boundary condition as Θxif = [x0 +xf , (xf − x0)/2, 0, 0, . . . , 0]T ∈ RN+1. Notice although the second order formula in (A.4)only solves for the position, the velocity solution can be computed conveniently by taking

the derivative of the position, which has been obtained in a form of Chebyshev polynomi-

als:

T =

⎡⎢⎢⎢⎢⎣T0(τ0) T1(τ0) · · · TN(τ0)

T0(τ1) T1(τ1) · · · TN(τ1)...

......

...

T0(τN) T1(τN) · · · TN(τN)

⎤⎥⎥⎥⎥⎦, (A.6)

and the two diagonal matrices W and V are defined as

W = diag

([1

2, 1, 1, . . . , 1, 1

]),

V = diag

([1

N,

2

N,

2

N, . . . ,

2

N,

1

N

]).

(A.7)


The definition of SB as shown below is more complicated than others

SB(k + 1, k − 1) = 1/4/k(k − 1), k = 2, 3,−2, . . . ,N − 2,

SB(k + 1, k + 1) = −1/2/(k2 − 1

), k = 2, 3,−2, . . . ,N − 2,

SB(k + 1, k + 3) = 1/4/k/(k + 1), k = 2, 3,−2, . . . ,N − 2,

SB(N,N) = −1/2/((N − 1)2 − 1

),

SB(N,N − 2) = 1/4/(N − 1)/(N − 2),

SB(N + 1,N + 1) = −1/2/(N2 − 1

),

SB(N + 1,N − 1) = 1/4/N/(N − 1),

SB(1, k + 1) = −3(

1 + (−1)k)/(k2 − 4

)/(k2 − 1

), k = 4, 5, . . . ,N − 2,

SB(2, k + 1) = −3/2(

1 − (−1)k)/(k2 − 4

)/(k2 − 1

), k = 4, 5, . . . ,N − 2,

SB(1, 1) = −1

4,

SB(1, 2) = 0,

SB(1, 3) =7

24,

SB(1, 4) = 0,

SB(2, 1) = 0,

SB(2, 2) = − 1

24,

SB(2, 3) = −0,

SB(2, 4) =1

20,

SB(1, k + 1) =(

1 + (−1)k)/4(k − 5)/

(k2 − 1

)/(k − 2), k =N − 1,

SB(2, k + 1) =(

1 − (−1)k)/8(k − 5)/

(k2 − 1

)/(k − 2), k =N − 1,

SB(1, k + 1) =(

1 + (−1)k)/4(k − 5)/

(k2 − 1

)/(k − 2), k =N,

SB(2, k + 1) =(

1 − (−1)k)/8(k − 5)/

(k2 − 1

)/(k − 2), k =N,

(A.8)

where S(i, j) represents the element in the ith row and jth column.

Acknowledgment

This work is supported by the Air Force Office of Scientific Research (Contract number

FA9550-11-1-0279).


References

[1] R. W. Farquhar, “Station-keeping in the vicinity of collinear libration points with an application to alunar communications problem,” in Proceedings of the Space Flight Mechanics Specialist Symposium, AASScience and Technology Series, pp. 519–535, Denver, Colo, USA, July 1966.

[2] R. W. Farquhar and A. A. Kamel, “Quasi-periodic orbits about the translunar libration point,” CelestialMechanics, vol. 7, no. 4, pp. 458–473, 1973.

[3] R. W. Farquhar, D. P. Muhonen, C. Newman, and H. Heuberger, “Trajectories and orbital maneuversfor the first libration-point satellite,” Journal of Guidance, Control and Dynamicss, vol. 3, no. 6, pp. 549–554, 1980.

[4] J. V. Breakwell, A. A. Kamel, and M. J. Ratner, “Station-keeping for a translunar communication sta-tion,” Celestial Mechanics, vol. 10, pp. 357–373, 1974.

[5] M. Xin, M. W. Dancer, S. Balakrishnan, and H. J. Pernicka, “Stationkeeping of an L2 Libration pointsatellite with θ − D technique,” in Proceedings of the American Control Conference, Boston, Mass, USA,2004.

[6] J. E. Kulkarni, M. E. Campbell, and G. E. Dullerud, “Stabilization of spacecraft flight in halo orbits: anH∞ approach,” IEEE Transactions on Control Systems Technology, vol. 14, no. 3, pp. 572–578, 2006.

[7] D. Cielaszyk and B. Wie, “New approach to halo orbit determination and control,” Journal of Guidance,Control, and Dynamics, vol. 19, no. 2, pp. 266–273, 1996.

[8] D. W. Dunham and C. E. Roberts, “Stationkeeping techniques for libration-point satellites,” Journal ofthe Astronautical Sciences, vol. 49, no. 1, pp. 127–144, 2001.

[9] G. Gomez, K. Howell, K. Howell, C. Simo, and J. Masdemont, “Station-keeping strategies for trans-lunar libration point orbits,” in Proceedings of the AAS/AIAA Space Mechanics Meeting, Monterey, Calif,USA, 1998.

[10] K. C. Howell and H. J. Pernicka, “Stationkeeping method for libration point trajectories,” Journal ofGuidance, Control, and Dynamics, vol. 16, no. 1, pp. 151–159, 1993.

[11] C. A. Renault and D. J. Scheeres, “Statistical analysis of control maneuvers in unstable orbital environ-ments,” Journal of Guidance, Control, and Dynamics, vol. 26, no. 5, pp. 758–769, 2003.

[12] E. D. Gustafson and D. J. Scheeres, “Optimal timing of control law updates for unstable systems withcontinuous control,” in Proceedings of the American Control Conference, Seattle, Wash, USA, June 2008.

[13] X. Bai and J. L. Junkins, “Modified Chebyshev-Picard iteration methods for solution of initial valueproblems,” Advances in the Astronautical Sciences, vol. 139, pp. 345–362, 2011.

[14] X. Bai and J. L. Junkins, “Modified Chebyshev-Picard iteration methods for solution of boundaryvalue problems,” Advances in the Astronautical Sciences, vol. 140, pp. 381–400, 2011.

[15] X. Bai, Modified Chebyshev-Picard iteration methods for solution of initial value and boundary value problems,Ph.D. dissertation, Texas A&M University, College Station, Tex, USA, 2010.

[16] D. L. Richardson, “Analytic construction of periodic orbits about the collinear points,” Celestial Mech-anics, vol. 22, no. 3, pp. 241–253, 1980.

[17] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, American Institute of Aeronauticsand Astronautics, Reston, Va, USA, 1st edition, 2003.

[18] J. V. Breakwell and J. V. Brown, “The “Halo” family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem,” Celestial Mechanics, vol. 20, no. 4, pp. 389–404, 1979.

[19] X. Bai, J. D. Turner, and J. L. Junkins, “Optimal thrust design of a mission to apophis based on a homo-topy method,” in Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting, Savannah, Ga, USA, 2009.


Research ArticleAssessment of the Ionospheric and TroposphericEffects in Location Errors of Data CollectionPlatforms in Equatorial Region during Highand Low Solar Activity Periods

Aurea Aparecida da Silva,1 Wilson Yamaguti,1Helio Koiti Kuga,2 and Claudia Celeste Celestino3

1 Space Systems Division (DSE), National Institute for Space Research (INPE),12227-010 Sao Jose dos Campos, SP, Brazil

2 Space Mechanics and Control Division (DMC), National Institute for Space Research (INPE),12227-010 Sao Jose dos Campos, SP, Brazil

3 Centro de Engenharia, Modelagem e Ciencias Sociais Aplicadas (CECS),Universidade Federal do ABC (UFABC), 09210-170 Santo Andre, SP, Brazil

Correspondence should be addressed to Aurea Aparecida da Silva, [email protected]

Received 15 August 2011; Accepted 3 November 2011


Copyright q 2012 Aurea Aparecida da Silva et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The geographical locations of data collection platforms (DCP) in the Brazilian Environmental DataCollection System are obtained by processing Doppler shift measurements between satellites andDCP. When the signals travel from a DCP to a satellite crossing the terrestrial atmosphere, theyare affected by the atmosphere layers, which generate a delay in the signal propagation, andcause errors in its final location coordinates computation. The signal propagation delay due tothe atmospheric effects consists, essentially, of the ionospheric and tropospheric effects. This workprovides an assessment of ionospheric effects using IRI and IONEX models and tropospheric delaycompensation using climatic data provided by National Climatic Data Center. Two selected DCPswere used in this study in conjunction with SCD-2 satellite during high and low solar activityperiods. Results show that the ionospheric effects on transmission delays are significant (abouthundreds of meters) in equatorial region and should be considered to reduce DCP location errors,mainly in high solar activity periods, while in those due to tropospheric effects the zenith errorsare about threemeters. Therefore it is shown that the platform location errors can be reduced whenthe ionospheric and tropospheric effects are properly considered.

1. Introduction

The Brazilian Environmental Data Collection System is currently composed by SCD-1 and

SCD-2 satellites, hundreds of data collection platforms (DCPs), receiving ground stations


located at Cuiaba and Alcantara, and Data Collection Mission Center located at Natal,

Brazil.

In this system, a satellite works as a message relay transponder. A DCP to receiving

ground station communication link is established when a simultaneous DCP to satellite

(uplink) and satellite to receiving station (downlink) occurs simultaneously. UHF frequency

bands are used for uplink and S band for the downlink. The uplink Doppler shifts are

measured and time-stamped at a receiving ground station while the downlink Doppler shift

is compensated for by the receiving equipments. The DCPs messages transmitted by satellites

and received by Cuiaba or Alcantara ground station are sent to the Data Collection Mission

Center, now located at INPE (Brazilian National Institute for Space Research) Northeast

Regional Center in Natal city, Brazil, where the data are processed, stored, and distributed

to users. The users receive data at most 30 minutes after the satellite pass.

The DCP geographical location can be determined using the uplink (around

401,635 MHz) Doppler measurements, the statistical least squares method, and knowledge

of the satellite orbit. As the S band (2.26752 GHz) downlink is compensated by the ground

station receiver, it is not considered in this study.

The signal transmitted from a DCP to a satellite is affected during its travelling

across the atmosphere. Among other factors the signal path is influenced by the chemical

elements that make up the extensive Earth’s atmosphere resulting in propagation delays. As

a consequence, errors are present in the final coordinates provided to system users.

As such the ionosphere and the troposphere in vertical direction can be an important

source of errors in the platforms location. The errors due to ionosphere can vary from tens to

hundreds of meters depending on frequency, while in the troposphere the zenith errors are

generally between two and three meters [1–3].In Celestino et al. [3] the simulated and real effects of ionosphere and troposphere

in DCPs geographical location in the Brazilian Environmental Data Collection System were

evaluated. In this work, in order to evaluate the effects associated with the ionosphere, the

values of total electron content (TEC) were obtained from the standard IRI (International

Reference Ionosphere) model. The troposphere values, dry and wet components, were

obtained from data of CPTEC-INPE (Weather Forecast and Climate Studies Center). The

study did not consider the effects of solar activity periods, but only the time of a day. Results

of the analysis indicated that correction of the ionosphere and troposphere effects can, on the

average, reduce the location errors to the scale between 10 to 250 meters.

In the absence of the selective availability (signal corruption which was turned offon May 1st, 2000) the largest source of error in GPS positioning and navigation, using

L1 frequency receiver, has been the ionosphere. The ionospheric effects were investigated

by Camargo et al. [4] in the determination of point positioning and relative positioning

using single frequency data. The model expressed by a Fourier series and the parameters

were estimated from data collected at the active stations of RBMC (Brazilian Network for

Continuous Monitoring of GPS satellites). Experiments were carried out in the equatorial

region, using data collected from dual frequency receivers. In order to validate the model, the

estimated values were compared with “ground truth”. Results show, for point positioning

and relative positioning, a reduction of error better than 80% and 50%, respectively, compared

to the processing without the ionosphere model. These results give an indication that more

research must be done to provide support to L1 GPS users in the equatorial region.

Hence, the present work is divided into two phases: the first is an analysis of the

DCP location errors generated by the delay of the signal propagated in the ionosphere.


Therefore, the ionosphere is considered in two different periods according to their intensity

(periods of high and low solar activity). The ionosphere representation through TEC values

(Total Electron Content) was obtained from IRI model—International Reference Ionosphere

[5]—and the one derived from GPS measurements in IONEX version—IONosphere map

EXchange [6]. These analyses show the relevance of the errors due to the ionospheric delay in

different periods and with use of different models. Doppler shift measurements are simulated

for ideal cases in order to measure the influence of the ionosphere at the location. A Doppler

shift measurement corresponds to the difference between a received signal frequency at the

satellite and the nominal platform transmitter frequency. This shift is caused by the relative

velocity between the satellite and platform. In this work, the DCP no. 32590 (15.55293◦S and

56.06875◦W) was selected for geographical location studies due to its location in Cuiaba,

Brazil, in the equatorial region, and data availability. The period of data covers the high (2001)and the low solar activity (2009).

The second phase of the work consisted of the analysis of the best model to express

the ionosphere in the equatorial region and the use of the selected model for DCP location

with Doppler measurements. This work shows that the IONEX file models the ionosphere

with a better accuracy, especially in periods of high solar activity, increasing sufficiently the

density of the atmosphere in low latitudes, that is, in the equatorial anomaly region. This

motivated the selection of the IONEX model for the second phase of the work. In addition to

the correction of the ionosphere, the effects of the troposphere were considered by the use of

climate data provided from National Climatic Data Center—NOAA Satellite and Information

Service.

Real Doppler shift measurements are also used for DCP no. 109 location (located

in French Guyana, at 5.1860◦S and 52.687◦W) and SCD-2 satellite, taking into account the

following conditions: time of the satellite pass from 15:00 to 22:00 UT (Universal Time),periods of high solar activity in 2001, and of low solar activity in 2008.

2. Characteristics of the Ionosphere and Troposphere

The ionosphere is roughly located between 50 to 1000 kilometers above the terrestrial surface,

being the electromagnetic radiation the largest agent of the ionization process. Furthermore,

the Sun inserts great amount of free electrons that contributes for the composition of

ionosphere. The solar wind and the solar electromagnetic radiation can have drastic change

during a solar storm, which implies changes in the conditions of the magnetic field and the

terrestrial ionosphere.

The location error is associated with the ionosphere which is directly proportional to

total electron content (TEC) contained along the signal trajectory in the ionosphere, and it

is inversely proportional to the square of signal frequency. TEC and, consequently, the error

due the ionosphere vary in time and space, and they are affected by several variables, such

as: solar cycle, local time of the day, season, geographical location, geomagnetic activity, and

others [4]. Thus, it is very important to know the behavior of the ionosphere, taking into

account their dependence on the location and time.

Besides the ionosphere temporal variation over the years, it presents variation of the

electron density in cycles of about 11 years. During these peaks, there is a higher incidence

of sun radiation (solar activity) with increase of ionization in terrestrial atmosphere layers.

TEC values are proportional to the increase of solar activity, that is, the increase of sunspots.

In periods of maximum activity, TEC can reach values about twice as larger than in periods


Sunspot number

Time (years)

Daily

Monthly

SM predictions

CM predictions

Smoothed

20000

50

100

150

200

250

300

350

2002 2004 2006 2008 2010 2012

RI

Figure 1: Sunspot numbers of cycle 23 and beginning of cycle 24. The daily (yellow), monthly(blue), and monthly smoothed (red) sunspot numbers since 2000, together with predictions for12 months ahead: SM (red dots): classical prediction method, based on an interpolation of Wald-meier’s standard curves; CM (red dashes): combined method proposed by K. Denkmayr. Source:http://sidc.oma.be/html/wolfjmms.html—Access in October 2011.

of low solar activity. Currently, the Sun is starting the cycle 24 that should have its maximum

period between 2012 and 2014. The previous cycle, 23, had its maximum between 2000 and

2002, when there was a considerable increase of the sunspot numbers and, consequently, the

number of electrons in the ionospheric layer.

Figure 1 shows the sunspot numbers of solar cycle 23 and the beginning of cycle 24,

in a period from 2000 to the first semester of 2011 and a prediction of the new cycle for 12

months ahead.

Figure 2 shows three different geographical regions of the ionosphere on terrestrial

globe known as high-latitude region, medium-latitude region, and equatorial region. Figure 3

shows that in the equatorial region, where a great part of South America and Africa region

are located, the equatorial anomaly occurs, increasing the TEC values, mainly in the most

active period of Sun.

TEC is given along the direction between the transmitting DCP and the satellite. TEC

quantities in vertical direction (VTEC) are given by

TEC = VTEC secZ, (2.1)

where secZ is the mapping function, and Z is the signal path zenithal angle in relation to the

plan of a mean reference altitude, estimated by means of so-called Bent Ionospheric Model.

For IRI model an altitude of 350 km was used, whereas for IONEX, 450 km was used. The


180◦W 150◦W 120◦W 90◦W 60◦W 30◦W 0◦ 30◦E 60◦E 90◦E 120◦E 150◦E 180◦E

180◦W 150◦W 120◦W 90◦W 60◦W 30◦W 0◦ 30◦E 60◦E 90◦E 120◦E 150◦E 180◦E

90◦N

60◦N

30◦N

0◦

30◦S

60◦S

90◦S

90◦N

60◦N

30◦N

0◦

30◦S

60◦S

90◦S

High latitude

High latitude

Medium latitude

Medium latitude

Equatorial region

Figure 2: Geographical regions of the ionosphere.

ionosphere signal delay in DCP/satellite direction is calculated, according to Aksnes et al.

[7] by

RI =40.3 VTEC secZ

f2, (2.2)

where f is the DCP frequency transmission. Considering VTEC constant for short times, the

temporal variation of the signal delay due to the ionosphere is given by

RI = −k1 VTEC cos γ sin γ

f2(1 − k2 cos2γ

)3/2γ , (2.3)

where VTEC is the total electron content in vertical direction, γ is the satellite elevation

angle with respect to DCP, and γ is the satellite elevation angle rate. Depending on the

mean reference altitude h, one has for h = 350 km, then k1 = 36.21 and k2 = 0.8985 and

for h = 450 km, then k1 = 35.16 and k2 = 0.8723.

The last equation can be applied to the ionosphere correction based on Doppler shift

measurements. The deviation due to the ionosphere is sensitive to the values of VTEC, which

can be obtained from IGS (International GNSS Service) with free access to any users in IONEX

format [6]. Such data correspond to vertical TECU that means vertical TEC unitary with the

following relationship: a unit of TEC (1 TECU) corresponds to 1016 electrons per square meter

[8]. Figures 3(a), 3(b), and 3(c) show a set of vertical TECU global maps on October 22, 2001.

The figures are presented in different times, 9:00, 13:00, and 17:00 UT and show the equatorial

anomaly evolution along low-latitude regions (from 30◦S to 30◦N).Tropospheric effect depends on atmosphere density and the satellite elevation

angle. This effect can be observed from the terrestrial surface up to about 50 km [9].The representation of the troposphere deviation depends on the atmospheric pressure,


88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

Vertical TECU for 22/10/2001 0900 UT

Geo

gra

ph

ic l

ati

tud

e

Geographic longitude

(a)

(a)

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

Geo

gra

ph

ic l

ati

tud

e


Vertical TECU for 22/10/2001 1300 UT(b)

(b)

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

−180◦ −120◦ −60◦ 0◦ 60◦ 120◦ 180◦

Geo

gra

ph

ic l

ati

tud

e


Vertical TECU for 22/10/2001 1700 UT(c)

0 38 75 113 150

(c)Figure 3: Vertical TECU from IONEX model for October 22, 2001. Source: Produced by LeicaGeosystems/GNSS QC software.


atmospheric temperature, and water vapor pressure. According to the IERS (International

Earth Rotation Service) model the tropospheric signal delay is given by [10]

RT =A + B

sin γ + B/(A + B)/(sin γ + 0.01

) . (2.4)

Thus, the temporary variation of the signal delay due to the troposphere is given by

RT = (A + B)2cos γ

[B − (A + B) ·

(sin γ + 0.01

)2]

[sin γ

(sin γ + 0.01

)· (A + B) + B

]2 γ , (2.5)

where A = 0.002357Po + 0.000141eo, B = (1.084 × 10−8)PoToK + (4.734 × 10−8)(P 2o/To)(2/(3 −

1/K)), K = 1.163 − 0.00968 cos 2φ − 0.00104 To + 0.00001435Po, γ = satellite elevation

angle, γ = satellite elevation angle rate, Po = atmospheric pressure (in 10−1 kPa, equivalent to

millibars), To = atmospheric temperature (in Kelvin), eo = water vapor pressure (in 10−1 kPa,

equivalent to millibars), and φ = DCP geodetic latitude.

Values of Po, To, and eo are monthly averages obtained for different geographical

latitudes and longitudes from National Climatic Data Center—NOAA Satellite and

Information Service [11].

3. Results

The geographical location is computed by the in-house developed GEOLOC (geographical

location) program in FORTRAN [12], which processes the uplink measured Doppler shift

suffered by the signal transmitted from the DCPs, together with a statistical least squares

method and satellite orbit ephemeris [2, 3].Given the complex dependence of the ionosphere with the solar cycle, local time of the

day, season, geographical location, geomagnetic activity, and others, it is important to make

an analysis of different models that describe VTECU values for the geographical location,

date, and time for the DCP to be located. First we introduce the ionosphere effect using either

IRI or IONEX model and through simulation one verifies its impact on location computation.

Second we use real data to check the consistency of the findings in the former simulation.

3.1. Ideal Doppler Data for DCP no. 32590 with Ionosphere Influence

Two different models of the ionosphere were analyzed: IRI (International Reference

Ionosphere) [5] and IONEX (IONosphere map EXchange) [6].The chosen periods for the location were in October 2001 and October 2009, periods

that are characterized by high and low solar activity, respectively. The month chosen was

October, because it is within the time that VTECU has its maximum values (spring equinox),mainly in 2001. Furthermore, it was possible to find a set of SCD-2 satellite pass data in a

period of the day that VTECU data are most intense (18-19 UT) for that month, in 2001 and

2009. This plays an important role for the location with real data.

In the first set of results VTECU values on October 19–22, 2001 and 2009 are shown

for geographical location of DCP no. 32590 (15.55293◦S and 56.06875◦W). Doppler shift


88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦

−180◦ −120◦ −60◦ 0◦

Geo

gra

ph

ic l

ati

tud

e



(a)

0 38 75 113 150

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦

−180◦ −120◦ −60◦ 0◦

Geo

gra

ph

ic l

ati

tud

e



(b)

Figure 4: VTECU maps on October 19 and 22, 2001 at 19 UT showing the equatorial anomaly crests in theregion close to Brazil for maximum values of VTECU (geographic equator in solid line and the magneticequator in dashed line). Source: Produced by Leica Geosystems/GNSS QC software.

measurements are simulated from this DCP real location, and the ionospheric effect is added

to the model that generates such data. Thus, it is possible to verify in the simulation how

far the influence of ionosphere in the location error is. VTECU values obtained from IRI and

IONEX models are used.

Figure 4 shows a sequence of VTECU maps of the region in latitude between 88◦ South

and 88◦ North and longitudes between −180◦ and 0◦, where American continent is located.

This figure shows maps for two days (October 19 and 22, 2001), a period that is characterized

by high solar activity, and it can be seen the equatorial anomaly effect [13], when VTECU


88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦

−180◦ −120◦ −60◦ 0◦

Geo

gra

ph

ic l

ati

tud

e



(a)

0 38 75 113 150

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

88◦

58◦

29◦

0◦

−29◦

−58◦

−88◦

−180◦ −120◦ −60◦ 0◦

−180◦ −120◦ −60◦ 0◦

Geo

gra

ph

ic l

ati

tud

e



(b)

Figure 5: VTECU maps on October 19 and 22, 2009 at 18 UT for minimum values of VTECU in lowsolar activity period (geographic equator in solid line and the magnetic equator in dashed line). Source:Produced by Leica Geosystems/GNSS QC software.

values are maximum in Brazilian region, around 18-19 UT (universal time) or 15 h local time.

Through the diagram of colors it is possible to verify that in this period VTECU reaches

values close to 150 × 1016 electrons/m2 in the equatorial anomaly region (red). On the other

hand, in Figure 5, which represents the period of October 19 and 22, 2009, there is not a

considerable difference in VTECU values for the same time, 18-19 UT. The diagram of colors

is maintained between the dark and the light blue, which represents a value about 10 to 40 ×1016 electrons/m2. This fact is given in low solar activity.


Vertical TECU for 19/10/2001—DCP #32590

Time (hours)

IONEX

IRI

SCD-2

VT

EC

U

0

0

20

40

60

80

100

120

140

2 4 6 8 10 12 14 16 18 20 22 24

(a)


Time (hours)

IONEX

IRI

SCD-2

VT

EC

U

0

0

20

40

60

80

100

120

140

2 4 6 8 10 12 14 16 18 20 22 24

(b)

Figure 6: VTECU values on October 19 and 22, 2001 in a period of one day at geographical location of DCPno. 32590 (IONEX (blue), IRI (pink), and triangles (green) represent the time of SCD-2 satellite passes).


Time (hours)

IONEX

IRI

SCD-2

VT

EC

U

0

0

20

40

60

80

100

120

140

2 4 6 8 10 12 14 16 18 20 22 24

(a)


Time (hours)

IONEX

IRI

SCD-2

VT

EC

U

0

0

20

40

60

80

100

120

140

2 4 6 8 10 12 14 16 18 20 22 24

(b)

Figure 7: VTECU values on October 19 and 22, 2009 in a period of one day at geographical location of DCPno. 32590 (IONEX (blue), IRI (pink), and triangles (green) represent the time of SCD-2 satellite passes).

Figures 6 and 7 show VTECU values in geographical location where DCP no. 32590

is located, for the same periods mentioned above, 2001 and 2009, using IRI (pink line) and

IONEX (blue line). The triangle symbols (green) in the figures represent the time of SCD-

2 satellite passes, which span a period between 15 and 22 UT. VTECU is showed with an

interval of two hours and in a full period of a day. In Figure 6 it is verified a discrepancy

between VTECU values of IRI and IONEX, mainly from 14 to 24 UT, and such values differ

by up to 3 times in the comparison between the models; that is, IONEX represents better the

total electron content in high solar activity periods. This difference results in an increase of

location error in up to 50% when the ionospheric delays are added to the simulated Doppler

data. Figure 7 has the same representation as the previous one, however, in 2009 (low solar

activity). In this case, it appears that both (IRI and IONEX) express the total electron content

in a similar way. Thus, the increase in DCP location error is consistent to any chosen model

for obtaining VTECU values. It depicts the importance of the ionospheric correction in high


Ionosphere: IONEX

Time

Location of the DCP no.32590—2001, October ionosphere effect evaluation

Ionosphere: IRI

Without correction

Err

or(k

m)

0

10/

19/

01

12:0

0

10

/19/

01 1

8:0

0

10/

20/

01 0

:00

10/

20/

01 6

:00

10

/20/

01 1

2:0

0

10

/20/

01 1

8:0

0

10/

21/

01 0

:00

10/

21/

01 6

:00

10

/21/

01 1

2:0

0

10

/21/

01 1

8:0

0

10/

22/

01

0:0

0

10/

22/

01 6

:00

10

/22/

01 1

2:0

0

10

/22/

01 1

8:0

0

10/

23/

01 0

:00

0.2

0.4

0.6

0.8

1

Figure 8: Ionospheric effect evaluation in DCP no. 32590 location error on October 19–22, 2001 (locationerrors without ionospheric effect (blue), with the addition of the ionospheric delay using IRI model (pink),and with the addition of the ionospheric delay using IONEX model (green)).

solar activity periods, because according to Figure 6, VTECU values jump from about 40 to

more than 120 × 1016 electrons/m2 at the end of the day.

With the data of VTECU from different models (IONEX e IRI), in high and low solar

activity periods, geographical location error for DCP no. 32590 due to ionosphere delay is

generated.

A first analysis of these results shows that the location error due to ionospheric effect

cannot be neglected because the error can increase to around 50% and the model which must

be used depends mainly on the intensity of solar activity. In fact, in low solar activity, both

models yield VTECU values with similar values. However, in high solar activity, IONEX has

yielded the values in a best way, mainly in the region that is called equatorial anomaly, close

to magnetic equator, which can be easily observed in Figure 4.

Figures 8 and 9 show DCP no. 32590 location errors in 2001 and 2009, respectively,

including the ionospheric effect and making use of VTECU values from both models. These

results are compared with the ideal location, that is, without the inclusion of ionospheric

effect.

3.2. Real Doppler Data for DCP no. 109 with Ionospheric and TroposphericEffects Correction

This second phase of the study aimed at applying the ionosphere correction in high and

low solar activity periods making use of IONEX model, which was previously studied, as

well as the tropospheric delay correction with climatic data obtained from National Climatic

Data Center—NOAA Satellite and Information Service [11]. Due to the availability of real


Ionosphere: IONEX

Time

Location of the DCP no.32590—2009, October ionosphere effect evaluation

Ionosphere: IRI

Without correction

Err

or(k

m)

0

10/

19/

09

12:0

0

10/

19/

09

18:0

0

10/

20/

09

0:0

0

10/

20/

09

6:0

0

10/

20/

09

12:0

0

10/

20/

09

18:0

0

10/

21/

09

0:0

0

10/

21/

09

6:0

0

10/

21/

09

12:0

0

10/

21/

09

18:0

0

10/

22/

09

0:0

0

10/

22/

09

6:0

0

10/

22/

09

12:0

0

10/

22/

09

18:0

0

10/

23/

09

0:0

0

0.2

0.4

0.6

0.8

1

Figure 9: Ionospheric effect evaluation in DCP no. 32590 location error on October 19–22, 2009 (locationerrors without ionospheric effect (blue), with the addition of the ionospheric delay using IRI model (pink)and with the addition of the ionospheric delay using IONEX model (green)).

Doppler shift data, it has been used the DCP no. 109, located in French Guyana (5.1860◦S and

52.687◦W), on October 06–15, 2001 for high solar activity and on October 01–14, 2008 for low

solar activity.

Figure 10 shows DCP no. 109 location error without ionospheric and tropospheric

corrections (blue), with the ionospheric correction using IONEX (green) and with both

corrections, ionospheric and tropospheric (pink) for the high solar activity period. Figure 11

makes use of the same color code and shows DCP no. 109 location errors for the low solar

activity period.

As in the simulated results, it is observed in real cases of DCP location that both

ionospheric and tropospheric effect should be inserted in the signal delay correction to

minimize the location error. It also observed that the ionospheric delay is very representative,

especially in maximum solar activity period, resulting in location error around 1 km, while

the tropospheric delay takes a smaller error scale, about some dozen of meters. In low

solar activity period, the ionosphere has much less influence on location error while the

troposphere is in the same order as mentioned above.

4. Conclusions and Recommendation

Due to high solar radiation in the equatorial region and Earth electric and magnetic field, the

electron density in the ionosphere layers suffers sensitive consequences mainly in Brazilian

regions or any areas close to magnetic equator. After some tests it was observed that, in such

regions, it is mandatory the use of an ionosphere correction model that takes into account

this effect called equatorial anomaly, depending on the solar activity intensity during the

period in which the location is accomplished. Then, a detailed study shows VTECU values


Ionosphere (IONEX)

Time

Ionosphere + troposphere (NOAA)

Without correction

Err

or(k

m)

0

10/

6/

01

12:0

0

10/

6/

01

21:3

510/

6/

01

19:5

2

10/

7/

01

20:4

9

10/

8/

01

20:0

4

10/

9/

01

19:2

2

10/

10/

01

18:3

3

10/

11/

01

19:3

5

10/

12/

01

18:4

6

10/

13/

01

18:0

2

10/

15/

01

18:1

6

10/

7/

01

0:0

0

10/

7/

01 1

2:0

0

10/

8/

01 0

:00

10/

8/

01 1

2:0

0

10/

9/

01 0

:00

10/

9/

01 1

2:0

0

10/

10/

01 0

:00

10/

10/

01 1

2:0

0

10/

11/

01 0

:00

10/

11/

01 1

2:0

0

10/

12/

01 0

:00

10/

12/

01 1

2:0

0

10/

13/

01 0

:00

10/

13/

01 1

2:0

0

10/

14/

01 0

:00

10/

14/

01 1

2:0

0

10/

15/

01 0

:00

10/

15/

01 1

2:0

0

10/

16/

01 0

:00

0.5

1

1.5

2

2.5

3

3.5

Location of the DCP no.109—2001, October ionospheric and tropospheric effects correction

Figure 10: Evaluation of DCP no. 109 error location on October 06–15, 2001 (higher solar activity).

Ionosphere (IONEX)

Time

Location of the DCP no.109—2008, October ionospheric and tropospheric effects correction

Ionosphere + troposphere (NOAA)

Without correction

10/

1/

08 0

:00

10/

2/

08 0

:00

10/

1/

08 1

9:0

510/

2/

08 1

8:1

7

10/

2/

08 2

1:5

4

10/

3/

08 1

7:3

0

10/

4/

08 2

0:1

8

10/

5/

08 1

9:3

0

10/

7/

08 1

9:4

5

10/

8/

08 1

8:5

7

10/

10/

08 1

7:2

2

10/

12/

08 1

7:3

8

10/

13/

08 1

6:4

9

10/

14/

08 1

6:0

1

10/

3/

08 0

:00

10/

4/

08 0

:00

10/

5/

08 0

:00

10/

6/

08 0

:00

10/

7/

08 0

:00

10/

8/

08 0

:00

10/

9/

08 0

:00

10/

10/

08 0

:00

10/

11/

08 0

:00

10/

12/

08 0

:00

10/

13/

08 0

:00

10/

14/

08 0

:00

10/

15/

08 0

:00

Err

or(k

m)

0

0.5

1

1.5

2

2.5

3

3.5

Figure 11: Evaluation of DCP no. 109 error location on October 01–14, 2008 (low solar activity).


for each DCP location in order to have its location errors minimized as much as possible. A

comparison was made between two mentioned models (IRI and IONEX) taking into account

the local time of the day, the year of high or low solar activity, and the region where the

ionospheric effect can cause high location errors.

Through VTECU analysis, mainly in the equatorial region, it is observed the influence

on DCPs location, whereas better the ionospheric model better the platform location.

According to the results, it is verified that the ionospheric data provided by IONEX are,

generally, better than IRI data, especially in high solar activity periods. In this case, VTECU

values from IONEX model reach values 2-3 times higher than IRI model, resulting in an

ionospheric correction difference of up to 1 km.

When IRI and IONEX model data are used in high solar activity periods, the VTECU

values may be quite different and consequently DCPs location. In low solar activity periods

both models have shown similar performance.

Simulated ideal Doppler shift measurements were used for the first case. It is shown

that the geographical location has its error increased due to the ionospheric effect, and such

effect cannot be neglected when the platform location is made with real data. For this first

case, IONEX model allows an improvement of about 1 km in location (2001) around 18:00 UT

compared to IRI model.

After analyzing VTECU values and its influence on DCP no. 32590 location for the

simulated ideal case, it was chosen IONEX data to model the ionosphere. In addition to

the ionospheric correction, the second phase of the work considered the tropospheric effect

whose data depend largely on climatic conditions in the regions where DCPs are. Such data

can be found easily by the users through the National Climatic Data Center [11]. The work is

based on real Doppler data of DCP no. 109, taking into account high and low solar activity,

in 2001 and 2008, respectively. The analysis is made under the following conditions: first, no

correction was applied; then only the ionospheric correction is applied to the location process;

finally ionospheric and tropospheric delays are corrected according to the conditions listed

above.

Results show that DCPs location error is minimized when the ionospheric effect is

considered, mainly in high solar activity period, when VTECU values are about 3 times larger

than low solar activity period. The tropospheric delays correction also decreases location

error, but in smaller proportions.

Finally, it is important to note that both delays, ionospheric and tropospheric, should

be corrected to obtain a better platform location, mainly when mobile platforms were used.

The choice of ionospheric model depends largely on the intensity of solar activity, because

the level of these activities influences the ionosphere density, due to high VTECU values.

References

[1] J. A. Klobuchar, “Ionospheric effects on GPS,” in Global Positioning System: Theory and Applications,B. W. Parkinson and J. J. Spilker, Eds., vol. 1, pp. 485–515, American Institute of Aeronautics andAstronautics, Cambridge, UK, 1996.

[2] C. C. Celestino, C. T. De Sousa, H. Koiti Kuga, and W. Yamaguti, “Errors due to the troposphericand ionospheric effects on the geographic location of data collection platforms,” Revista Brasileira deGeofisica, vol. 26, no. 4, pp. 427–436, 2008.

[3] C. C. Celestino, C. T. Sousa, W. Yamaguti, and H. K. Kuga, “Evaluation of tropospheric andionospheric effects on the geographic localization of data collection platforms,” Mathematical Problemsin Engineering, vol. 2007, Article ID 32514, 11 pages, 2007.


[4] P. D. O. Camargo, J. F. G. Monico, and L. D. D. Ferreira, “Application of ionospheric corrections in theequatorial region for L1 GPS users,” Earth, Planets and Space, vol. 52, no. 11, pp. 1083–1089, 2000.

[5] D. Bilitza, “International Reference Ionospheric Model–IRI,” http://modelweb.gsfc.nasa.gov/models/iri.html.

[6] “CDDIS – NASA’s Archive of Space Geodesy Data,” ftp://cddis.gsfc.nasa.gov/pub/gps/products/ionex/.

[7] K. Aksnes, P. H. Andersen, and E. Haugen, “A precise multipass method for satellite Dopplerpositioning,” Celestial Mechanics, vol. 44, no. 4, pp. 317–338, 1988.

[8] S. Schaer and W. Gurtner, “IONEX: the IONosphere map eXchange format version 1,” in Proceedingsof the IGS AC Workshop, Darmstadt, Germany, February 1998.

[9] J. F. G. Monico, Posicionamento pelo GNSS: Descricao, Fundamentos e Aplicacoes, UNESP, Sao Paulo,Brazil, 2000.

[10] D. D. McCarthy and G. Petit, “Tropospheric Model. International Earth Rotation and ReferenceSystem Service (IERS),” IERS Technical Note, no. 32, pp. 99–103, 2003.

[11] “National Climatic Data Center – NOAA Satellite and Information Service,” http://www7.ncdc.noaa.gov/IPS/mcdw/mcdw.html.

[12] C. T. de Sousa, Geolocation of transmitters by satellites using Doppler shifts in near real time, Ph.D. thesis,Space Engineering and Technology, Space Mechanics and Control Division, INPE - Instituto Nacionalde Pesquisas Espaciais, Sao Jose dos Campos, Brazil, 2000.

[13] A. DasGupta, A. Paul, and A. Das, “Ionospheric total electron content (TEC) studies with GPS in theequatorial region,” Indian Journal of Radio & Space Physics, vol. 36, pp. 278–292, 2007.


Research ArticleStudy of Stability of Rotational Motion ofSpacecraft with Canonical Variables

William Reis Silva,1 Maria Cecılia F. P. S. Zanardi,1Regina Elaine Santos Cabette,2, 3

and Jorge Kennety Silva Formiga3, 4

1 Group of Orbital Dynamics and Planetology, Sao Paulo State University (UNESP),Guaratingueta 12516-410, SP, Brazil

2 Department of Mathematics, Sao Paulo Salesian University (UNISAL), Lorena 12600-100,SP, Brazil

3 Space Mechanic and Control Division, National Institute for Space Research (INPE),Sao Jose dos Campos 12227-010, SP, Brazil

4 Department of Aircraft Maintenance and Aeronautical Manufacturing, Faculty ofTechnology (FATEC), Sao Jose dos Campos 12247-004, SP, Brazil

Correspondence should be addressed to William Reis Silva, [email protected]

Received 16 July 2011; Revised 10 October 2011; Accepted 23 October 2011


Copyright q 2012 William Reis Silva et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This work aims to analyze the stability of the rotational motion of artificial satellites in circular orbitwith the influence of gravity gradient torque, using the Andoyer variables. The used method in thispaper to analyze stability is the Kovalev-Savchenko theorem. This method requires the reductionof the Hamiltonian in its normal form up to fourth order by means of canonical transformationsaround equilibrium points. The coefficients of the normal Hamiltonian are indispensable in thestudy of nonlinear stability of its equilibrium points according to the three established conditionsin the theorem. Some physical and orbital data of real satellites were used in the numericalsimulations. In comparison with previous work, the results show a greater number of equilibriumpoints and an optimization in the algorithm to determine the normal form and stability analysis.The results of this paper can directly contribute in maintaining the attitude of artificial satellites.

1. Introduction

Stability analysis of the rotational motion of a satellite taking into account the influence of

external torques is very important in maintaining the attitude to ensure the success of a space

mission.

Recently, some studies on the subject have been developed, and they motivated the

development of this paper. In [1, 2] is presented a study on the stability of the rotational

motion of artificial satellites in an elliptic orbit, considering disturbance due to gravity


gradient torque, using canonical formulations and the Andoyer canonical variables. These

studies use the procedure presented in [3] to determine a normal form of the Hamiltonian up

to 4th order.

In [4] is developed a numerical-analytical method for normalization of Hamiltonian

systems with 2 and 3 degrees of freedom. The normal form is obtained using the Lie-Hori

method [5]. The stability analysis of the system is made by the Kovalev-Savchenko theorem

[6]. The most important role of this work is the results obtained analytically for the generating

function of 3rd order, necessary for determining the coefficients of the normal Hamiltonian

of 4th order.

Thus, the objective of this work is to optimize the stability analysis developed in

[2] to determine the equilibrium points, the normal form for dynamical systems with two

degrees of freedom, and the applications of the Kovalev-Savchenko theorem [6]. It will be

done by applying the expressions obtained in [4] for the coefficients of the normal 4th-order

Hamiltonian.

In this paper equilibrium points and/or regions of stability are established when

parcels associated with gravity gradient torque acting on the satellite are included in the

equations of rotational motion.

The Andoyer variables are used to describe the rotational motion of the satellite

in order to facilitate the application of methods of stability of Hamiltonian systems.

The Andoyer canonical variables [7] are represented by generalized moments (L1, L2, L3)and by generalized coordinates (1, 2, 3) that are outlined in Figure 1. The angular

variables 1, 2, 3 are angles related to the satellite system Oxyz (with axes parallel to the

spacecraft’s principal axes of inertia) and equatorial system OXYZ (with axes parallel to the

axis of the Earth’s equatorial system). Variables metrics L1, L2, L3 are defined as follows: L2

is the magnitude of the angular momentum of rotation �L2, L1 is the projection of �L2 on the

z-axis of principal axis system of inertia (L1 = L2 cos J2, where J2 is the angle between the

z-satellite axis and �L2), and L3 is the projection of �L2 on the Z-equatorial axis (L3 = L2 cos I2,

where I2 is the angle between Z-equatorial axis and �L2).The nonlinear stability of equilibrium points of the rotational motion is analyzed here

by the Kovalev-Savchenko theorem [6], which requires the normalized Hamiltonian up to

terms of fourth order around the equilibrium points.

The equilibrium points are found from the equations of motion. With the application

of the Kovalev-Savchenko theorem, it is possible to verify if they remain stable under the

influence of terms of higher order of the normal Hamiltonian.

In this paper numerical simulations were made for two hypothetical groups of

artificial satellites, considering them in a circular orbit and with symmetric shape in relation

to their physical and geometric characteristics. The satellites are classified as medium and

small sized; they have orbital data and physical characteristics similar to real satellites.

2. Equations of Motion

The Andoyer variables, defined above, are used to characterize the rotational motion of a

satellite around its center of mass [7], and the Delaunay variables describe the translational

motion of the center of mass of the satellite around the Earth [8].The Delaunay variables (L,G,H, l, g, h) are defined as [8] L = M

√μa,G =

L√

1 − e2, H = G cos I, l is the mean anomaly, g is the argument of perigee, h is the longitude


Plane perpendicular

Equatorial plane

X

Z

xz

y

principal axis of inertia

to the angular

m om entum vector

Plane perpendicular to the

Y

L1

L2

L3

J2

J2I2

I2

l1

l2

l3

ym

zm

xm

O

Figure 1: The Andoyer canonical variables.

of the ascending node, M is the mass of the satellite, μ is the Earth gravitational parameter, I

is the inclination of the orbit, a is the semimajor axis, and e is the orbital eccentricity.

Here it is assumed that the satellites are in a circular orbit, which differs from the

study presented in [2] which considers satellites in eccentric orbits. This consideration was

adopted to simplify the Hamiltonian of the problem, which is extensive [1], and to facilitate

the stability analysis of the equilibrium points.

Thus, assuming that the satellites have well-defined circular orbit, the main focus of

the work is only aimed at stability analysis of the rotational movement of the satellites in

study.

In this case the Hamiltonian of the problem is expressed in terms of the Andoyer and

Delaunay variables [9] as follows:

F(L1, L2, L3, 2, 3, L, G,H, l, g, h

)= Fo(L, L1, L2) + F1

(L1, L2, L3, 2, 3, L, G,H, l, g, h

), (2.1)

where Fo is the unperturbed Hamiltonian and F1 is the term of the Hamiltonian associated

with the disturbance due to the gravity gradient torque, both are described respectively by

[10]

Fo = −μ2M3

2L2+

1

2

((1

C− 1

2A− 1

2B

)L1

2 +1

2

(1

A+

1

B

)L2

2

)+

1

4

(1

B− 1

A

)(L2

2 − L12)

cos 21,

F1 =μ4M6

L6

[2C −A − B

2H1(m, Ln) +

A − B4

H2(m, Ln)],

(2.2)


where m = 2, 3 and n = 1, 2, 3; A,B, and C are the principal moments of inertia of the satellite

on x-axis, y-axis, and z-axis, respectively; H1 and H2 are functions of the variables (m, Ln),where 2 and 3 appear in the arguments of cosines. The complete analytical expression is

presented in [1] for eccentric orbits. In this paper F1 will be simplified considering the satellite

in a circular orbit; its complete analytical expression is given in the Appendix.

Thus, the equations of motion associated with the Hamiltonian (2.1) are given by

didt

=∂F

∂Li,

dLidt

= −∂F∂i

(i = 1, 2, 3). (2.3)

These equations are used to find the possible equilibrium points of the rotational

movement.

In this study, it is also considered that the satellite has two of its principal moments

of inertia equal, B = A. With this relationship, the variable 1 will not be present in the

Hamiltonian, reducing the dynamic system to two degrees of freedom, a necessary condition

for applying the stability theorem chosen for analysis of equilibrium points.

3. Determining the Normal Form of the Hamiltonian ofthe Rotational Motion

3.1. Normal Form of the Hamiltonian of 2nd Order andLinear Stability Analysis

Because the variable 1 is a cyclic coordinate, the Hamiltonian (2.1) is written as

F(L2, L3, 2, 3) and the equations of motion (2.3) can be written in vector form as [1]

w = JFw, (3.1)

where w is the state vector and Fw is the matrix of partial derivatives with respect to their

respective variables given by

w =

⎛⎜⎜⎝2

L2

3

L3

⎞⎟⎟⎠, Fw=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂F

∂2

∂F

∂L2

∂F

∂3

∂F

∂L3

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(3.2)

and J a symplectic matrix given by

J =

⎛⎜⎜⎝0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

⎞⎟⎟⎠. (3.3)


Now, it is necessary to linearize the system around the equilibrium point. However,

it is convenient to make a translation of the coordinates in the Hamiltonian (2.1) so that the

origin coincides with the equilibrium point under study; it means that

L1 = L1e ,

2 = 2e + q1,

L2 = L2e + p1,

3 = 3e + q2,

L3 = L3e + p2,

(3.4)

where L1e, 2e, L2e, 3e, and L3e are the coordinates of equilibrium points.

With this translation we can expand the Hamiltonian in Taylor series around the

new origin that is nothing more than expand the Hamiltonian in the neighborhoods of the

equilibrium point when q1 = p1 = q2 = p2 = 0. Then,

F(m, Ln) =∞∑k=2

Fk(m, Ln) = F2(m, Ln) + F3(m, Ln) + F4(m, Ln) + · · · , (3.5)

where m = 2, 3, n = 1, 2, 3, and Fk is the Hamiltonian expanded up to terms of order k, with

k = 2, 3, 4, . . . .

The Hessian P of the problem is calculated from the Hamiltonian expanded up to 2nd

order around the equilibrium point:

P =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂2F

∂q21

∂2F

∂q1∂p1

∂2F

∂q1∂q2

∂2F

∂q1∂p2

∂2F

∂p1∂q1

∂2F

∂p21

∂2F

∂p1∂q2

∂2F

∂p1∂p2

∂2F

∂q2∂q1

∂2F

∂q2∂p1

∂2F

∂q22

∂2F

∂q2∂p2

∂2F

∂p2∂q1

∂2F

∂p2∂p1

∂2F

∂p2∂q2

∂2F

∂p22

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (3.6)

Thus, the system of linearized equations can be written as

W = JPW, (3.7)

where W is the state vector in the new variables

W =

⎛⎜⎜⎝q1

p1

q2

p2

⎞⎟⎟⎠. (3.8)


So, to find the eigenvalues of the reduced linear system, we should diagonalize the

matrix JP present in (3.7) by means of the determinant det(λI−JP) = 0, where I is an identity

matrix and λ are the eigenvalues of the matrix JP . If these eigenvalues are pure imaginary

λj = ±iωj , the normal form of 2nd-order Hamiltonian can be expressed as [2]

H∗2

(qj , pj

)=

2∑j=1

ωj

2

(qj

2 + pj2). (3.9)

3.2. Extension of the Normal Form of the Hamiltonian to Higher Orders

Extending the process of normalization of the Hamiltonian to higher orders (H3,H4, . . .), the

Hamiltonian normal in terms of variables qi and pi can be expressed by

H(qj , pj

)=

2∑j=1

ωi

2

(qj

2 + pj2)+H∗

3

(qj , pj

)+H∗

4

(qj , pj

)+ · · · . (3.10)

To obtain the Hamiltonian normal H∗3 ,H

∗4 , . . ., it is necessary to use the method of Lie-

Hori. According to [11], it is desirable that the normal Hamiltonian is written in complex

variables before using this method.

This transformation of variables is given by

qj =1√2

(xj + iyj

), pj =

1√2

(ixj + yj

)(3.11)

with inverse given by

xj =1√2

(qj − ipj

), yj =

1√2

(−iqj + pj

). (3.12)

Thus, the normal Hamiltonian of 2nd order is given by

H∗2

(xj , yj

)=

2∑j=1

iωjxjyj (3.13)

and normal Hamiltonian until the 4th order in complex variables is represented by

H∗(xj , yj) = 2∑j=1

iωjxjyj +H∗3

(xj , yj

)+H∗

4

(xj , yj

)+ · · · , (3.14)


where H∗3(x, y) and H∗

4(x, y) can be expressed briefly as

H∗3

(x, y)=∑

|α|+|β|=3

h3,α,βxαyβ, (3.15)

H∗4

(x, y)=∑

|α|+|β|=4

h4,α,βxαyβ. (3.16)

In (3.15) and (3.16), is always assumed it α, β > 0. For (3.15), |α| + |β| = 3 means that

the sum of the exponents of the new variables must obey the order of the Hamiltonian H∗3 ,

and, for (3.16), |α| + |β| = 4 means that the sum of the exponents of the new variables must

obey the order of the Hamiltonian H∗4 . These equations are presented entirely in [1, 4].

Thus, we have the Hamiltonian written in complex variables. It is important to note

that H2 is already normalized, and now we can apply the method of Lie-Hori to find the

normal form to higher orders H∗3 ,H

∗4 .

3.2.1. Series Method of Lie-Hori

The normal form is obtained from the expansion of the Hamiltonian in terms of the Lie series

given by [12, 13]

Hnew = H∗ + {H∗, G} + 1

2!{{H∗, G}, G} + 1

3!{{{H∗, G}, G}, G} + · · · , (3.17)

where Hnew is normal Hamiltonian obtained by canonical transformations around the

equilibrium point,G = G(xi, yi) is the generating function developed as a power series, where

each degree of Gn is a sum of homogeneous polynomials of degree n, H∗ = H∗(x, y) is the

original Hamiltonian, and {H∗, G} is the Poisson brackets of order r + s − 2, where r and s

represent the order of the polynomials H∗ and G, respectively.

The Poisson bracket is defined as

{H∗, G} =∂H∗

∂x

∂G

∂y− ∂H∗

∂y

∂G

∂x. (3.18)

The new Hamiltonian Hnew after the new canonical transformation in the neighbor-

hoods of identity can be expressed as [2]

Hnew = Hnew2 +Hnew

3 +Hnew4 + · · · . (3.19)


So the new Hamiltonian, ordered by degree up to 4th order, can be expressed following

the next form [11, 13]:

2nd order: Hnew2 = H∗

2 , (3.20)

3rd order: Hnew3 = H∗

3 +{H∗

2 , G3

}, (3.21)

4th order: Hnew4 = H∗

4 +{H∗

3 , G3

}+

1

2!

{{H∗

2 , G3

}, G3

}+{H∗

2 , G4

}. (3.22)

As Hnew2 is already given in its normal form, the next step is to calculate the generating

function which leads to the normal form up to 4th-order terms. This procedure allows to

obtain the minimum number of nonresonant monomials defined in [12]; thus, the generating

function Gn is chosen to eliminate the complex variables in the total Hamiltonian (3.5) in

which the monomials xj and yj have different exponents of the desired degree, leaving

only the monomials that carry on the resonance of the intrinsic Hamiltonian systems. The

following is a procedure to obtain the normal Hamiltonian to 4th order [1, 4].(I) It is known that Hamiltonian systems are naturally resonant in even orders, 2nd,

4th, and so forth [10, 12], so to find the normal Hamiltonian until the 4th order, the 3rd

disturbance Hnew3 must first be removed from Hnew, it means

Hnew3 = 0. (3.23)

Assuming H∗3 is briefly defined in (3.15) to eliminate terms of 3rd order, we consider

the generating function G3(x, y) expressed in abbreviated form as

G3

(x, y)=∑

|α|+|β|=3

g3,α,βxαyβ. (3.24)

Then it is necessary to find G3(x, y) of the third-order homological equation (3.21),where

{H∗

2 , G3

}= −H∗

3 . (3.25)

After some algebraic manipulations, we find G3(x, y) given by [4]

G3

(x, y)=∑

|α|+|β|=3

−h3,α,β⟨

β − α,W⟩xαyβ. (3.26)

Note that this equation is well defined if 〈β − α,W〉/= 0, where 〈β − α,W〉 represents

an inner product between β − α = (β1 − α1, β2 − α2, . . . , βn − αn) and W = (λ1, λ2, . . . , λn) =(iω1, iω2, . . . , iωn) which are pure imaginary eigenvalues obtained from the quadratic part of

normal Hamiltonian. The complete analytical representation for 〈β − α,W〉 was presented in

[4].


For the resonance condition it is not accepted that 〈β − α,W〉/= 0, which is always true

for monomials of third-order Hamiltonian when considering only the natural resonance of

the Hamiltonian.

(II) This process is repeated for the polynomial of degree 4 finding G4(x, y) that will

eliminate some of monomials H∗4 in (3.16). The remaining monomials in Hnew

4 are called

resonant monomials [3]. Then (3.22) can be reduced as [1]

{G4,H

∗2

}+Hnew

4 = H∗4 +{H∗

3 , G3

}+

1

2!

{{H∗

2 , G3

}, G3

}, (3.27)

{G4,H

∗2

}+Hnew

4 = F4. (3.28)

Equation (3.28) is called homological equation of 4th order, and it can be rewritten as

{G4,H

∗2

}= F4 −Hnew

4 ,{G4,H

∗2

}= R4.

(3.29)

Assume that R4(x, y) and G4(x, y) will be expressed in short form as

R4

(x, y)=∑

|α|+|β|=4

r4,α,βxαyβ, (3.30)

G4

(x, y)=∑

|α|+|β|=4

g4,α,βxαyβ. (3.31)

After some algebraic manipulations, we find G4(x, y) given by [4]

G4

(x, y)=∑

|α|+|β|=4

r4,α,β⟨α − β,W

⟩xαyβ, (3.32)

where 〈α − β,W〉/= 0 represents an inner product as mentioned above.

Thus, using (3.34) in (3.29) it is possible to find Hnew4 after some algebraic

manipulations.

This procedure can be repeated for any order that is required in order to find a normal

form of the new Hamiltonian Hnew. In this paper it is considered the 4th order of the Hnew.

3.3. Normal Form of the Hamiltonian of 4th Order andNonlinear Stability Analysis

As we get the normal form of 2nd-order Hamiltonian (3.9), we can apply the method

described previously to find the Hamiltonian normal until the 4th order using the

Hamiltonian in complex variables (3.14) represented by [1, 4]

H∗(xj , yj) = iω1x1y1 + iω2x2y2 +H∗3

(x1, y1, x2, y2

)+H∗

4

(x1, y1, x2, y2

)+ · · · , (3.33)


where H∗3(x1, y1, x2, y2) and H∗

4(x1, y1, x2, y2) are obtained from expressions

H3∗(x1, y1, x2, y2

)=∑

|α|+|β|=3

h3,α1,β1,α2,β2x1

α1y1β1x2

α2y2β2 ,

H∗4

(x1, y1, x2, y2

)=∑

|α|+|β|=4

h4,α1,β1,α2,β2x1

α1y1β1x2

α2y2β2 .

(3.34)

Using the method of Lie-Hori presented before, it is possible to determine the

generating function G3(x1, y1, x2, y2) in order to satisfy the homological equation (3.25),where G3(x1, y1, x2, y2) was determined analytically in [4]. This generating function is

responsible for the elimination of the 3rd-order terms of the Hamiltonian (3.33).The generating function G3(x1, y1, x2, y2) is used to find the terms of 4th order, F4,

and perform the separation described in (3.28), obtaining H4 and G4(x1, y1, x2, y2). The

coefficients of the normal form Hnew4 can be calculated using the terms obtained in H4 and

the eigenvalues of H2.

Thus, the normal Hamiltonian in complex variables separated by degree can be

expressed as follows[4]:

2nd order: Hnew2 = iω1x1y1 + iω2x2y2, (3.35)

3 rd order: Hnew3 = 0, (3.36)

4th order: Hnew4 = δ11

(x1y1

)2 + δ12

(x1y1x2y2

)+ δ22

(x2y2

)2, (3.37)

where ωj(j = 1, 2) is the imaginary part of eigenvalues associated with the matrix defined by

the product of a matrix symplectic of order 4 with the Hessian of the Hamiltonian expanded

in Taylor series up to 2nd order around the equilibrium point; δij are real coefficients obtained

from the combination of hk,α,βxαyβ (k = 3, 4).

However, the Kovalev-Savchenko theorem requires that the normal Hamiltonian is in

real variables, and with the applications of the transformation of variables (3.12) the normal

form Hamiltonian can been expressed as follows:

2nd order: Hnew2 =

ω1

2

(q1

2 + p12)+ω2

2

(q2

2 + p22), (3.38)

3 rd order: Hnew3 = 0, (3.39)

4th order: Hnew4 = δ11

(q1

4 + p14 + 2q1

2p12)+ δ12

(q1

2q22 + q1

2p22 + p1

2q22 + p1

2p22)

+ δ22

(q2

4 + p24 + 2q2

2p22),

(3.40)

where δij are the coefficients of the Hamiltonian normal 4th order which are expressed

analytically in [4].


4. Kovalev-Savchenko Theorem

To use the stability theorem, the normal Hamiltonian of the problem is necessary as it was

discussed above.

Considering the normal Hamiltonian Ho, an analytics function of coordinates (qν) and

generalized moments (pν) to a fixed point P is expressed by

Ho =2∑ν=1

ωoν

2Rν +

2∑ν,υ=1

δoν,υ

4RνRυ +O5, (4.1)

where O5 represents higher-order terms; ωoν is the imaginary part of eigenvalues associated

with the matrix defined by the product of a 4th-order matrix symplectic with the Hessian

of the Hamiltonian expanded in Taylor series up to 2nd order around the equilibrium point;

δoν,υ depend on the eigenvaluesωo

ν and the coefficients of the Hamiltonian expanded in Taylor

series of 3rd and 4th order around the equilibrium point and they are presented analytically

in Formiga [4], and

Rm = qm2 + pm2 with m = 1, 2. (4.2)

The stability analysis is performed here by the Kovalev-Savchenko theorem [6] which

ensures that the motion is Lyapunov stable if the following conditions are satisfied.

(i) The eigenvalues of the reduced linear system are pure imaginary ±iωo1 and ±iωo

2.

(ii) The condition

k1ωo1 + k2ω

o2 /= 0 (4.3)

is valid for all k1 and k2 integers satisfying the inequality

|k1| + |k2| ≤ 4. (4.4)

(iii) The determinant Do must satisfy the inequality

Do = −(δo11ω

o2

2 − 2δo12ωo1ω

o2 + δo22ω

o1

2)/= 0, (4.5)

where δoν,υ are the coefficients of the normal 4th-order Hamiltonian.

This theorem says that a Hamiltonian reduced in its normal form up to 4th order, in

the absence of the resonance condition of the eigenvalues associated and if the condition (4.5)is satisfied, it is guaranteed the existence of tori invariant in a neighborhood small enough of

equilibrium position [14–16].


Figure 2: Flowchart representative of the stability analysis of equilibrium points.

5. Computational Algorithm for the Normal Form of the Hamiltonianof Rotational Motion and Analysis of Stability

In order to synthetize the process to analyze the stability of equilibrium points, the logical

sequence of the algorithm is now presented.

5.1. Stability Analysis of Equilibrium Points

See Figure 2.


Table 1: Quantitative summary of the classification of equilibrium points.

Satellite Stable points Unstable points

MP 2

58

Failed 1st condition Failed 2nd condition Failed 3rd condition

58 0 0

PP-1 7

43


43 0 0

PP-2 10

40


40 0 0

6. Numerical Simulations

As already mentioned, we consider two types of satellites: medium sized (MP), which

has similar orbital characteristics with the American satellite PEGASUS [17], and small

sized (PP), which has similar orbital characteristics of the Brazilian data collection satellite

SCD-1 and SCD-2 [18]. All the numerical simulations were developed using the software

MATHEMATICA.

Table 1 shows a quantitative summary of the equilibrium points found and their

stability, according to the criteria of Kovalev-Savchenko theorem [6]. It can be observed that,

when the first condition is not satisfied, the eigenvalues associated with the matrix JP are

real or are not pure imaginary. This equilibrium point is not linearly stable.

There were found totals of 60 equilibrium points for the MP satellite, 50 equilibrium

points to the PP-1 satellite and 50 equilibrium points for the PP-2 satellite. Tables 2, 3 and 4,

show two equilibrium points found in the simulations, one being Lyapunov stable and the

other unstable, to satellites MP, PP1 and PP2, respectively.

For the Lyapunov stable equilibrium points of Tables 2, 3, and 4 respectively, Table 5

shows the values of the angles I2, J2, the rotation speed ω, and the rotation period T .

These values characterize the nonexistence of singularities in these points; it means that

the angles I2 and J2 are not null or close to zero. By Figure 1, when these angles I2 and J2

are null or close to zero, the Andoyer variables l1, l2, and l3 are indeterminate, because it is

difficult to determine the intersection between the involved planes in the definitions of these

variables. This analysis was performed for all equilibrium points found in the simulations.

Table 6 shows the same conditions for the unstable points of Tables 2, 3, and 4.

7. Conclusion

In this paper we presented a semianalytical stability of the rotational motion of artificial

satellites, considering the influence of gravity gradient torque for symmetric satellite

in a circular orbit. Applications in numerical simulations, performed with the software

MATHEMATICA, were made to two types of satellites: medium (MP) and small (PP).Initially the points of equilibrium were determined using the physical, orbital, and

attitude characteristics of each satellite. Then the algorithm for stability analysis was applied

and it was obtained 2 stable equilibrium points for the satellite MP, 7 stable points for the

satellite PP-1, and 10 stable points for the satellite PP-2.


Table 2: Satellite MP: A = B = 3, 9499 ∗ 10−1 kg·(km)2, C = 1, 0307 ∗ 10−1 kg·(km)2.

Equilibrium points Lyapunov stable Unstable

L1(kg · km2/s) 2, 50725 ∗ 10−4 −1, 34202 ∗ 10−11

L2(kg · km2/s) 3, 85388 ∗ 10−4 −4, 19645 ∗ 10−10

L3(kg · km2/s) −2, 28562 ∗ 10−4 −3, 53991 ∗ 10−10

l2(rad) −1, 6 ∗ 10−7 0,0934286

l3(rad) 0,5235 0,41354

Table 3: Satellite PP-1: A = B = 9.20 ∗ 10−6 kg·(km)2, C = 13 ∗ 10−6 kg·(km)2.


L1(kg · km2/s) 2, 69365 ∗ 10−7 7, 13725 ∗ 10−11

L2(kg · km2/s) 2, 07921 ∗ 10−5 2, 14416 ∗ 10−10

L3(kg · km2/s) 1, 05633 ∗ 10−5 1, 07843 ∗ 10−10

l2(rad) 0,066560 −0,090962

l3(rad) 0,4 0,07

Table 4: PP-2: A = B = 28, 7 ∗ 10−4 kg·(km)2, C = 36, 9 ∗ 10−4 kg·(km)2.


L1(kg · km2/s) 6, 25509 ∗ 10−5 2, 48081 ∗ 10−11

L2(kg · km2/s) 7, 24797 ∗ 10−4 2, 77796 ∗ 10−11

L3(kg · km2/s) 2, 17564 ∗ 10−4 −2, 54792 ∗ 10−11

l2(rad) 0,098581 −1,68042

l3(rad) 1,42527 1,05253

For the satellite MP it was gotten only two equilibrium points because this satellite

has similar characteristics to the satellite PEGASUS, which is tumbling [17]. For satellites PP-

1 and PP-2 were obtained many other equilibrium points, but most of them were discarded,

because they lead to the Andoyer variables, a condition of uniqueness (it means that the

angles I2 and J2 are null or close to zero, and the Andoyer variables l1, l2, and l3 are

indeterminate).It can be observed that a larger number of stable equilibrium points were determined

in comparison with the results of [1, 2], which show the stability analysis of the rotational

motion with the gravity gradient torque, but with the satellite in an elliptic orbit.

An optimization was done in the algorithm of determining the normal form of the

Hamiltonian and the stability analysis, using expressions obtained by [4] for the coefficients

of the normal 4th-order Hamiltonian. The introduction of these expressions enabled a more

effective stability analysis for equilibrium points in comparison with the results of [1, 2]. It is

possible to say that the numerical simulations have become less laborious allowing analysis

of data in more numbers .

This paper presents results that can directly contribute in maintaining the attitude

of artificial satellites. Once the regions of stability are known for the rotational motion, a

smaller number of maneuvers to maintain the desired attitude can be accomplished. In this

case, a fuel economy can be generated to the satellite with propulsion system, increasing the

satellite’s lifetime.


Table 5: Analysis of possible singularities of the Anloyer variables for satellites MP, PP1 of and PP2.

Lyapunov stable equilibrium point of Table 2, 3 and 4, respectively.

Satellite I2 (rad) J2 (rad) ω (rad/s) T (s) Situation

MP 2,20566 0,862451 0,00373909 1680,4 Accepted

PP-1 1,03788 1,55784 1,59939 3,92848 Accepted

PP-2 1,26592 1,48439 0,196422 31,9882 Accepted

Table 6: Analysis of possible singularities of the Anloyer variables for satellites MP, PP1 and PP2.

Unstable equilibrium point of Table 2, 3, and 4, respectively.

Satellite I2 (rad) J2 (rad) ω (rad/s) T (s) Situation

MP 0,56693 1,53881 −4, 071 ∗ 10−9 −1, 543 ∗ 109 Accepted

PP-1 1,04377 1,23145 1, 64936 ∗ 10−5 380948,0 Accepted

PP-2 2,73176 0,46676 7, 52836 ∗ 10−9 8, 346 ∗ 108 Accepted

Appendix

The disturbance Hamiltonian F1, due to the gravity gradient torque, for a satellite in a circular

orbit with two of its principal moments of inertia equal, B = A, is given by [19]

F1 =μ4M7

L6

{C −AM

{((3

2

(L1

L2

)2

− 1

2

)

×(−1

2+

3

8

(1 +(H

G

)2

+(L3

L2

)2

− 3

(H

G

)2(L3

L2

)2)

− 3

8sin[2I] sin[2I2] cos[h − 3]

−3

8sin2[I]sin2[I2] cos[2h − 23]

)

− 3

16

(1 − 3

(H

G

)2)

sin[2I2] sin[2J2] cos[2]

+3

16

(1 − 3

(H

G

)2)

sin2[I2]sin2[J2] cos[22]

+3

16

(1 − L3

L2

)(1 + 2

L3

L2

)sin[2I] sin[2J2] cos[h − 3 − 2]

+3

16

(1 +

L3

L2

)(1 − 2

L3

L2

)sin[2I] sin[2J2] cos[h − 3 + 2]

+3

16

(1 − L3

L2

)sin2[I] sin[I2] sin[2J2] cos[2h − 23 + 2]

− 3

16

(1 − L3

L2

)sin2[I] sin[I2] sin[2J2]

]cos[2h − 3 + 22]


− 3

16

(1 − L3

L2

)sin[2I] sin[I2]sin2[J2] cos[h − 3 + 22]

+3

16

(1 +

L3

L2

)sin[2I] sin[I2]sin 2 [J2] cos[h − 3 − 22]

− 3

32

(1 − L3

L2

)2

sin2[I]sin2[J2] cos[2h − 23 + 22]

− 3

32

(1 +

L3

L2

)2

sin2[I]sin2[J2] cos[2h − 23 − 22]

)

+

(3

2

(L1

L2

)2

− 1

2

){3

8sin2[I]

(1 − 3

(L3

L2

)2)

× cos[2l + 2g

]− 3

8

(1 − H

G

)sin[I] sin[I2]

× cos[2l + 2g − (h − 3)

]− 3

16

(1 +

H

G

)2

sin2[I2]

× cos[2l + 2g + (2h − 23)

]− 3

16

(1 − H

G

)2

sin2[I2] cos[2l + 2g − (2h − 23)

]}

+9

32sin2[I] sin[2I2] sin[2J2] cos

[2l + 2g + 2

]+

9

32sin2[I] sin[2I2] sin[2J2] cos

[2l + 2g − 2

]− 3

16

(1 +

H

G

)(1 − L3

L2

)(1 + 2

L3

L2

)sin[I] sin[2J2]

× cos[2l + 2g + (h − 3) + 2

]− 3

16

(1 +

H

G

)(1 +

L3

L2

)(1 − 2

L3

L2

)sin[I] sin[2J2]

× cos[2l + 2g + (h − 3) − 2

]+

3

16

(1 − H

G

)(1 − L3

L2

)(1 + 2

L3

L2

)sin[I] sin[2J2]

× cos[2l + 2g − (h − 3) − 2

]+

3

16

(1 − H

G

)(1 +

L3

L2

)(1 − 2

L3

L2

)sin[I] sin[2J2]

× cos[2l + 2g − (h − 3) + 2

]


+3

32

(1 +

H

G

)2(1 − L3

L2

)sin[I2] sin[2J2]

× cos[2l + 2g + (2h − 23) − 2

]− 3

32

(1 +

H

G

)2(1 +

L3

L2

)sin[I2] sin[2J2]

× cos[2l + 2g + (2h − 23) + 2

]+

3

32

(1 − H

G

)2(1 − L3

L2

)sin[I2] sin[2J2]

× cos[2l + 2g − (2h − 23) + 2

]− 3

32

(1 − H

G

)2(1 +

L3

L2

)sin[I2] sin[2J2]

× cos[2l + 2g − (2h − 23) − 2

]− 9

32sin2[I]sin2[I2]

× sin2[J2] cos[2l + 2g + 22

]− 9

32sin2[I]sin2[I2]sin2[J2] cos

[2l + 2g − 22

]+

3

16

(1 +

H

G

)(1 − L3

L2

)sin[I] sin[I2]sin2[J2]

× cos[2l + 2g + (h − 3) + 22

]− 3

16

(1 +

H

G

)(1 +

L3

L2


× cos[2l + 2g + (h − 3) − 22

]− 3

16

(1 − H

G

)(1 − L3

L2


× cos[2l + 2g − (h − 3) − 22

]+

3

16

(1 − H

G

)(1 +

L3

L2


× cos[2l + 2g − (h − 3) + 22

]− 3

64

(1 − H

G

)2(1 − L3

L2

)2

sin2[J2]

× cos[2l + 2g + (2h − 23) + 22

]+

3

64

(1 − H

G

)2(1 +

L3

L2

)2

sin2[J2]


× cos[2l + 2g + (2h − 23) − 22

]+

3

64

(1 +

H

G

)2(1 − L3

L2

)2

sin2[J2]

× cos[2l + 2g − (2h − 23) − 22

]+

3

64

(1 +

H

G

)2(1 +

L3

L2

)2

sin2[J2]

× cos[2l + 2g − (2h − 23) + 22

]}},

(A.1)

where: A = B and C are the principal moments of inertia on x-axis, y-axis and z-axis

respectively,M is the mass of the satellite, μ is the gravitational constant, I is the inclination of

the orbit, I2 is the inclination of the plane of angular momentum with the plane of the equator,

and J2 is the inclination of the principal plane with the plane of the angular momentum.

In (A.1) the generalized moments (L1, L2, L3) are implicit in some terms, using the

definitions of generalized moments present in the introduction and trigonometric properties,

they can be explicit replacing.

sin[I2] =

√1 − L3

2

L22,

sin2[I2] = 1 − L32

L22,

sin[2I2] =2L3

L2

√1 − L3

2

L22,

sin[J2] =

√1 − L1

2

L22,

sin2[J2] = 1 − L12

L22,

sin[2J2] =2L1

L2

√1 − L1

2

L22.

(A.2)

Acknowledgment

This present work was supported by CAPES.

References

[1] R. E. S. Cabette, Estabilidade do movimento rotacional de satelites artificiais, Dissertation, National Institutefor Space Research, Sao Paulo, Brazil, 2006.


[2] R. V. de Moraes, R. E. S. Cabette, M. C. Zanardi, T. J. Stuchi, and J. K. Formiga, “Attitude stability ofartificial satellites subject to gravity gradient torque,” Celestial Mechanics & Dynamical Astronomy, vol.104, no. 4, pp. 337–353, 2009.

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[4] J. K. Formiga, Formas normais no estudo da estabilidade para L4 no problema fotogravitacional, Dissertation,National Institute for Space Research, Sao Jose dos Campos, Brazil, 2009.

[5] G. Hori, “Theory of general perturbations for non-canonical system,” Astronomical Society of Japan,vol. 23, no. 4, pp. 567–587, 1971.

[6] A. M. Kovalov and A. Ia. Savchenko, “Stability of uniform rotations of a rigid body about a principalaxis,” Prikladnaia Matematika i Mekhanika, vol. 39, no. 4, pp. 650–660, 1975.

[7] H. Kinoshita, “First order perturbation of the two finite body problem,” Publications of the AstronomicalSociety of Japan, vol. 24, pp. 423–439, 1972.

[8] C. D. Murray and S. F. Dermott, Solar system dynamics, Cambridge University Press, Cambridge, 1999.[9] M. C. Zanardi, Movimento rotacional e translacional acoplado de satelites artificiais, Dissertacao de

mestrado, Instituto de Tecnologia Aeronautica, Sao Jose dos Campos, Sao Paulo, Brazil, 1983.[10] M. C. Zanardi, “Study of the terms of coupling between rotational and translational motions,” Celestial

Mechanics, vol. 39, no. 2, pp. 147–158, 1986.[11] T. J. Stuchi, “KAM tori in the center manifold of the 3-D Hill problem,” in Advanced in Space, O. C.

Winter and A. F. B. Prado, Eds., vol. 2, pp. 112–127, INPE, Sao Paulo, Brazil, 2002.[12] G. Hori, “Theory of general perturbations with unspecified canonical variables,” Publications of the

Astronomical Society of Japan, vol. 18, pp. 287–299, 1966.[13] S. Ferraz-Mello, Canonical Perturbation Theories-Degenerate Systems and Resonance, Springer, New York,

NY, USA, 2007.[14] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Di erential Equations, vol. 250 of Grundlehren

der MathematischenWissenschaften [Fundamental Principles of Mathematical Science], Springer, New York,NY, USA, 1983.

[15] V. I. Arnold, Equacoes Diferenciais Ordinarias, Mir, Moscow, Russia, 1985.[16] V. I. Arnold, “Metodos matematicos da mecanica classica,” Editora Mir Moscou, p. 479, 1987.[17] J. U. Crenshaw and P. M. Fitzpatrick, “Gravity effects on the rotational motion of a uniaxial artificial

satellite,” AIAAJ, vol. 6, article 2140, 1968.[18] H. K. Kuga, V. Orlando, and R. V. F. Lopes, “Flight dynamics operations during leap for the INPE’s

second environmental data collecting satellite SCD 2,” Journal of the Brazilian Society of MechanicalSciences, vol. 21, pp. 339–344, 1999.

[19] W. R. Silva, Estudo da estabilidade do movimento rotacional de satelites artificiais com variaveis canonicas,Qualificacao de mestrado, UNESP—Universidade Estadual Paulista, Sao Paulo, Brazil, 2011.


Research ArticleAnalysis of Filtering Methods for SatelliteAutonomous Orbit Determination UsingCelestial and Geomagnetic Measurement

Xiaolin Ning,1, 2, 3 Xin Ma,1, 2, 3 Cong Peng,1, 2, 3

Wei Quan,1, 2, 3 and Jiancheng Fang1, 2, 3

1 School of Instrumentation Science & Opto-electronics Engineering, Bei Hang University (BUAA),Beijing 100191, China

2 Science and Technology on Inertial Laboratory, Beijing 100191, China3 Fundamental Science on Novel Inertial Instrument & Navigation System Technology Laboratory,Beijing 100191, China

Correspondence should be addressed to Xiaolin Ning, [email protected]

Received 15 July 2011; Revised 30 September 2011; Accepted 5 October 2011


Copyright q 2012 Xiaolin Ning et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Satellite autonomous orbit determination (OD) is a complex process using filtering method tointegrate observation and orbit dynamic equations effectively and estimate the position andvelocity of a satellite. Therefore, the filtering method plays an important role in autonomous orbitdetermination accuracy and time consumption. Extended Kalman filter (EKF), unscented Kalmanfilter (UKF), and unscented particle filter (UPF) are three widely used filtering methods in satelliteautonomous OD, owing to the nonlinearity of satellite orbit dynamic model. The performance ofthe system based on these three methods is analyzed under different conditions. Simulations showthat, under the same condition, the UPF provides the highest OD accuracy but requires the highestcomputation burden. Conclusions drawn by this study are useful in the design and analysis ofautonomous orbit determination system of satellites.

1. Introduction

Orbit determination (OD) of satellite plays a significant role in satellite missions, aiming at

estimating the ephemeris of a satellite at a chosen epoch accurately. To date, the conventional

OD system is dominated by measurements based on (1) ground tracking approaches [1]such as range, range rate, and angle, and (2) Global Position System measurement [2, 3]. The

orbit determination technologies have shown fair performance on various space missions.

However, its high cost, lack of robustness to loss of contact, space segment degradation, and


Sensor subsystem

Measurement

Dynamic orbitmodel

Measurementmodel

Earth sensor

Magnetometer

Earth

Star sensor

Star

vector

Earth

vector

Position andvelocity

Magneticvector

OD system

Filter subsystem

Time update

Measurement

update

Estimator

Star

Model subsystem

SN

Figure 1: The process of orbit determination.

other factors promote the application of autonomous OD system, which is less costly and less

vulnerable in hostile environment [4].In general, orbit determination is the process of estimating the satellite’s state variables

(position and velocity) by comparing (in statistical sense) the difference between the

measurement data and the estimated data. Orbit determination system, as shown in Figure 1,

usually includes sensor subsystem, model subsystem, and filter subsystem. Sensor subsystem

contains sensing instruments, such as star sensor, earth sensor, and magnetometer, in order

to measure and process the original measurements which are functions of state variables.

Model system generates estimated data including state model and measurement model. In

the filter subsystem, the optimal algorithms (filtering methods) process both data from sensor

subsystem and from model subsystem and then estimate state variables.

Owing to the nonlinear dynamic model of satellite orbit motion, the filtering method

applied in OD system should be appropriate for nonlinear system [5, 6]. Extended Kalman

filter (EKF), unscented Kalman filter (UKF), and unscented particle filter (UPF) are three

main methods used in satellite OD system. The EKF is based on the analytical Taylor series

expansion of the nonlinear systems and measurement equations. It works on the principle

that the state distribution is approximated by a Gaussian random variable. However, the

Taylor series approximations in EKF introduce large errors due to the neglected nonlinearities

[7]. The UKF uses the true nonlinear model and a set of sigma sample points produced by

the unscented transformation to capture the mean and covariance of state, but the UKF has

the limitation that it does not apply to general non-Gaussian distribution [8, 9]. The particle

filter (PF) is a computer-based method for implementing a recursive Bayesian filter by Monte

Carlo simulations. The performance of the PF largely depends on the choices of importance

sampling density and resampling scheme [10, 11]. Among many improved PF methods, UPF

is a hybrid of the UKF and the particle filter which uses the UKF to get better importance

sampling density [12, 13]. It combines the merits of unscented transformation and particle

filtering and avoids their limitations.

A variety of autonomous orbit determination methods have been proposed and

explored, including a magnetometer-based OD method [14, 15], a celestial OD method

[16, 17], a landmark OD method [18, 19], and an X-ray pulsar OD method [20, 21]. The first

two methods can be used in low earth orbit (LEO) satellite autonomous orbit determination

system. Thus, in this paper, these two OD methods are selected for analysis.


This paper is divided into five sections. After this introduction, the basic descriptions

of three filtering methods in autonomous OD system are given in Section 2. Then, the state

model and measurement models in OD model subsystem are described in detail in Section 3.

In Section 4, simulations are shown for analyzing and comparing three filtering methods.

Finally, conclusions are drawn in Section 5.

2. Filtering Methods

The best known algorithms to solve the problem of autonomous satellite orbit determination

are the EKF, UKF, and UPF. In this section, we shall present the theories of the three filter

algorithms. These algorithms will be incorporated into the filtering framework based on the

dynamic state-space model as follows:

xk = f(xk−1, k − 1) +wk,

zk = h(xk, k) + vk,(2.1)

where xk−1 denotes the state of the system at time k − 1, zk denotes the observations at step k,

wk denotes the process noise, and vk denotes the measurement noise. The mappings f and h

represent the process and measurement models. E(wkwTj ) = Qk, E(vkvTj ) = Rk, for all k, j, and

Qk is the process noise covariance at step k, Rk is the measurement noise covariance at step k.

2.1. Extended Kalman Filter

A Kalman filter that linearizes about the current mean and covariance is referred to as an

extended Kalman filter or EKF. The EKF is the minimum mean-square-error estimator based

on the Taylor series expansion of the nonlinear functions. For example,

f(xk) = f(xk|k−1

)+∂f(xk)∂xk

|xk=xk|k−1

(xk − xk|k−1

)+ · · · . (2.2)

Using only the linear expansion terms, it is easy to derive the update equations for the mean

and covariance of the Gaussian approximation to the distribution of the states [12].The equations for the extended Kalman filter fall into two groups: time update

equations and measurement update equations. The specific equations for the time and

measurement updates are presented below as shown in (2.3)∼(2.8) [22].

(1) Time Update

Predicted state estimate:

xk|k−1 = f(xk−1, k − 1). (2.3)

Predicted estimate covariance:

P−k = ΦkPk−1ΦT

k +Qk−1. (2.4)


The time update equations project the state, xk, and covariance, Pk, estimates from the

previous time step k − 1 to the current time step k, Φk is the state transition matrix at step k,

which is defined to be the following Jacobians:

Φk =∂f

∂xk|xk=xk|k−1

. (2.5)

(2) Measurement Update

Near-Optimal Kalman gain:

Kk = P−1k HT

k

(HkP−1

k HTk + R−1

k

)−1. (2.6)

Updated state estimate:

xk = xk|k−1 +Kk

(zk − h

(xk|k−1, k

)). (2.7)

Updated estimate covariance:

Pk = (I −KkHk)P−k, (2.8)

where Kk is known as the Kalman gain. The measurement update equations correct the state

and covariance estimates with the measurement zk. Hk is the observation matrix at step k,which is defined to be the following Jacobians:

Hk =∂h

∂xk|xk=xk|k−1

. (2.9)

The major drawback of EKF is that it only uses the first order terms in the Taylor series

expansion. Sometimes it may introduce large estimation errors in a nonlinear system and lead

to poor representations of the nonlinear functions and probability distributions of interest. As

a result, this filter can diverge [23].

2.2. Unscented Kalman Filter

The unscented Kalman filter (UKF) [8, 24] uses the unscented transformation to capture

the mean and covariance estimates with a minimal set of sample points. The UKF process

is identical to the standard EKF process with the prediction-estimation recursive loop. The

exception is that the UKF uses the sigma points and the nonlinear equations to compute the

predicted states and measurements and the associated covariance matrices. If the dimension

of state is n × 1, the 2n + 1 sigma point and their weight are computed by [9]

X0,k = xk, W0 =τ

(n + τ),

Xi,k = xk +√n + τ

(√P(k | k)

)i

, Wi =1

[2(n + τ)],

Xi+n,k = xk −√n + τ

(√P(k | k)

)i

, Wi+n =1

[2(n + τ)],

i = 1, 2, . . . , n, (2.10)


where τ ∈ R, (√P(k | k))i is the ith column of the matrix square root. The UKF process can

be described as follows.

(1) Time = 0, initialize the UKF with x0 and P0 as follows:

x0 = E[x0],

P0 = E[(x0 − x0)(x0 − x0)

T].

(2.11)

(2) Time = k, define 2n + 1 sigma points from

Xk−1 =[X0,k Xi,k Xi+n,k

], i = 1, 2, . . . , n. (2.12)

The equations for the UKF fall into two groups the same as EKF: time update equations

and measurement update equations. The specific equations for the time and measurement

updates are presented below.

(1) Time Update

Xk|k−1 = f(Xk−1, k − 1),

x−k =2n∑i=0

WiXi,k|k−1,

P−k =

2n∑i=0

Wi

[Xi,k|k−1 − x−k

]·[Xi,k|k−1 − x−k

]T +Qk,

Zk|k−1 = h(Xk|k−1, k

),

z−k =2n∑i=0

WiZi,k|k−1.

(2.13)

(2) Measurement Update

Pzk zk =2n∑i=0

Wi

[Zi,k|k−1 − z−k

][Zi,k|k−1 − z−k

]T + Rk,

Pxk zk =2n∑i=0

Wi

[Xi,k|k−1 − x−k

][Zi,k|k−1 − z−k

]T,

Kk = Pxk zkP−1zk zk

,

xk = x−k +Kk

(zk − z−k

),

Pk = P−k −KkPzk zkK

Tk .

(2.14)


2.3. Unscented Particle Filter

The unscented particle filter (UPF) is a hybrid of the UKF and the particle filter which uses

the UKF to get better importance sampling density. A pseudo-code description of UPF is as

follows [11–13].(1) Initialization: Time = 0.

Generate N samples xi0, (i = 1, 2, . . . ,N) from the prior p(x0), and set the importance

weight wi0 of each sample 1/N:

xi0 = E[xi0], Pi0 = E

[(xi0 − xi0

)(xi0 − xi0

)T], wi

0 =1

N. (2.15)

(2) Time = k.

(I) (a) Update the particles with the UKF:

(i) calculate sigma points from {xik−1,Pi

k−1} using (2.12),

(ii) propagate particle into future by (2.13),(iii) incorporate new observation to update the measurement by (2.14) and obtain

{xik,Pi

k}.

(b) Sample a new particle xik

and make xik∼ q(xi

k| xi

k−1, zk) =N(xi

k,Pi

k).

(II) Compute the importance weight wik and normalize the importance weights wi

k:

wik = wi

k−1 ·p(zk | xi

k

)p(xik| xi

k−1

)q(xik| xi

k−1, zk−1

) ,

wik =

wik∑N

i=1 wik

,

(2.16)

where p(zk | xik) is likelihood probability distribution, which is given by measurement model

zk = h(xk, k)+vk, p(xik| xi

k−1) is the forward transition probability distribution, which is given

by process model xk = f(xk−1, k − 1) +wk, q(xik| xi

k−1, zk−1) is the proposal distribution [12].

(III) Resampling step:

The basic idea of resampling is to eliminate particles with small weights and to con-

centrate on particles with large weights. Multiply/suppress particles {xik, Pi

k} with high/low

importance weights wik, respectively, to obtain N random particles {xi

k, Pi

k}.

(IV) Output step:

The overall state estimation and covariance are

xk =N∑i=1

wikx

ik,

Pk =N∑i=1

wikP

ik =

N∑i=1

wik

(xik − xk

)(xik − xk

)T.

(2.17)


3. System Models

3.1. State Model

The state model (dynamical model) of the celestial OD system for a near-Earth satellite based

on the orbital dynamics in the Earth-Centered Inertial (ECI) frame (J2000.0) is

dx

dt= vx,

dy

dt= vy,

dz

dt= vz,

dvxdt

= −μxr3

·[

1 − J2

(Re

r

)2(

7.5z2

r2− 1.5

)]+ ΔFx,

dvy

dt= −μy

r3·[

1 − J2

(Re

r

)2(

7.5z2

r2− 1.5

)]+ ΔFy,

dvzdt

= −μzr3

·[

1 − J2

(Re

r

)2(

7.5z2

r2− 4.5

)]+ ΔFz,

r =√x2 + y2 + z2.

(3.1)

Equation (3.1) can be written in a general state equation as

X(t) = f(X(t), t) +w(t), (3.2)

where X = [x y z vx vy vz]T is the state vector. x, y, z, vx, vy, vz are satellite positions

and velocities of the three axes, respectively, μ is the gravitational constant of earth, J2 is the

second zonal coefficient and has the value 0.0010826269 [25], and Re is the earth’s radius.

ΔFx, ΔFy, ΔFz are the perturbations including high order nonspherical earth perturbations,

third-body perturbations, atmospheric drag perturbations, solar radiation perturbations, and

other perturbations, which are considered as process noises w(t).

3.2. Celestial Orbit Determination and Its Measurement

The celestial OD method is based on the fact that the position of a celestial body in the inertial

frame at a certain time is known and that its position measured in the spacecraft body frame

is a function of the satellite’s position. To earth satellite, stars are distributed all over the sky,

and the positions of Earth are fixed at a certain time. The geometric relationship among stars,

the Earth, and satellite enables us to determine the position of the satellite [26].Satellite celestial OD methods can be broadly separated into two major approaches:

directly sensing horizon method and indirectly sensing horizon method. In this paper, the

directly sensing horizon method is used.

The angle between a star and the earth, α, as shown in Figure 2, is a kind of directly

sensing horizon measurement of satellite celestial OD system, which is measured by star

sensor and earth sensor. The measurement model using the star-earth angle is given by [27]

α = arccos

(−s · rr

)+ να, (3.3)


Satellite

Earth

Star

α

Figure 2: The measurement of celestial OD system.

where r is the position vector of the satellite, which is the same as that in (3.2), s is the position

vector of the star in the earth-centered inertial frame, να is the measurement noise.

Assuming a measurement Z1 = [α] and measurement noise V1 = [vα], (3.2) can be

written as a general measurement equation

Z1(t) = h1[X(t), t] +V1(t). (3.4)

3.3. Geomagnetic Orbit Determination and Its Measurement

Geomagnetic OD system relies on measurements from a three-axis magnetometer to

determine satellite position and orbit. It uses a model of Earth’s magnetic field and a model of

orbital dynamics to predict the time-varying magnitude of Earth’s magnetic field vector at the

space. OD system compares the time history of the predicted magnitude and the measured

magnitude time history in filter sense to obtain the optimal estimated state (position and

velocity) [14].

3.3.1. Magnetic Model

Two main models used for describing Earth’s magnetic vector in the geodetic reference frame

are World Magnetic Model (WMM) and International Geomagnetic Reference Field (IGRF)[28]. The WMM 2005 is selected in this paper for geomagnetic orbit determination [29].

According to the WMM model 2005, the vector field B can be written as the gradient

of a potential function

B(r, λ, θ, t) = −∇V (r, λ, θ, t), (3.5)

where (r, λ, θ) represent the radius, the longitude, and the colatitude in a spherical, geocentric

reference frame, respectively.

This potential V can be expanded in terms of spherical harmonics:

V (r, λ, θ, t) =N∑n=1

n∑m=0

Vmn = a

N∑n=1

(a

r

)n+1 n∑m=0

[gmn (t) cos(mλ) + hmn (t) sin(mλ)

]Pmn (cos θ),

(3.6)


Satellite

Satellite

Earth

Magneticnorth pole

Geographical

north pole

Magneticnorth pole

Geographicalnorth pole

Satelliteorbit

S

N

v

v

t

e

e

n

n

B

B

B

B

B

B

B

Figure 3: The measurement of geomagnetic OD system.

where N = 12 is the degree of the expansion of the WMM, a is the standard Earth’s magnetic

reference radius, gmn (t) and hmn (t) are the time-dependent Gauss coefficients of degree n and

order m, and Pmn (cos θ) are the Schmidt normalized associated Legendre polynomials.

3.3.2. Magnetic Measurement Model

Based on the relationship between magnetic vector, which is obtained by the magnetometer,

and the earth magnetic model, the measurement model can be written as

Bs = AsbAbiAitBt + vB, (3.7)

where Bs is the magnetic vector of local position in sensor coordinates, which can be

obtained from vector magnetometer system consisting of three mutually orthogonal, single-

axis magnetometers. Bt = [Bn Be Bv]T is the magnetic vector of local position in geocentric

coordinates, and it can be obtained from WMM according to local longitude, latitude, and

height, as shown in Figure 3; Asb, Abi, and Ait are the transformation matrices from satellite

body coordinates to sensor coordinates, from earth inertial coordinates to satellite body

coordinates, and from earth inertial coordinates to geocentric coordinates, respectively. vBis the measurement noise.

Assuming a measurement Z2 = Bs and measurement noise V2 = vB, (3.7) can be

written as a general measurement equation as

Z2(t) = h2[X(t), t] +V2(t). (3.8)

4. Analysis and Comparison

4.1. Simulation Condition

The trajectory used in the following simulation is a LEO satellite whose orbital parameters

are semimajor axis a = 7136.635444 km, eccentricity e = 1.809 × 10−3, inclination i = 65◦, right

ascension of the ascending node Ω = 30◦, and the argument of perigee ω = 30◦. The orbit


0 100 200 300 400 500 6000

500

1000

1500

2000

2500

3000

3500Position estimation error

Time (min)

Po

siti

on

err

or

(m)

EKF

UKF

UPF

(a)

0 100 200 300 400 500 6000

1

2

3

4

5

6Velocity estimation error

Time (min)

Vel

oci

ty e

rro

r (m

/s)

EKF

UKF

UPF

(b)

Figure 4: Three filtering methods results of celestial OD system (T = 3 s).

0 100 200 300 400 500 6000

500

1000

1500

2000

2500


Time (min)

Po

siti

on

err

or

(m)

EKF

UKF

UPF

(a)

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

4

4.5Velocity estimation error

Time (min)

Vel

oci

ty e

rro

r (m

/s)

EKF

UKF

UPF

(b)

Figure 5: Three filtering methods results of geomagnetic OD system (T = 3 s).

and attitude data of the satellite are produced by the Satellite Tool Kit (STK) software [30].The accuracy of star sensor and earth sensor is selected 3′′ and 0.02◦, respectively. The stellar

database used in simulation is the Tycho stellar catalog [31]. The magnetometer measurement

and geomagnetic model accuracy is considered as 100 nT [32].

4.2. Performances under Different Sampling Intervals

Figures 4 and 5 show the performances comparison among the EKF, UKF, and UPF methods

of celestial OD system and geomagnetic OD system, respectively. Data is obtained with a 3 s


Table 1: Performance of celestial OD system under different sampling intervals.

RMS (after convergence) Maximum (after convergence)Sampling interval

Position error/m Velocity error/m/s Position error/m Velocity error/m/s

T = 3 s

EKF 203.318211 0.196622 532.987272 0.486705

UKF 161.312723 0.162900 357.545600 0.407503

UPF 159.756079 0.160780 354.555359 0.402378

T = 15 s

EKF 271.640953 0.287641 703.000581 0.613708

UKF 245.939302 0.219864 593.829098 0.563749

UPF 245.229683 0.219566 593.802048 0.563396

T = 60 s

EKF 934.238939 0.976641 2457.895170 2.348656

UKF 736.876288 0.699942 2322.291896 2.302674

UPF 735.166932 0.698808 2314.761249 2.294924

Table 2: Performance of geomagnetic OD system under different sampling intervals.

RMS (after convergence) Maximum (after convergence)Sampling interval


T = 3 s

EKF 591.253628 0.601946 1129.365510 1.061735

UKF 376.894372 0.371566 877.909403 0.799659

UPF 376.516863 0.366538 861.513975 0.783449

T = 15 s

EKF 1481.673752 1.358867 2851.660177 2.607626

UKF 705.765450 0.648228 1325.501267 1.276198

UPF 705.161263 0.647876 1323.976107 1.274585

T = 60 s

EKF 4343.783162 4.308921 14643.275741 11.712547

UKF 3904.890544 3.633798 10403.528541 10.328902

UPF 3904.747892 3.633460 10401.279139 10.328554

sampling interval during the 600 min period (6 orbits). Tables 1 and 2 present the details of

the simulation results of celestial OD system and geomagnetic OD system under different

sampling intervals, respectively.

The simulations in Figures 4 and 5 suggest that the EKF-based OD system performance

is the worst. In contrast, UPF-based OD system provides the highest OD accuracy. As the

details in Tables 1 and 2, regardless the celestial OD and geomagnetic OD system, the

different sampling intervals can strongly affect the OD accuracy. OD performance is degraded

remarkably with increasing sample interval. However, under the same sampling interval, the

EKF method is the most sensitive to the sampling interval, for the nonlinear error increases

rapidly with the longer sampling interval. In contrast, the UKF and UPF perform distinctly

better.

4.3. Performance under Different Noise Distributions

This subsection reports how different noise distributions affect the OD performances using

three filters. We selected three common noise distributions in navigation, and they are normal

distribution, student’s t distribution, and uniform distribution [33].


0 20 40 60 80 100 1200

500

1000

1500

2000

2500

3000

3500

4000

4500


Time (min)

Po

siti

on

err

or

(m)

EKF

UKF

UPF

(a)

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8Velocity estimation error

Time (min)

Vel

oci

ty e

rro

r (m

/s)

EKF

UKF

UPF

(b)

Figure 6: Celestial OD results of three filtering methods under Student’s t noise distributions.

0 20 40 60 80 100 120

0

1000

2000

3000

4000

5000

6000

7000


Time (min)

Po

siti

on

err

or

(m)

EKF

UKF

UPF

(a)

EKF

UKF

UPF

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

Time (min)

Vel

oci

ty e

rro

r (m

/s)

Velocity estimation error

(b)

Figure 7: Geomagnetic OD results of three filtering methods under Student’s t noise distributions.

Figures 6 and 7 show the OD results of celestial OD system and geomagnetic OD

system using three filters under student’s t noise distributions, respectively. All performance

curves were obtained with 15 s sampling interval during the 600 min period (6 orbits). Tables

3 and 4 present the details of the simulation results of celestial OD system and geomagnetic

OD system under three different noise distributions, respectively.

As the results in Figures 6 and 7 showed, the UPF-based geomagnetic OD system

provides the highest OD accuracy. As the details in Tables 3 and 4 demonstrated, OD

performance under different noise distribution is similar. In general, the EKF performance


Table 3: Performance of celestial OD system under different noise distributions.

RMS (after convergence) Maximum (after convergence)Noise distribution


Normaldistribution

EKF 271.640953 0.287641 703.000581 0.613708

UKF 245.939302 0.219864 593.829098 0.563749

UPF 245.229683 0.219566 593.802048 0.563396

Student’s tdistribution

EKF 387.618044 0.411669 827.358300 0.972190

UKF 257.104360 0.261611 604.325265 0.664447

UPF 246.294197 0.250511 587.154229 0.639735

Uniformdistribution

EKF 385.803609 0.398842 994.714264 0.902353

UKF 350.079516 0.339869 888.790677 0.788940

UPF 349.537325 0.338473 880.417206 0.782641

Table 4: Performance of geomagnetic OD system under different noise distributions.

RMS (after convergence) Maximum (after convergence)Noise distribution


Normaldistribution

EKF 1481.673752 1.358867 2851.660177 2.607626

UKF 705.765450 0.648228 1325.501267 1.276198

UPF 705.161263 0.647876 1323.976107 1.274585

Student’s tdistribution

EKF 1059.874849 0.828895 2900.800178 3.184135

UKF 670.321661 0.604835 1594.060692 1.436411

UPF 669.959004 0.604513 1593.085809 1.436468

Uniformdistribution

EKF 1246.529130 1.207728 4029.187774 3.253167

UKF 776.622486 0.767562 1985.870577 1.781823

UPF 776.526144 0.767452 1985.704699 1.781594

is the worst and the UPF performance is the best, no matter what measurement errors are

chosen.

4.4. Computation Cost of Three Methods

Besides the accuracy, the computation cost is another essential requirement to evaluate the

performance of filtering methods. Table 5 gives the computation cost of the three methods

for the celestial orbit determination system and the geomagnetic orbit determination system,

respectively. As in the theoretical value of computation cost, where Φ is the process Jacobian,

n is the order of the Φ, and in the simulation n equals 6. The simulation results presented

here were run on a 2.66 GHz Inter Core2 Duo CPU with 32-bit Windows 7 system. The

simulation time of celestial OD system in Table 5 demonstrates that the UPF demands the

highest computation time, which is almost twenty times (= sample number) higher than UKF,

and EKF requires almost a quarter of the computation time of UKF. However, the simulation

time of geomagnetic OD system is not the same amount as celestial OD system, and the EKF-

based geomagnetic OD system takes significantly longer time, since the time for computing

measurement Jacobians takes a lot of computer resource.


Table 5: Comparison of computation cost.

Filtermethod

Theoretical value of computation cost Simulation value of computation cost

Φ is full matrix Φ is diagonal matrix

Computationtime per orbit

of celestialOD system(s)

Computationtime per orbit

of geomagneticOD system (s)

EKF CEKF = 3n3 + 3n2 + 4n C′EKF = 6n2 + 4n 0.3 9.5

UKF CUKF = 2n3 + 12n2 + 14n + 5 C′UKF = 2n3 + 12n2 + 14n + 5 1.6 11.7

UPF(samplenum = 20)

CUPF = sample num · CUKF C′UPF = sample num · C′

UKF 39.5 313.9

5. Conclusion

The problem of choosing a suitable filtering method for the orbit determination application

has been studied here. Three filtering methods for the autonomous orbit determination using

either celestial or geomagnetic measurements have been studied and their performances have

been compared for the estimation problem.

The algorithms are tested with STK satellite orbit data, and the simulation results

demonstrate that UPF yields the best OD accuracy and the EKF yields the worst under the

same condition. The main reason is that the state equations and measurement equations

for autonomous orbit determination system are significantly nonlinear as well as the non-

Gaussian errors.

In addition, the paper analyzed the computation cost of the three filtering methods,

and UPF-based OD system can provide the highest OD accuracy, though it requires the largest

computation time. However, the UPF can finally meet the real-time requirements, as with the

development of computer technology.

Acknowledgments

The work described in this paper was supported by the National Natural Science Foundation

of China (60874095) and Hi-Tech Research and Development Program of China. The

authors would like to thank all members of Science and Technology on Inertial Laboratory

and Fundamental Science on Novel Inertial Instrument & Navigation System Technology

Laboratory, for their useful comments regarding this work effort.

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Research ArticleThe Orbital Dynamics of Synchronous Satellites:Irregular Motions in the 2 : 1 Resonance

Jarbas Cordeiro Sampaio,1 Rodolpho Vilhena de Moraes,2and Sandro da Silva Fernandes3

1 Departamento de Matematica, Universidade Estadual Paulista (UNESP),12516-410 Guaratingueta-SP, Brazil

2 Instituto de Ciencia e Tecnologia, Universidade Federal de Sao Paulo (UNIFESP),12231-280 Sao Jose dos Campos, SP, Brazil

3 Departamento de Matematica, Instituto Tecnologico de Aeronautica (ITA),12228-900 Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Jarbas Cordeiro Sampaio, [email protected]

Received 7 July 2011; Accepted 27 September 2011


Copyright q 2012 Jarbas Cordeiro Sampaio et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The orbital dynamics of synchronous satellites is studied. The 2 : 1 resonance is considered; in otherwords, the satellite completes two revolutions while the Earth completes one. In the developmentof the geopotential, the zonal harmonics J20 and J40 and the tesseral harmonics J22 and J42 are con-sidered. The order of the dynamical system is reduced through successive Mathieu transforma-tions, and the final system is solved by numerical integration. The Lyapunov exponents are usedas tool to analyze the chaotic orbits.

1. Introduction

Synchronous satellites in circular or elliptical orbits have been extensively used for naviga-

tion, communication, and military missions. This fact justifies the great attention that has

been given in literature to the study of resonant orbits characterizing the dynamics of these

satellites since the 60s [1–14]. For example, Molniya series satellites used by the old Soviet

Union for communication form a constellation of satellites, launched since 1965, which have

highly eccentric orbits with periods of 12 hours. Another example of missions that use eccen-

tric, inclined, and synchronous orbits includes satellites to investigate the solar magneto-

sphere, launched in the 90s [15].The dynamics of synchronous satellites are very complex. The tesseral harmonics

of the geopotential produce multiple resonances which interact resulting significantly in

nonlinear motions, when compared to nonresonant orbits. It has been found that the orbital


elements show relatively large oscillation amplitudes differing from neighboring trajectories

[11].Due to the perturbations of Earth gravitational potential, the frequencies of the lon-

gitude of ascending node Ω and of the argument of pericentre ω can make the presence of

small divisors, arising in the integration of equation of motion, more pronounced. This

phenomenon depends also on the eccentricity and inclination of the orbit plane. The impor-

tance of the node and the pericentre frequencies is smaller when compared to the mean

anomaly and Greenwich sidereal time. However, they also have their contribution in the reso-

nance effect. The coefficients l, m, p which define the argument φlmpq in the development of

the geopotential can vary, producing different frequencies within the resonant cosines for the

same resonance. These frequencies are slightly different, with small variations around the

considered commensurability.

In this paper, the 2 : 1 resonance is considered; in other words, the satellite completes

two revolutions while the Earth carries one. In the development of the geopotential, the zonal

harmonics J20 and J40 and the tesseral harmonics J22 and J42 are considered. The order of the

dynamical system is reduced through successive Mathieu transformations, and the final sys-

tem is solved by numerical integration. In the reduced dynamical model, three critical angles,

associated to the tesseral harmonics J22 and J42, are studied together. Numerical results show

the time behavior of the semimajor axis, argument of pericentre and of the eccentricity. The

Lyapunov exponents are used as tool to analyze the chaotic orbits.

2. Resonant Hamiltonian and Equations of Motion

In this section, a Hamiltonian describing the resonant problem is derived through successive

Mathieu transformations.

Consider (2.1) to the Earth gravitational potential written in classical orbital elements

[16, 17]

V =μ

2a+

∞∑l=2

l∑m=0

l∑p=0

−∞∑q=+∞

μ

a

(aea

)lJlmFlm(I)Glpq(e) cos

(φlmpq(M,ω,Ω, θ)

), (2.1)

where μ is the Earth gravitational parameter, μ = 3.986009 × 1014 m3/s2, a, e, I, Ω, ω, M are

the classical keplerian elements: a is the semimajor axis, e is the eccentricity, I is the in-

clination of the orbit plane with the equator, Ω is the longitude of the ascending node,ω is the

argument of pericentre, and M is the mean anomaly, respectively; ae is the Earth mean

equatorial radius, ae = 6378.140 km, Jlm is the spherical harmonic coefficient of degree l and

order m, Flmp(I) and Glpq(e) are Kaula’s inclination and eccentricity functions, respectively.

The argument φlmpq(M,ω,Ω, θ) is defined by

φlmpq(M,ω,Ω, θ) = qM +(l − 2p

)ω +m(Ω − θ − λlm) + (l −m)

π

2, (2.2)

where θ is the Greenwich sidereal time, θ = ωet (ωe is the Earth’s angular velocity, and t is

the time), and λlm is the corresponding reference longitude along the equator.


In order to describe the problem in Hamiltonian form, Delaunay canonical variables

are introduced,

L = √μa, G =

√μa(1 − e2), H =

√μa(1 − e2) cos(I),

=M, g = ω, h = Ω.(2.3)

L, G, and H represent the generalized coordinates, and , g, and h represent the conjugate

momenta.

Using the canonical variables, one gets the Hamiltonian F,

F =μ2

2L2+

∞∑l=2

l∑m=0

Rlm, (2.4)

with the disturbing potential Rlm given by

Rlm =l∑

p=0

+∞∑q=−∞

Blmpq(L,G,H) cos(φlmpq

(, g, h, θ

)). (2.5)

The argument φlmpq is defined by

φlmpq(, g, h, θ

)= q +

(l − 2p

)g +m(h − θ − λlm) + (l −m)

π

2, (2.6)

and the coefficient Blmpq(L,G,H) is defined by

Blmpq =∞∑l=2

l∑m=0

l∑p=0

−∞∑q=+∞

μ2

L2

(μae

L2

)lJlmFlmp(L,G,H)Glpq(L,G). (2.7)

The Hamiltonian F depends explicitly on the time through the Greenwich sidereal

time θ. A new term ωeΘ is introduced in order to extend the phase space. In the extended

phase space, the extended Hamiltonian H is given by

H = F −ωeΘ. (2.8)

For resonant orbits, it is convenient to use a new set of canonical variables. Consider

the canonical transformation of variables defined by the following relations:

X = L, Y = G − L, Z = H −G, Θ = Θ,

x = + g + h, y = g + h, z = h, θ = θ,(2.9)

where X,Y, Z,Θ, x, y, z, θ are the modified Delaunay variables.


The new Hamiltonian H ′, resulting from the canonical transformation defined by (2.9),is given by

H ′ =μ2

2X2−ωeΘ +

∞∑l=2

l∑m=0

R′lm, (2.10)

where the disturbing potential R′lm

is given by

R′lm =

l∑p=0

+∞∑q=−∞

B′lmpq(X,Y, Z) cos

(φlmpq

(x, y, z, θ

)). (2.11)

Now, consider the commensurability between the Earth rotation angular velocity ωe

and the mean motion n = μ2/X3. This commensurability can be expressed as

qn −mωe∼= 0, (2.12)

considering q and m as integers. The ratio q/m defining the commensurability will be de-

noted by α. When the commensurability occurs, small divisors arise in the integration of the

equations of motion [9]. These periodic terms in the Hamiltonian H ′ with frequencies qn −mωe are called resonant terms. The other periodic terms are called short- and long-period

terms.

The short- and long-period terms can be eliminated from the Hamiltonian H ′ by ap-

plying an averaging procedure [12, 18]:

⟨H ′⟩=

1

4π2

∫2π

0

∫2π

0

H ′dξspdξlp. (2.13)

The variables ξsp and ξlp represent the short- and long-period terms, respectively, to be elimi-

nated of the Hamiltonian H ′.The long-period terms have a combination in the argument φlmpq which involves only

the argument of the pericentre ω and the longitude of the ascending node Ω. From (2.10) and

(2.11), these terms are represented by the new variables in the following equation:

H ′lp=

∞∑l=2

l∑m=0

l∑p=0

+∞∑q=−∞


((l − 2p

)(y − z

)+mz

). (2.14)

The short-period terms are identified by the presence of the sidereal time θ and mean

anomalyM in the argument φlmpq; in this way, from (2.10) and (2.11), the term H ′sp in the new

variables is given by the following equations:

H ′sp =

∞∑l=2

l∑m=0

l∑p=0

+∞∑q=−∞


(q(x − y

)−mθ + ζp

). (2.15)


The term ζp represents the other variables in the argument φlmpq, including the argument of

the pericentre ω and the longitude of the ascending node Ω, or, in terms of the new variables,

y − z and z, respectively.

A reduced Hamiltonian Hr is obtained from the Hamiltonian H ′ when only secular

and resonant terms are considered. The reduced Hamiltonian Hr is given by

Hr =μ2

2X2−ωeΘ +

∞∑j=1

B′2j,0,j,0(X,Y, Z)

+∞∑l=2

l∑m=2

l∑p=0

B′lmp(αm)(X,Y, Z) cos

(φlmp(αm)

(x, y, z, θ

)).

(2.16)

Several authors, [11, 15, 19–22], also use this simplified Hamiltonian to study the resonance.

The dynamical system generated from the reduced Hamiltonian, (2.16), is given by

d(X,Y, Z,Θ)dt

=∂Hr

∂(x, y, z, θ

) , d(x, y, z, θ

)dt

= − ∂Hr

∂(X,Y, Z,Θ). (2.17)

The equations of motion dX/dt, dY/dt, and dZ/dt defined by (2.17) are

dX

dt= −α

∞∑l=2

l∑m=2

l∑p=0

mB′lmp(αm)(X,Y, Z) sin

(φlmp(αm)

(x, y, z, θ

)), (2.18)

dY

dt= −

∞∑l=2

l∑m=2

l∑p=0

(l − 2p −mα

)B′lmp(αm)(X,Y, Z) sin

(φlmp(αm)

(x, y, z, θ

)), (2.19)

dZ

dt=

∞∑l=2

l∑m=2

l∑p=0

(l − 2p −m

)B′lmp(αm)(X,Y, Z) sin

(φlmp(αm)

(x, y, z, θ

)). (2.20)

From (2.18) to (2.20), one can determine the first integral of the system determined by

the Hamiltonian Hr .

Equation (2.18) can be rewritten as

1

α

dX

dt= −

∞∑l=2

l∑m=2

l∑p=0


(φlmp(αm)

(x, y, z, θ

)). (2.21)

Adding (2.19) and (2.20),

dY

dt+dZ

dt= (α − 1)

∞∑l=2

l∑m=2

l∑p=0


(φlmp(αm)

(x, y, z, θ

)), (2.22)


and substituting (2.21) and (2.22), one obtains

dY

dt+dZ

dt= −(α − 1)

1

α

dX

dt. (2.23)

Now, (2.23) is rewritten as

(1 − 1

α

)dX

dt+dY

dt+dZ

dt= 0. (2.24)

In this way, the canonical system of differential equations governed by Hr has the first

integral generated from (2.24):

(1 − 1

α

)X + Y + Z = C1, (2.25)

where C1 is an integration constant.

Using this first integral, a Mathieu transformation

(X,Y, Z,Θ, x, y, z, θ

)−→(X1,Y1, Z1,Θ1, x1, y1, z1, θ1

)(2.26)

can be defined.

This transformation is given by the following equations:

X1 = X, Y1 = Y, Z1 =(

1 − 1

α

)X + Y + Z, Θ1 = Θ,

x1 = x −(

1 − 1

α

)z, y1 = y − z, z1 = z, θ1 = θ.

(2.27)

The subscript 1 denotes the new set of canonical variables. Note that Z1 = C1, and the z1 is an

ignorable variable. So the order of the dynamical system is reduced in one degree of freedom.

Substituting the new set of canonical variables, X1, Y1, Z1, Θ1, x1, y1, z1, θ1, in the

reduced Hamiltonian given by (2.16), one gets the resonant Hamiltonian. The word “reso-

nant” is used to denote the Hamiltonian Hrs which is valid for any resonance. The periodic

terms in this Hamiltonian are resonant terms. The Hamiltonian Hrs is given by

Hrs =μ2

2X21

−ωeΘ1 +∞∑j=1

B2j,0,j,0(X1,Y1, C1)

+∞∑l=2

l∑m=2

l∑p=0

Blmp,(αm)(X1,Y1, C1) cos(φlmp(αm)

(x1, y1, θ1

)).

(2.28)


The Hamiltonian Hrs has all resonant frequencies, relative to the commensurability α,

where the φlmp(αm) argument is given by

φlmp(αm) = m(αx1 − θ1) +(l − 2p − αm

)y1 − φlmp(αm)0, (2.29)

with

φlmp(αm)0 = mλlm − (l −m)π

2. (2.30)

The secular and resonant terms are given, respectively, by B2j,0,j,0(X1,Y1, C1) and

Blmp(αm)(X1,Y1, C1).Each one of the frequencies contained in dx1/dt, dy1/dt, dθ1/dt is related, through

the coefficients l, m, to a tesseral harmonic Jlm. By varying the coefficients l, m, p and keeping

q/m fixed, one finds all frequencies dφ1,lmp(αm)/dt concerning a specific resonance.

From Hrs, taking, j = 1, 2, l = 2, 4, m = 2, α = 1/2, and p = 0, 1, 2, 3, one gets

H1 =μ2

2X21

−ωeΘ1 + B1,2010(X1,Y1, C1) + B1,4020(X1,Y1, C1)

+ B1,2201(X1,Y1, C1) cos(x1 − 2θ1 + y1 − 2λ22

)+ B1,2211(X1,Y1, C1) cos

(x1 − 2θ1 − y1 − 2λ22

)+ B1,2221(X1,Y1, C1) cos

(x1 − 2θ1 − 3y1 − 2λ22

)+ B1,4211(X1,Y1, C1) cos

(x1 − 2θ1 + y1 − 2λ42 + π

)+ B1,4221(X1,Y1, C1) cos

(x1 − 2θ1 − y1 − 2λ42 + π

)+ B1,4231(X1,Y1, C1) cos

(x1 − 2θ1 − 3y1 − 2λ42 + π

).

(2.31)

The Hamiltonian H1 is defined considering a fixed resonance and three different criti-

cal angles associated to the tesseral harmonic J22; the critical angles associated to the tesseral

harmonic J42 have the same frequency of the critical angles associated to the J22 with a dif-

ference in the phase. The other terms in Hrs are considered as short-period terms.

Table 1 shows the resonant coefficients used in the Hamiltonian H1.

Finally, a last transformation of variables is done, with the purpose of writing the

resonant angle explicitly. This transformation is defined by

X4 = X1, Y4 = Y1, Θ4 = Θ1 + 2X1,

x4 = x1 − 2θ1, y4 = y1, θ4 = θ1.(2.32)


Table 1: Resonant coefficients.

Degree (l) Order (m) p q

2 2 0 1

2 2 1 1

2 2 2 1

4 2 1 1

4 2 2 1

4 2 3 1

So, considering (2.31) and (2.32), the Hamiltonian H4 is found to be

H4 =μ2

2X24

−ωe(Θ4 − 2X4) + B4,2010(X4,Y4, C1) + B4,4020(X4,Y4, C1)

+ B4,2201(X4,Y4, C1) cos(x4 + y4 − 2λ22

)+ B4,2211(X4,Y4, C1) cos

(x4 − y4 − 2λ22

)+ B4,2221(X4,Y4, C1) cos

(x4 − 3y4 − 2λ22

)+ B4,4211(X4,Y4, C1) cos

(x4 + y4 − 2λ42 + π

)+ B4,4221(X4,Y4, C1) cos

(x4 − y4 − 2λ42 + π

)+ B4,4231(X4,Y4, C1) cos

(x4 − 3y4 − 2λ42 + π

),

(2.33)

with ωeΘ4 constant and

B4,2010 =μ4

X64

ae2J20

(−3

4

(C1 + 2X4)2

(X4 + Y4)2+

1

4

)(1 +

3

2

−Y42 − 2X4Y4

X42

), (2.34)

B4,4020 =μ6

X104

ae4J40

⎛⎝105

64

(1 − (C1 + 2X4)2

(X4 + Y4)2

)2

− 3

2+

15

8

(C1 + 2X4)2

(X4 + Y4)2

⎞⎠×(

1 + 5−Y4

2 − 2X4Y4

X42

),

(2.35)

B4,2201 =21

8X74

μ4ae2J22

(1 +

C1 + 2X4

X4 + Y4

)2√−Y4

2 − 2X4Y4, (2.36)

B4,2211 =3

2X74

μ4ae2J22

(3

2− 3

2

(C1 + 2X4)2

(X4 + Y4)2

)√−Y4

2 − 2X4Y4, (2.37)


B4,2221 = − 3

8X74

μ4ae2J22

(1 − C1 + 2X4

X4 + Y4

)2√−Y4

2 − 2X4Y4, (2.38)

B4,4211 =9

2X114

μ6ae4J42

(35

27

(1 − (C1 + 2X4)2

(X4 + Y4)2

)(C1 + 2X4)

×(

1 +C1 + 2X4

X4 + Y4

)(X4 + Y4)−1

−15

8

(1 +

C1 + 2X4

X4 + Y4

)2)√

−Y42 − 2X4Y4,

(2.39)

B4,4221 =5

2X114

μ6ae4J42

(105

16

(1 − (C1 + 2X4)2

(X4 + Y4)2

)(1 − 3

(C1 + 2X4)2

(X4 + Y4)2

)

+15

4− 15

4

(C1 + 2X4)2

(X4 + Y4)2

)√−Y4

2 − 2X4Y4,

(2.40)

B4,4231 =μ6

X104

ae4J42

(−35

27

(1 − (C1 + 2X4)2

(X4 + Y4)2

)(C1 + 2X4)

×(

1 − C1 + 2X4

X4 + Y4

)(X4 + Y4)−1 − 15

8

(1 − C1 + 2X4

X4 + Y4

)2)

×

⎛⎜⎝1

2

√−Y4

2 − 2X4Y4

X4+

33

16

−Y42 − 2X4Y4

X42

⎞⎟⎠.

(2.41)

Since the term ωeΘ4 is constant, it plays no role in the equations of motion, and a new

Hamiltonian can be introduced,

H4 = H4 +ωeΘ4. (2.42)

The dynamical system described by H4 is given by

d(X4,Y4)dt

=∂H4

∂(x4, y4

) , d(x4, y4

)dt

= − ∂H4

∂(X4,Y4). (2.43)


Table 2: The zonal and tesseral harmonics.

Zonal harmonics Tesseral harmonics

J20 = 1.0826 × 10−3 J22 = 1.8154 × 10−6

J40 = −1.6204 × 10−6 J42 = 1.6765 × 10−7

The zonal harmonics used in (2.34) and (2.35) and the tesseral harmonics used in (2.36)to (2.41) are shown in Table 2.

The constant of integration C1 in (2.34) to (2.41) is given, in terms of the initial values

of the orbital elements, ao, eo, and Io, by

C1 =√μao

(√1 − e2

o cos(Io) − 2

)(2.44)

or, in terms of the variables X4 and Y4,

C1 = X4(cos(Io) − 2) + Y4 cos(Io). (2.45)

In Section 4, some results of the numerical integration of (2.43) are shown.

3. Lyapunov Exponents

The estimation of the chaoticity of orbits is very important in the studies of dynamical

systems, and possible irregular motions can be analyzed by Lyapunov exponents [23].In this work, “Gram-Schmidt’s method,” described in [23–26], will be applied to

compute the Lyapunov exponents. A brief description of this method is presented in what

follows.

The dynamical system described by (2.43) can be rewritten as

dX4

dt= P1

(X4,Y4, x4, y4;C1

),

dY4

dt= P2

(X4,Y4, x4, y4;C1

),

dx4

dt= P3

(X4,Y4, x4, y4;C1

),

dy4

dt= P4

(X4,Y4, x4, y4;C1

).

(3.1)


Introducing

z =

⎛⎜⎜⎜⎜⎜⎝X4

Y4

x4

y4

⎞⎟⎟⎟⎟⎟⎠,

Z =

⎛⎜⎜⎜⎜⎜⎝P1

P2

P3

P4

⎞⎟⎟⎟⎟⎟⎠.

(3.2)

Equations (3.2) can be put in the form

dz

dt= Z(z). (3.3)

The variational equations, associated to the system of differential equations (3.3), are given

by

dζ

dt= Jζ, (3.4)

where J = (∂Z/∂z) is the Jacobian.

The total number of differential equations used in this method is n(n+ 1), n represents

the number of the motion equations describing the problem, in this case four. In this way,

there are twenty differential equations, four are motion equations of the problem and sixteen

are variational equations described by (3.4).The dynamical system represented by (3.3) and (3.4) is numerically integrated and the

neighboring trajectories are studied using the Gram-Schmidt orthonormalization to calculate

the Lyapunov exponents.

The method of the Gram-Schmidt orthonormalization can be seen in [25, 26] with more

details. A simplified denomination of the method is described as follows.

Considering the solutions to (3.4) as uκ(t), the integration in the time τ begins from

initial conditions uκ(t0) = eκ(t0), an orthonormal basis.

At the end of the time interval, the volumes of the κ-dimensional (κ = 1, 2, . . . ,N)produced by the vectors uκ are calculated by

Vκ =

∥∥∥∥∥∥κ∧j=1

uj(t)

∥∥∥∥∥∥, (3.5)

where∧

is the outer product and ‖ · ‖ is a norm.


26554

26556

26558

26560

26562

26564

26566

26568

0 1000 2000 3000 4000 5000 6000 7000

a(k

m)

t (days)

a(0)= 26555 km

a(0)= 26561.7 km

a(0)= 26562.4 km

a(0)= 26563.5 km

a(0)= 26565 km

Figure 1: Time behavior of the semimajor axis for different values of C1 given in Table 3.

−200

0

200

400

600

800

1000

1200

1400

1600

1800

0 1000 2000 3000 4000 5000 6000 7000

x4(d

egre

es)

t (days)

a(0)= 26555 km

a(0)= 26561.7 km

a(0)= 26562.4 km

a(0)= 26563.5 km

a(0)= 26565 km

Figure 2: Time behavior of x4 angle for different values of C1 given in Table 3.

In this way, the vectors uκ are orthonormalized by Gram-Schmidt method. In other

words, new orthonormal vectors eκ(t0 + τ) are calculated, in general, according to

eκ =uκ −

∑κ−1j=1

(uκ · ej

)ej∥∥∥uκ −∑κ−1

j=1

(uκ · ej

)ej∥∥∥ . (3.6)


Table 3: Values of the constant of integration C1 for e = 0.001, I = 55◦ and different values for semimajoraxis.

a(0) × 103(m) C1 × 1011(m2/s)26555.000 −1.467543158

26561.700 −1.467728282

26562.400 −1.467747623

26563.500 −1.467778013

26565.000 −1.467819454

−200

−150

−100

−50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

ω(d

egre

es)

t (days)

a(0)= 26555 km

a(0)= 26561.7 km

a(0)= 26562.4 km

a(0)= 26563.5 km

a(0)= 26565 km

Figure 3: Time behavior of the argument of pericentre for different values of C1 given in Table 3.

The Gram-Schmidt method makes invariant the κ-dimensional subspace produced by

the vectors u1,u2,u3, . . . ,uκ in constructing the new κ-dimensional subspace spanned by the

vectors e1, e2, e3, . . . , eκ.

With new vector uκ(t0 + τ) = eκ(t0 + τ), the integration is reinitialized and carried

forward to t = t0 + 2τ . The whole cycle is repeated over a long-time interval. The theorems

guarantee that the κ-dimensional Lyapunov exponents are calculated by [25, 26]:

λ(κ) = limn→∞

1

nτ

n∑j=1

ln(Vκ(t0 + jτ

))ln(Vκ(t0 +(j − 1)τ)) . (3.7)

The theory states that if the Lyapunov exponent tends to a positive value, the orbit is

chaotic.

In the next section are shown some results about the Lyapunov exponents.


0

0.005

0.01

0.015

0.02

0.025

0 1000 2000 3000 4000 5000 6000 7000

e

t (days)

a(0)= 26555 km

a(0)= 26561.7 km

a(0)= 26562.4 km

a(0)= 26563.5 km

a(0)= 26565 km

Figure 4: Time behavior of the eccentricity for different values of C1 given in Table 3.

26540

26545

26550

26555

26560

26565

26570

26575

26580

0 1000 2000 3000 4000 5000 6000 7000

a(k

m)

t (days)

J22 and J42

J22


4. Results

Figures 1, 2, 3, and 4 show the time behavior of the semimajor axis, x4 angle, argument of

perigee and of the eccentricity, according to the numerical integration of the motion equa-

tions, (2.43), considering three different resonant angles together: φ2201, φ2211, and φ2221 associ-

ated to J22, and three angles, φ4211, φ4221, and φ4231 associated to J42, with the same frequency

of the resonant angles related to the J22, but with different phase. The initial conditions corre-

sponding to variables X4 and Y4 are defined for eo = 0.001, Io = 55◦, and ao given in Table 3.


−3000

−2500

−2000

−1500

−1000

−500

0

500

1000

0 1000 2000 3000 4000 5000 6000 7000

x4(d

egre

es)

t (days)

J22 and J42

J22


−700

−600

−500

−400

−300

−200

−100

0

100

0 1000 2000 3000 4000 5000 6000 7000

ω(d

egre

es)

t (days)

J22 and J42

J22


Table 4: Values of the constant of integration C1 for e = 0.05, I = 10◦, and different values for semimajoraxis.

a(0) × 103(m) C1 × 1011 (m2/s)26555.000 −1.045724331

26565.000 −1.045921210

26568.000 −1.045980267

26574.000 −1.046098370

The initial conditions of the variables x4 and y4 are 0◦ and 0◦, respectively. Table 3 shows the

values of C1 corresponding to the given initial conditions.


0.044

0.046

0.048

0.05

0.052

0.054

0.056

0.058

0 1000 2000 3000 4000 5000 6000 7000

e

t (days)

J22 and J42

J22


26552

26554

26556

26558

26560

26562

26564

26566

26568

26570

0 1000 2000 3000 4000 5000 6000 7000

a(k

m)

t (days)

a(0)= 26555 km

a(0)= 26558 km

a(0)= 26562 km

a(0)= 26564 km

a(0)= 26568 km


Figures 5, 6, 7, and 8 show the time behavior of the semimajor axis, x4 angle, argument

of perigee and of the eccentricity for two different cases. The first case considers the critical

angles φ2201, φ2211, and φ2221, associated to the tesseral harmonic J22, and the second case

considers the critical angles associated to the tesseral harmonics J22 and J42. The angles associ-

ated to the J42, φ4211, φ4221, and φ4231, have the same frequency of the critical angles associated

to the J22 with a different phase. The initial conditions corresponding to variables X4 and Y4

are defined for eo = 0.05, Io = 10◦, and ao given in Table 4. The initial conditions of the varia-

bles x4 and y4 are 0◦ and 60◦, respectively. Table 4 shows the values of C1 corresponding to

the given initial conditions.


−1000

−500

0

500

1000

1500

2000

0 1000 2000 3000 4000 5000 6000 7000

x4(d

egre

es)

t (days)

a(0)= 26555 km

a(0)= 26558 km

a(0)= 26562 km

a(0)= 26564 km

a(0)= 26568 km


−140

−120

−100

−80

−60

−40

−20

0

20

40

60

80

0 1000 2000 3000 4000 5000 6000 7000

ω(d

egre

es)

t (days)

a(0)= 26555 km

a(0)= 26558 km

a(0)= 26562 km

a(0)= 26564 km

a(0)= 26568 km


Analyzing Figures 5–8, one can observe a correction in the orbits when the terms re-

lated to the tesseral harmonic J42 are added to the model. Observing, by the percentage, the

contribution of the amplitudes of the terms B4,4211, B4,4221, and B4,4231, in each critical angle

studied, is about 1,66% up to 4,94%. In fact, in the studies of the perturbations in the arti-

ficial satellites motion, the accuracy is important, since adding different tesseral and zonal

harmonics to the model, one can have a better description about the orbital motion.

Figures 9, 10, 11, and 12 show the time behavior of the semimajor axis, x4 angle, argu-

ment of perigee and of the eccentricity, according to the numerical integration of the motion

equations, (2.43), considering three different resonant angles together; φ2201, φ2211, and φ2221


0

0.005

0.01

0.015

0.02

0.025

0 1000 2000 3000 4000 5000 6000 7000

e

t (days)

a(0)= 26555 km

a(0)= 26558 km

a(0)= 26562 km

a(0)= 26564 km

a(0)= 26568 km


1e−007

1e−006

1e−005

0.0001

0.001

0.01

0.1

10 100 1000 10000 100000

Time (days)

Ly

ap

un

ov

ex

po

nen

tsλ(κ)

λ (2)(a(0)= 26565 km)

λ (2)(a(0)= 26563.5 km)λ (1)(a(0)= 26563.5 km)

λ (1)(a(0)= 26565 km)

Figure 13: Lyapunov exponents λ(1) and λ(2), corresponding to the variables X4 and Y4, respectively, forC1 = −1.467778013 × 1011 m2/s and C1 = −1.467819454 × 1011 m2/s, x4 = 0◦ and y4 = 0◦.

associated to J22 and three angles φ4211, φ4221, and φ4231 associated to J42. The initial conditions

corresponding to variables X4 and Y4 are defined for eo = 0.01, Io = 55◦, and ao given in Table

5. The initial conditions of the variables x4 and y4 are 0◦ and 60◦, respectively. Table 5 shows

the values of C1 corresponding to the given initial conditions.

Analyzing Figures 1–12, one can observe possible irregular motions in Figures 1–4,

specifically considering values for C1 = −1.467778013 × 1011 m2/s and C1 = −1.467819454 ×1011 m2/s, and, in Figures 9–12, for C1 = −1.467765786 × 1011 m2/s and C1 = −1.467821043 ×1011 m2/s. These curves will be analyzed by the Lyapunov exponents in a specified time

verifying the possible regular or chaotic motions.


1e−007

1e−006

1e−005

0.0001

0.001

0.01

0.1

10 100 1000 10000 100000

Time (days)

Ly

ap

un

ov

ex

po

nen

tsλ(κ)

λ (1)(a(0)= 26562 km)

λ (1)(a(0)= 26564 km) λ (2)(a(0)= 26564 km)λ (2)(a(0)= 26562 km)


Table 5: Values of the constant of integration C1 for e = 0.01, I = 55◦, and different values for semimajoraxis.

a(0) × 103(m) C1 × 1011(m2/s)26555.000 −1.467572370

26558.000 −1.467655265

26562.000 −1.467765786

26564.000 −1.467821043

26568.000 −1.467931552

Figures 13 and 14 show the time behavior of the Lyapunov exponents for two different

cases, according to the initial values of Figures 1–4 and 9–12. The dynamical system in-

volves the zonal harmonics J20 and J40 and the tesseral harmonics J22 and J42. The method

used in this work for the study of the Lyapunov exponents is described in Section 3. In

Figure 13, the initial values for C1, x4, and y4 are C1 = −1.467778013 × 1011 m2/s and C1 =−1.467819454 × 1011 m2/s, x4 = 0◦ and y4 = 0◦, respectively. In Figure 14, the initial values for

C1, x4, and y4 areC1 = −1.467765786 × 1011 m2/s andC1 = −1.467821043 × 1011 m2/s, x4 = 0◦

and y4 = 60◦, respectively. In each case are used two different values for semimajor axis

corresponding to neighboring orbits shown previously in Figures 1–4 and 9–12.

Figures 13 and 14 show Lyapunov exponents for neighboring orbits. The time used in

the calculations of the Lyapunov exponents is about 150.000 days. For this time, it can be ob-

served in Figure 13 that λ(1), corresponding to the initial value a(0) = 26565.0 km, tends to a

positive value, evidencing a chaotic region. On the other hand, analyzing the same Figure 13,

λ(1), corresponding to the initial value a(0) = 26563.5 km, does not show a stabilization

around the some positive value, in this specified time. Probably, the time is not sufficient for

a stabilization in some positive value, or λ(1), initial value a(0) = 26563.5 km, tends to a nega-

tive value, evidencing a regular orbit. Analyzing now Figure 14, it can be verified that λ(1),corresponding to the initial value a(0) = 26564.0 km, tends to a positive value, it contrasts


λ (2)(a(0)= 26565 km)

λ (2)(a(0)= 26563.5 km)λ (1)(a(0)= 26563.5 km)

λ (1)(a(0)= 26565 km)

1e−007

1e−006

1e−005

0.0001

10000 100000

Time (days)

Ly

ap

un

ov

ex

po

nen

tsλ(κ)


1e−007

1e−006

1e−005

0.0001

Ly

ap

un

ov

ex

po

nen

tsλ(κ)

10000 100000Time (days)

λ (1)(a(0)= 26562 km)

λ (1)(a(0)= 26564 km) λ (2)(a(0)= 26564 km)λ (2)(a(0)= 26562 km)


with λ(1), initial value a(0) = 26562.0 km. Comparing Figure 13 with Figure 14, it is observed

that the Lyapunov exponents in Figure 14 has an amplitude of oscillation greater than the Ly-

apunov exponents in Figure 13. Analyzing this fact, it is probable that the necessary time for

the Lyapunov exponent λ(2), in Figure 14, to stabilize in some positive value is greater than

the necessary time for the λ(2) in Figure 13.

Rescheduling the axes of Figures 13 and 14, as described in Figures 15 and 16, respec-

tively, the Lyapunov exponents tending to a positive value can be better visualized.


5. Conclusions

In this work, the dynamical behavior of three critical angles associated to the 2 : 1 resonance

problem in the artificial satellites motion has been investigated.

The results show the time behavior of the semimajor axis, argument of perigee and e-

ccentricity. In the numerical integration, different cases are studied, using three critical angles

together: φ2201, φ2211, and φ2221 associated to J22 and φ4211, φ4221, and φ4231 associated to the

J42.

In the simulations considered in the work, four cases show possible irregular motions

for C1 = −1.467778013 × 1011 m2/s, C1 = −1.467819454 × 1011 m2/s, C1 = −1.467765786 ×1011 m2/s, and C1 = −1.467821043 × 1011 m2/s. Studying the Lyapunov exponents, two cases

show chaotic motions forC1 = −1.467819454 × 1011 m2/s andC1 = −1.467821043 × 1011 m2/s.

Analyzing the contribution of the terms related to the J42, it is observed that, for the

value ofC1 = −1.045724331 × 1011 m2/s, the amplitudes of the terms B4,4211, B4,4221, and B4,4231

are greater than the other values of C1. In other words, for bigger values of semimajor axis, it

is observed a smaller contribution of the terms related to the tesseral harmonic J42.

The theory used in this paper for the 2 : 1 resonance can be applied for any resonance

involving some artificial Earth satellite.

Acknowledgments

This work was accomplished with support of the FAPESP under the Contract no. 2009/00735-

5 and 2006/04997-6, SP Brazil, and CNPQ (Contracts 300952/2008-2 and 302949/2009-7).

References

[1] M. B. Morando, “Orbites de resonance des satellites de 24h,” Bulletin of the American AstronomicalSociety, vol. 24, pp. 47–67, 1963.

[2] L. Blitzer, “Synchronous and resonant satellite orbits associated with equatorial ellipticity,” Journal ofAdvanced Robotic Systems, vol. 32, pp. 1016–1019, 1963.

[3] B. Garfinkel, “The disturbing function for an artificial satellite,” The Astronomical Journal, vol. 70, no.9, pp. 688–704, 1965.

[4] B. Garfinkel, “Tesseral harmonic perturbations of an artificial satellite,” The Astronomical Journal, vol.70, pp. 784–786, 1965.

[5] B. Garfinkel, “Formal solution in the problem of small divisors,” The Astronomical Journal, vol. 71, pp.657–669, 1966.

[6] G. S. Gedeon and O. L. Dial, “Along-track oscillations of a satellite due to tesseral harmonics,” AIAAJournal, vol. 5, pp. 593–595, 1967.

[7] G. S. Gedeon, B. C. Douglas, and M. T. Palmiter, “Resonance effects on eccentric satellite orbits,”Journal of the Astronautical Sciences, vol. 14, pp. 147–157, 1967.

[8] G. S. Gedeon, “Tesseral resonance effects on satellite orbits,” Celestial Mechanics, vol. 1, pp. 167–189,1969.

[9] M. T. Lane, “An analytical treatment of resonance effects on satellite orbits,” Celestial Mechanics, vol.42, pp. 3–38, 1988.

[10] A. Jupp, “A solution of the ideal resonance problem for the case of libration,” The Astronomical Journal,vol. 74, pp. 35–43, 1969.

[11] T. A. Ely and K. C. Howell, “Long-term evolution of artificial satellite orbits due to resonant tesseralharmonics,” Journal of the Astronautical Sciences, vol. 44, pp. 167–190, 1996.

[12] D. M. Sanckez, T. Yokoyama, P. I. O. Brasil, and R. R. Cordeiro, “Some initial conditions for disposedsatellites of the systems GPS and galileo constellations,” Mathematical Problems in Engineering, vol.2009, Article ID 510759, 22 pages, 2009.


[13] L. D. D. Ferreira and R. Vilhena de Moraes, “GPS satellites orbits: resonance,” Mathematical Problemsin Engineering, vol. 2009, Article ID 347835, 12 pages, 2009.

[14] J. C. Sampaio, R. Vilhena de Moraes, and S. S. Fernandes, “Artificial satellites dynamics: resonant ef-fects,” in Proceedings of the 22nd International Symposium on Space Flight Dynamics, Sao Jose dos Cam-pos, Brazil, 2011.

[15] A. G. S. Neto, Estudo de Orbitas Ressonantes no Movimento de Satelites Artificiais, Tese de Mestrado, ITA,2006.

[16] J. P. Osorio, Perturbacoes de Orbitas de Satelites no Estudo do Campo Gravitacional Terrestre, ImprensaPortuguesa, Porto, Portugal, 1973.

[17] W. M. Kaula, Theory of Satellite Geodesy: Applications of Satellites to Geodesy, Blaisdel, Waltham, Mass,USA, 1966.

[18] A. E. Roy, Orbital Motion, Institute of Physics Publishing Bristol and Philadelphia, 3rd edition, 1988.[19] P. H. C. N. Lima Jr., Sistemas Ressonantes a Altas Excentricidades no Movimento de Satelites Artificiais, Tese

de Doutorado, Instituto Tecnologico de Aeronautica, 1998.[20] P. R. Grosso, Movimento Orbital de um Satelite Artificial em Ressonancia 2:1, Tese de Mestrado, Instituto

Tecnologico de Aeronautica, 1989.[21] J. K. S. Formiga and R. Vilhena de Moraes, “Dynamical systems: an integrable kernel for resonance

effects,” Journal of Computational Interdisciplinary Sciences, vol. 1, no. 2, pp. 89–94, 2009.[22] R. Vilhena de Moraes, K. T. Fitzgibbon, and M. Konemba, “Influence of the 2:1 resonance in the orbits

of GPS satellites,” Advances in Space Research, vol. 16, no. 12, pp. 37–40, 1995.[23] F. Christiansen and H. H. Rugh, “Computing lyapunov spectra with continuous gram-schmidt

orthonormalization,” Nonlinearity, vol. 10, no. 5, pp. 1063–1072, 1997.[24] L. Qun-Hong and T. Jie-Yan, “Lyapunov exponent calculation of a two-degree-of-freedom vibro-

impact system with symmetrical rigid stops,” Chinese Physics B, vol. 20, no. 4, Article ID 040505, 2011.[25] I. Shimada and T. Nagashima, “A numerical approach to ergodic problem of dissipative dynamical

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pp. 3686–3697, 1993.


Research ArticleApplication of Periapse Maps forthe Design of Trajectories Near the SmallerPrimary in Multi-Body Regimes

Kathleen C. Howell, Diane C. Davis, and Amanda F. Haapala

School of Aeronautics and Astronautics, Purdue University, Armstrong Hall, West Lafayette,IN 47907, USA

Correspondence should be addressed to Kathleen C. Howell, [email protected]



Copyright q 2012 Kathleen C. Howell et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The success of the Genesis spacecraft, as well as the current Artemis mission, continue to generateinterest in expanding the trajectory options for future science and exploration goals throughout thesolar system. Incorporating multi-body dynamics into the preliminary design can potentially offerflexibility and influence the maneuver costs to achieve certain objectives. In the current analysis,attention is focused on the development and application of design tools to facilitate preliminarytrajectory design in a multi-body environment. Within the context of the circular restricted three-body problem, the evolution of a trajectory in the vicinity of the smaller primary is investigated.Introduced previously, periapse Poincare maps have emerged as a valuable resource to predictboth short- and long-term trajectory behaviors. By characterizing the trajectories in terms of radiusand periapse orientation relative to the P1-P2 line, useful trajectories with a particular set of desiredcharacteristics can be identified and computed.

1. Introduction

Spacecraft exploration activities increasingly involve trajectories that reach the vicinity of

the libration points in various types of three-body systems. Libration point trajectories and

low-energy transfers, in particular, have garnered much recent attention because of their

potential to incorporate the natural dynamics, to generate new types of design options and,

for some spacecraft applications, reduce propellant. A number of successful missions within

the last decade have successfully exploited the natural dynamics of multiple gravity fields,

for example, NASA’s Genesis mission, launched in 2001 with a return to Earth in 2004

[1, 2]. In a Sun-Earth rotating view in Figure 1, the trajectory for the Genesis spacecraft

leveraged the gravity of the Earth, Sun, and the Moon to supply a gravitational balance and


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Sun

Millions of kilometers

Mil

lio

ns

of

kil

om

eter

s

−1

−0.5

0.5

1

L1 Earth

Moon

Lunar orbitPositioning for

daylight reentry

Total flight time

(37.3 mos.)

Halo

orbits

(5)

Outward leg

(2.7 mos.)

Return and

recovery

(5.3 mos.)

Solar wind collection

in halo orbit about L1

(29.3 mos.)

Figure 1: Genesis trajectory as viewed in the Sun-Earth rotating frame. Samples of solar material werecollected on the spacecraft over two years in an L1 libration point orbit and returned to Earth. During thereturn, the lunar gravity was also leveraged as the spacecraft shifted to the L2 region prior to Earth reentry.CoutesyNASA/JPL-Caltech, http://genesismission.jpl.nasa.gov/gm2/mission/halo.htm.

deliver a trajectory that met the goals and satisfied the constraints with a path that does

not emerge within the context of a two-body problem. In another example from an ongoing

mission, Artemis involves two identical spacecraft identified as P1 and P2, originally two

of the five Themis spacecraft [3–5]. Employing the Sun-Earth-Moon multi-body dynamical

environment, these two vehicles were directed from the outermost of five elliptical Earth

orbits to eventually arrive in Earth-Moon libration point orbits on August 25, 2010, and

October 10, 2010. The transfer phase appears in Figure 2(a) as viewed in the Sun-Earth

rotating fame; both spacecraft are to be inserted into elliptical lunar orbits (see Figure 2(b))on June 27 and July 17, 2011, respectively.

Although much has been learned about the design space for such missions in the

last few decades, as is evident from Genesis and Artemis as well as a number of other

libration point missions, trajectory design in this type of regime remains a nontrivial problem.

Typical challenges in the use of non-Keplerian orbits in a multi-body environment include (i)complexity: there are many destinations and competing goals; (ii) the search for solutions

in new dynamical environments with frequent attempts to blend arcs from various models

with different levels of fidelity; (iii) new types of scenarios that are explored to offer options

for extended missions and contingencies. To exploit the gravity of multiple bodies requires a

capability to deliver the trajectory characteristics that meet the requirements for a particular

mission. Without analytical solutions, increasing insight into the dynamical structure in the

three-body problem has been developed, beginning with Henon and resulting in a wide

variety of investigations, frequently with a focus toward applications [6–27]. In many of

these analyses, the invariant manifolds associated with the L1 and L2 Lyapunov orbits have

been increasingly used to predict the behavior of trajectories that originate near the smaller

primary in the circular restricted three-body problem. In addition, the use of Poincare maps to

identify trajectories with various short- and long-term behaviors is effective. As noted in some

of these recent publications, however, a major barrier to the development of a simple orbit


EarthSun-Earth line

Flyby number 2

(Feb 13, 2010) Flyby number 1

(Jan 31, 2010)

Libration orbit (1-rev)

Postraisedelliptical orbit

Libration orbit (Aug 2010)

Transfer trajectory

Earth BBR axes25 Jun 2011 10:33:36

Moon

Time step: 57600 s

(a)

Directionto Earth

Artemis L1 orbit

L2

L1

L2 to L1 transfer orbit

Direction of

motion

Artemis L2 orbit

Artemis-P1

here on

August 25th

Moon

22 Aug 2010 07:10:00 Time step: 600 s

Moon inertial axesMoon’s orbit

(b)

Figure 2: (a) Artemis trajectory for one spacecraft during transfer from Earth to lunar vicinity viewed inthe Sun-Earth rating frame. The trajectory reaches the vicinity of the Sun-Earth L1 libration point after tworelatively high energy lunar flybys (a trajectory “backflip”) to eventually reach a low-energy trajectoryin the vicinity of the Moon. http://www.nasa.gov/mission pages/artemis/news/lunar-orbit.html. (b)Artemis trajectory at arrival in lunar vicinity viewed in Earth-Moon rotating frame. Both spacecraft employEarth-Moon L1 and L2 libration point orbits to modify the path and eventually insert into lunar orbit laterthis year. http://www.nasa.gov/mission pages/artemis/news/lunar-orbit.html.


design process is the overwhelming nature of the design space. The trajectory design process

remains challenging due to the varied, and sometimes chaotic, nature of trajectories that are

simultaneously influenced by two gravitational bodies. To effectively select a trajectory to

satisfy a given requirement, it is necessary to simplify and organize the design space as

much as possible. The invariant manifold structure associated with the collinear libration

points, in particular, has offered a geometrical framework for this dynamical environment.

Yet, harnessing this information to supply relatively quick and efficient options for trajectory

designers is a formidable challenge. Thus, the design difficulties remain significant, but a

wider range of tools is emerging.

The motivation for this investigation originated from a general need to repeatedly

develop design concepts for potential applications. A major challenge involved in orbit

design within the context of the circular restricted three-body problem (CR3BP) is the

organization of the vast set of options that is available within the design space; it is difficult

to locate the specific initial conditions that lead to a trajectory with a particular set of

characteristics. The invariant manifolds and the corresponding phase space yield a rich

dynamical structure, and one method for visualizing the space involves the use of Poincare

maps, which reduce the dimensionality of the problem. Such maps are successfully employed

in a number of analyses including Koon et al. [11, 12], Gomez et al. [13, 17], Howell et al.

[14], Topputo et al. [18], and Anderson and Lo [27]. However, an alternative representation

is sought to capture this knowledge and further facilitate trajectory design. A different

parameterization of a Poincare map involves the surface of section at the plane corresponding

to periapsis. Its advantage is based on the fact that the map is viewed within position

space. This type of map is denoted by a periapse Poincare map and was first defined and

introduced by Villac and Scheeres [28] to relate a trajectory escaping the vicinity of P2 back

to its previous periapsis in the planar Hill problem. Paskowitz and Scheeres [29, 30] extend

this analysis, using periapse Poincare maps to define lobes corresponding to the first four

periapse passages after capture into an orbit about P2. For application to the Europa orbiter

problem, the authors define “safe zones” where a spacecraft is predicted to neither escape

nor impact the surface of the satellite for a specified period of time. Davis and Howell

[31, 32] build on the analysis for short- and long-term trajectories to illustrate some of the

structures associated with manifold tubes corresponding to the L1 and L2 Lyapunov orbits

as viewed in terms of periapse maps. Long-term periapsis Poincare maps aid in organizing

large numbers of trajectories and can deliver initial conditions that yield trajectories with

specific characteristics. Davis and Howell [32] as well as Haapala and Howell [33] employ

such maps to compute specific types of trajectories in the region near P2. For trajectory design

in this regime, good initial guesses are critical and a technique that supplies geometrical

insight concerning the structure and delivers good approximate solutions is a practical

alternative to construct trajectory options. Poincare maps generally do require large amounts

of computation, but such capabilities are expanding very quickly. In addition, as more

design is accomplished within an interactive visual environment, techniques that are easily

adaptable to visual interfaces are appealing and likely to be incorporated into the next

generation of design tools. Ultimately, any of the design approaches that successfully leverage

computational speed and visual interfaces possess advantages.

This analysis is focused on exploring periapse maps to construct trajectory options

in multi-body regimes. The CR3BP frequently serves as a basis for the preliminary analysis

in such problems, and, thus, for this investigation, the primary focus is the region near the

smaller primary, P2. The goal is a strategy that facilitates preliminary trajectory design in

the CR3BP but still embraces the invariant manifold framework. Periapse Poincare maps are


defined and the structure that is apparent in such types of maps is summarized. The map

appears in position space, and different parameters at periapsis are represented depending

on the application. Examples serve to demonstrate how the maps can be exploited to deliver

different types of solutions. Periapse maps can also be used in conjunction with other types of

analysis tools. Such an approach has proved useful to isolate specific arcs and, in some cases,

blend them with other arcs for design.

2. Background

2.1. Dynamical Model

The dynamical model that is assumed for this analysis is consistent with the formulation

in the circular restricted three-body problem (CR3BP), where the motion of a particle of

infinitesimal mass, P3, is examined as it moves in the vicinity of two larger primary bodies, P1

and P2. A rotating frame, centered at the system barycenter, B, is defined such that the rotating

x-axis is directed from the larger primary (P1) to the smaller (P2), the z-axis is parallel to the

direction of the angular velocity of the primary system with respect to the inertial frame, and

the y-axis completes the dextral orthonormal triad. Let the nondimensional mass μ be the

ratio of the small mass of P2 to the total system mass. The system then admits five equilibrium

solutions comprised of the three collinear points (L1, L2, and L3) and two equilateral points

(L4 and L5) as depicted in Figure 3(a). Note that the magnitude of the Hill radius [8] is

rH =(μ

3

)1/3

, (2.1)

and is nearly equal to the distance between P2 and the Lagrange points L1 and L2. A single

integral of motion exists in the CR3BP. Known as the Jacobi integral, it is evaluated as

C = x2 + y2 +2μ

r23+

2(1 − μ

)r13

− v2, (2.2)

where v is the magnitude of the spacecraft velocity relative to the rotating frame. The scalar

nondimensional distances r13 and r23 reflect the distance to P3 from the primaries P1 and

P2, respectively. Then, the Jacobi constant restricts the motion of the spacecraft to regions in

space, where v2 ≥ 0; these regions are bounded by surfaces of zero velocity. In the planar

problem, the surfaces reduce to the zero velocity curves (ZVCs). For values of the Jacobi

integral higher than that associated with the L1 libration point, the ZVCs form closed regions

around the two primaries. As the energy of the spacecraft is increased, the Jacobi value

decreases until, at the L1 value of the Jacobi integral, that is, CL1, the ZVCs open at the L1

libration point, exposing a gateway. The particle P3 is now free to move between the two large

primaries. Similarly, when the value of the Jacobi integral decreases to the value associated

with L2,CL2, the ZVCs open at L2 and the particle, that is, spacecraft, may escape entirely from

the vicinity of the primaries. For values of such that CL3< C < CL2

, the ZVCs appear as in

Figure 3(b); the gray area cannot be reached by P3 at this Jacobi level and is thus “forbidden.”


L3x

y

L2

L5

L4

B

P1 P2

L1

(a) Schematic of libration points

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

x (dimensionless)

y(d

imen

sio

nle

ss)

Forbidden region

Interior region

Exterior region

P1P2

P2 region

∗∗

(b) Sample ZVC

Figure 3: Regions of position space delineated by the zero velocity curves.

The exterior and interior regions in the figure are denoted to be consistent with Koon et al.

[11].

2.2. Invariant Manifolds

For unstable periodic orbits in the CR3BP, in particular the periodic Lyapunov orbits in the

planar problem, higher-dimensional manifold structures exist and supply a framework for

this region via the stable and unstable manifolds. For L1 and L2 Lyapunov orbits to exist at a

given level of Jacobi constant, both the L1 and L2 gateways are open. Let to be the initial time

and the symbol T identify the time for one period. Assume that λs < 1 and λu = 1/λ2 are the

stable and unstable eigenvalues from the monodromy matrix, Φ(to + T, to), as determined for


−0.2

−0.1

0

0.1

0.2

0.8 1 1.2

x (dimensionless)

y(d

imen

sio

nle

ss)

Figure 4: Stable (blue) and unstable (red) manifolds associated with periodic L1 and L2 Lyapunov orbitsfor C = 3.1672 in the Earth-Moon three-body problem.

a given Lyapunov orbit. Let ws and wu be the associated eigenvectors, computed by solving

the equation Φ(to + T, to) ws = λsws, Φ(to + T, to) wu = λuwu. Define w+, w− as the

two directions associated with an eigenvector. The local half-manifolds, WU−loc

and WS−loc

, are

approximated by introducing a perturbation relative to a fixed point, x∗, along the periodic

orbit in the direction w−u and w−

s , respectively. Likewise, a perturbation relative to x∗ in the

direction w+u and w+

s , respectively, produces the local half-manifolds WU+loc

and WS+loc

. The

step along the direction of the eigenvector is denoted d and the initial states along the local

stable and unstable manifolds are evaluated as x±s = x∗ ± d · w±

s . The local stable manifolds

are globalized by propagating the states x+s and x−

s in reverse time in the nonlinear model.

This process yields the numerical approximation for the global manifolds WS+x∗

and WS−x∗

,

respectively. Propagating the state x±u = x∗ ± d ·w±

u in forward-time yields the unstable global

manifolds WU+x∗

and WU−x∗

. The collection of the stable and unstable manifolds corresponding

to each fixed point along sample L1 and L2 Lyapunov orbits in the Earth-Moon system appear

in Figure 4 in configuration space.

3. Periapse Poincare Maps

3.1. Creation of Periapse Maps

Connecting arcs in the CR3BP by exploiting the invariant manifold structures and the use of

Poincare maps has been successfully demonstrated by Koon et al., Gomez et al., and others

[11–14, 16, 17]. A Poincare map is commonly used to interpret the behavior of groups of

trajectories, relating the states at one point in time to a future state forward along the path. By

fixing the value of the Jacobi integral and selecting a surface of section, the dimensionality of

the problem is reduced by two; the four-dimensional planar problem is thus reduced to two

dimensions. Poincare maps, with various parameterizations, have proven to be a useful tool

for trajectory analysis and design. An alternative parameterization that facilitates exploration

of the design space and selection of certain types of characteristics is the periapse Poincare

map, first defined and applied by Villac and Scheeres [28]. In this type of map, the surface of

section is the plane of periapse passage. Villac and Scheeres, [28] as well as Paskowitz and


Scheeres, [29] employ periapse Poincare maps to explore the short-term behavior of escaping

trajectories as well as captured trajectories within the context of the Hill three-body problem

with applications to the Jupiter-Europa system. Building on these results, the short- and long-

term behavior of trajectories in the CR3BP as viewed in terms of periapse maps is explored

by Davis and Howell [31, 32] as well as Haapala and Howell [33]. Periapsis and apoapsis

relative to P2 are defined such that the radial velocity, r, of P3 relative to P2 is zero and are

distinguishable by the direction of radial acceleration, r ≥ 0 at periapsis and r ≤ 0 at apoapsis

[34].The periapse maps are relatively simple to create. Each point within the periapse

region corresponds to an initial condition that is associated with a specific trajectory about

P2 at a specified level of Jacobi constant. The initial condition always reflects a periapsis.

Given an initial position, velocity can be selected to produce a prograde or retrograde path;

all initial velocities are assumed prograde in this analysis. The initial state corresponding

to each trajectory is then propagated forward in time for a specified number of revolutions

to generate a series of subsequent periapse points. After the first revolution, the state is

evaluated against four possible outcomes: the particle impacts P2, the particle escapes out

the L1 gateway; the particle escapes through the L2 gateway, or the particle remains captured

near P2, that is, it continues to evolve within the ZVCs. Impact trajectories are defined as

those possessing a position vector, at any time, that passes on or within the radius of the body

P2. Escape trajectories are identified by an x-coordinate lying more than 0.01 nondimensional

units beyond either L1 or L2. Finally, the initial periapse position is colored consistent with the

outcome. Any states that continue to evolve are evaluated after the return to the next periapse

condition and the process continues until a predetermined time or number of revolutions.

Thus, maps are created to isolate certain types of behavior. Maps are produced in the Sun-

Saturn system for both short- and long-term propagations in Davis and Howell [32] and

Haapala and Howell [33]. Once a region is isolated, relationships between other periapsis

parameters are also exploited [32, 33].

3.2. Defining Regions in the Maps

As an example, the periapse structures in the Sun-Saturn system, associated with the

invariant manifolds corresponding to the planar Lyapunov orbits and a specified Jacobi

constant value, appear in Figure 5. Note that the Sun and Saturn are simply a representative

system. The manifolds in Figure 5(a) are propagated through their first periapses which

are indicated as blue points along the manifold trajectories. Observe that the periapses

along the manifold tubes define well-defined lobes that identify the escaping trajectories.

These lobes are analogous to the lobes defined for the Hill restricted three-body problem in

Villac and Scheeres [28] and Paskowitz and Scheeres [29]. Consistent with the nature of the

stable/unstable manifolds that are associated with the Lyapunov orbits as separatrices for the

flow, these lobes represent regions in which a periapsis occurs just prior to direct escape from

the vicinity of Saturn; any trajectory with a periapsis within one of these lobes escapes prior

to reaching its next periapsis. Conversely, a trajectory with periapsis lying outside a lobe

does not escape before its next periapse passage. These lobes can, therefore, be considered

gateways to escape: all escaping trajectories pass through one of these regions at the final

periapse passage prior to escape. (Note that, for some trajectories, the first periapsis actually

occurs in the vicinity of the libration point orbit near L1 or L2, however, these periapses are

neglected in this investigation.) The boundaries of the lobes, that is, the periapses along the

manifolds, appear as contours in Figure 5(a). To isolate the structures in the figure, a naming


∗∗

0.94 0.96 0.98 1 1.02 1.04 1.06 1.06−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

WS−L2

WS+L1

SL1 ,1

SL2 ,1

x (dimensionless)

y(d

imen

sio

nle

ss)

(a) Map of first periapses along WL1S+ and WL2S- forC=3.0173 (blue dots)

0.94 0.96 0.98 1 1.02 1.04 1.06

x (dimensionless)

y(d

imen

sio

nle

ss)

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

SL1 ,1

SL1 ,2

SL1 ,3

SL2 ,1

SL2 ,2

SL2 ,3

t+L1 ,1t+L1 ,2t+L1 ,3

t+L2 ,1t+L2 ,2t+L2 ,3

∗ ∗

= 0r

(b) First three periapses along WS+L1,WS−

L2, and along L1

forward-time escapes, and L2 forward-time escapes for C =3.0174

0.94 0.96 0.98 1 1.02 1.04 1.06

x (dimensionless)

y(d

imen

sio

nle

ss)

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

UL1 ,1

UL1 ,2

UL1 ,3

UL2 ,1

UL2 ,2 U

L2 ,3

∗ ∗

t−L1 ,1t−L1 ,2t−L1 ,3

t−L2 ,1t−L2 ,2t−L2 ,3

= 0r

(c) First three periapses along WU+L1,WU−

L2and L1 reverse-

time escapes, and L2 reverse-time escapes for C = 3.0174

0.94 0.96 0.98 1 1.02 1.04 1.06

y(d

imen

sio

nle

ss)

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

∗∗

x (dimensionless)

(d) Long-term periapse map associated with WS+L1

(blue)and WU−

L2(red) for C = 3.0174

Figure 5: Manifold and periapse structures in the Sun-Saturn system.

convention similar to one that appears in Koon et al. and Gomez et al. [11, 17] is employed.

Let ΓSLi,m denote the periapse contour formed by the mth intersection of the stable manifold

tube associated with the Li Lyapunov orbit in the P2 region, and ΓULj ,n denote the periapse

contour formed by the nth intersection of the unstable manifold tube associated with the LjLyapunov orbit in the P2 region. Then, to examine a periapse Poincare map, consider the

map in Figure 5(b). For the Lyapunov orbits at the given value of Jacobi constant, the first

three periapses along each manifold WS+L1

and WS−L2

appear, as marked by blue dots in the

figure, and distinct regions appear. Recall that the delineation between regions of allowed

periapses and apoapses occur where r = 0, and this boundary is plotted as a dotted black

line in the figures. Then, for a large number of arbitrary initial periapse locations, escaping

trajectories are examined in both forward and reverse time. Those trajectories that cross


−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.94 0.96 0.98 1 1.02 1.04 1.06

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(a) Sun-Saturn system; μ = 2.858 × 10−4, C = 3.016

−0.1

−0.05

0

0.05

0.1

0.15

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(b) Earth-Moon system; μ = 1.215 × 10−2, C = 3.160

Figure 6: Comparison of periapse regions in different systems.

the boundary x = xL1− 0.01 are defined as forward-time L1 escape trajectories, while those

that cross x = xL2+ 0.01 are defined as forward-time L2 escapes. Colors represent the fate

of these initial periapse states. The colored areas in Figure 5(b) identify the locations of the

first three periapses along these L1 and L2 forward-time escape trajectories and are denoted

Πt+L1,1→ 3, and Πt+

L2,1→ 3, respectively, where superscript t+ indicates a forward-time escape. The

map associated with the first three periapses along the unstable manifolds (WU+L1

and WU−L2

)is simply the reflection of the stable manifold contours from Figure 5(b) across the x-axis,

as demonstrated in Figure 5(c), and represent entry or capture into the specified region.

The colored areas in Figure 5(c) represent locations of the first three periapses along the L1

and L2 reverse-time escape trajectories and are denoted Πt−L1,1→ 3 and Πt−

L2,1→ 3, respectively,

where superscript t− indicates reverse-time escape. Propagating WS+L1

and WU−L2

for a longer

interval and plotting all the manifold periapses together, the periapse structures appearing

in Figure 5(d) emerge. The patterns apparent in the colored periapse regions are a function

of the mass ratio as well as the value of the Jacobi integral. However, due entirely to the

structure of the invariant manifolds associated with Lyapunov orbits, the patterns reappear

in different systems as is apparent in Figure 6. The Sun-Saturn mass ratio is μ = 2.858 × 10−5

as compared to the Earth-Moon system for which μ = 1.215 × 10−2. The values of C differ, of

course, but similar patterns are apparent in the two different systems as expected.

Ultimately, these maps represent pathways through the system. To highlight the paths

that are available in this type of map, consider Figure 7, where both the Sun-Saturn and

Earth-Moon systems are represented. In Figures 7(a) and 7(b), the regions corresponding

to the first six periapses along L1 forward-time escape trajectories are identified by color.

Contours ΓSLi,m appear in blue for m within ∼33 orbits of the primaries. The values of C in

each system are selected for visual comparison and simply represent a similar opening of

the gateway at L2. Sample paths are over plotted on the maps and periapses are marked by

dots in Figures 7(c) and 7(d). Assume that the initial periapsis occurs in the yellow region

Πt+

L1,6. The propagated path then moves to the next periapses in the green region (Πt+

L1,5),

and so on. The final periapsis occurs in the orange Πt+

L1,1region with a subsequent escape

through L1. In Figures 7(a) and 7(b), it is also apparent that some white regions exist. These

are regions outside the lobes, that is, outside of the manifolds. Because the initial states in

these white regions lie outside the manifolds, periapses in these regions can correspond to

long-term behavior in the system. Not all white regions in Figure 7 correspond to long-term


0.94 0.96 0.98 1 1.02 1.04 1.06−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

∗∗

x (dimensionless)

y(d

imen

sio

nle

ss)

t+L1 ,1t+L1 ,2t+L1 ,3

t+L1 ,4t+L1 ,5t+L1 ,6

(a) Sun-Saturn escape map for six periapses at a valueof Jacobi constant C = 3.0174

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

−0.1

−0.05

0

0.05

0.1

0.15

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

t+L1 ,1t+L1 ,2t+L1 ,3

t+L1 ,4t+L1 ,5t+L1 ,6

(b) Earth-Moon escape map for six periapses at a valueof Jacobi constant C = 3.17212

0.94 0.96 0.98 1 1.02 1.04 1.06−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(c) Sun-Saturn sequence originates with Πt+L1 ,6

through

Πt+L1 ,1

to escape through the L1 gateway

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

−0.1

−0.05

0

0.05

0.1

0.15

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(d) Earth-Moon sequence originates with Πt+L1 ,6

to

escape through the L1 gateway

Figure 7: Comparison of periapse maps for different Jacobi values and different systems.

capture in the vicinity of Saturn. At this value of Jacobi in the Sun-Saturn system, a relatively

large white region appears near P2 and a zoom of the same region near P2 at C = 3.0173

appears in Figure 8. In Figure 8, the periapses along the manifold associated with the L1

Lyapunov orbit appear in blue (WS+L1

) and those associated with the L2 orbit appear in red

(WS−L2

). The lobes reflecting escapes appear in white in Figure 8. The black dots correspond to

periapses representing periodic orbits and other trajectories that evolve in this system for as

long as 1000 years and it is apparent that there is significant structure in this long-term map.

Numerous orbits and quasiperiodic trajectories that yield certain characteristics are selected

directly from the map in Davis [35].

4. Applications of Periapse Maps

Exploiting Poincare maps to construct trajectories has certainly been accomplished by others.

Producing trajectories with certain characteristics can be facilitated with these types of

periapse maps as well. Maps at different energy levels can also be employed together to blend


−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

yp(rH)

xp (rH)

Figure 8: Long-term trajectories in a periapse Poincare map; Sun-Saturn system; C = 3.0173.

arcs. Examples illustrating the process appear below. In some examples, known solutions

emerge quickly.

4.1. Example: Transit through Both L1 and L2 Gateways

The unstable manifold tubes corresponding to the L1 and L2 Lyapunov orbits delineate

regions in the periapse maps that correspond to trajectories that enter the vicinity of P2

through the L1 or L2 gateways. As previously noted, the periapses of the unstable manifold

trajectories are the mirror image (reflected across the x-axis) of the stable manifold apses.

A trajectory that lies both within a stable L1 tube and an unstable L2 tube can represent a

“double transit” trajectory, that is, a trajectory that transits through both gateways. Such a

trajectory enters the P2 vicinity through L2 and subsequently escapes, after an unspecified

number of revolutions about P2, through L1 [11, 13, 17, 36]. For a particle to move from

the exterior to the interior region requires such a path. Similarly, a transit trajectory may

enter through the L1 gateway and depart through L2. A sample transit trajectory (L2 → L1)appears in Figure 9 in the Sun-Saturn system. The first two lobes representing periapses

within the L2 unstable manifold appear in red; the first two lobes associated with the L1 stable

manifold appear in the figure in blue. A periapse state is selected that lies within both of the

tubes; it appears as a black dot. Thus, the two lobes Πt−L2,2

and Πt+L1,2

overlap and the selected

point appears in the intersection. The result is a transit trajectory that enters the vicinity of

Saturn through L2 and completes three periapse passages before escaping through the L1

gateway, passing through three periapse lobes in sequence that highlight its passage. One

advantage of the periapse Poincare map for trajectory design applications is that the maps

exist in configuration space, allowing the selection of initial conditions based on the physical

location of periapsis. This type of application is explored in Davis and Haapala [35, 36].Haapala and Howell [33] further explore the use of periapse Poincare maps as a transit

trajectory design tool. By selecting initial conditions that correspond to periapses within the

region inside both contours ΓSLi,m and ΓULj ,n, that is, within the intersections of regions Πt+Li,m


−0.5

0

0.5

−1 −0.5 0 0.5 1

y(rH)

x (rH)

Figure 9: Transit from the exterior region to the interior region through gateways at both L2 and L1.

0.94 0.96 0.98 1 1.02 1.04 1.06−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

∗ ∗

t+L1 ,1t+L1 ,2t+L1 ,3

t+L1 ,4t+L1 ,5t+L1 ,6

x (dimensionless)

y(d

imen

sio

nle

ss)

Figure 10: Arrival contour ΓUL2 ,1in red and contours ΓSL1 ,1→ 10 in blue; C = 3.0174 (Saturn at 10x).

and Πt−Lj ,n

, and propagating in both forward- and reverse-time, a transit orbit passing through

the Li gateway in forward time, and the Lj gateway in reverse time is produced. Defining

one revolution about P2 as consisting of one periapsis and one apoapsis, the transit trajectory

experiences a number of revolutions about P2 equal to p = m + n − 3/2. Thus, to design

transit trajectories with some desired behavior in the vicinity of P2, the contours and/or

intersections are selected such that m + n = p + 3/2, where for an Interior-to-Interior (I-to-

I) transit i = j = 1, for an exterior-to-exterior (E-to-E) transit i = j = 2, for an Interior-

to-Exterior (I-to-E) transit i = 2, j = 1, and lastly for an Exterior-to-Interior (E-to-I) transit

i = 1, j = 2. Reconsider an E-to-I transfer using the map in Figure 10. After entering the

L2 gateway, all E-to-I transit trajectories reach their first periapses within the first contour

ΓUL2,1. Likewise, the last periapsis before exiting through the L1 gateway occurs within the first


0.992 0.993 0.994 0.995 0.996

×10−3

0.5

1

1.5

2

2.5

3

3.5

SL1 ,4

SL1 ,5

SL1 ,3

UL2 ,1

x (dimensionless)

y(d

imen

sio

nle

ss)

(a) Zoom view of ΓUL2 ,1

×107

×109

5

0

−5

1.35 1.4 1.45 1.5

x (km)

y(k

m)

∗ ∗

(b) p = 2.5 (tP2= 4.51 revs = 21.14 years)

×107

×109

5

0

−5

1.35 1.4 1.45 1.5

x (km)

y(k

m)

∗ ∗

(c) p = 3.5 (tP2= 6.10 revs = 28.59 years)

×107

×109

5

0

−5

1.35 1.4 1.45 1.5

x (km)

y(k

m)

∗ ∗

(d) p = 4.5 (tP2= 8.71 revs = 40.81 years)

Figure 11: Transit trajectories of varying numbers of revolutions about Saturn appear with periapsesmarked in red.

contour ΓSL1,1. Selecting n = 1, m = 10, it is possible to obtain trajectories with a maximum of

p = 9.5 revolutions about P2, although increasing n, m, or both could certainly render p ≥ 9.5.

The scenario from Figure 7(a) is repeated in Figure 10 but the manifold periapses are plotted

for fewer crossings. The figure includes contours ΓSL1,1→ 10 as blue dots, and ΓUL2,1as a contour

in red (indicated by a red arrow). The six regions Πt+L1,1→ 6 prior to escape out L1 are colored

appropriately. Within the red L2 entry lobe, a certain structure is apparent when considering

the intersections with the L1 escaping contours as noted by Haapala and Howell [33] as well

as Haapala [36]. Identifying the overlaps of the lobe regions yields transits that correspond

to p = 2.5, 3.5, 4.5 and are colored in magenta, navy, and green in Figure 11(a). Note that

the manifold contours at this particular value of Jacobi constant further split the arrival lobe

into different subregions that reflect different types of E-to-I transit paths in terms of p, that

is, the number of revolutions about P2. Representative E-to-I transit trajectories are generated

from the initial conditions marked as red points in Figure 11(a) and are plotted in Figures

11(b)–11(d). The time interval in the figures corresponds to the time required to pass from

x = xL2to x = xL1

and appears in the captions.


4.2. Example: Earth-Moon Transfers

Periapse Poincare maps are also applied to the problem of designing a low-energy ballistic

lunar transfer from low Earth orbit. Examined by various researchers [10, 12, 24, 37, 38], a

ballistic lunar transfer utilizes the gravity of the Sun to naturally raise the periapsis of an

Earth-centered trajectory, lowering the Δv required to reach the orbital radius of the Moon as

compared to a Hohmann transfer. Initial condition or periapse maps simplify the problem of

determining both the Δv and the orientation in the Sun-Earth frame that yield the appropriate

periapse raise.

Consider a spacecraft in a 167 km circular parking orbit centered at Earth. At this

energy level, the ZVCs are completely closed and the Sun has little effect on the orbit. A

maneuver applied at an appropriate location in the parking orbit decreases Jacobi Constant

and shifts the spacecraft to periapsis of a large Earth-centered orbit. This larger orbit is

affected significantly by the Sun, and the subsequent periapsis is raised to the radius of the

lunar orbit. If RE is the radius of the Earth, to reach the Moon’s orbital radius, the periapsis

must be raised from RE + 167 km to 384,400 km, corresponding to Δrp = 377, 855 km =0.2525 rH , in terms of the Sun-Earth Hill radius. To determine the required Δv, periapse maps

are created for a series of post-Δv Jacobi values. For each value of Δv, a Δrp map is created.

These maps allow a quick visualization of the orientation that produces the largest increase in

periapse radius for each Δv. The ZVCs and the trajectories corresponding to approximately

the largest periapse increase for a set of eight values of Δv appear in Figure 12(a). For

Δv < 3.199 km/s, the ZVCs constrain the apoapsis to a radius too small to allow solar gravity

to raise periapsis sufficiently. For Δv ≥ 3.2 km/s, however, the ZVCs are sufficiently open

(that is, the low value of Jacobi constant allows open gateways) to result in a periapse raise

sufficiently large to reach the lunar orbit. This value agrees well with a theoretical minimum

Δv determined by Sweetser [39] as 3.099 km/s for transfer from a 167-km parking orbit at

Earth, as well as with optimized Earth-Moon transfer Δv values calculated by Parker and

Born [38], who computes Δv ≈ 3.2 km/s for a transfer from a 185 km parking orbit in a Sun-

Earth-Moon gravity model.

A maneuver of 3.2 km/s shifts the value of Jacobi Constant from C = 3.068621, the

value of C corresponding to the low-Earth orbit, to C = 3.000785 in the Sun-Earth system,

the value of C associated with the transfer orbit. The postmaneuver initial condition map

corresponding to Δv = 3.2 km/s, or C = 3.000785, appears in a full view in Figure 12(b) and a

zoomed view in Figure 12(c); both are in the Sun-Earth rotating frame. The colors in Figures

12(b) and 12(c) simplify the options over the next revolution, that is, at this Jacobi constant,

C = 3.00078518, depending on the location of the post-Δv periapsis along the parking orbit,

the spacecraft can impact Earth (black) or escape the vicinity of the Earth entirely (red or

blue). However, in this application, the focus is on the trajectories that remain in the vicinity

of the Earth for at least one revolution, but with a significant rise in radius at the second

periapsis. To locate the orientation for the appropriate periapse raise, a Δrp map is produced.

The map is recolored in Figure 12(d) to reflect the value of Δrp over one revolution. Note

that black indicates a decrease in Δrp and blue→ red indicates an increasing magnitude of

Δrp. The 167 km parking orbit is again marked in green on the map in Figure 12(d). Four

bands of initial periapse angles exist that correspond to Δrp = 0.2525 rH ; these bands are

marked by purple rays. By selecting an initial periapsis at the intersection of a purple ray

with the green parking orbit, the orientation is determined that will result in the desired

raise in periapsis. Four such trajectories appear in Figure 12(e) corresponding to the four

initial conditions marked on the map in Figure 12(d). Clearly, each trajectory originates from


−1 0 1

−1

−0.5

0

0.5

1

y(rH)

x (rH)

v = 3.194 km/s

v = 3.195 km/s

v = 3.196 km/s

v = 3.197 km/s

v = 3.198 km/s

v = 3.199 km/s

v = 3.2 km/s

v = 3.201 km/s

(a) Trajectories at a series of Δv values, eachoriented for the largest periapse raise; lunar orbitalradius marked in green

−1 −0.5 0 0.5 1

y(rH)

x (rH)

−0.5

0

0.5

(b) Sun-Earth initial condition map correspondingto C = 3.000785

y(rH)

x (rH)

×10−3

−5 0 5−5

0

5

(c) Zoom near Earth; 167-km altitude; Earth orbitis green

y(rH)

x (rH)

×10−3

−5 0 5−5

0

5

IIa

IIb

IVa

IVb

(d) Initial condition map colored in terms of Δrpover one rev; desired Δrp for Earth-Moon transferdenoted by purple rays; four selected ICs

−1 0 1

−1

−0.5

0

0.5

1

y(rH)

x (rH)

IIa

IIb

IVa

IVb

(e) Four Earth-Moon transfer trajectories correlatedto selected points in map (d); Moon’s orbit is green

Figure 12: Low-energy Earth-Moon transfer.


the 167-km parking orbit; after a Δv = 3.2 km/s, the trajectory reaches apoapsis well beyond

the radius of the Moon’s orbit. But the subsequent periapsis along each orbit is precisely at

the radius of the Moon’s orbit. Given that the data for the maps is available, computed ahead

or in real time, the appropriate point is selected directly from the map; in a simple three-body

model, the map is a fast easy method for preliminary design.

4.3. Example: Arrival through the L1 Gateway

Sun-Saturn System

Consider arrivals in the vicinity of Saturn either by natural objects or spacecraft. The arrival

lobe from either gateway can serve to deliver different types of arrival trajectories. Consider

first an arrival through the L1 gateway in the Sun-Saturn system as represented in Figure 13.

The goal is an arrival periapsis that will produce a path that remains in the P2 vicinity.

In Figure 13(a), the arrival contour ΓUL1,1appears in red. Also plotted in the figure are the

contours representing ΓSL2,1→n. In this example, the points within Πt−L1,1

are colored based on

time to escape from P2 through either gateway, although not all trajectories escape within

the time of propagation. Thus, periapses within the arrival region, Πt−L1,1

, are colored such

that red→ blue indicates increasing time to escape. Selection of a periapsis point that is in a

region of the deepest blue color yields a trajectory that will remain in the P2 vicinity for an

extended time. Such a point is indicated in white with a white arrow in Figure 13(a). The

corresponding path appears in Figure 13(b). Once propagated, it remains near Saturn for at

least 1000 years. Arrival through L2 could be accomplished with the same process.

Earth-Moon System

The same type of arrival lobe can yield a path for a different application in the Earth-Moon

system in Figure 14. Many investigators have examined transfers from Earth to a periodic L1

Lyapunov or halo orbit including Perozzi and Di Salvo [37] and Parker and Born [23]. The L1

arrival contour for a given value of Jacobi Constant (C = 3.17212) appears in Figure 14(a). A

periapsis is selected within this lobe (as indicated by the red dot and arrow) that also lies on

a stable invariant manifold associated with the Lyapunov orbit. The selected lunar periapse

point represents the gold trajectory arc in Figures 14(b) and 14(c). Earth departure occurs as

a result of a maneuver, Δv1 = 3.105 km/s, from a 200 km circular Earth parking orbit in black.

The black arc then intersects the manifold (gold) corresponding to an L1 Lyapunov orbit

(Δv2 = 0.630 km/s). After a close pass by the Moon at 100 km altitude, the path eventually

asymptotically approaches the L1 orbit as seen in Figures 14(b) and 14(c). The final result

is a total Δv = 3.735 km/s in the CR3BP. The periapsis is selected specifically to be along a

manifold and is located on the blue contour ΓSL1,1, however, this path could also produce a

capture into a 100-km lunar orbit by including a 0.631 km/s maneuver at periapsis.

In Figure 15, the same objective is achieved with more excursions about the Moon

prior to arrival in the L1 orbit. Again, consider the arrival contour ΓUL1,1as plotted as the

red lobe in Figure 14(a). Within this contour, select a periapsis from the green region, which

represents the periapses corresponding to escaping trajectories from Πt+L1,5

. Propagating an

initial condition from this region yields a path with four revolutions about the Moon prior

to departure again through L1. If the periapsis initial condition that is selected in the green

region also lies on the contour ΓSL1,5, as noted in the figure, the green trajectory arc in Figure 15


1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1.009−6

−5

−4

−3

−2

−1

0

×10−3SL1 ,1

SL1 ,2

SL1 ,3

SL1 ,4

SL1 ,6

UL1 ,1

x (dimensionless)

y(d

imen

sio

nle

ss)

(a) Sun-Saturn system C = 3.0174; arrival contour ΓUL1 ,1overlaps with

contours ΓSL1 ,1→m; escape periapses colored by time to escape

0.94 0.96 0.98 1 1.02 1.04 1.06−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

∗ ∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(b) Selection of arrival periapsis in deep blue region results in a path thatremains in the vicinity of P2 for over 1000 years

Figure 13: Arrival through the Sun-Saturn L1 gateway; periapsis selected to yield a trajectory that remainsin the P2 vicinity for an extended time.

is generated. In comparison to the path in Figures 14(b) and 14(c), the green path in Figure 15

completes four revolutions about the Moon, passing the Moon at a higher altitude with a

result that the total Δv = 3.794 km/s with a corresponding increase in the time of flight.

Other periapses yield opportunities to insert into alterative lunar trajectories as well.

5. Summary and Concluding Remarks

Trajectory design in the multi-body regime remains a nontrivial problem. The goal in

any design process is to exploit the gravity of multiple bodies and deliver a trajectory

with characteristics that meet the requirements for a particular mission. Without analytical

solutions, increasing insight into the dynamical structure in the three-body problem


0.98 0.99 1 1.01 1.02 1.03 1.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005SL1 ,1

SL1 ,2

SL1 ,3

SL1 ,4

SL1 ,6S

L1 ,5 UL1 ,1

Periapsis

Periapsis

Arrivaltrajectory

arc

x (dimensionless)

y(d

imen

sio

nle

ss)

(a) Arrival contour ΓUL1 ,1and points within the lobe Πt−

L1,1colored in terms

of overlaps with regions Πt+L1 ,1→ 6 as determined from contours ΓSL1 ,1→ 6. A

periapse point is selected (red dot) and the trajectory computed in forwardand backward time from that periapsis (in gold)

∗ ∗

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x (dimensionless)

y(d

imen

sio

nle

ss)

(b) Earth departure as a result of a maneuver from a200-km circular parking orbit onto the black arc thatintersects an L1 Lyapunov orbit manifold

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

−0.1

−0.05

0

0.05

0.1

0.15

∗∗

x (dimensionless)

y(d

imen

sio

nle

ss)

(c) Zoom: close pass by the Moon and approach to theLyapunov orbit

Figure 14: Earth-Moon system C = 3.17212; manifold trajectory connects to the black arc that departs from200-km altitude Earth orbit; lunar periapsis at 100-km altitude; asymptotic approach to L1 Lyapunov orbit.

introduces more options for mission design as well as some new and exotic trajectories

that may enable future opportunities. However, the design space is very large and the

tradeoffs are not well defined. Many investigators are working to develop techniques to

create viable solutions for applications. The invariant manifold structure associated with the

collinear libration points, in particular, has supplied a geometrical framework and a number

of approaches are now available to represent this information. This current analysis explores


−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

x (dimensionless)

y(d

imen

sio

nle

ss)

(a)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

−0.1

−0.05

0

0.05

0.1

0.15

x (dimensionless)

y(d

imen

sio

nle

ss)

(b)

Figure 15: Earth-Moon system; C = 3.17212; manifold trajectory connects to the black arc that departs from200-km altitude Earth orbit; asymptotic approach to L1 Lyapunov orbit after four revolutions about theMoon.

an additional type of map to add another perspective for selecting arcs in support of the

design process. Within the context of the CR3BP, the periapse maps are an efficient design

tool. Using a combination of strategies to manage the information and deliver the results

is almost always necessary, however, automating these techniques is a priority; a visual

interface is a longer-term goal.

Acknowledgments

The authors wish to thank the Purdue University Graduate School, the College of

Engineering, and the School of Aeronautics and Astronautics for support of this effort.

Additional support is also appreciated through a GAANN Fellowship (Graduate Assistance

in Areas of National Need), a Zonta International Amelia Earhart Fellowship, and a Purdue

Forever Fellowship. Assistance for the computational facilities is provided through the

Eliasen Visualization Laboratory. The paper has been presented at the 6th International

Workshop and Advanced School “Spaceflight Dynamics and Control,” Covilha, Portugal,

March 28-30, 2011, http://www.aerospace.ubi.pt/workshop2011/.

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Research ArticleUnscented Kalman Filter Applied to the SpacecraftAttitude Estimation with Euler Angles

Roberta Veloso Garcia,1 Helio Koiti Kuga,1and Maria Cecilia F. P. S. Zanardi2

1 Space Mechanic and Control Division, INPE, 12227-010 Sao Jose dos Campos, SP, Brazil2 Department of Mathematics, FEG, UNESP, 12516-410 Guaratingueta, SP, Brazil

Correspondence should be addressed to Roberta Veloso Garcia, [email protected]



Copyright q 2012 Roberta Veloso Garcia et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The aim of this work is to test an algorithm to estimate, in real time, the attitude of an artificialsatellite using real data supplied by attitude sensors that are on board of the CBERS-2 satellite(China Brazil Earth Resources Satellite). The real-time estimator used in this work for attitudedetermination is the Unscented Kalman Filter. This filter is a new alternative to the extendedKalman filter usually applied to the estimation and control problems of attitude and orbit.This algorithm is capable of carrying out estimation of the states of nonlinear systems, withoutthe necessity of linearization of the nonlinear functions present in the model. This estimationis possible due to a transformation that generates a set of vectors that, suffering a nonlineartransformation, preserves the same mean and covariance of the random variables before thetransformation. The performance will be evaluated and analyzed through the comparison betweenthe Unscented Kalman filter and the extended Kalman filter results, by using real onboard data.

1. Introduction

The attitude of a spacecraft is defined by its orientation in space related to some reference

system. The importance of determining the attitude is related not only to the performance

of attitude control but also to the precise usage of information obtained by payload

experiments performed by the satellite. The attitude estimation is the process of calculating

the orientation of the spacecraft in relation to a reference system from data supplied by

attitude sensors. Chosen the vector of reference, an attitude sensor measures the orientation

of these vectors with respect to the satellite reference system [1]. Once these one or more

vectors measurements are known, it is possible to compute the orientation of the satellite

processing these vectors, using methods of attitude estimation. There are several methods


for determining the attitude of a satellite. Each method is appropriate to a particular type of

application and meets the needs such as available time for processing and accuracy to be

attained. However, all methods need observations that are obtained by means of sensors

installed on the satellite. The sensors are essential for attitude estimation, because they

measure its orientation relative to some referential, for example, the Earth, the sun, or a star.

In this work, the satellite attitude is described by Euler angles, due to its easy geometric

interpretation, and the method to estimate the attitude used is the Unscented Kalman Filter.

This method is capable of performing state estimation in nonlinear systems, besides taking

into account measurements provided by different attitude sensors. In this work there were

considered real data supplied by gyroscopes, infrared Earth sensors, and digital sun sensors.

These sensors are on board of the CBERS-2 satellite (China-Brazil Earth Resources Satellite),and the measurements were collected by the Satellite Control Centre of INPE (Brazilian

Institute for Space Research).

2. Representation of Attitude by Euler Angles

The attitude of an artificial satellite is directly related to its orientation in space. Through

the attitude one can know the spatial orientation of the satellite, since in most cases it can

be considered as a rigid body, where the attitude is expressed by the relationship between

two coordinate systems, one fixed on the satellite and another associated with a reference

system, for example, inertial system [2]. For a good performance of a mission it is essential

that the satellite be stabilized in relation to a specified attitude. The attitude stabilization is

achieved by the on-board attitude control, which is designed to acquire and maintain the

satellite in a predefined attitude. The CBERS-2 attitude is stabilized in three axes nominally

geo-pointed and can be described with respect to the orbital system. In this reference system,

the movement around the direction of the orbital velocity is called roll. The movement around

the direction normal to the orbit is called pitch, and finally the movement around the direction

Nadir/Zenith is called yaw. To transform a vector represented in a given reference to another

it is necessary to define a matrix of direction cosines (R), where its elements are written in

terms of Euler angles (φ, θ, ψ) [3]. The rotation sequence used in this work for the Euler

angles was the 3-2-1, where the coordinate system fixed in the body of the satellite (x, y,

z) is related to the orbital coordinate system (xo, yo, zo) through the following sequence of

rotations:

(i) 1st rotation of an angle ψ (yaw angle) around the zo axis,

(ii) 2nd rotation of an angle θ (roll angle) around an intermediate axis y′,

(iii) 3rd rotation of an angle φ (pitch angle) around the x-axis.

The matrix obtained through the 3-2-1 rotation sequence is given by

R =

⎡⎣ C(θ)C(ψ)

C(θ)S(ψ)

−S(θ)S(φ)S(θ)C

(ψ)− S(ψ)C(φ)S(φ)S(θ)S

(ψ)+ C(φ)C(ψ)S(φ)C(θ)

S(θ)C(ψ)C(φ)+ S(φ)S(ψ)C(φ)S(θ)S

(ψ)− S(φ)C(ψ)C(θ)C

(φ)⎤⎦, (2.1)

where R is the matrix of direction cosines with S = sin, C = cos, and T = tan.


By representing the attitude of a satellite with Euler angles, the set of kinematic

equations are given by [4]

⎡⎣φθψ

⎤⎦ =

⎡⎢⎢⎣1 S(φ)T(θ) C

(φ)T(θ)

0 C(φ)

−S(φ)

0S(φ)

C(θ)C(φ)C(θ)

⎤⎥⎥⎦⎧⎨⎩⎡⎣wx

wy

wz

⎤⎦ − R

⎡⎣ 0

−ω0

0

⎤⎦⎫⎬⎭, (2.2)

whereω0 is the orbital angular velocity and wx, wy, and wz are the components of the angular

velocity on the satellite system.

3. The Measurements System of Satellite

In order to estimate the satellite attitude accurately, several types of sensors, including gyros,

earth sensors, and solar sensors, are used in the attitude determination system. The equations

of these sensors are introduced here.

3.1. The Model for Gyros

The advantage of a gyro is that it can provide the angular displacement and/or angular

velocity of the satellite directly. However, gyros have an error due to drifting, meaning that

their measurement error increases with time. In this work, the rate-integration gyros (RIGs)are used to measure the angular velocities of the roll, pitch, and yaw of the satellite. The

mathematical model of the RIGs is [4]

ΔΘi =∫Δt

0

(ωi + εi)dt(i = x, y, z

), (3.1)

where ΔΘ are the angular displacement of the satellite in a time interval Δt, and εi are

components of bias of the gyroscope.

Thus, the measured components of the angular velocity of the satellite are given by

�ω =

(Δ�ΘΔt

)− �ε − �η1 = �g − �ε − �ηatt, (3.2)

where �g(t) is the output vector of the gyroscope, and �η1(t) represents a Gaussian white noise

process covering all the remaining unmodelled effects:

E[ηattη

Tatt

]= σ2

att. (3.3)

3.2. The Measurement Model for Infrared Earth Sensors (IRESs)

One way to compensate for the drifting errors present in gyros is to use the earth sensors.

These sensors are located on the satellite and aligned with their axes of roll and pitch. In the


work, two earth sensors are used, with one measuring the roll angle and the other measuring

the pitch angle. In principle, an earth sensor cannot measure the yaw angle. The measurement

equations for the earth sensors are given as [5]

φH = φ + ηφH ,

θH = θ + ηθH ,(3.4)

where ηφH and ηθH are the white noise representing the small remaining misalignment,

installation, and/or assembly errors assumed to be gaussian:

E[ηφHη

TφH

]= E[ηθHη

TθH

]= σ2

IRES. (3.5)

3.3. The Measurement Model for Digital Solar Sensors (DSSs)

Since an earth sensor is not able to measure the yaw angle, the solar sensors are used by the

Attitude Control System in order to overcome this problem. However, these sensors do not

provide direct measurements but coupled angle of pitch (αθ) and yaw (αψ). The measurement

equations for the solar sensor are obtained as follows [5]:

αψ = tan−1

( −SySx cos(60◦) + Sz cos(150◦)

)+ ηαψ (3.6)

when |Sx cos(60◦) + Sz cos(150◦)| ≥ cos(60◦), and

αθ = 24◦ − tan−1

(SxSz

)+ ηαθ (3.7)

when |24◦ − tan−1(Sx/Sz)| < 60◦, where ηαψ and ηαθ are the white noise representing the small

remaining misalignment, installation, and/or assembly errors assumed Gaussian:

E[ηαψη

Tαψ

]= E[ηαθη

Tαθ

]= σ2

DSS. (3.8)

The conditions are such that the solar vector is in the field of view of the sensor, and Sx,

Sy, and Sz are the components of the unit vector associated to the sun vector in the satellite

system and given by

Sx = S0x + ψS0y − θS0z,

Sy = S0y − ψS0x + φS0z,

Sz = S0z − φS0y + θS0z,

(3.9)

where S0x, S0y, and S0z are the components of the sun vector in the orbital coordinate system

and φ, θ, and ψ are the Euler angles-estimated attitude.


4. Attitude Estimation Methods

The goal of an estimator is to calculate the state vector (attitude) based on a set of observations

(sensors) [6]. In other words, it is an algorithm capable of processing measurements to

produce, according with a given criterium, a minimum error estimate of the state of a system.

In this paper, the real-time estimator used to estimate the satellite attitude is a variant of the

Kalman filter, applied to problems that present some nonlinearity. This estimator is described

as follows.

4.1. Unscented Kalman Filter

The basic premise behind the Unscented Kalman Filter (UKF) is that it is easier to

approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function.

Instead of linearizing to first order using Jacobian matrices, the UKF uses a rational

deterministic sampling approach to capture the mean and covariance estimates with a

minimal set of sample points. The nonlinear function is applied to each point, in turn, to yield

a cloud of transformed points. The statistics of the transformed points can then be calculated

to form an estimate of the nonlinearly transformed mean and covariance. We present an

algorithmic description of the UKF omitting some theoretical considerations, left to [7, 8].Consider the system model given by

xk+1 = f(xk, k) +Gkwk,

yk = h(xk, k) + νk,(4.1)

where xk is the n × 1 state vector and yk is the m × 1 measurement vector. We assume that the

process noise wk and measurement-error noise νk are zero-mean Gaussian noise processes

with covariances given by Qk and Rk, respectively. In this work the state vector at time k is

defined by the Euler angles and gyro biases:

xk =[φ, θ, ψ, εx, εy, εz

]T. (4.2)

Performing the necessary simplifications (small Euler angles) in the set of (2.2), the

attitude angles and gyro angular velocity biases are modelled as follows:

φ(t) = ω0 sin ψ + ωx + θωz,

θ(t) = ω0 cos ψ + ωy + φωz,

ψ(t) = ω0

(θ sin ψ − φ cos ψ

)+ ωz + φωy,

�ε(t) = 0.

(4.3)

Given the state vector at step k − 1, we compute a collection of sigma-points, stored in

the columns of the n × (2n + 1) sigma point matrix χk−1, where n is the dimension of the state


vector. In our case, n = 6, so χk−1 is a 6 × 13 matrix. The columns of χk−1 are computed by

(χk−1

)0= xk−1,(

χk−1

)i= xk−1 +

(√(n + λ)Pk−1

)i

, i = 1, . . . , n,

(χk−1

)i= xk−1 −

(√(n + λ)Pk−1

)i−n, i = n + 1, . . . , 2n,

(4.4)

in which λεR , (√(n + λ)Pk−1)i is the ith column of the matrix square root of (n + λ)Pk−1.

Once χk−1 computed, we perform the prediction step by first predicting each column

of χk−1 through time by Δt using

(χk)i= f((χk−1

)i

), i = 0, . . . , 2n, (4.5)

where f is differential equation defined in (2.2) or (4.3). In our formulation, we perform a

numerical Runge-Kutta integration.

With (χk)i being calculated, the a priori state estimate is

x−k =

2n∑i=0

Wmi

(χk)i, (4.6)

where Wmi are weights defined by

Wm0 =

λ

(n + λ),

Wmi =

1

2(n + λ)i = 1, . . . , 2n.

(4.7)

As the last part of the prediction step, we calculate the a priori error covariance with

P−k =

2n∑i=0

Wci

[(χk)i− x−

k

][(χk)i− x−

k

]T +Qk, (4.8)

where Qk is the process error covariance matrix, and the weights are defined by

Wc0 =

λ

(n + λ)+(

1 − α2 + β2),

Wci =

1

2(n + λ)i = 1, . . . , 2n,

(4.9)

where α is a scaling parameter which determines the spread of the sigma points and β is a

parameter used to incorporate any prior knowledge about the distribution of x [9].


To compute the correction step, we first must transform the columns of χk through the

measurement function to Yk. In this way

(Yk)i = h((χk)i

), i = 0, . . . , 2n,

y−k =

2n∑i=0

Wmi (Yk)i.

(4.10)

With the mean measurement vector y−k

, we compute the a posteriori state estimate using

xk = x−k +Kk

(yk − y−

k

), (4.11)

where Kk is the Kalman gain. In the UKF formulation, Kk is defined by

Kk = Pxk,ykP−1yk ,yk

, (4.12)

where

Pyk,yk =2n∑i=0

W(c)i

[(Yk)i − y−

k

][(Yk)i − y−

k

]T + Rk,

Pxk,yk =2n∑i=0

W(c)i

[(χk)i− x−

k

][(Yk)i − y−

k

]T,

(4.13)

where Rk represents the measurement error covariance matrix.

Finally, the last calculation in the correction step is to compute the a posteriori estimate

of the error covariance given by

Pk = P−k −KkPyk,ykK

Tk . (4.14)

5. Results

Here, the results and the analysis from the algorithm developed to estimate the attitude are

presented. To validate and to analyze the performance of the estimators, real sensors data

from the CBERS-2 satellite were used (see [10, 11]). The CBERS-2 satellite was launched

on October 21, 2003. The measurements are for the month of April 2006, available to the

ground system at a sampling rate of about 8.56 sec. The algorithm was implemented through

MATLAB software. To check the performance the UKF, their results were compared with the

estimated attitude by the more conventional EKF (Extended Kalman Filter), considering the

following set of initial conditions:

(i) initial attitude: φ = θ = ψ = 0 deg;

(ii) initial bias of gyro: εx = 5.56 deg/hour, and εy = 0.87 deg/hour, εz = 6.12 deg/hour;

(iii) initial covariance (P): σ2φ,θ,ψ

= (0.5 deg)2 (error related to the attitude), and σ2bg

=(1 deg/hour)2 (error related to the drift of gyro);


13.76 13.78 13.8 13.82 13.84 13.86 13.88 13.9 13.92 13.94−20

0

20

40

Time (hours)M

easu

rem

ents

(deg)

s sDigital un ensors

DSS1

DSS2

(a)

13.76 13.78 13.8 13.82 13.84 13.86 13.88 13.9 13.92 13.94

Time (hours)

Mea

sure

men

ts(d

eg)

Infrared earth sensor

IRES1

IRES2

−0.55

−0.45

−0.35

(b)

13.76 13.78 13.8 13.82 13.84 13.86 13.88 13.9 13.92 13.94

Time (hours)

Mea

sure

men

ts(d

eg)

Incremental measures of gyroscope

Gyrox

Gyroy ( 102)

Gyroz

×10−4

−5

0

5

10

·

(c)

Figure 1: Real measurements supplied by attitude sensors.

(iv) observation noise covariance (R): σ2DSS

= (0.3 deg)2 (sun sensor), are σ2IRES

=(0.03 deg)2 (earth sensor);

(v) dynamic noise covariance (Q): σ2att = (0.1 deg)2 (noise related to the attitude),

σ2Dbgx

= σ2Dbgy

= (0.01 deg/hour)2, and σ2Dbgz

= (0.005 deg/hour)2 (noise related to

the drifting of gyro).

The real measurements obtained by the attitude sensors (digital sun sensors, infrared

Earth sensors, and gyroscopes) are shown in Figure 1.

In Figures 2 and 3 are observed the behavior of attitude and the biases of gyro during

the period analyzed. The average estimated values for the axes of roll and pitch, considering

the Unscented Kalman Filter, are in the order of −0.47 deg and −0.45 deg, respectively, and

their standard deviations are about 0.02 deg. For the yaw axis the estimate seems not to behave

randomly and its average estimated value is about −1.47 deg with standard deviation 0.3 deg.

The attitude estimated by the extended Kalman filter had their values for the axes roll and

pitch in the order of about −0.49 deg and −0.43 deg with standard deviation about 0.05 deg.

For the axis pitch, its average value is −1.42 deg and standard deviation is 0.36 deg.


13.76 13.8 13.84 13.88 13.92−0.7

−0.6

−0.5

−0.4

−0.3

Time (hours)

Att

itu

de(d

eg) Estimated roll angle

(a)

13.76 13.8 13.84 13.88 13.92

Time (hours)

Att

itu

de(d

eg) −0.8

−0.6

−0.4

−0.2

0

Estimated pitch angle

(b)

13.76 13.8 13.84 13.88 13.92

Time (hours)

Att

itu

de(d

eg)

−2

−1.5

−1

−0.5

0

UKF

EKF

Estimated yaw angle

(c)

Figure 2: Attitude estimated by the Unscented Kalman filter end Extended Kalman filter.

13.76 13.8 13.84 13.88 13.92

Time (hours)

5.7

5.75

5.8

ε x(d

eg/

ho

ur)

Estimated gyro bias in x

(a)

13.76 13.8 13.84 13.88 13.92

Time (hours)

ε y(d

eg/

ho

ur)

0.8

1

1.2

1.4Estimated gyro bias in y

(b)

13.76 13.8 13.84 13.88 13.92

Time (hours)

6.1

6.2

6.3

6.4

UKF

EKF

ε z(d

eg/

ho

ur)

Estimated gyro bias in z

(c)

Figure 3: Bias estimated by the Unscented Kalman filter end Extended Kalman filter.


13.76 13.8 13.84 13.88 13.920

0.05

0.1

0.15

0.2

Time (hour)

σφ(d

eg)

Attitude error estimation

(a)

13.76 13.8 13.84 13.88 13.920

0.05

0.1

0.15

0.2

Time (hour)

σθ(d

eg)


(b)

13.76 13.8 13.84 13.88 13.920.2

30.

40.

50.

Time (hour)

σψ(d

eg)

UKF

EKF


(c)

Figure 4: Attitude errors (Standard Deviation) estimated.

13.76 13.8 13.84 13.88 13.92

Time (hour)

Biases errors estimation in

0.98

0.99

1

1.01

1.02

σεx(d

eg/

ho

ur)

x

(a)

13.76 13.8 13.84 13.88 13.92

Time (hour)

σεy(d

eg/

ho

ur)

Biases estimation errors in

0.98

0.99

1

1.01

1.02y

(b)

13.76 13.8 13.84 13.88 13.92

Time (hour)

Biases estimation errors in

0.98

0.99

1

1.01

1.02

UKF

EKF

σεz(d

eg/

ho

ur) z

(c)

Figure 5: Bias errors (Standard Deviation) estimated.


13.76 13.8 13.84 13.88 13.92−1

−0.8

−0.6

−0.4

−0.20

0.2

0.4

Time (hour)

Res

idu

als(d

eg)

Digital sun sensor (DSS1)

UKF

EKF

(a)

13.76 13.8 13.84 13.88 13.92

Time (hour)

Res

idu

als(d

eg)

Digital sun sensor (DSS2)

−0.2

0

0.2

0.4

0.6

0.8

UKF

EKF

(b)

Figure 6: Residuals related to DSS attitude sensors.

13.76 13.8 13.84 13.88 13.92

Time (hour)

Res

idu

als(d

eg)

−0.6

−0.4

−0.2

0

0.2

UKF

EKF

Infrared earth sensor (IRES1)

(a)

13.76 13.8 13.84 13.88 13.92

Time (hour)

Res

idu

als(d

eg)

−0.5

−0.4

−0.3

−0.2

−0.10

0.1

0.2

UKF

EKF

Infrared earth sensor (IRES2)

(b)

Figure 7: Residuals related to IRES attitude sensors.

Figures 4 and 5 present the standard deviations for both estimators for the attitude

and the bias of the gyro. It is observed that the attitude standard deviations and the standard

deviations of the gyro bias decrease with a tendency to stabilize around a value for both

filters. However, the graphs show the superiority of UKF, because in most cases it works

within a range of protection better than the EKF.

In Figures 6 and 7, we can see that the residues of sun sensors and Earth sensors have

the same behavior for both estimators. For the Earth sensors, the residuals obtained by the

both estimators are smaller and show a tendency to zero mean. However, the residues of

UKF are still lower, being at about −0.009 deg for IRES1 and 0.004 deg for IRES2. Already

the residues of the EKF are approximately 0.01 deg and −0.027 deg for IRES1 and IRES2,

respectively.

These results seem consistent because in this case it is not possible to compare the

estimated values with true values, since these values are not known.

6. Final Comments

In this paper, the Unscented Kalman Filter estimator applied in nonlinear systems was

presented for use in real-time attitude estimation. The main objective was to estimate the


attitude of a CBERS-2 like satellite, using real data provided by sensors that are on board

of the satellite. To verify the consistency of the estimator, the attitude was estimated by two

different methods. The usage of real data from on-board attitude sensors poses difficulties

like mismodelling, mismatch of sizes, misalignments, unforeseen systematic errors, and

postlaunch calibration errors. However, it is observed that, although the EKF and UKF have

roughly the same accuracy, the UKF leads to a convergence of the state vector faster than the

EKF. This fact was expected, since the UKF prevents the linearizations needed for EKF, when

the system has some nonlinearity in their equations.

References

[1] V. L. Pisacane and R. C. Moore, Fundamentals of Space Systems, Oxford University Press, New York,NY, USA, 1994.

[2] M. D. Shuster, “Survey of attitude representations,” Journal of the Astronautical Sciences, vol. 41, no. 4,pp. 439–517, 1993.

[3] M. C. Zanardi, Dinamica de atitude de satelites artificiais, Ph.D. thesis, FEG-UNESP, Guaratingueta, SaoPaulo, Brazil, 2005.

[4] J. R. Wertz, Spacecraft Attitude Determination and Control, D. Reidel, Dordrecht, The Netherlands, 1978.[5] H. Fuming and H. K. Kuga, “CBERS simulator mathematical models,” CBTT Project, CBTT /2000

/MM /001. INPE, S ao Jose dos Campos, 1999.[6] R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filterin, Wiley,

New York, NY, USA, 1996.[7] S. J. Julier and J. K. Uhlmann, “Reduced sigma point filters for the propagation of means and

covariances through nonlinear transformations,” in Proceedings of the American Control Conference, vol.2, pp. 887–892, Anchorage, Alaska, USA, May 2002.

[8] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proceedings of the IEEE,vol. 92, no. 3, pp. 401–422, 2004.

[9] S.J. Julier and J. K. Uhlmann, “A new extension of the kalman filter for nonlinear systems,” inProceedings of the International Symposium on Aerospace/Defense Sensing, Simulation and Controls, (SPIE’99), SPIE, Orlando, Fla, USA, April 1997.

[10] H. K. Kuga, R. V. F. Lopes, and A. R. Silva, “On board attitude reconstitution of CBERS-2 Satellite,” inProceedings of the Proceedings of XIII Brazilian Coloquium on Orbital Dynamics, p. 109, 2006.

[11] R. V. F. Lopes and H. K. Kuga, “CBERS-2: on ground attitude determination from telemetry data,”Internal report C-ITRP, INPE, 2005.


Research ArticleOptimal Two-Impulse Trajectories withModerate Flight Time for Earth-Moon Missions

Sandro da Silva Fernandesand Cleverson Maranhao Porto Marinho

Departamento de Matematica, Instituto Tecnologico de Aeronautica,12228-900 Sao Jose dos Campos, SP, Brazil


Received 10 June 2011; Accepted 4 September 2011


Copyright q 2012 S. da Silva Fernandes and C. Maranhao Porto Marinho. This is an open accessarticle distributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properly cited.

A study of optimal two-impulse trajectories with moderate flight time for Earth-Moon missions ispresented. The optimization criterion is the total characteristic velocity. Three dynamical modelsare used to describe the motion of the space vehicle: the well-known patched-conic approximationand two versions of the planar circular restricted three-body problem (PCR3BP). In the patched-conic approximation model, the parameters to be optimized are two: initial phase angle of spacevehicle and the first velocity impulse. In the PCR3BP models, the parameters to be optimized arefour: initial phase angle of space vehicle, flight time, and the first and the second velocity impulses.In all cases, the optimization problem has one degree of freedom and can be solved by means ofan algorithm based on gradient method in conjunction with Newton-Raphson method.

1. Introduction

In the last two decades, new types of trajectories have been proposed to transfer a spacecraft

from an Earth orbit to a Moon orbit which reduce the cost of the traditional Hohmann transfer

based on the two-body dynamics [1]. The new trajectories are designed using more realistic

models of the motion of the spacecraft such as the PCR3BP [2, 3] or the planar bicircular four

body problem [4]. These models describing the motion of the spacecraft exhibit very complex

dynamics that are used to design new Earth-to-Moon trajectories [5–10]. The most of the

proposed approaches to calculate the new trajectories are based on the concept of the weak

stability boundary introduced by Belbruno [11], and, usually, involve large flight time. Only

few works consider the minimization of the total cost or the time for two-impulse trajectories

[8–10, 12, 13].In this paper, the problem of transferring a space vehicle from a circular low Earth orbit

(LEO) to a circular low Moon orbit (LMO) with minimum fuel consumption is considered.


It is assumed that the propulsion system delivers infinite thrusts during negligible times

such that the velocity changes are instantaneous, that is, the propulsion system is capable of

delivering impulses. The class of two impulse trajectories is considered: a first accelerating

velocity impulse tangential to the space vehicle velocity relative to Earth is applied at a

circular low Earth orbit and a second braking velocity impulse tangential to the space vehicle

velocity relative to Moon is applied at a circular low Moon orbit [12]. The minimization of

fuel consumption is equivalent to the minimization of the total characteristic velocity which

is defined by the arithmetic sum of velocity changes [14].Three dynamical models are used to describe the motion of the space vehicle: the well-

known patched-conic approximation [1] and two versions of the planar circular restricted

three-body problem (PCR3BP). One version of PCR3BP assumes the Earth is fixed in space;

this version will be referred as simplified version of PCR3BP, and it is same one used by

Miele and Mancuso [12]. The second version of PCR3BP assumes the Earth moves around the

center of mass of the Earth-Moon system, that is, it corresponds to the classical formulation [2,

3]. In the patched-conic approximation model, the parameters to be optimized are two: initial

phase angle of space vehicle and the first velocity impulse. In this approach, the two-point

boundary value problem involves only one final constraint. In this model, the flight time and

the second velocity impulse are determined from the two-body dynamics after solving the

two-point boundary value problem. In the PCR3BP models, the parameters to be optimized

are four: initial phase angle of space vehicle, flight time, and the first and the second

velocity impulses. In these formulations, the two-point boundary value problem involves

three final constraints. In all cases, the optimization problem has one degree of freedom

and can be solved by means of an algorithm based on gradient method [15] in conjunction

with Newton-Raphson method [16]. The analysis of optimal trajectories is then carried out

considering several final altitudes of a clockwise or counterclockwise circular low Moon orbit.

All trajectories departure from a counterclockwise circular low Earth orbit corresponding

to the altitude of the Space Station. Maneuvers with direct ascent and multiple revolutions

around the Earth are considered in the analysis. The results for maneuvers with direct ascent

are compared to the ones obtained by Miele and Mancuso [12] who used the sequential

gradient-restoration algorithm for solving the optimization problem [15]. The results for

maneuvers with multiple revolutions show that fuel can be saved if a lunar swing-by occurs.

2. Optimization Problem Based on Patched-Conic Approximation

In this section, the optimization problem based on the patched-conic approximation is

formulated. A detailed presentation of the patched-conic approximation can be found in

Bate et al. [1].The following assumptions are employed.

(1) The Earth is fixed in space.

(2) The eccentricity of the Moon orbit around the Earth is neglected.

(3) The flight of the space vehicle takes place in the Moon orbital plane.

(4) The gravitational fields of Earth and Moon are central and obey the inverse square

law.

(5) The trajectory has two distinct phases: geocentric and selenocentric trajectories. The

geocentric phase corresponds to the portion of the trajectory which begins at the

point of application of the first impulse and extends to the point of entering the


v0

r0

γ0ϕ0

γ1

r1

Earth

Moonat time t1

Moonat time t0

Moon’s sphere ofinfluence

D

r2

λ1

Figure 1: Geometry of the geocentric phase.

Moon’s sphere of influence. The selenocentric phase corresponds to the portion of

trajectory in the Moon’s sphere of influence and ends at the point of application

of the second impulse. In each one of these phases, the space vehicle is under the

gravitational attraction of only one body, Earth or Moon.

(6) The class of two impulse trajectories is considered. The impulses are applied

tangentially to the space vehicle velocity relative to Earth (first impulse) and Moon

(second impulse).

An Earth-Moon trajectory is completely specified by four quantities: r0—radius of

circular LEO; v0—velocity of the space vehicle at the point of application of the first impulse

after the application of the impulse; ϕ0—flight path angle at the point of application of the

first impulse; γ0—phase angle at departure. These quantities must be determined such that

the space vehicle is injected into a circular LMO with specified altitude after the application of

the second impulse. It is particularly convenient to replace γ0 by the angle λ1 which specifies

the point at which the geocentric trajectory crosses the Moon’s sphere of influence.

Equations describing each phase of an Earth-Moon trajectory are briefly presented in

what follows. It is assumed that the geocentric trajectory is direct and that lunar arrival occurs

prior to apoapsis of the geocentric orbit. Figure 1 shows the geometry of the geocentric phase.

For a given set of initial conditions (r0, v0, ϕ0), energy and angular momentum of the

geocentric trajectory can be determined from the equations

E =1

2v2

0 −μE

r0,

h = r0v0 cosϕ0,

(2.1)

where μE is the Earth gravitational parameter.


Moon

at time t1

Moon’s sphere of

influence

r1

λ1

r2

v1

v2

ϕ2

−vMoon

Figure 2: Geometry of the selenocentric phase.

From the geometry of the geocentric phase (Figure 1), one finds

r1 =√D2 + R2

S − 2DRS cosλ1,

sin γ1 =RS

r1sinλ1,

(2.2)

where D is the distance from the Earth to the Moon and RS is the radius of the Moon’s sphere

of influence. Subscript 1 denotes quantities of the geocentric trajectory calculated at the edge

of the Moon’s sphere of influence.

From energy and angular momentum of the geocentric trajectory, one finds

v1 =

√2

(E +

μE

r1

),

cosϕ1 =h

r1v1.

(2.3)

The selenocentric phase begins at the point at which the geocentric trajectory crosses

the Moon’s sphere of influence. Figure 2 shows the geometry of the selenocentric phase for a

clockwise arrival to LMO. Thus,

r2 = RS, (2.4)

v2 = v1 − vM, (2.5)

where vM is the velocity vector of the Moon relative to the center of the Earth. Subscript 2

denotes quantities of the selenocentric trajectory calculated at the edge of the Moon’s sphere

of influence.


From (2.5), one finds

v2 =√v2

1 + v2M − 2v1vM cos

(ϕ1 − γ1

),

tan(λ1 ± ϕ2

)= −

v2 sin(ϕ1 − γ1

)vM − v2 cos

(ϕ1 − γ1

) . (2.6)

The upper sign refers to clockwise arrival to LMO and the lower sign refers to counter-

clockwise to LMO.

The semimajor axis af and eccentricity ef of the selenocentric trajectory are given by

af =r2

2 −Q2,

ef =√

1 +Q2(Q2 − 2)cos2ϕ2,

(2.7)

where Q2 = r2v22/μM and μM is the Moon gravitational parameter.

The second impulse is applied at the periselenium of the selenocentric trajectory such

that the terminal conditions, before the impulse, are defined by

rpM = af(1 − ef

),

vpM =

√√√√μM(1 + ef

)af(1 − ef

) . (2.8)

Equations (2.1)–(2.8) lead to the following two-point boundary value problem: for a

specified value of λ1 and a given set of initial parameters r0 and ϕ0 = 0◦ (the impulse is

applied tangentially to the space vehicle velocity relative to Earth), determine v0 such that

the final condition rpM = rf is satisfied, where r0 is the radius of LEO and rf is the radius of

LMO (both orbits, LEO and LMO, are circular). This boundary value problem can be solved

by means of Newton-Raphson method [16].After computing v0, the velocity changes at each impulse can be determined

Δv1 = v0 −√μE

r0,

Δv2 =

√√√√μM(1 + ef

)af(1 − ef

) −√μM

rf.

(2.9)

The total characteristic velocity is then given by

ΔvTotal = Δv1 + Δv2. (2.10)

Note that the total characteristic velocity is a function of λ1 for a given set of parameters

(r0, ϕ0 = 0◦, rf). Accordingly, the following optimization problem can be formulated:


determine λ1 to minimize ΔvTotal. This minimization problem was solved using a classic

gradient method [15]. The results are presented in Section 5.

The total flight time of an Earth-Moon trajectory is given by

T = ΔtE + ΔtM, (2.11)

where ΔtE is the flight time of the geocentric trajectory and ΔtM is the flight time of the

selenocentric trajectory. These flight times are calculated from the well-known time of flight

equations of two-body dynamics as follows:

ΔtE =

√√√ a30

μE(E1 − e0 sinE1),

ΔtM =

√√√√(−af)3

μM

(ef sinhF2 − F2

),

(2.12)

with eccentric anomaly E1 and hyperbolic eccentric anomaly F2 obtained, respectively, from

cosE1 =1

e0

(1 − r1

a0

),

coshF2 =1

ef

(1 − r2

af

).

(2.13)

Since lunar arrival occurs prior to apoapsis of the geocentric trajectory, 0 < E1 ≤ 180◦, and F2

is positive. So, equations above define E1 and F2 without ambiguity. The semimajor axis and

eccentricity of the geocentric trajectory are given by

a0 =r0

2 −Q0,

e0 =√

1 +Q0(Q0 − 2)cos2ϕ0,

(2.14)

where Q0 = r0v20/μE. Recall that the impulses are applied at the periapses of the geocentric

and selenocentric trajectories.

3. Optimization Problem Based on the Simplified Version of PCR3BP

In this section, the optimization problem based on the simplified PCR3BP is formulated. A

detailed presentation of this problem can be found in Miele and Mancuso [12].The following assumptions are employed.

(1) The Earth is fixed in space.

(2) The eccentricity of the Moon orbit around the Earth is neglected.

(3) The flight of the space vehicle takes place in the Moon orbital plane.


(4) The space vehicle is subject to only the gravitational fields of Earth and Moon.

(5) The gravitational fields of Earth and Moon are central and obey the inverse square

law.

(6) The class of two impulse trajectories is considered. The impulses are applied

tangentially to the space vehicle velocity relative to Earth (first impulse) and Moon

(second impulse).

Consider an inertial reference frame Exy contained in the Moon orbital plane: its origin

is the Earth center; the x-axis points towards the Moon position at the initial time t0 = 0 and

the y-axis is perpendicular to the x-axis. Figure 3 shows the inertial reference frame Exy.

In the Exy reference frame, the motion of the space vehicle (P) is described by the

following differential equations:

xP = − μEr3EP

xP − μM

r3MP

(xP − xM),

yP = − μEr3EP

yP − μM

r3MP

(yP − yM

),

(3.1)

where rEP and rMP are, respectively, the radial distances of space vehicle from Earth (E) and

Moon (M), that is, r2EP = (xP −xE)2+(yP −yE)2 and r2

MP = (xP −xM)2+(yP −yM)2. Because the

origin of the inertial reference frame Exy is the Earth center, the position vector of the Earth

is defined by rE = (0, 0). The position vector of the Moon in the inertial reference frame Exy is

defined by rM = (xM, yM). Since the eccentricity of the Moon orbit around Earth is neglected,

the Moon inertial coordinates are given by

xM(t) = D cos(ωMt),

yM(t) = D sin(ωMt),(3.2)

where ωM =√μE/D3 is the angular velocity of the Moon.

The initial conditions of the system of differential equations correspond to the position

and velocity vectors of the space vehicle after the application of the first impulse. The initial

conditions (t0 = 0) can be written as follows:

xP (0) = xEP (0) = rEP (0) cos θEP (0),

yP (0) = yEP (0) = rEP (0) sin θEP (0),

xP (0) = xEP (0) = −[√

μE

rEP (0)+ Δv1

]sin θEP (0),

yP (0) = yEP (0) =

[√μE

rEP (0)+ Δv1

]cos θEP (0),

(3.3)


Earth

Moon

at time t0x

y

Space vehicle

Moon’s trajectory

rEP rMP

rMθEP

Figure 3: Inertial reference frame Exy.

where Δv1 is the velocity change at the first impulse, rEP (0) = rEP0, and θEP (t) is the angle

defining the position of the space vehicle in the inertial reference frame Exy at time t, more

precisely the angle which the position vector rP forms with x-axis. It should be noted that

rEP (0) and vEP (0) or, equivalently, rP (0) and vP (0) are orthogonal, because the impulse is

applied tangentially to the circular LEO.

The final conditions of the system of differential equations correspond to the position

and velocity vectors of the space vehicle before the application of the second impulse. The

final conditions (tf = T) can be written as follows:

xP (T) = xMP (T) + xM(T) = rMP (T) cos θMP (T) + xM(T), (3.4)

yP (T) = yMP (T) + yM(T) = rMP (T) sin θMP (T) + yM(T), (3.5)

xP (T) = xMP (T) + xM(T) = ±[√

μM

rMP (T)+ Δv2

]sin θMP (T) + xM(T), (3.6)

yP (T) = yMP (T) + yM(T) = ∓[√

μM

rMP (T)+ Δv2

]cos θMP (T) + yM(T), (3.7)

where Δv2 is the velocity change at the second impulse, rMP (T) = rMPf , and θMP (t) is the

angle which the position vector rMP forms with x-axis. The upper sign refers to clockwise

arrival to LMO and the lower sign refers to counterclockwise to LMO. Since the eccentricity

of the Moon orbit around Earth is neglected, it follows from (3.2) that the components of the

Moon inertial velocity at time T are given by

xM(T) = −DωM sin(ωMT),

yM(T) = DωM cos(ωMT).(3.8)


The angle θMP (T) is free and can be eliminated. After the problem has been solved,

the angle θMP (T) can be calculated from (3.4) and (3.5). So, combining (3.4)–(3.7), the final

conditions can be put in the following form:

(xP (T) − xM(T))2 +(yP (T) − yM(T)

)2 = (rMP (T))2, (3.9)

(xP (T) − xM(T))2 +(yP (T) − yM(T)

)2 =

[√μM

rMP (T)+ Δv2

]2

, (3.10)

(xP (T) − xM(T))(yP (T) − yM(T)

)−(yP (T) − yM(T)

)(xP (T) − xM(T))

= ∓ rMP (T)

[√μM

rMP (T)+ Δv2

].

(3.11)

As before, the upper sign refers to clockwise arrival to LMO and the lower sign refers to

counterclockwise to LMO. It should be noted that constraint defined by (3.11) is derived

from the angular momentum considering a direct (counterclockwise arrival) or a retrograde

(clockwise arrival) orbit around the Moon.

The problem defined by (3.1)–(3.11) involves four unknowns Δv1, Δv2, T , and θEP (0)that must be determined in order to satisfy the three final conditions (3.9)–(3.11). Since

this problem has one degree of freedom, an optimization problem can be formulated as

follows: determine Δv1, Δv2, T , and θEP (0) which satisfy the final constraints (3.9)–(3.11)and minimize the total characteristic velocity ΔvTotal = Δv1 + Δv2. This problem was

solved by Miele and Mancuso [12] using the sequential gradient-restoration algorithm for

mathematical programming problems developed by Miele et al. [15].In this paper, the optimization problem described above is solved by means of an

algorithm based on gradient method [15] in conjunction with Newton-Raphson method

[16], similarly to the one described in the previous section for the problem based on the

patched-conic approximation. The angle θEP (0) has been chosen as the iteration variable in

the gradient phase with Δv1, Δv2, and T calculated through Newton-Raphson method. The

results are presented in Section 5.

4. Optimization Problem Based on the Classical Version of PCR3BP

In this section, the optimization problem based on the classical version of PCR3BP is

formulated. The assumptions employed in this formulation are the same ones previously

presented in Section 3, except for assumption (1) which must be replaced by the following

one: Earth moves around the center of mass of the Earth-Moon system.

4.1. Problem Formulation in Inertial Frame

Consider an inertial reference frame Gxy contained in the Moon orbital plane: its origin is the

center of mass of the Earth-Moon system; the x-axis points towards the Moon position at the

initial time and the y-axis is perpendicular to the x-axis. Figure 4 shows the inertial reference

frame Gxy.


Space vehicle

Earth

Moon

at time t0x

y

Moon’s trajectory

Center of mass

Earth’s trajectory

rE

rE

P

rP

rMP

rMθEP

Figure 4: Inertial reference frame Gxy.

In the Gxy reference frame, the motion of the space vehicle (P) is described by the

following differential equations:

xP = − μEr3EP

(xP − xE) −μM

r3MP

(xP − xM),

yP = − μEr3EP

(yP − yE

)− μM

r3MP

(yP − yM

),

(4.1)

where rEP and rMP are, respectively, the radial distances of space vehicle from Earth (E) and

Moon (M), that is, r2EP = (xP − xE)2 + (yP − yE)2 and r2

MP = (xP − xM)2 + (yP − yM)2. Because

the origin of the inertial reference frame Gxy is the center of mass of Earth-Moon system,

the position vectors of the Earth and the Moon are, respectively, defined by rE = (xE, yE) and

rM = (xM, yM). Since the eccentricity of the Moon orbit around Earth is neglected, the Earth

and Moon inertial coordinates are given by

xE(t) = −μxM(t),

yE(t) = −μyM(t),

xM(t) =D

1 + μcos(ωt),

yM(t) =D

1 + μsin(ωt),

(4.2)

where μ = μM/μE and ω =√(μE + μM)/D3. Note that ω = ωM

√1 + μ.


The initial conditions of the system of differential equations correspond to the position

and velocity vectors of the space vehicle after the application of the first impulse. The initial

conditions (t0 = 0) can be written as follows:

xP (0) = xEP (0) + xE(0) = rEP (0) cos θEP (0) + xE(0),

yP (0) = yEP (0) + yE(0) = rEP (0) sin θEP (0) + yE(0),

xP (0) = xEP (0) + xE(0) = −[√

μE

rEP (0)+ Δv1

]sin θEP (0) + xE(0),

yP (0) = yEP (0) + yE(0) =

[√μE

rEP (0)+ Δv1

]cos θEP (0) + yE(0),

(4.3)

where Δv1, rEP (0), and θEP (t) have the same meaning previously defined in Section 3 and,

from (4.2),

xE(0) = − μD

1 + μ, yE(0) = 0, xE(0) = 0, yE(0) = −μDω

1 + μ. (4.4)

It should be noted that rEP and vEP are orthogonal because the impulse is applied tangentially

to LEO, assumed circular.

The final conditions of the system of differential equations correspond to the position

and velocity vectors of the space vehicle before the application of the second impulse and

they are given by (3.4)–(3.7), with the final position and velocity vectors of Moon obtained

from (4.2), that is, given by

xM(T) =D

1 + μcos(ωT), yM(T) =

D

1 + μsin(ωT),

xM(T) = − Dω

1 + μsin(ωT), yM(T) =

Dω

1 + μcos(ωT).

(4.5)

Accordingly, the final conditions can be put in the same form defined by (3.9)–(3.11).Therefore, the optimization problem is the same one defined in Section 3, and it is

solved by the same algorithm previously described. The results are presented in Section 5.

4.2. Transformation to Rotating Frame

Consider a rotating reference frame Gξη contained in the Moon orbital plane: its origin is the

center of mass of the Earth-Moon; the ξ-axis points towards the Moon position at any time

t and the η-axis is perpendicular to the ξ-axis. In this rotating reference frame the Earth and


the Moon are at rest. Figure 5 shows the inertial and rotating reference frames, Gxy and Gξη.

To write the differential equations of motion of the space vehicle (P) in the rotating reference

frame, consider the coordinate transformation equations:

xk = ξk cos(ωt) − ηk sin(ωt),

yk = ξk sin(ωt) + ηk cos(ωt),(4.6)

or

ξk = xk cos(ωt) + yk sin(ωt),

ηk = −xk sin(ωt) + yk cos(ωt),(4.7)

where k = E,M, P . Thus, the new coordinates of the Earth and Moon are

ξE = xE cos(ωt) + yE sin(ωt),

ηE = −xE sin(ωt) + yE cos(ωt),

ξM = xM cos(ωt) + yM sin(ωt),

ηM = −xM sin(ωt) + yM cos(ωt).

(4.8)

Substituting (4.2) into (4.8), one finds the fixed positions of the Earth and the Moon in

the rotating reference frame:

ξE = − μD

1 + μ,

ηE = 0,

ξM =D

1 + μ,

ηM = 0.

(4.9)

Now, consider the inertial coordinates of the space vehicle written in terms of the rotating

coordinates

xP = ξP cos(ωt) − ηP sin(ωt),

yP = ξP sin(ωt) + ηP cos(ωt).(4.10)

Differentiating each of these equations twice and substituting into (4.1), one finds the new

equations of motion:

ξP − 2ωηP −ω2ξP = − μEr3EP

(ξP − ξE) −μM

r3MP

(ξP − ξM),

ηP + 2ωξP −ω2ηP = − μEr3EP

(ηP − ηE

)− μM

r3MP

(ηP − ηM

),

(4.11)


Moon

at time t

x

y

Moon’s trajectory

Center of mass

η

ωt

ωt

ξrM

Figure 5: Rotating reference frame Gxy.

where

r2EP = (ξP − ξE)2 +

(ηP − ηE

)2,

r2MP = (ξP − ξM)2 +

(ηP − ηM

)2.

(4.12)

The system of differential equations above has a constant of motion, the so-called Jacobiintegral. In order to determine it, procede as follows. Multiply the first of (4.11) by ξP , the

second by ηP and add. It results in

ξP ξP + ηP ηP −ω2(ξP ξP + ηPηP

)= − μE

r3EP

ξP (ξP − ξE) −μM

r3MP

ξP (ξP − ξM)

− μE

r3EP

ηP(ηP − ηE

)− μM

r3MP

ηP(ηP − ηM

).

(4.13)

This equation can be rewritten as

1

2

d

dt

(ξ2P + η2

P

)− 1

2ω2 d

dt

(ξ2P + η2

P

)=d

dt

(μE

rEP+μM

rMP

). (4.14)

Therefore,

v2 = 2Φ(ξP , ηP

)− C, (4.15)


Table 1: Lagrange points for the Earth-Moon system.

ξLk ηLk Ck

(km) (km) (km2/s2)L1 3.2171 × 105 0 3.3468

L2 4.4424 × 105 0 3.3298

L3 −3.8635 × 105 0 3.1618

L4 1.8753 × 105 3.3290 × 105 3.1365

L5 1.8753 × 105 −3.3290 × 105 3.1365

where

v2 = ξ2P + η2

P ,

Φ(ξP , ηP

)=

1

2ω2(ξ2P + η2

P

)+μE

rEP+μM

rMP,

(4.16)

and C is the so-called Jacobi constant. Therefore, from (4.15) it is seen that the Jacobi integral,

J(ξP , ηP , ξP , ηP

)= 2Φ

(ξP , ηP

)− v2, (4.17)

is equal to C during the motion [17].The system of differential equations (4.11) has five equilibrium points. They are called

Lagrange points and are labelled L1, . . . , L5. The points L1, L2, L3 are located on the ξ-axis while

L4 and L5 form two equilateral triangles with the Earth and Moon in the plane of rotation.

See Figure 6. At each Lk, the corresponding value of Jacobi constant Ck is given by (4.15)substituting rLk = (ξLk , ηLk) and vLk = (ξLk , ηLk) = (0, 0). These Ck are related to the regions in

the rotating reference frame Gξη where the spacecraft can move. See in Table 1 the position

and the correspondent value of the Jacobi constant per unit mass of each Lagrange point.

Observe that the right-hand side of the (4.15) must be nonnegative, since v2 ≥ 0. Thus,

an initial position (ξP0, ηP0) and an initial velocity (ξP0, ηP0) yield a Jacobi constant value C

and the motion of the spacecraft is possible only in positions satisfying the relation

2Φ(ξP , ηP

)≥ C. (4.18)

The set of points in the (ξ, η)-plane defined by the inequality (4.18) is called Hill’s regions. See

in Figure 6 how the Hill’s regions (white areas) are related to the Ck values. The shaded areas

are the forbidden regions.

Finally, we note that the transformation to the rotating frame gives a better insight

about swing-by maneuvers with the Moon, as described in the results presented in Section 5.

5. Results

In this section, results are presented for some lunar missions using the three formulations

described in the preceding sections. Analysis of the results is discussed in two subsections:

in the first one, direct ascent maneuvers with flight time about 4.7 days are considered; in


−5

−4

−3

−2

−1

0

1

2

3

4

5

−5 −4 −3 −2 −1 0 1 2 3 4 5

Earth Moon

P

L1 L2L3

L4

L5

η(1

05

km)

ξ (105 km)

(a) Lagrange points

Earth Moon

P

L1 L2L3

L4

L5

(b) C1 < C

Earth Moon

P

L1 L2L3

L4

L5

(c) C2 < C < C1

Earth Moon

P

L1 L2L3

L4

L5

(d) C3 < C < C2

Earth Moon

P

L1 L2L3

L4

L5

(e) C4 < C < C3

Earth Moon

P

L1 L2L3

L4

L5

(f) C < C4

Figure 6: Lagrange points and Hill’s regions for the Earth-Moon system.


Table 2: Lunar mission, counterclockwise LMO arrival, major parameters.

LMO altitudeModel

ΔvTotal Δv1 Δv2 T θEP (0) Feasibility(km) (km/s) (km/s) (km/s) (days) (deg)

100

Patched-conic 3.8482 3.0655 0.7827 4.794 −115.723 Yes

Simplified PCR3BP 3.8758 3.0649 0.8109 4.564 −116.800 Yes

Classical PCR3BP 3.8777 3.0658 0.8119 4.573 −116.410 Yes

Miele and Mancuso [12] 3.876 3.065 0.811 4.37 −118.98 —

200





300





the second subsection, some maneuvers with multiple revolutions and flight time about

14, 24, 32, 40, and 58 days are considered. Three final altitudes hLMO of a clockwise or

counterclockwise circular LMO and a specified altitude hLEO of a counterclockwise circular

LEO, which corresponds to the altitude of the Space Station [12], are considered. The

following data are used:

G = 6.672 × 10−20 km3 kg−1 s−2(universal constant of gravitation

),

ME = 5.9742 × 1024 kg (mass of the Earth),

MM = 7.3483 × 1022 kg (mass of the Moon),

REM = 384 400 km (mean distance from the Earth to the Moon),

RE = 6 378 km (Earth radius),

RM = 1 738 km (Moon radius),

hLEO = 463 km (altitude of circular LEO),

hLMO = 100, 200, 300 km (altitude of circular LMO).

(5.1)

5.1. Direct Ascent Maneuvers

Table 2 shows the results for lunar missions with counterclockwise arrival at LMO, and

Table 3 shows the results for lunar missions with clockwise arrival at LMO. Recall that the

departure from LEO is counterclockwise for all missions. The major parameters that are

presented in these tables are the velocity changes Δv1 and Δv2 at each impulse, the total

characteristic velocity ΔvTotal = Δv1 +Δv2, the flight time of lunar mission T , and the angular

position of the space vehicle with respect to Earth at the initial time defined by the angle

θEP (0).


Table 3: Lunar mission, clockwise LMO arrival, major parameters.

LMO altitudeModel

ΔvTotal Δv1 Δv2 T θEP (0) Feasibility(km) (km/s) (km/s) (km/s) (days) (deg)


100Simplified PCR3BP 3.8811 3.0677 0.8134 4.750 −114.215 Yes



200





300





Results in Tables 2 and 3 show good agreement. It should be noted the excellent results

were obtained using the patched-conic approximation model. In all missions, the patched-

conic approximation model yields very accurate estimate for the first impulse in comparison

to the results obtained using the PCR3BP models. For the second impulse, there exists a

small difference between the results given by the patched-conic approximation model and

the PCR3BP models. For all lunar missions, the values of the major parameters ΔvTotal, Δv1,

Δv2, and T obtained using the simplified PCR3BP model are a little lesser than the values

obtained using the classical PCR3BP. In all cases, the trajectories are feasible, that is, the

spacecraft does not collide with the Moon. As described in the next subsection, collisions

can occur for maneuvers with multiple revolutions.

Tables also show a small difference in the flight time T and in the angle θEP (0)calculated by the three approaches. We suppose that the difference between the values

obtained in this paper and the values presented by [12] for the flight time T and the angle

θEP (0) calculated using the simplified PCR3BP model should be related to the accuracy

in the integration of differential equations and in the solution of the terminal constraints.

The algorithm based on gradient algorithm in conjunction with Newton-Raphson method,

described in this paper, uses a Runge-Kutta-Fehlberg method of orders 4 and 5, with step-size

control and relative error tolerance of 10−10 and absolute error tolerance of 10−11, as described

in Stoer and Bulirsch [16] and Forsythe et al. [18]. The terminal constraints are satisfied with

an error lesser than 10−8. In all simulations the following canonical units are used: 1 distance

unit = RE and 1 time unit =√R3E/μE. On the other hand, the paper by Miele and Mancuso

[12] does not describe the accuracy used in the calculations.

According to the results presented in Tables 2 and 3, we note, regardless the dynamical

model used in the analysis, that

(1) lunar missions with clockwise LMO arrival spend more fuel than lunar missions

with counterclockwise LMO arrival;

(2) the flight time is nearly the same for all lunar missions with clockwise LMO arrival,

independently on the final altitude of LMO. The differences between the flight


Table 4: Lunar mission, major parameters, hLEO = 463 km, hLMO = 100 km.

Model ManeuverΔvTotal Δv1 Δv2 T θEP (0) Feasibility(km/s) (km/s) (km/s) (days) (deg)

3.8758 3.0649 0.8109 4.564 −116.800 Yes

3.8778 3.0653 0.8125 14.175 9.119 Yes

Simplified Counterclockwise 3.8746 3.0647 0.8099 23.832 135.661 Yes

PCR3BP 3.8411 3.0585 0.7826 32.108 232.269 Yes

3.8444 3.0591 0.7853 40.871 347.205 Yes

3.7936 3.0498 0.7438 58.701 220.730 Yes

3.8811 3.0677 0.8134 4.750 −114.215 Yes

3.8829 3.0681 0.8149 14.791 17.307 Yes

Simplified Clockwise 3.8785 3.0672 0.8113 24.916 149.879 Yes

PCR3BP 3.8320 3.0586 0.7734 32.087 232.573 Yes

3.8352 3.0592 0.7760 40.862 347.608 Yes

3.7842 3.0498 0.7444 58.551 220.998 Yes

time of each mission are small, they are approximately lesser tan 0.004 days (5.8

minutes);

(3) the flight time is nearly the same for all lunar missions with counterclockwise LMO

arrival, independently on the final altitude of LMO. The differences between the

flight time of each mission are small; they are approximately lesser than 0.010 days

(14.4 minutes);

(4) the first change velocity Δv1 is nearly independent of the LMO altitude;

(5) the second change velocity Δv2 decreases with the LMO altitude;

(6) the flight time for lunar missions with clockwise LMO arrival is larger than the

flight time for lunar missions with counterclockwise LMO arrival;

(7) for the PCR3BP and patched-conic approximation models, the angle θEP (0) varies

with the LMO altitude for all lunar missions.

We note that some of these general results are quite similar to the ones described by [12].For hLMO = 100 km, the trajectory is shown in Figure 7 for counterclockwise LMO

arrival and in Figure 8 for clockwise LMO arrival. In both figures, trajectories are shown in

the inertial reference frame Exy and in the rotating reference frame Gξη. Only results obtained

through the classical PCR3BP model are depicted.

5.2. Multiple Revolution Ascent Maneuvers

Tables 4, 5, and 6 show the results for lunar missions with clockwise and counterclockwise

arrival at LMO, for hLMO = 100, 200, 300 km, respectively, considering the simplified PCR3BP

model. Tables 7, 8, and 9 show similar results considering the classical PCR3BP model. The

major parameters that are presented in these tables are the same ones presented in Tables 3

and 2. The value of the Jacobi constant per unit mass for each mission is presented in Tables

7, 8, and 9.

For hLMO = 100 km, the trajectories are shown in Figures 9 to 13 for counterclockwise

LMO arrival and in Figures 14 to 18 for clockwise LMO arrival, considering the classical


−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

y(1

05

km)

x (105 km)

(a) Earth-Moon trajectory, inertial coordinateframe

−0.2 −0.1 0 0.1 0.2 0.3

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

y(1

05

km)

x (105 km)

(b) LEO departure, inertial coordinate frame

1.84 1.86 1.88 1.9 1.92 1.94 1.96

3.28

3.3

3.32

3.34

3.36

3.38

3.4

y(1

05

km)

x (105 km)

(c) LMO arrival, inertial coordinate frame

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

η(1

05

km)

ξ (105 km)

(d) Earth-Moon trajectory, rotating coordinateframe

−0.6 −0.4 −0.2 0 0.2 0.4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

η(1

05

km)

ξ (105 km)

(e) LEO departure, rotating coordinate frame

3.65 3.7 3.75 3.8 3.85 3.9 3.95

−0.15

−0.1

−0.05

0

0.05

0.1

η(1

05

km)

ξ (105 km)

(f) LMO arrival, rotating coordinate frame

Figure 7: Direct ascent, counterclockwise LMO arrival, Δv1 = 3.0658 km/s, Δv2 = 0.8119 km/s, T = 4.573days, θEP (0) = −116.41 deg.


−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

y(1

05

km)

x (105 km)


−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

y(1

05

km)

x (105 km)


1.6 1.65 1.7 1.75 1.8 1.85 1.9

3.3

3.35

3.4

3.45

3.5

3.55

y(1

05

km)

x (105 km)


−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

η(1

05

km)

ξ (105 km)


−0.6 −0.4 0 0.2 0.4

−0.4

−0.2

0

0.2

0.4

0.6

−0.2

η(1

05

km)

ξ (105 km)


3.65 3.7 3.75 3.8 3.85 3.9

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

η(1

05

km)

ξ (105 km)


Figure 8: Direct ascent, clockwise LMO arrival, Δv1 = 3.0686 km/s, Δv2 = 0.8143 km/s, T = 4.763 days,θEP (0) = −113.795 deg.




3.8614 3.0648 0.7966 4.562 −116.832 Yes

3.8635 3.0653 0.7983 14.168 9.002 Yes


PCR3BP 3.8263 3.0585 0.7678 32.110 232.270 Yes

3.8296 3.0591 0.7705 40.873 347.204 Yes

3.7826 3.0506 0.7320 58.398 216.284 Yes

3.8670 3.0677 0.7993 4.753 −114.187 Yes

3.8688 3.0681 0.8007 14.798 17.390 Yes


PCR3BP 3.8167 3.0586 0.7581 32.080 232.475 Yes

3.8200 3.0592 0.7608 40.863 347.613 Yes

3.7679 3.0498 0.7181 58.530 220.726 Yes



3.8483 3.0648 0.7835 4.560 −116.881 Yes

3.8504 3.0652 0.7852 14.157 8.851 Yes


PCR3BP 3.8127 3.0585 0.7542 32.111 232.260 Yes

3.8161 3.0591 0.7570 40.874 347.201 Yes

3.7633 3.0498 0.7135 58.685 220.409 Yes

3.8541 3.0678 0.7863 4.760 −114.110 Yes

3.8559 3.0681 0.7878 14.809 17.504 Yes


PCR3BP 3.8026 3.0586 0.7440 32.080 232.475 Yes

3.8060 3.0592 0.7467 40.864 347.620 Yes

3.7530 3.0498 0.7031 58.488 220.014 Yes

PCR3BP model. In both figures, trajectories are shown in the inertial reference frame Exy and

in the rotating reference frame Gξη.

For maneuvers with multiple revolutions, major comments are as follows.

(1) All trajectories, regardless of the flight time, are feasible in the simplified PCR3BP

model.

(2) In the classical PCR3BP model, all trajectories have Jacobi constant less than C4.

Therefore, there are not forbidden regions for the motion of the spacecraft.

(3) Trajectories with two revolutions around the Earth and flight time about 24.0

days collide with the Earth in the classical PCR3BP model. Figures 10 and 15



Model ManeuverΔvTotal Δv1 Δv2 T θEP (0) C

Feasibility(km/s) (km/s) (km/s) (days) (deg) (km2/s2)

3.8777 3.0658 0.8119 4.573 −116.410 2.4784 Yes

3.8732 3.0650 0.8082 14.330 12.466 2.4965 Yes

Classical Counterclockwise 3.8776 3.0658 0.8118 24.019 140.054 2.4788 No

PCR3BP 3.8428 3.0593 0.7835 31.910 232.464 2.6166 Yes

3.8379 3.0584 0.7795 40.742 348.882 2.6358 Yes

3.8300 3.0570 0.7730 58.415 229.239 2.6668 Yes

3.8829 3.0686 0.8143 4.763 −113.795 2.4187 Yes

3.8785 3.0678 0.8107 14.970 21.069 2.4363 Yes

Classical Clockwise 3.8821 3.0684 0.8136 25.103 154.315 2.4220 No

PCR3BP 3.8337 3.0595 0.7742 31.872 232.674 2.6138 Yes

3.8288 3.0586 0.7702 40.713 349.322 2.6331 Yes

3.7893 3.0513 0.7380 58.420 224.480 2.7871 Yes




3.8634 3.0658 0.7976 4.571 −116.451 2.4793 Yes

3.8587 3.0649 0.7938 14.322 12.340 2.4974 Yes


PCR3BP 3.8280 3.0593 0.7686 31.912 232.461 2.6166 Yes

3.8230 3.0584 0.7645 40.743 348.876 2.6358 Yes

3.8149 3.0570 0.7579 58.417 229.218 2.6669 Yes

3.8688 3.0686 0.8002 4.769 −113.742 2.4178 Yes

3.8643 3.0678 0.7965 14.981 21.198 2.4355 Yes


PCR3BP 3.8184 3.0595 0.7589 31.873 232.676 2.6137 Yes

3.8195 3.0597 0.7598 39.664 353.119 2.6094 Yes

3.7709 3.0509 0.7199 58.361 223.886 2.7957 Yes

depict the collision between the spacecraft and the Earth for lunar missions with

hLMO = 100 km and counterclockwise or clockwise arrival to Moon, respectively.

The remaining trajectories are feasible.

(4) According to the results in Tables 4–9, simplified and classical PCR3BP models

show that the group of trajectories with counterclockwise arrival to Moon is slightly

superior to the group of trajectories with clockwise arrival to Moon in terms of total

characteristic velocity and flight time for maneuvers with flight time smaller than

25.5 days.

(5) According to the results in Tables 4–9, simplified and classical PCR3BP models

show that the group of trajectories with clockwise arrival to Moon is slightly





3.8502 3.0657 0.7845 4.569 −116.491 2.4802 Yes

3.8455 3.0649 0.7806 14.313 12.213 2.4983 Yes


PCR3BP 3.8144 3.0593 0.7551 31.913 232.461 2.6166 Yes

3.8093 3.0584 0.7509 40.745 348.870 2.6359 Yes

3.8010 3.0570 0.7441 58.422 229.235 2.6671 Yes

3.8559 3.0687 0.7872 4.771 −113.716 2.4170 Yes

3.8513 3.0678 0.7834 14.992 21.336 2.4346 Yes


PCR3BP 3.8043 3.0595 0.7448 31.873 232.676 2.6137 Yes

3.8054 3.0597 0.7458 39.665 353.111 2.6094 Yes

3.7559 3.0509 0.7050 58.361 223.913 2.7957 Yes

superior to the group of trajectories with counterclockwise arrival to Moon in terms

of total characteristic velocity and flight time (excepting trajectories with flight time

about 58.5 days) for maneuvers with flight time larger than 30.0 days.

(6) For lunar missions with hLMO = 100 km, Figures 11, 12, 13, 16, 17, and, 18 show that

the spacecraft carries out one or two close approaches to the Moon before entering

into the low circular orbit around the Moon. Such maneuvers, known as swing-by

maneuvers, reduce the fuel consumption, although the flight time increases (see

results in Table 7).

(7) According to the results in Tables 4–9, the first change velocity Δv1 is nearly

independent of the LMO altitude, and, the second change velocity Δv2 decreases as

the LMO altitude increases, for all group of trajectories with similar flight time and

same sense of arrival to Moon.

(8) The first change velocity Δv1 is nearly the same for maneuvers with flight time

smaller than 25.5 days (the differences in Δv1 are approximately 0.6 m/s).

(9) For maneuvers with flight time larger than 30.0 days, the first change velocity Δv1

changes slightly (the differences in Δv1 are approximately 15.0 to 20.0 m/s).

6. Conclusion

In this paper, a systematic study of optimal trajectories for Earth-Moon flight of a space

vehicle is presented. The optimization criterion is the total characteristic velocity. The

optimization problem has been formulated using the patched-conic approximation and

two versions of the planar circular restricted three-body problem (PCR3BP) and has been

solved using a gradient algorithm in conjunction with Newton-Raphson method. Results for

direct ascent maneuvers and for maneuvers with multiple revolutions around the Earth are

presented. For direct ascent maneuvers, all models show that lunar missions with clockwise


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Figure 9: Multiple revolution ascent, counterclockwise LMO arrival, Δv1 = 3.065 km/s, Δv2 =0.8082 km/s, T = 14.33 days, θEP (0) = 12.466 deg.


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Figure 14: Multiple revolution ascent, clockwise LMO arrival, Δv1 = 3.0678 km/s, Δv2 = 0.8107 km/s,T = 14.97 days, θEP (0) = 21.069 deg.


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LMO arrival spend more fuel than lunar missions with counterclockwise LMO arrival. For

maneuvers with multiple revolutions, fuel can be saved if the spacecraft accomplishes a

swing-by maneuver with the Moon before the arrival at LMO.

Acknowledgments

This research was supported by CNPq under contract 302949/2009-7 and by FAPESP under

contract 2008/56241-8.

References

[1] R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics, Dover, 1971.

[2] A. E. Roy, Orbital Motion, Taylor & Francis, 4th edition, 2004.

[3] V. Szebehely, Theory of Orbits, Academic Press, New York, NY, USA, 1967.

[4] W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross, Dynamical Systems, the Three-Body Problem andSpace Mission Design, Springer, 2007.

[5] E. A. Belbruno and J. Miller, “Sun-perturbed earth-to-moon transfers with ballistic capture,” Journalof Guidance, Control, and Dynamics, vol. 16, no. 4, pp. 770–775, 1993.

[6] E. A. Belbruno and J. P. Carrico, “Calculation of weak stability boundary ballistic lunar transfertrajectories,” in Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, AIAA, Denver, Colo,USA, August 2000.

[7] W. S. Koon, M. W. Lo, J. E. Marsden, and S. D. Ross, “Low energy transfer to the moon,” CelestialMechanics & Dynamical Astronomy, vol. 81, no. 1-2, pp. 63–73, 2001.

[8] K. Yagasaki, “Computation of low energy Earth-to-Moon transfer with moderate flight time,” PhysicaD, vol. 197, no. 3-4, pp. 313–331, 2004.

[9] K. Yagasaki, “Sun-perturbed Earth-to-Moon transfers with low energy and moderate flight time,”Celestial Mechanics & Dynamical Astronomy, vol. 90, no. 3-4, pp. 197–212, 2004.

[10] E. A. Belbruno, “Lunar capture orbits, a method of constructing earth-moon trajectories and lunar gasmission,” in Proceedings of 19th AIAA/DGLR/JSASS International Electric Propulsion Conference, AIAA,Colorado Springs, Colo, USA, May 1987, number 87–1054.

[11] E. Belbruno, Capture Dynamics and Chaotic Motions in Celestial Mechanics, Princeton University Press,Princeton, NJ, USA, 2004.

[12] A. Miele and S. Mancuso, “Optimal trajectories for Earth-Moon-Earth flight,” Acta Astronautica, vol.49, pp. 59–71, 2001.

[13] S. A. Fazelzadeh and G. A. Varzandian, “Minimum-time Earth-Moon and Moon-Earth orbitalmaneuvers using time-domain finite element method,” Acta Astronautica, vol. 66, pp. 528–538, 2010.

[14] J. P. Marec, Optimal Space Trajectories, Elsivier, 1979.

[15] A. Miele, H. Y. Huang, and J. C. Heideman, “Sequential gradient-restoration algorithm for theminimization of constrained functions—ordinary and conjugate gradient versions,” Journal ofOptimization Theory and Applications, vol. 4, no. 4, pp. 213–243, 1969.

[16] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics,Springer, New York, NY, USA, 3rd edition, 2002.

[17] H. Pollard, Mathematical Introduction to Celestial Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA,1966.

[18] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations,Prentice-Hall, Englewood Cliffs, NJ, USA, 1977.


Research ArticleComparison between Two Methods to Calculate theTransition Matrix of Orbit Motion

Ana Paula Marins Chiaradia,1 Helio Koiti Kuga,2and Antonio Fernando Bertachini de Almeida Prado2

1 Grupo de Dinamica Orbital e Planetologia, Departament of Mathematics, FEG/UNESP, CEP 12516-410,Guaratingueta, SP, Brazil

2 DEM-INPE, Avenida dos Astronautas, 1758, CEP 12227-010, Sao Jose dos Campos, SP, Brazil

Correspondence should be addressed to Ana Paula Marins Chiaradia, [email protected]

Received 10 June 2011; Accepted 6 September 2011


Copyright q 2012 Ana Paula Marins Chiaradia et al. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Two methods to evaluate the state transition matrix are implemented and analyzed to verify thecomputational cost and the accuracy of both methods. This evaluation represents one of the highestcomputational costs on the artificial satellite orbit determination task. The first method is anapproximation of the Keplerian motion, providing an analytical solution which is then calculatednumerically by solving Kepler’s equation. The second one is a local numerical approximationthat includes the effect of J2. The analysis is performed comparing these two methods with areference generated by a numerical integrator. For small intervals of time (1 to 10 s) and when oneneeds more accuracy, it is recommended to use the second method, since the CPU time does notexcessively overload the computer during the orbit determination procedure. For larger intervalsof time and when one expects more stability on the calculation, it is recommended to use the firstmethod.

1. Introduction

The orbit determination process consists of obtaining values of the parameters which com-

pletely specify the motion of an orbiting body, like artificial satellites, based on a set of

observations of the body. It involves nonlinear dynamical and nonlinear measurement

systems, which depends on the tracking system, and estimation technique (e.g., the Kalman

filtering or least squares [1–6]). The dynamical system model consists of the description for

the dynamics of the orbital motion of a satellite, measurement models, Earth’s rotation effects,

and perturbation models. According to Montenbruck and Gill [7], these models depend on,

besides the state variables that define the initial conditions, a variety of parameters that


either affect the dynamical motion or the measurement process. Due to the complexity of

the applied models, it is hardly possible to have a direct solution for any of these parameters

from a given set of observations. To linearize the relation between the observables and the

independent parameters, one should obtain simplified expressions that can be handled more

easily. A linearization of the trajectory and measurement model requires a large number of

partial derivatives, among them, the state transition matrix [7–9].The function of the transition matrix is to relate the rectangular coordinate variations

for the times tk and tk+1. The evaluation of the state transition matrix presents one of the

highest computational costs on the artificial satellite orbit determination procedure, because

it requires the evaluation of the Jacobian matrix and the integration of the current variational

equations. This matrix can pose cumbersome analytical expressions when using a complex

force model [10]. Binning [11] suggests one method to avoid the problem of the high

computational cost and extended analytical expressions of the transition matrix. This method

consists of propagating the state vector using complete force model and, then, to compute the

transition matrix using a simplified force model. For sure, this simplification provides some

impact on the accuracy of the orbit determination. The analytical calculation of the transition

matrix of the Keplerian motion is a reasonable approximation when only short time intervals

of the observations and reference instant are involved. On the other hand, the inclusion of

the Earth-flattening (J2) effect in the transition matrix can be performed adopting Markley’s

method [12].In Chiaradia [13], Chiaradia et al. [14], and Gomes et al. [15], simplified and

compact algorithms with low computational cost are developed for artificial satellite orbit

determination in real time and onboard, using the global positioning system (GPS). The

state vector, composed of the position, velocity, bias, drift, and drift rate of the GPS receiver

clock, has been estimated by the extended Kalman filter in all three works. The fourth-

order Runge-Kutta numerical integrator has been used to integrate the state vector. The

equations of motion have considered only the perturbations due to the geopotential. The

state error covariance matrix has been propagated through the transition matrix, which

has been calculated considering the pure Keplerian motion. To improve the accuracy of

those algorithms, two methods have been compared for calculating the transition matrix

considering circular (1000 km of altitude) and elliptical (Molniya) orbits. Those methods

considered the pure Keplerian motion and perturbed only by the flattening effect of the Earth.

Markley’s method is used to include the flattening of the Earth in the transition matrix. It is a

method that allows the inclusion of more perturbations in a simple way.

In the present work, the spherical harmonic coefficients of degree and order up

to 50 and the drag effect are included in the reference orbit provided by the RK78

numerical integrator (Runge-Kutta with Fehlberg coefficients of order 7-8). For this study,

the atmospheric density is considered constant. For the simulations made here, circular and

elliptical low (up to 300 km) orbits were used. To analyze the results, the reference orbit is

compared with the two methods implemented in this work.

2. State Transition Matrix

The differential equation for the Keplerian motion is expressed by

r = −μrr3, (2.1)


with r = (x, y, z),

r =√x2 + y2 + z2, (2.2)

where r is the magnitude of the satellite position vector and μ is the Earth gravitational

constant. The equation that relates the state deviations in different times is given by(δr

δv

)= Φ(t, t0)

(δr0

δv0

), (2.3)

where r and v are the position and velocity vectors at the time t, respectively, r0 and v0 are the

initial position and velocity vectors at the time t0, respectively, and Φ is the transition matrix

given by

Φ =

(Φ11 Φ12

Φ21 Φ22

)=

⎛⎜⎜⎝∂r∂r0

∂r∂v0

∂v∂r0

∂v∂v0

⎞⎟⎟⎠. (2.4)

The submatrices Φ11, Φ12, Φ21, and Φ22 are calculated in accordance with the two methods

that will be shown in the next topics.

3. First Method: The Analytical Transition Matrix SolutionConsidering the Pure Keplerian Motion

Goodyear [16] published a method for the analytical calculation of a transition matrix for

the two-body problem. This method is valid for any kind of orbit. Kuga [10] implemented

this method using the same elegant and adequate formulation optimized for the Keplerian

elliptical orbit problem. He performed some simplifications in this method to increase its

numerical efficiency (processing time, memory, and accuracy). However, this method is used

to propagate the position and velocity of the satellite and can be used in any kind of two-body

orbits. Such equations are simple and easy to be developed.

According to Goodyear [16] and Shepperd [17], the four submatrices 3×3 are obtained

developing (2.4). Then, they are written as

Φ11 = fI +(r v)(M21 M22

M31 M32

)(r0 v0

)T,

Φ12 = gI +(r v)(M22 M23

M32 M33

)(r0 v0

)T,

Φ21 = fI −(r v)(M11 M12

M21 M22

)(r0 v0

)T,

Φ22 = gI −(r v)(M12 M13

M22 M23

)(r0 v0

)T,

(3.1)


where I is the identity matrix 3 × 3,(r v)

is matrix 3 × 2, and the Mi,j , i, j = 1, 2, 3, are the

components of a 3×3 matrix M, which will be shown later, and f , g, f , g are calculated in the

next topic [8, 9].

3.1. Calculating the Functions f , g, f , g

Given r0 = (x0, y0, z0), v0 = (x0, y0, z0), and the propagation interval Δt = t − t0,

r0 =√x2

0 + y20 + z

20,

h0 = x0x0 + y0y0 + z0z0,

v0 =√x2

0 + y20 + z

20,

α = v20 −

2μ

r0,

1

a= −α

μ,

(3.2)

where a is the semimajor axis. The eccentric anomaly E0 and the eccentricity e for the initial

orbit are calculated by

e sinE0 =h0√μa

,

e cosE0 = 1 − r0

a,

(3.3)

with E0 reduced to the interval 0 to 2π . The mean anomalies for the initial and propagated

orbits, M0 and M, are given by

M0 = E0 − e sinE0,

n =√

μ

a3,

M = nΔt +M0,

(3.4)

with M0 and M reduced to the interval 0 to 2π . The eccentric anomaly for the propagated

orbit is calculated by Kepler’s equation. Kepler’s equation is solved from an initial guess

based on an approximation series, then iterated by Newton-Raphson’s method, until

convergence to the level of 10−12 is achieved. The variation of the eccentric anomaly is

calculated and reduced to the interval 0 to 2π :

ΔE = E − E0. (3.5)


The transcendental functions s0, s1, s2 for the elliptical orbit, according to Goodyear [16], are

calculated by

s0 = cosΔE,

s1 =

√a

μsinΔE,

s2 =a

μ(1 − s0).

(3.6)

Therefore, the functions f , g, f , g are valid only for elliptical orbits and are calculated by

r = r0s0 + h0s1 + μs2, (3.7)

f = 1 − μs2

r0, (3.8)

g = r0s1 + h0s2, (3.9)

f = −μs1

rr0, (3.10)

g = 1 − μs2

r. (3.11)

The propagated vectors r e v are given by

r = r0f + v0g,

v = r0f + v0g.(3.12)

There is no singularity problem and Kepler’s equation is solved through Newton Raphson’s

method in double precision. The classical parameters a, e, i, Ω,ω, are constant in the Keplerian

motion, and, therefore, there is no different subscript for them.

3.2. The Evaluation of the Matrix M3×3

First of all, it is necessary to calculate the secular component U including the effect of multi-

revolutions in the case where the orbit propagation time, Δt, is larger than one orbital period:

ΔE = ΔE + INT

(Δt

n

2π

)2π,

s′4 = cosΔE − 1,

s′5 = sinΔE −ΔE,

U = s2Δt +

√(a

μ

)5(ΔEs′4 − 3s′5

),

(3.13)


where INT(x) provides the truncated integer number of the real argument x, and the variables

s′4 and s′5 are related with the transcendental functions s4 and s5 (Goodyear [16]). Therefore,

the components of the matrix M are related to the partial derivatives of (3.8)–(3.11) with

respect to the r0 and v0, given by

M11 =

(s0

rr0+

1

r20

+1

r2

)f − μ2 U

r3r30

,

M12 =

(fs1

r+

(g − 1

)r2

),

M13 =

(g − 1

)s1

r− μU

r3,

M21 = −(fs1

r0+

(f − 1

)r2

0

),

M22 = −fs2,

M23 = −(g − 1

)s2,

M31 =

(f − 1

)s1

r0− μU

r30

,

M32 =(f − 1

)s2,

M33 = gs2 −U.

(3.14)

3.3. The Property of the Transition Matrix

Sometimes the inverse matrix is required, such as in backward filters. It is also easily accom-

plished as follows. The inverse matrix Φ−1, which propagates deviations backward from t to

t0, is given by

Φ−1 =

(ΦT

22 −ΦT12

−ΦT21 ΦT

11

), (3.15)

which results from the canonic nature of the original equations [18, 19]. This also applies

to the second method (Markley’s) if the perturbations are derived from a potential (e.g., J2

effect).

4. Second Method: Markley’s Method

Markley’s method uses two states, one at the tk−1 time and the other at the tk time, and

calculates the transition matrix between them by using μ, J2, Δt, the radius of the Earth, and

the two states. In this case, the effect of the Earth’s flattening is the most influent factor in the

process. Markley’s method consists of making one approximation for the transition matrix


of the state vector based on the Taylor series expansion for short intervals of propagation,

Δt. This method can be used by any kind of orbit and the equations are simple and easily

implemented, as shown next. The state transition’s differential equation is defined by

dΦ(t, t0)dt

= A1(t)Φ(t, t0) =

[0 I

G(t) 0

]Φ(t, t0), (4.1)

where Φ(t0, t0) ≡ I is the initial condition, r = (x y z)T and v = ( x y z )T are the Cartesian

state at the instant t, r0 and v0 are the Cartesian state at the instant t0, 0 ≡ the matrix 3 × 3 of

zeros, I ≡ the identity matrix 3 × 3, G(t) ≡ ∂a(r, t)/∂r ≡ the gradient matrix, and a(r, t) = the

accelerations of the satellite.

Performing successive derivatives of (4.1), followed by substitutions, gives the deriva-

tive of the transition matrix:

diΦdti

= Ai(t)Φ(t, t0), (4.2)

where

Ai(t) = Ai−1(t) +Ai−1(t)A1(t). (4.3)

The dot represents the derivative with respect to the time. Developing Φ(t, t0) in Taylor’s

series at t = t0, using the matrices Ai(t0) for i = 1, . . . , 4 and the initial condition Φ(t0, t0) ≡ I,the transition matrix of the position and velocity obtained after some simplifications is given

by [12]

Φ(t, t0) ≈[Φrr Φrv

Φvr Φvv

]6×6

, (4.4)

where

Φrr ≡ I + (2G0 +G)(Δt)2

6,

Φrv ≡ IΔt + (G0 +G)(Δt)3

12,

Φvr ≡ (G0 +G)(Δt)

2,

Φvv ≡ I + (G0 + 2G)(Δt)2

6,

Δt ≡ t − t0, G0 ≡ G(t0).

(4.5)

The calculations of these matrices pose no problems, since the gradient matrix G in the

end of the propagating interval is a function of the final Cartesian state which should be


calculated during the data processing and is available at no extra cost. The G and, therefore,

Φrr ,Φrv,Φvr ,Φvv are symmetric if the perturbation derives from a potential. The G gradient

matrix including only the central force and the J2 is given by

G(t) =∂a(r, t)∂r

=

⎡⎢⎢⎢⎢⎢⎢⎣

∂ax∂x

∂ax∂y

∂ax∂z

∂ay

∂x

∂ay

∂y

∂ay

∂z∂az∂x

∂az∂y

∂az∂z

⎤⎥⎥⎥⎥⎥⎥⎦. (4.6)

The accelerations due the central force and the Earth’s flattening are given by

ax =−μxr3

[1 +

3

2

J2r2e

r2

(1 − 5z2

r2

)],

ay =y

xax,

az =−μzr3

[1 +

3

2

J2r2e

r2

(3 − 5z2

r2

)].

(4.7)

The partial derivatives are [20]

∂ax∂x

=μ

r5

[3x2 − r2 − 3

2J2r

2e +

15

2

J2r2e

r2

(x2 + z2

)− 105

2

J2r2e

r4x2z2

],

∂ax∂y

=3μxy

r5

[1 +

5

2

J2r2e

r2− 35

2

J2r2e

r4z2

],

∂ax∂z

=3μxz

r5

[1 +

15

2

J2r2e

r2− 35

2

J2r2e

r4z2

],

∂ay

∂x=∂ax∂y

,

∂ay

∂y=y

x

∂ax∂y

+axx,

∂ay

∂z=y

x

∂ax∂z

,

∂az∂x

=∂ax∂z

,


∂az∂y

=∂ay

∂z,

∂az∂z

=μ

r5

[−r2 + 3

(z2 − 3

2J2r

2e + 15

J2r2e

r2z2 − 35

2

J2r2e

r4z4

)].

(4.8)

5. Analyses of the Methods

First of all, the reference orbit is integrated using the numerical integrator Runge-Kutta of

eighth order with automatic step size control (RK78). This integrator is implemented to

integrate simultaneously the stated vectors considering the spherical harmonic coefficients

up to 50th order and degree and the transition matrix considering the Earth’s flattening and

the atmospheric drag effects.

The transition matrix generated by the numerical integrator is used as the reference

to compare the transition matrix generated by the two methods. To compare the accuracy of

them, let us define the global relative error as [10]

εGlobal =1

36

6∑i=1

6∑j=1

∣∣∣∣∣∣∣(Φij −Φij

)Φij

∣∣∣∣∣∣∣, (5.1)

where Φij is the component i, j of the transition matrix calculated by the RK78 considered

as reference and Φij is the component i, j of the transition matrix calculated by one of the

methods. It gives a rough idea of the number of the common significant figures retained after

the computations.

The first method considers the pure Keplerian motion and the second one considers

the Keplerian motion and the J2 effect. A whole day of integration is divided in intervals of 1

to 60 seconds, which produces from 86,400 to 1,440 steps of integration, respectively, as shown

in Table 1. The total processing time of each method is also shown in Table 1, although it

depends on the code, the computer, and the programmer skills (the evaluation used a regular

PC Pentium II processor, 512 MB, running FORTRAN codes). However, it illustrates roughly

the comparative expected performance. One can note that the CPU time difference at any

step of integration and for any method is very small. The first method is slightly faster than

the second one for small steps of time, like 1 and 10 seconds.

To analyze the accuracy of the methods, six comparisons are done, using two kinds

of orbits: one circular and one elliptical. The transition matrix generated by each of the

methods is compared with the reference generated by the RK78. The circular orbit is from

the Topex/Poseidon satellite [21] and the elliptical one is from the Molniya satellite [12].All those tests are performed for a period of 24 hours. The data of these orbits are shown in

Table 2.

Besides, the RK4 (Runge-Kutta of fixed 4th order) numerical method for computing

the transition matrix is also implemented and the transition matrix generated by the RK4 is

also included in the comparison. The RK4 used the same dynamical model of the reference

RK78. The results of those three comparisons for the circular orbit are shown in Table 3, and

the results for the elliptical orbit are shown in Table 4. The errors are shown in terms of per


Table 1: Processing time for analytical calculation of the transition matrix.

Δt (s) StepsCPU TIME(s)

First method Second method

1 86,400 19.8 26.2

10 8,640 2.1 2.7

30 2,880 0.7 0.9

60 1,440 0.4 0.4

Table 2: Keplerian parameters of the test orbits.

Parameter Topex/Poseidon Molniya

Semiaxis major 7714423.46 m 26563000.0 m

Eccentricity 1.13458 × 10−4 0.75

Inclination 66◦.039 63◦.435

Longitude of node 236◦.72 0◦.0

Argument of perigee 102◦.83 270◦.0

Mean anomaly 153◦.54 0◦.0

Table 3: Comparison for a circular orbit (Topex).

Δt (s) Global errorRK4


1 3.7 × 10−3 ± 6.0 × 10−3 1.2 × 10−6 ± 1.3 × 10−5 6.3 × 10−8 ± 8.9 × 10−7

10 3.8 × 10−3 ± 8.3 × 10−3 1.0 × 10−4 ± 5.4 × 10−4 5.0 × 10−6 ± 4.0 × 10−5

30 3.9 × 10−3 ± 9.5 × 10−3 7.6 × 10−4 ± 2.1 × 10−3 3.5 × 10−5 ± 1.4 × 10−4

60 3.6 × 10−3 ± 4.9 × 10−3 2.7 × 10−3 ± 5.5 × 10−3 1.3 × 10−4 ± 4.0 × 10−4

300 3.2 × 10−3 ± 1.7 × 10−3 6.6 × 10−2 ± 1.1 × 10−1 3.8 × 10−3 ± 8.2 × 10−3

600 3.1 × 10−3 ± 1.4 × 10−3 2.3 × 10−1 ± 2.3 × 10−1 2.1 × 10−2 ± 2.5 × 10−2

Table 4: Comparison for an elliptical orbit (Molniya).

Δt (s) Global errorRK4


1 3.7 × 10−4 ± 9.8 × 10−4 8.8 × 10−7 ± 2.3 × 10−5 6.2 × 10−8 ± 1.7 × 10−6

10 3.6 × 10−4 ± 8.2 × 10−4 5.7 × 10−5 ± 1.1 × 10−3 4.0 × 10−6 ± 7.6 × 10−6

30 3.5 × 10−4 ± 7.4 × 10−4 2.4 × 10−4 ± 2.7 × 10−3 1.7 × 10−5 ± 1.9 × 10−4

60 3.4 × 10−4 ± 6.9 × 10−4 5.1 × 10−4 ± 3.1 × 10−3 3.6 × 10−5 ± 2.3 × 10−4

300 2.7 × 10−4 ± 3.8 × 10−4 1.5 × 10−2 ± 2.3 × 10−1 1.6 × 10−3 ± 1.3 × 10−2

600 3.5 × 10−4 ± 6.7 × 10−4 4.3 × 10−2 ± 1.9 × 10−1 4.6 × 10−3 ± 1.9 × 10−2

step mean and standard deviations considering the whole day sample. The comparison is

done for only these two kinds of orbits, where the Keplerian approximation for the transition

matrix provides already reasonable accuracy. For large time interval, the second method is

always disadvantageous because it is a local numerical approximation, as depicted in Tables

3 and 4.


6. Conclusions

The goal of this research was to compare and choose the most suited method to calculate the

transition matrix used to propagate the covariance matrix of the position and velocity of the

state estimator, (e.g., Kalman filtering or least squares), within the procedure of the artificial

satellite orbit determination. The methods were evaluated according to accuracy, processing

time, and handling complexity of the equations for two kinds of orbits: circular and elliptical.

The processing time of the second method is around 30% larger than the first one; however,

this difference is not considered enough to harm the computer load in the orbit determination

tasks. The second method is more accurate for short intervals, as Δt = 1 and 10 seconds, for

the orbits considered. For other intervals of propagation, the first method shows to be more

stable in the sense of keeping the same accuracy regardless of the step size. The equations

of the second method are easier to be handled, which means that it is possible to include

more perturbations easily. However, the first method is a closed analytical solution for the

two-body problem only. The second method has no singularity and no restriction about the

type of orbit; the first one is optimized for elliptical orbits (not optimized for parabolic and

hyperbolic orbits). The second method performs an approximation in Taylor’s series, whereas

in the first there is an analytical solution for the Keplerian motion. Therefore, for small

intervals of time (1 to 10 seconds) and when one expects more accuracy, it is recommended

to use the second method, since the CPU time does not overload excessively the computer in

the orbit determination procedure. For larger intervals of time and when one expects more

stability on the calculation, it is recommended to use the first method.

Acknowledgments

The authors wish to express their appreciation for the support provided by UNESP

(Universidade Estadual Paulista “Julio de Mesquita Filho”) of Brazil and INPE (Brazilian

Institute for Space Research).

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[4] P. S. Maybeck, Stochastic Models, Estimation, and Control, vol. 1, Academic Press, New York, NY, USA,1979.

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[20] H. K. Kuga, Adaptive orbit estimation applied to low altitude satellites, M.S. thesis, Instituto Nacional dePesquisas Espaciais, Sao Paulo, Brazil, 1982.

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Research ArticleHigher-Order Analytical Attitude Propagation of anOblate Rigid Body under Gravity-Gradient Torque

Juan F. San-Juan,1 Luis M. Lopez,2 and Rosario Lopez1

1 Departamento de Matematicas y Computacion, Universidad de La Rioja, 26004 Logrono, Spain2 Departamento de Ingenierıa Mecanica, Universidad de La Rioja, 26004 Logrono, Spain

Correspondence should be addressed to Juan F. San-Juan, [email protected]

Received 3 March 2011; Revised 25 June 2011; Accepted 3 August 2011


Copyright q 2012 Juan F. San-Juan et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A higher-order perturbation theory for the rotation of a uniaxial satellite under gravity-gradienttorque demonstrates that known special configurations of the attitude dynamics at which thesatellite rotates, on average, as in a torque-free state, are only the result of an early truncationof the secular frequencies of motion. In addition to providing a deeper insight into the dynamics,the higher order of the analytical solution makes it competitive when compared with the long-termnumerical integration of the equations of motion.

1. Introduction

The torque-free rotation of artificial satellites may be perturbed by a variety of effects [1].Due to the complexity of the force models to integrate, the problem of attitude propagation

is commonly approached numerically, with a variety of available algorithms [2, 3]. Nev-

ertheless, the problem can also be approached analytically, in which the analytical alternative

is usually based on perturbation methods: the Euler-Poinsot problem is taken as the un-

perturbed part of the problem, and the other effects are perturbations of the torque-free

rotation.

Among the perturbing torques that drive the rotational motion of an artificial satellite,

a research topic of current interest is the study of the effects produced by external torques

due to the satellite’s interaction with the Earth’s magnetic field [4, 5]. On the other hand,

the gravity-gradient torque is often identified as one of the more important perturbations

affecting the torque free rotation [6, 7]. Therefore, the model of a free rigid body perturbed

by gravity-gradient torque appears in this literature on this matter as one of the basic,

nonintegrable models used in the study of attitude propagation of artificial satellites [8, 9].The usefulness of this model is not restricted to the case of artificial satellites and also fits the

description of the rotational motion of natural satellites [10, 11].


Whereas bodies whose attitude propagation may be of interest are triaxial in general,

many celestial bodies have symmetry or near-symmetry rotation, which justifies the use of

the uniaxial model in understanding the perturbed dynamics in this case. The uniaxial model

is not limited to natural bodies, and this has also attracted the attention of aerospace engineers

in the study of the attitude of artificial satellites [12].The uniaxial model still remains of interest in the study of the attitude propagation,

either due to its direct application to actual problems [13] or because it can be taken as a

truncation of a whole triaxial model as long as the departure from axisymmetry is small. In

this last case, the uniaxial model is considered to be the zeroth order part in a perturbation

approach in which the small triaxiality plays the role of a perturbation [14, 15].In the case of natural celestial bodies, the work of external moments derived from the

gravitational attraction of other bodies is normally negligible when compared with the kinetic

energy of their torque-free rotation. This is why only first-order effects of the gravity-gradient

torque are usually taken into account in the study of the rotation of most solar system bodies.

However, even when the perturbation model only includes first order effects, extending the

perturbation solution to higher orders provides a clearer insight into the dynamics involved.

Thus, the secular terms of a first-order approach reveal special configurations of the satellite

[13, 16] in which the satellite’s attitude under gravity-gradient torque evolves, on average, as

in the torque-free state, but with a slightly modified angular momentum. However, we will

demonstrate that these special configurations do not survive when considering higher-order

terms in calculating the solution. In addition, higher-order solutions would be definitely

useful in increasing computational accuracy, thus extending the validity of an analytical

approach for much longer intervals. For artificial satellites, the gravity-gradient effect may

be much more important than in the case of natural celestial bodies, and the inclusion of

second-order effects would be imperative when investigating the attitude propagation in the

long term. In these case,s when the gravity-gradient second-order effects are nonnegligible,

other effects such as a small triaxiality of the rigid body, or the orbit eccentricity or other

external torques, may introduce observable frequencies in the rotation solution.

We will deal with the problem of the attitude propagation of a satellite under gravity-

gradient torque. In order to get insight into the contribution of higher-order terms in cal-

culating the solution, we will make some simplifying assumptions and focus on the specific

case of a uniaxial satellite under the disturbing gravitation of a perturber in circular orbit. The

problem is solved by perturbation theory up to higher order effects in the gravity-gradient,

thus extending the applicability of analytical results of [13]. The successive approximation

method used by [13] now becomes too intricate for computing an analytical solution in which

higher-order effects are involved. Therefore, from the beginning, we resort to the method

of averaging using Lie transforms [17–19] to solve the nonintegrable perturbed problem. It

deserves to be mentioned that approaching attitude dynamics problems with Lie transforms

is not new and has been used from long ago in finding either analytical or semianalytical

solutions of perturbed rotational motion (see, for instance [6, 7, 20, 21]).We use Deprit’s method [18], which is specifically designed for automatic machine

computation which allows the perturbation theory to be easily computed to higher orders,

although it is enough to compute the second-order effects introduced by the gravity-

gradient torque to show how the secular behavior changes with respect to that arising from

lower-order truncation theories. Specifically, we will show that special configurations of the

attitude dynamics at which the satellite has been established as rotating as in a torque-

free state except for periodic terms are only the result of an early truncation of the secular


frequencies. The new solution not only provides the required insight into the long-term dy-

namics, but also reveals it is highly competitive when compared with the numerical in-

tegration of the equations of motion.

2. Hamiltonian Formulation

The Hamiltonian of the rigid-body rotation is

H = T + V, (2.1)

where T is the kinetic energy of rotation of a rigid body around its center of mass

T =1

2

(Aω2

1 + Bω22 + Cω

23

), (2.2)

ω1, ω2, and ω3 are the components of the instantaneous rotation vector in the frame of the

principal axis of the body, and A ≤ B ≤ C denote the principal moment of inertia. In the case

which concerns us, the gravity-gradient torque, within the accuracy of higher-order terms in

the ratio of the linear dimensions of the satellite to those of the orbit, the potential energy V

of the external torques is taken from MacCullagh’s approximation [22],

V = −Gm1m

r

(1 +

A + B + C − 3D

2mr2

), (2.3)

where m is the mass of the rotating body, m1 is the mass of the disturbing body, r is the

distance between the centers of mass of both bodies, G is the gravitational constant, and

D = Aα2 + Bβ2 + Cγ2 is the moment of inertia of the rigid body with respect to an axis in

the direction of the line joining its center of mass with the disturbing body’s, with direction

cosines α, β, and γ , where the constraint α2 + β2 + γ2 = 1 is applied.

In order to get insight into the contribution of higher-order terms of the analytical

solution, we make some simplifying assumptions. First, we assume that the rotation does

not affect the orbital motion; hence, we neglect the Keplerian term in the potential. Then, we

assume circular orbital motion with r = a constant. Finally, we assume that the rotating body

is oblate; hence, A = B. Therefore,

H =1

2

[A(ω2

1 +ω22

)+ Cω2

3

]− Gm1

2a3(C −A)

(1 − 3γ2

). (2.4)

In the Hamiltonian setting, the components of the instantaneous rotation ω1, ω2, ω3,

and the direction cosine γ in (2.4) are assumed to be state functions of any specific set of

canonical variables. Although Euler angles provide an immediate description of the attitude,

the Hamiltonian of the free rigid body H = T does not reveal all the symmetries of the Euler-

Poinsot problem when using Euler canonical variables. In contrast, the integrable character

of the free rigid body problem is evident when using Andoyer variables [23, 24], which

take advantage of the angular momentum integral so as to use the plane perpendicular to

it, leaving aside the center of mass of the rigid body. The invariant plane is defined by the


canonical variables that link the inertial and rotating frames, in which is the latter defined by

the principal axes of the rotating body with the x and y axes which define its equatorial plane.

Using Andoyer variables (λ, μ, ν,Λ,M,N), the kinetic energy takes the form

T =1

2

(sin2ν

A+

cos2ν

B

)(M2 −N2

)+N2

2C, (2.5)

where M is the modulus of the angular momentum vector, N is its projection on the z axis of

the body frame, and ν is the angle encompassed by the x axis of the body frame and the axis

defined by the intersection of the equatorial plane of the body and the invariant plane. The

variables μ, the angle encompassed by the intersections of the invariant plane with both the

inertial plane and the equatorial plane of the rigid body, λ, the angle encompassed by the x

axis of the inertial plane and the intersection of the inertial and invariant planes, and Λ, the

projection of the angular momentum vector on the axis perpendicular to the inertial plane,

are cyclic in T showing that the torque-free rotation is a problem of one degree of freedom

and, therefore, integrable. Furthermore, in the case of axisymmetric bodies A = B, the kinetic

energy is simply

T =M2 −N2

2A+N2

2C, (2.6)

which is a trivial Hamiltonian integration problem.

In order to express the disturbing function introduced by the gravity-gradient in

Andoyer variables, it is convenient to refer the inertial motion to the orbital plane. Then, after

the usual rotations that match the body and inertial frames, we obtain the direction cosine γ

in Andoyer variables,

γ = sin J cos(λ − nt) sinμ +(cos J sin I + sin J cos I cosμ

)sin(λ − nt), (2.7)

where t is time, n is the mean orbital motion, J = arccos(N/M) is the inclination angle

between the equatorial and invariant planes, and I = arccos(Λ/M) is the inclination angle

between the invariant and inertial planes (Figure 1).The explicit appearance of time in the Hamiltonian is easily avoided by moving to a

rotating frame at the same rotation rate of the orbital motion. Thus, we introduce the new

variable = λ − nt; then,

d

dt=

dλ

dt− n =

∂H∂Λ

− n =∂

∂Λ(H− nΛ), (2.8)

shows that this change of the reference frame will also require the introduction of the Coriolis

term −nΛ in the Hamiltonian. As a result, we now deal with the conservative Hamiltonian

K(μ, ν, ,M,N, L) = H(μ, ν, ,M,N, L)−nL, where L = Λ is the conjugate momentum to the

new variable .


Equatorial plane

Plane perpendicular to

the angular m om entum

Inertial plane

M

NJ

λ

ν

μ

I

Figure 1: Relations between the Andoyer angles and the inclination angles I and J .

Therefore, the Hamiltonian of the rotation of the oblate rigid body in the Andoyer

variables in the rotating frame takes the form

K =a1

2M2 − a1 − a3

2N2 − nL

+κ

8

{(2 − 6 cos2J

)(1 − 3 cos2I − 3 sin2I cos 2

)− 6 sin I sin 2J

[2 cos I cosμ − (1 + cos I) cos

(2 + μ

)+ (1 − cos I) cos

(2 − μ

)]+3 sin2J

[2 sin2I cos 2μ + (1 + cos I)2 cos

(2 + 2μ

)+ (1 − cos I)2 cos

(2 − 2μ

)]},

(2.9)

where we use the notation κ = −Gm1(C−A)/(2a3), a1 = 1/A, a3 = 1/C. Note that ν is cyclic,

and, therefore, N remains constant in the perturbed problem.

The equations of motion of the perturbed problem are obtained from the Hamilton

equations

d(, ν, μ

)dt

=∂K

∂(L,M,N),

d(L,M,N)dt

= − ∂K∂(, ν, μ

) . (2.10)

Therefore,

dM

dt=

3

4κ{

sin2J[2sin2I sin 2μ + (1 + cos I)2 sin

(2 + 2μ

)− (1 − cos I)2 sin

(2 − 2μ

)]− sin 2J

[sin 2I sinμ − (1 + cos I) sin I sin

(2 + μ

)− (1 − cos I) sin I sin

(2 − μ

)]},

dL

dt=

3

4κ{

2 sin I sin 2J[(1 + cos I) sin

(2 + μ

)− (1 − cos I) sin

(2 − μ

)]−(

2 − 6 cos2J)

× sin2I sin 2 + sin2J[(1 + cos I)2 sin

(2 + 2μ

)+ (1 − cos I)2 sin

(2 − 2μ

)]},


dμ

dt= a1M

+3κ

4M

{[3 cos2J +

(1 − 6 cos2J

)cos2I

](2 − 2 cos 2) − 4 cos2J

−(

cot2J + cot2I − 2)

sin 2I sin 2J cosμ + 2(

cos2Jsin2I + cos2Isin2J)

cos 2μ

− sin I sin 2J[(1 − cos I)

(cot2J + cot2I

)− 1 + 2 cos I

]cos(2 − μ

)+ sin I sin 2J

[(1 + cos I)

(cot2J + cot2I

)− 1 − 2 cos I

]cos(2 + μ

)+ (1 + cos I)

(cos2J + cos I cos 2J

)cos(2 + 2μ

)+(1 − cos I)

(cos2J − cos I cos 2J

)cos(2 − 2μ

)},

dν

dt= −(a1 − a3)N

+3κ

4Mcos J

{4 − 2 sin2I

(6 sin2 + cos 2μ

)− (1 + cos I)2 cos

(2 + 2μ

)− (1 − cos I)2 cos

(2 − 2μ

)+ 2 sin I(cotJ − tan J)

×[2 cos I cosμ − (1 + cos I) cos

(2 + μ

)+ (1 − cos I) cos

(2 − μ

)]},

d

dt= −n +

3κ

4M

{4 cos I

(2 − 3 sin2J

)sin2 − 2 sin I

(1 − cot2I

)sin 2J cosμ

+ sin2J[(1 + cos I) cos

(2 + 2μ

)− 2 cos I cos 2μ − (1 − cos I) cos

(2 − 2μ

)]+ sin 2J[sin I + (1 − cos I)cotI] cos

(2 − μ

)+ sin 2J[sin I − (1 + cos I)cotI] cos

(2 + μ

)}.

(2.11)

We must note that Andoyer variables are singular for zero inclination of the in-

termediate plane with respect to either the inertial or the equatorial planes of the body, or

both. These singularities may be avoided when using other sets of variables [8, 25].

3. Perturbation Approach

In general, one can resort to the classical double averaging method to find the secular terms

of the disturbing function (see [26], for instance). In our case, the averaging would remove

the fast angles μ and from (2.9). However useful the classical double averaging may

be in finding the relevant long-term evolution of a dynamical system, in the case of the

Hamiltonian equation (2.9), it would be limited to providing the known first order terms

in the pertinent literature [13, equation (22)].In order to reach the higher orders required in this work, the perturbation solution

is computed analytically by Lie transforms using Deprit’s algorithm [18]. This method is

based on solving the homological equation L0(Wn) = Kn − Hn, which, in general, is a partial

differential equation where the term Hn comes from previous computations, the term Kn of

the new Hamiltonian is chosen at our convenience, and L0 is the Lie derivative, such that

now the term Wn of the generating function can be solved.


We constrain the solution to the case when the spin rate of the satellite is much faster

than its orbital rate, thus precluding the problem of resonances, and then we place the Coriolis

term at the first order. Besides, we assume that the gravity-gradient torque is a second-order

effect. Then, the Hamiltonian equation (2.9) is ordered as K =∑

m≥0 Km,0/m!, where

K0,0 =1

2a1M

2 − 1

2(a1 − a3)N2,

K1,0 = −nL,

K2,0 = 2!κ

8

{(2 − 6 cos2J

)(1 − 3 cos2I − 3 sin2I cos 2

)− 6 sin I sin 2J

[2 cos I cosμ − (1 + cos I) cos

(2 + μ

)+ (1 − cos I) cos

(2 − μ

)]+3 sin2J

[2 sin2I cos 2μ + (1 + cos I)2 cos

(2 + 2μ

)+ (1 − cos I)2 cos

(2 − 2μ

)]},

Km,0 = 0, m > 2.

(3.1)

This order allows us to proceed stepwise to eliminate first the rotation angle μ by

means of a Lie transform and then the angle by means of a second Lie transform. Moreover,

this splitting of averages has the added advantage that the homological equation of the

method can be solved by quadrature in both canonical transformations.

Note that if the spin and orbital rates were of the same order, the Lie derivative of the

homological equation would be the operator

L0 = ω∂

∂− a1M

∂

∂μ(3.2)

which involves the solution of a partial differential equation, contrary to a quadrature, for

computing the generating function of the canonical transformation. As shown with the

generating function W just below (22) of [13], this way of proceeding explicitly shows the

denominators that would be small when close to resonances, thus preventing the convergence

of the series solution.

To avoid the explicit appearance of square roots in the automatic evaluation of Poisson

brackets required by the method, it is easier for us to handle circular functions of the

inclination angles I and J as state functions of the Andoyer variables. Their nonvanishing

partial derivatives with respect to the Andoyer variables are

∂(cosQ)∂M

= − 1

McosQ,

∂(sinQ)∂M

=1

McotQ cosQ,

∂(cosQ)∂N

=1

M,

∂(sinQ)∂N

= − 1

McotQ,

(3.3)

either for Q = J or Q = I.


Averaging of μ

The first canonical transformation (μ, ν, ,M,N, L) → (μ′, ν′, ′,M′,N, L′) has the effect of

averaging the Hamiltonian over the rotation angle μ′. The Lie derivative is now the operator

L0 = −a1M∂

∂μ, (3.4)

and the homological equation can be solved by a simple quadrature to compute the

generating function of the canonical transformation.

The Hamiltonian in the new variables is K =∑

m≥0 K0,m/m!, which, up to the fourth

order approximation,

K0,0 =1

2a1M

′2 − 1

2(a1 − a3)N2,

K0,1 = −nL′,

K0,2 = 2!κ

4

(1 − 3 cos2J ′

)(1 − 3 cos2I ′ − 3 sin2I ′ cos 2′

),

K0,3 = 0,

K0,4 = 4!a1M

2

2

3

64

×{

32n2

a21M

′2κ

a1M′2

(1 + cos2I ′

)sin2J ′ cos 2′

− 3κ2

a21M

′4

[1 + 26 cos2J ′ + 5 cos4J ′ +

(6 − 356 cos2J ′ + 414 cos4J ′

)cos2I ′

−(

5 − 126 cos2J ′ + 153 cos4J ′)(

3 cos4I ′ + sin4I ′ cos 4′)

− 4[1 + 26 cos2J ′ − 27 cos4J ′ +

(5 − 126 cos2J ′ + 153 cos4J ′

)cos2I ′

]× sin2I ′ cos 2′

]},

(3.5)

that only depends on one angle, ′. As a consequence, up to the truncation order, the

averaging makes M′ constant, and, therefore J ′.

Elimination of ′

A second canonical transformation (μ′, ν′, ′,M′,N, L′) → (μ′′, ν′′, ′′,M′,N, L′′) is now

computed so as to eliminate ′. Now, the Lie derivative is

L0 = ω∂

∂′, (3.6)


and again the homological equation is solved by quadrature. From this second transformation

we obtain the double averaged Hamiltonian K =∑

m≥0 Km/m! given by

K0 =1

2a1M

′2 − 1

2(a1 − a3)N2,

K1 = −nL′′,

K2 = 2!κ

4

(1 − 3 cos2J ′

)(1 − 3 cos2I ′′

),

K3 = −3!9κ2

16M′ncos I ′′sin2I ′′

(1 − 3 cos2J ′

)2,

K4 = −4!a1M

′2

2

9

64

{3κ3

a1M′4n2

(1 − 3 cos2J ′

)3sin2I ′′

(1 − 5 cos2I ′′

)

+κ2

a21M

′4

[1 + 26 cos2J ′ + 5 cos4J ′ +

(6 − 356 cos2J ′ + 414 cos4J ′

)cos2I ′′

−3(

5 − 126 cos2J ′ + 153 cos4J ′)

cos4I ′′]},

(3.7)

which, up to the truncation order, only depends on momenta and, therefore, is easily in-

tegrated to give the linear motion

μ′′ = μ′′0 + nμt, ν′′ = ν′′0 + nνt, ′′ = ′′0 + nt =⇒ λ′′ = λ′′0 + (n + n)t, (3.8)

in which the secular frequencies of the motion are

nμ = a1M′ +

3κ

2M′

[cos2J ′ +

(1 − 6 cos2J ′

)cos2I ′′

]+

9κ2

8M′2ncos I ′′

(1 − 3 cos2J ′

)[1 − 9 cos2J ′ − 2

(1 − 6 cos2J ′

)cos2I ′′

]+

9κ2

64M′3a1

[1 + 52 cos2J ′ + 15 cos4J ′ + 12

(1 − 89 cos2J ′ + 138 cos4J ′

)cos2I ′′

−9(

5 − 168 cos2J ′ + 255 cos4J ′)

cos4I ′′]+

27κ3

64M′3n2

(1 − 3 cos2J ′

)2

×[1 − 12 cos2J ′ −

(12 − 90 cos2J ′

)cos2I ′′ + 15

(1 − 6 cos2J ′

)cos4I ′′

],

nν = −(a3 − a1)N − 3κ

2M′

(1 − 3 cos2I ′′

)cos J ′ +

27κ2

4M′2ncos I ′′sin2I ′′ cos J ′

(1 − 3 cos2J ′

)− 9κ2

32M′3a1

cos J ′[13 + 5 cos2J ′ −

(178 − 414 cos2J ′

)cos2I ′′ + 27

(7 − 17 cos2J ′

)cos4I ′′

]+

243κ3

64M′3n2cos J ′

(1 − 3 cos2J ′

)2(1 − 5 cos2I ′′

)sin2I ′′,


n = −n − 3κ

2M′ cos I ′′(

1 − 3 cos2J ′)− 9κ2

16M′2n

(1 − 3 cos2I ′′

)(1 − 3 cos2J ′

)2

− 9κ2

32a1M′3cos I ′′

[3 − 178 cos2J ′ + 207 cos4J ′ − 3

(5 − 126 cos2J ′ + 153 cos4J ′

)cos2I ′′

]+

27κ3

32M′ 3n2

(1 − 3 cos2J ′

)3(3 − 5 cos2I ′′

)cos I ′′.

(3.9)

We note that the special configurations mentioned in [13]—critical inclinations such

as cos2I = cos2J = 1/3 or cos I = cos J = 0 in which the rigid body evolves, on average, as in

torque-free rotation, although at a slightly different rotation rate from the unperturbed case—

are exactly the result of the order of approximation used, whose perturbation solution only

considered first-order effects on the gravity-gradient perturbation (which are equivalent to

the second-order of our present approach by Lie transforms). These special configurations no

longer exist when truncating the perturbation approach at higher orders. Thus, while special

inclinations such as cos2I = cos2J = 1/3 are preserved up to the third-order truncation of

(3.9), where

nμ = a1M′, nν = −(a1 − a3)N, n + n = 0, (3.10)

and the frequencies of the averaged problem correspond to a torque-free rotation state, this

is no longer true for the case cos I = cos J = 0, where

nμ = a1M′, nν = −(a1 − a3)N, n + n = − 9κ2

16M′2n(3.11)

show that λ is no longer fixed, on average, and suffers from a small precessional motion.

Furthermore, the unperturbed-type state at special configurations cos2I = cos2J = 1/3 is also

destroyed at the fourth order of the theory, where

nμ = a1M′ − 3κ2

2M′3a1

, nν = −(a1 − a3)N −√

3κ2

2M′3a1

, n + n =5√

3κ2

4M′3a1

. (3.12)

These higher-order terms may explain the observed behavior in Figure 6 of [13].

4. Numerical Comparisons

So as to evaluate the performance of the analytical solution, we take a fictitious Earth satellite

with moments of inertiaA = B = 400 kg km2, C = 600 kg km2, which we assume to be rotating

at a rate of 1 rotation per minute. The satellite is assumed to be in a circular orbit of the MEO

region with a semimajor axis a = 13 000 km.

In the first test case, we take the satellite in a high-inclination orbit with I = 70 deg

and assume that it is rotating close to the axis of maximum inertia with J = 10−3. Besides,

internal units such that the initial value of the modulus of the momentum is M = 1 are


0.4

0−0.4

Orbital periods

0 5 10 15 20 25 30

10

8×δ

M

(a)

0.2−0.2

−0.6

Orbital periods

0 5 10 15 20 25 30

10

13×δ

M

(b)

0.2

0−0.2

Orbital periods

0 5 10 15 20 25 30

10

2×δ

L

(c)

−0.2

0.2

Orbital periods

0 5 10 15 20 25 30

10

10×δ

L

(d)

0.30.1

−0.3

10

3×

λra

d

Orbital periods

0 5 10 15 20 25 30

−0.1

(e)

0123

10

9×

λra

d

Orbital periods

0 5 10 15 20 25 30

(f)

0.60.2

−0.2−0.6

10

2×

μra

d

Orbital periods

0 5 10 15 20 25 30

(g)

10

7×

μra

d

Orbital periods

0 5 10 15 20 25 30−2.5−1.5−0.5

0.5

(h)

0.60.2

−0.2−0.6

10

2×

νra

d

Orbital periods

0 5 10 15 20 25 30

(i)

−1012

10

7×

νra

d

Orbital periods

0 5 10 15 20 25 30

(j)

Figure 2: Errors of the fourth-order analytical solution versus the numerical integration for initialconditions μ = ν = 0, λ = 1, I = 70 deg, J = 10−3 rad. (Left): only the secular terms of the analyticalsolution are considered. (Right): the analytical solution includes secular and periodic terms. The notationδ is used for relative errors, whereas Δ means absolute differences.

chosen for the integration; the other initial conditions are μ = ν = 0 and = 1 radian.

For these initial conditions, the nonaveraged equations of motion, (2.11), are integrated

for 30 orbital periods, a time interval in which the satellite completes more than 11 000

rotation cycles. Results of the numerical integration are then compared with the analytical

attitude propagation. In the latter case, the initial conditions must be transformed to the

double averaged phase space, resulting in μ′′0 = 0.005824457466, ν′′0 = −0.005932985721,

′′0 = 1.0003166358, M′0 = 1.0000000025, and L′′

0 = 0.34164643181.

Results of the comparison between the analytical solution and the numerical

integration of the nonaveraged equations are presented in Figure 2, where the left-hand

column presents the differences between the numerical integration and the analytical

propagation of the secular terms of the fourth order truncation defined by the frequencies

in (3.9). The right-hand column of Figure 2 shows the differences obtained when using the


4

0

−4

×10−4

Orbital periods

0 5 10 15 20 25 30

−2

2

μ−μ

(a)

×10−4

62

−2−6

ν−ν

Orbital periods

0 5 10 15 20 25 30

(b)

4

0

−4

λ−λ

×10−6

Orbital periods

0 5 10 15 20 25 30

2

−2

(c)

Figure 3: Attitude evolution under gravity-gradient torque for the special configuration cos2I = cos2J =1/3 when compared with the corresponding torque-free rotation μ = μ′′

0 + a1M′t, ν = ν′′0 − (a1 − a3)Nt, and

λ = λ′′0.

full fourth-order theory, which, in addition to the secular terms propagation, includes the

third-order transformation equations for recovering the medium period terms related to the

averaging over , and the fourth-order transformation equations which allow us to recover

the short-period terms related to the averaging of μ.

As shown in the plotting on the left of Figure 2, the periodic errors have a noticeable

amplitude when compared with the secular trend. The amplitude of the errors due to periodic

terms is of about 2 arc minutes for λ, although this increases up to about one degree for μ and

ν. When the full fourth order analytical theory is used (right-hand column of Figure 2), the

periodic errors are confined to very small values, revealing a secular trend in the order of

10−10 radians per cycle in angular-variables errors, even though the amplitude of the periodic

errors of μ and ν mask this linear trend to some extent. The computation of higher orders in

the perturbation approach should improve the behavior of the analytical theory for both the

secular and periodic terms.

The second test case is for a satellite of the same physical characteristics and initial

configuration, except that now we take cos I = cos J =√

1/3, one of the special configurations

of the lower-order theories. The propagation of these initial conditions in the nonaveraged

model shows that this configuration is very close to an unperturbed state. As shown in

Figure 3, the time history of the difference between μ and the unperturbed state μ = μ′′0+a1M

′tseems to consist of only periodic terms, as well as what happens to the difference between

ν and the unperturbed analog ν = ν′′0 − (a1 − a3)Nt. In fact, there is a small linear trend of

the order of 3.1 · 10−7t for μ and 7.7 · 10−7t for ν, that is masked by the amplitude of periodic


−0.4−0.2

00.2

Orbital periods

0 5 10 15 20 25 30

10

5×δ

M

(a)

−0.2

0.2

0.6

Orbital periods

0 5 10 15 20 25 30

10

12×δ

M

(b)

1

0

−1

Orbital periods

0 5 10 15 20 25 30

10

5×δ

L

(c)

Orbital periods

0 5 10 15 20 25 30

−0.6−0.2

0.2

10

11×δ

L

(d)

0.4

0

−0.4

10

5×

λra

d

Orbital periods

0 5 10 15 20 25 30

(e)

−1−0.5

0

10

9×

λra

d

Orbital periods

0 5 10 15 20 25 30

(f)

0.4

0

−0.4

10

3×

μra

d

Orbital periods

0 5 10 15 20 25 30

(g)

−1.2−0.8−0.4

0

10

8×

μra

d

Orbital periods

0 5 10 15 20 25 30

(h)

0.60.2

−0.2−0.6

10

3×

νra

d

Orbital periods

0 5 10 15 20 25 30

(i)

10

9×

νra

d

Orbital periods

0 5 10 15 20 25 30

−1−0.6−0.2

(j)

Figure 4: Errors of the fourth-order analytical solution versus the numerical integration for initial con-

ditions μ = ν = 0, λ = 1, cos I = cos J =√

1/3 rad. (Left): only the secular terms of the analytical solutionare considered. (Right): the analytical solution includes secular and periodic terms. The notation δ is usedfor relative errors, whereas Δ means absolute differences.

oscillations. In contrast, the linear trend of the differences, in spite of being of the order of

2.9 · 10−8t, is better appreciated in the evolution of λ because of the notably smaller amplitude

of the periodic oscillations. To highlight this difference, we superimposed a linear fit to the

differences λ − λ′′0 to their time history presented in the inferior plotting in Figure 3, thus

revealing a clear departure from this fit (represented by the straight dashed white line) from

zero value.

The initial state in the double-averaged phase-space is now μ′′0 = 0.00030577890713,

ν′′0 = −0.0005329981843, ′′0 = 0.9999991669, M′0 = 1.0000016387, and L′′

0 = 0.577352484,

which are the initial conditions that feed the analytical solution. The comparison between

the numerical integration of the nonaveraged equations, and the perturbation solution is

presented in Figure 4. As in the previous example, we note that the propagation of the

secular terms alone introduces important periodic errors (left-hand column of Figure 4). In


contrast, when recovering the periodic terms removed from in averaging process, the errors

are reduced to quite acceptable values. As presented in the right-hand column of Figure 4, the

momenta obtained from the fourth-order analytical theory are mainly affected by very small

periodic errors, whilst the secular ones seem to be negligible. On the other hand, the angular

variables are affected by both periodic and secular errors that are apparent in their time

histories. Nevertheless, the secular errors grow at the very low rates of just a few microarc

seconds during an orbital period for λ and ν, and tens of microarc seconds during an orbital

period in the case of μ.

5. Conclusions

The rotation of an oblate rigid body under gravity-gradient torque is a nonintegrable problem

that may be approached by perturbations. Lower-order approaches to the solution that

only consider first-order effects in the gravity-gradient perturbation are normally considered

enough in some applications. However, analytical solutions that consider the effects of the

gravity-gradient torque up to the second-order may be required in the engineering problem

of attitude propagation. Carrying the perturbation approach up to this higher-order provides

a complete insight into the long term dynamics, and the perturbation solution continues

being competitive when compared with the numerical integration of the equations of motion.

The use of Andoyer variables resulted crucial for the perturbation approach used, because

their use notably facilitates the solution of the partial differential equations that provide the

generating functions of the Lie transforms. Finally, since modern perturbation methods are

designed for automatic computation, the analytical solution may be easily extended to even

higher orders if required.

Acknowledgments

Support is acknowledged from grant Gobierno de La Rioja Fomenta 2010/16. The authors

would like to thank Dr. M. Lara and the two anonymous reviewers for providing us with

constructive comments and suggestions.

References

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[4] M. C. Zanardi, I. M.P. Quirelli, and H. K. Kuga, “Analytical attitude prediction of spin stabilizedspacecrafts perturbed by magnetic residual torque,” Advances in Space Research, vol. 36, no. 3, pp.460–465, 2005.

[5] R. V. Garcia, M. C. Zanardi, and H. K. Kuga, “Spin-stabilized spacecrafts: analytical attitudepropagation using magnetic torques,” Mathematical Problems in Engineering, vol. 2009, Article ID242396, 18 pages, 2009.

[6] J. E. Cochran, “Effects of gravity-gradient torque on the rotational motion of A triaxial satellite in aprecessing elliptic orbit,” Celestial Mechanics, vol. 6, no. 2, pp. 127–150, 1972.

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