A Low-Cost Attitude Determination and Control System

for the UYS-1 NanosatelliteGabriel F. de Oliveira, Joao Y. Ishihara, Renato A. Borges, Henrique C. Ferreira

Electrical and Control EngineeringUniversity of BrasiliaBrasilia, DF - Brazil

gabrielfigueiro@lara.unb.br

Anatolyi M. Kulabukhov, Vladimir A. Larin, Vladimir V. BelikovDepartment of Guidance and ControlDnepropetrovsk National University

Dnepropetrovsk - Ukraine

Abstract—This paper considers the development of a Low-CostAttitude Determination and Control Subsystem (LCADCS) forthe first Ukrainian University Nanosatellite, UYS-1. For theattitude determination part, an attitude determination frame-work is implemented to combine all available data at each timesample using a modified Unscented Kalman Filter, based onthe Unscented Quaternion Estimator (USQUE). For the attitudecontrol part, the subsystem should rely only on magnetic actua-tion with magnetorquers operating in a relay mode. A proposedcontrol strategy is presented. The choice of hardware andalgorithms is addressed observing the LCADCS requirements,theoretical and practical considerations, ease of implementation,and time and cost budgets inherent to small-satellite projects.The validity and performance of the general proposed systemsare evaluated using computer simulations.

TABLE OF CONTENTS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 UYS-1 NANOSATELLITE . . . . . . . . . . . . . . . . . . . . . . . . 2

3 SYSTEM MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 ATTITUDE DETERMINATION . . . . . . . . . . . . . . . . . . . . 4

5 ATTITUDE CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1. INTRODUCTION

Boosted mainly by commercial applications, the last decadesexperimented massive advances in integrated circuits tech-nologies and microelectromechanical systems. As a result,more and more computability and sophisticated solutionswere available at smaller costs concerning power consump-tion, volume, mass, and commercial prices. These ef-fects, combined with the adoption of commercial-of-the-shelf(COTS) microelectronics to space applications were a majorbreakthrough to the small-satellite field [1]. The continuouslygrowing number of small-satellite projects in developmenthas relevant impact in the world-wide aerospace engineeringdevelopment. A remarkable aspect is that the aerospaceapplication development was brought in an inclusive way to

978-1-4673-1813-6/13/$31.00 c©2013 IEEE.1 IEEEAC Paper #2358, Version 5, Updated 01/19/2013.

the academic environment, finding its major use in educa-tional and training purposes, and in the evaluation of newengineering solutions [2].

Even though project specifications with relatively low re-quirements are recurrent in the small-satellite domain, theseprojects usually give rise to relevant engineering problems.The technologies developed, besides the contribution to fu-ture regular spacecraft engineering, gradually increase theapplicability of small-spacecrafts. Today, there are small-spacecrafts capable of delivering functionalities (such asEarth imagery [1], [3], [4]) at drastically smaller costs com-pared to conventional satellites.

Among the enabling technologies to the field, attitude deter-mination and control can be pointed out as one of the bot-tlenecks of small-spacecraft development [2]. Several workscan be found in the literature addressing attitude subsystemsfor small-spacecrafts [5], [6], [7]. In this work, the design ofa Low-Cost Attitude Determination and Control Subsystem(LCADCS) for the First Ukrainian University NanosatelliteUYS-1 is considered. The employment of low-cost limitedhardware to fulfill the spacecraft attitude requirements posesan engineering challenge that is in resonance with the small-spacecraft philosophy.

For the attitude determination part, the restrictions and re-quirements of the LCADCS resemble most general spacecraftattitude determination problems. Thus, it finds a great dealof works and documentation in literature [8], [9], [10]. Thechoice for an unscented formulation for spacecraft attitudefilter has been evaluated in several works [10], [9], [11].In this context, applications of the unscented filtering haveincreased among small-satellites and big space applications[12]. In this work, the USQUE filter is implemented withthe addition of the angular velocities in the state vector, toimprove the control response of the LCADCS.

Concerning the attitude control part, the system should relyonly in magnetic actuation with actuators operating in arelay mode. This restrictions compromise the overall con-trol performance, and the direct application of usual ap-proaches found in the literature [13], [14] do not guaranteethe LCADCS with stability and accuracy required. In [15] theauthors used the same type of relay actuation as the availablefor the LCADCS. However, the system also made use of amomentum wheel. Nonetheless, the steady state actuationstill was not adequate to the LCADCS requirements. Re-cently, several works have approached the 3-axis magneticstabilization of spacecrafts using Model Predictive Control

1

(MPC) techniques [16], [17], [18]. However, the computa-tional burden for the MPC computation and the discretizedform of the actuation make these solutions unsuitable for theLCADCS. In this context, the paper proposes a control strat-egy with the goal of stabilizing a 3-axis relay magneticallyactuated spacecraft in near polar orbits.

The paper is organized as follows: In Section 2, a generaloverview of the nanosatellite and its implications to theLCADCS are presented. In Section 3 the main conceptsconcerning the system modeling are discussed. Section 4concerns the attitude determination. In Section 5 the attitudecontrol is considered. Section 6 shows the simulated resultsfor the LCADCS. Section 7 concludes the work.

2. UYS-1 NANOSATELLITE

The acronym UYS-1 can be directly translated from Russianas “First Satellite of the Ukrainian Youth”. It designatesthe first Ukrainian University Nanosatellite, being developedby different Ukrainian Universities under the leadership ofDnepropetrovsk National University. This is a non-standardnanosatellite designed in a project developed by Ukrainianuniversities with wide student participation. This project isfinanced by the Ukrainian government, the universities andexternal sponsors. The satellite’s launch is supposed to bedone for free by a Ukrainian launcher currently under design,however, to the best of the author’s knowledge, no final deci-sions have yet been made about the launch. The present workwas developed in partnership with the University of Brasilia,as a result of the recent aerospace cooperation between Braziland Ukraine.

Overall Spacecraft Description

The UYS-1 fits in the category of nanosatellites, designedto weight about 10kg (no more than 12kg). The nanosatel-lite body consists of a rectangular shaped mounting withdimensions 150 × 150 × 400mm. Within this structure, thespacecraft equipment is allocated trying to achieve near evenmass distribution and optimize the magnetic balance. Outsidethis structure, solar panels are attached making an angle withthe spacecraft body. Figure 1 shows a simplistic sketch of thenanosatellite structure summarizing the above information.

Figure 1. Representation of the nanosatellite structure, bodycoordinate frame, and solar panels identification.

The spacecraft will be traveling in a sun-synchronous orbitwith a revolution period of approximately 99min, and its

Figure 2. Schematic showing the sensors, boards andthe data flow to the CPU, for all the available data for theLCADCS.

main function is Earth imagery. All the spacecraft subsys-tems are developed around a Central Processing Unit (CPU),which should handle all the computation concerning theLCADCS algorithms. To obtain required navigational data,the spacecraft uses a satellite aided navigation board equippedwith a GPS receiver, and a sensor board providing the data forthe CPU via an USB connection.

Attitude Determination and Control System

The spacecraft Attitude Determination and Control System(ADCS) will have two distinct subsystems. The first one isa High Precision Attitude Determination and Control Sub-system (HPADCS), which employs reaction wheels and astar tracker to achieve short time attitude maneuvers andstabilization with high precision (less than 0.5 deg in eachaxis). The second one is the LCADCS, which uses primarilylow-cost and low-power-consuming sensors and actuatorsto provide time permissible attitude maneuvers and attitudestabilization with moderate precision. The pointing accuracyfor the LCADCS is of 3 deg in each axis.

Each attitude subsystem should operate separately dependingon the spacecraft state and given mission operation modes.They can also work in conjunction using the magnetic actua-tion for unloading of the mechanical actuators. As the energyrestrictions are severe in the nanosatellite, the HPADCS willbe used only when strictly needed and most of the spacecraftoperation will rely on the LCADCS. In the present work, onlythe LCADCS is considered.

For the LCADCS, the determination part can make use of one3-axis MEM gyros, one 3-axis magnetometer, and four lighttransducers. The light transducers are fixed in the solar panelsmaking a given angle with the satellite body, and togetherthey compose a sun Line-Of-Sight (LOS) sensor. The sensorsare connected to a Sensor Board equipped with a microcon-troller which is responsible to handle data requisition fromthe CPU. In Figure 2 a schematic view of the LCADCSdetermination hardware is presented.

The attitude control part of the LCADCS should rely onlyon magnetic actuation, using three magnetorquers alignedwith the principal axes of inertia of the body. More thanthat, the actuators should operate in a relay mode. Eightbits connect the CPU to the actuation board, responsible forthe magnetic actuation. Every 2 bits handle the actuationof one of the three magnetorquers. The unused pair of bits

2

Table 1. Bit Addressing and Configuration for Actuationin the LCADCS

Bits Actuator Configuration Induction

0-1 Magnetorquer 1 0-0 Null

2-3 Magnetorquer 2 0-1 Max. Positive

4-5 Magnetorquer 3 1-0 Max. Negative

6-7 Unused 1-1 Forbidden

can be allocated for an extra backup magnetorquer or forsynchronization purposes. Table 1 outlines this information.

The magnetic actuation can produce magnetic inductions thatwould be measured by the magnetometers, interfering withthe attitude determination system. To avoid this effect, themagnetometers and magnetorquers are mechanically isolatedas much as possible within the nanosatellite small volume. Asthe physical separation is not enough, the magnetic actuationand sensing are also separated in time. The controller cycleis of one second, but the eventual activation of a givenmagnetorquer should occur for only a portion of that cycle,leaving the rest of the time window for the magnetometeroperation.

Hardware Selection

At the present stage of the development, the system havehardware proposals there are used to simulate and improvethe design considering COTS equipment. Some sets ofsensors were proposed during the LCADCS development andthe selection was continuously improved given the algorithmscharacterization and simulation results. Table 2 depicts thehardware proposal for the attitude determination used in thispaper.

3. SYSTEM MODELING

Nanosatellite Attitude Motion

The first step to model the attitude motion of the spacecraftis the definition of the coordinate systems of interest and aproper attitude representation. As the LCADCS is to estimateand control the global attitude, the main coordinate frames arethe body frame and the inertial reference frame. The bodyframe, depicted in Figure 1, is considered aligned with theprincipal axes of inertia. The z-axis is considered collinearwith the axis of minor inertia. For an inertial reference frame,was adopted the Earth Centered Inertial (ECI) coordinate sys-tem following the J2000 standard [19]. Secondary coordinatesystems as ECEF WG84 and NED (defined in [20]) were usedto handle sensor and environmental models.

The attitude representation more suitable for this type ofproject has been vastly discussed [21], [22]. The use ofquaternions is justified by the absence of singularities and theadequacy of this representation to embedded computing [23].In this design, the quaternion and some useful identities aredefined as in [9]

q = [ q q4 ]T

q = [ q1 q2 q3 ]T= e sin

(

θ2

)

q4 = cos(

θ2

)

, (1)

where e is a unit vector around which a rotation of θ radians isperformed (Euler vector) to describe a rotation. The identity

[

a×]

=

[

0 −a3 a2a3 0 −a1−a2 a1 0

]

(2)

is defined so that for 2 vectors a and b

[

a×]

b = a× b. (3)

The Direction Cosine Matrix (DCM ) can be computed fromthe quaternion using

DCM (q) = ΞT (q)Ψ (q) , (4)

Ξ (q) =

[

q4I3×3 + [q×]−qT

]

,

Ψ(q) =

[

q4I3×3 − [q×]−qT

]

.

(5)

Mathematical models for the spacecraft dynamics and kine-matics are necessary for determination and control calcu-lations. For simplicity and without loss of functionality,the system is considered as a rigid body and it is assumedthat the principal axes of inertia are known. In the futurephases of the nanosatellite development, its structure is to befully evaluated experimentally to come up with a trustworthyinertia matrix value. Thus, the classical Euler equationsdescribe spacecraft dynamics

T = Jω + [ω×]Jω

J =

[

J1 0 00 J2 00 0 J3

]

, (6)

where J is the inertia matrix, T is the vector of torques beenapplied to the spacecraft, and ω is the angular velocity vector.By considering the nanosatellite structure as presented inSection 2 and assuming the mass equally distributed, one cancalculate J2 = J1 = 0.1521kg ·m2 and J3 = 0.0375kg ·m2.

The angular velocities are related to the nanosatellite attituderate of change by the relation

q =1

2Ξ (q)ω. (7)

Space Environment

The modeling of the space environment in which a spacecraftwill operate is crucial to the design process. More than thepassive conditions, some environmental aspects play an im-portant role in the attitude determination and control systemsitself. In this work, only the environment in the nanosatelliteoperational phase is considered.

3

Table 2. Attitude Determination Hardware Proposed for the LCADCS Design and Simulation

Sensor Model Description

Magnetometer Honeywell HMC2300 3-axis digital magnetometer

sun LOS sensor - General Light Transducers

Gyros Analog Devices ADIS16485 Low mass, low power, low noise gyros

GPS Surrey-SST SGR-05U COTS based, suitable for university projects

A geomagnetic field model is required to the LCADCSfor embedded implementation in the nanosatellite. TheLCADCS attitude determination algorithm models the ex-pected geomagnetic measurements in a inertial frame. Forsuch, geomagnetic models maintained by scientific organi-zations are used. The World Magnetic Model (WMM) [20]is used to predict geomagnetic field measures for a givennanosatellite positioning and time. This model is open andwell documented, and is also used in the simulations. Furtherdetails on the model and its derivation can be found in [20].

Similarly, the sun LOS should be known at a referencecoordinate system, given the satellite position. This is a ratherknown problem, and for the LCADCS is enough to employapproximated analytic models described and presented in[24].

Besides the environmental models needed for embedded im-plementation, the environmental torques to which the space-craft will be subjected play an important role in the controlsystem specification and in the evaluation with simulations.Given the characteristics of the UYS-1 and its orbit, thedominant effect during normal operation of the spacecraftwill be magnetic residual dipole moments [25]. The residualmagnetic dipole has a variety of sources and its characteristicsare quite hard to model. In future phases of the nanosatellitedevelopment, the magnetic characteristics of the satellite willbe physically evaluated. More than that, this type of informa-tion is not well documented in the literature [26]. Concerningthe LCADCS requirements, the accuracy specification wereto be attained without considering the disturbances. However,to roughly evaluate the effects of the residual magnetic distur-bances we use information of other spacecrafts presented in[26].

4. ATTITUDE DETERMINATION

Sensor Models

Gyros and magnetometers are subject to errors arising frommisalignments, biases and noises. Misalignments and somebiases can be modeled and compensated by calibration infuture phases of the development. But random noises and biasrandom drifts should be dealt with using a stochastic filter.For simplicity, the calibrated parameters are not consideredhere. The estimation state vector is augment with the biases,and other effects are considered as Gaussian noise. For aquantity m being measured, the measurement value at instantk is given by

mk = mk + βmk+ νmk

, (8)

where mk is the actual physical value of the quantity, mk isthe measured value, βmk

is the bias of the measurement, andνmk

a Gaussian noise. These notations will be used from now

on in this work.

The sun LOS sensor is composed by a set of general lighttransducers. The sensors output is proportional to the lightcomponent normal to the sensors surface. If ϕi is the anglebetween the sun LOS and the vertical defining the sensitivesurface of the sensor placed in the solar panel i, the intensitymeasured is

Ii = Imax cos (ϕi) , (9)

where the index i can assume values from 1 to 4 correspond-ing to which solar panel the light transducer is fixed, and Imax

is the intensity received when the sun LOS is completelynormal to the sensor. It was stipulated that the panel i = 1is the one fixed on the solar panel i = 1 pointed by the x-axis, and the indexes increase in the direction from x to y, asshown in 1.

In order to recover the sun LOS from the sensor readings ageometric model is worked out to combine the information.Consider that sensors i and k are at opposite panels, beingk the one with lower intensity. If j is the sensor in thedirection pointed by the y-axis, the sun LOS in the bodyframe (superscript b) is then given by

sbLOS =

[

vxvyvz

]

,

vz = Iksin(αk)

,

vy =Ii−

(

Ik

sin(αk)

)

sin(αi)

cos(αi),

vx =Ij−

(

Ik

sin(αk)

)

sin(αj)

cos(αj),

(10)

where αi is the angle between the panel i and the satel-lite body. If there are 4 sensors excited, the problem isoverdetermined. As a first strategy a simple mean of the 2possible determinations is used. There will be also situationsin which only 1 or 2 transducers will be enlighten. Inthese particular cases, it is impossible to retrieve the sunLOS without considering Imax in (9) known. In Figure 3simulation results showing the number of active sensors forthe possible sun LOS’s are presented. For the simulationthe nanosatellite is considered with its z-axis pointing upand x-axis pointing east. In the figure, the horizontal axisis the angle of the LOS in degrees from north to east. Thevertical axis is the angle from up to down. Darker colors meanless active sensors (dark being 0 sensors, and white being 4

4

Figure 3. Simulation result showing the number of activesensors for each possible sun LOS. Darker colors denote lessactive sensors. Black: 0 active sensors, and white: 4 activesensors.

Figure 4. A schematic for the adapted version of the USQUEin the LCADCS is presented.

sensors). The simulations took in account a value of 30 degof inclination between the panels and satellite body.

Attitude Determination Scheme

The attitude determination is to be as accurate as possible inorder not to compromise the control part of the LCADCS.Fortunately, the task of attitude determination for a rigidsmall-spacecraft with characteristics similar to the ones of theLCADCS, is a well-established and well-known problem [8],[9], [10], [25], [11], [23].

A general overview of the attitude determination frameworkof the LCADCS is presented in Figure 4. The core of thesystem is a nonlinear filter based in the Unscented QuaternionEstimator (USQUE) proposed in [11].

Given that the LCADCS disposes at most of only two vectorobservations at each time sample, one of the most used andstraight forward solutions is the adoption of some variationof the TRIAD algorithm [27] to recover the body attitude.However, this approach can correct the attitude only when thesun LOS is available. Some works toggle between differentcorrection models in order to employ the TRIAD when thesun LOS is available and other method to correct when onlywith magnetometer measures [4]. In the proposed solution forthe LCADCS, the filter incorporates the vector informationdirectly in the correction model whenever a given set ofvectorial measures is available.

The choice for an unscented formulation for spacecraft at-titude filter has been evaluated in several works [10], [9],[11], [28], [29]. Despite the EKF reliability and successful

use in past missions [9], the Unscented filtering has shown tobe a very suitable solution to spacecraft attitude estimation,usually presenting better accuracy and faster convergencecompared to other the EKF. In this context, applications ofthe Unscented filtering have increased among small-satellitesand big space applications [12].

Since the control part of the LCADCS should rely onmagnetic actuation, the relevance of the angular velocitycomponent in the required control torque is dominant [13],[14]. For this reason, as the magnetorquers will operatein a relay fashion, the quality of the controller will have ahigh dependence with the accuracy of the velocity estimates.Thus, the nonlinear filter used in the LCADCS augments thestate vector of the original formulation of the USQUE [11]to incorporate information from the dynamic model in theestimates. The state vector becomes

x+k ≡

δp+k

ω+k

β+k

, (11)

where the hat marker (ˆ) denotes estimated variables, and theplus and minus superscripts (+ and −) denote corrected andpredicted variables, respectively. The δp is correspondent tothe attitude, ω corresponds to the angular velocities, and βdenotes the bias. The USQUE has the main structure of ageneral UKF with multiplicative error quaternion approach,but it uses a stereographic projection to map the quaterniondeviation into a vector space whenever the mathematicaloperations of the filter require the attitude to be in a vectorspace. The attitude error parameterization δp in the statevector commutates with the attitude error quaternion δq =

[ δq δq4 ]T

through the relations

δp = fδq

a+ δq4, (12)

δq4 =−a‖δp‖2+f

√f2+(1−a2)‖δp‖2

f2+‖δp‖2 ,

δq = f−1 (a+ δq4) δp,

(13)

where a and f are projection parameters. The direct appli-cation of the UKF algorithm over an attitude state vector inquaternion domain, would pose mathematical inconsistencies[30]. In this work only the main aspects of the filter and itsadaptation to the LCADCS are addressed.

The sigma points are generated in the LCADCS filter usingthe same method as presented [11]

χk (i) ≡

χδpk (i)

χωk (i)

χβk (i)

, (14)

where χδpk is the part of state vector related to the the attitude

deviation estimated in a multiplicative error formulation [8].

5

The component χβk is related to the bias estimates and χω

k isrelated to the angular velocity estimation.

The sigma points χδpk are converted to their quaternion coun-

terpart and the attitude deviations are applied to the currentquaternion estimate in order to assemble a set of scatteredquaternion attitudes. In the prediction step of the filter, theseattitudes are propagated using a discretized model of (7)

q−k+1 = Ω

(

ω+k

)

q+k

Ω(

ω+k

)

≡[

cos(

0.5‖ω+k ‖∆t

)

I3x3 −[

ψ+×

k

]

ψ+k

−ψ+k cos

(

0.5‖ω+k ‖∆t

)

]

ψ+k ≡ sin(0.5‖ω+

k‖∆t)ω+

k

‖ω+

k‖

.

(15)

This discretization, presented in [8], is necessary due to theconstrained nature of the quaternions. The propagation ofthe χω

k sigma points is done using a first order discretizedapproximation of (6)

ω−k+1 = J−1∆t

(

Tk −[

ω+×

k

]

Jω+k

)

+ ω+k , (16)

where Tk is the applied torque at instant k, and ∆t is theelapsed time. The bias propagation is assumed an identity

β−k+1 = β+

k . (17)

A vector observation in the body frame can be described as

vb = DCM (q)vi + ν, (18)

where vb is the measured vector in body frame, vi is thevector in inertial frame, and ν is a random noise. Thecorrection model is slightly different from that presented in[11], once the angular velocities measurement is incorporated

yk =

ω− + β−

DCM (q−)vi1

...DCM (q−)vi

m

k

+

νων1...νm

k

. (19)

In Section 6 simulations concerning the discussed system arepresented.

5. ATTITUDE CONTROL

The attitude control of spacecrafts using only magnetic ac-tuation is a challenging and interesting problem, and hasbeen subject of research of many previous works [31], [32],[33]. The magnetic attitude control poses a severe restriction,intrinsic to the actuation principle. Any induction generated

by the magnetic actuators will interact with the geomagneticlocal field to produce an actuation torque

T =[

M×]

B, (20)

where, M is the magnetic induction generated by the mag-netorquers, and B denotes the geomagnetic field. The space-craft experiences local uncontrollability around the directionof the Earths magnetic field. However, the direction of thegeomagnetic field has almost periodic variation with respectto a given inertial frame, once the spacecraft changes itsposition along the orbit. It can be shown that the meancontrollability can be achieved for orbits with high inclination[13].

One of the most common approaches to the 3-axis stabi-lization of a magnetically actuated spacecraft consists in thedefinition of a desired torque actuation (Tdes), and posteriordefinition of a optimal magnetic induction that best approx-imates the desired torque [31], [13], [34], [14]. In order tooptimize energy savings, the desired magnetic induction canbe obtained from a simple geometric analysis as

Mdes =[B×]Tdes

‖B‖2 . (21)

This approach, sometimes referred to as torque projectionapproach, is one of the most common in literature. In [13] theauthors propose a control law to determine Tdes resembling aPD-like controller. PD-like controllers have long been shownto solve the attitude control problem when a unrestrictedactuation is available [35]. It is noteworthy that the controlgain matrices in [13] are subject to very low scaling factors,which are crucial to the system stability. Even using differentapproaches to obtain Tdes, this design direction is observed.In [14], the author uses a modified sliding mode approach forthe control law, and it is cited that an increase in the gains,improving the spacecraft response, would eventually breakthe stability. In fact, as pointed out in [31], if the closed-loopspacecraft dynamics are sufficiently slow, the time-varyingsystem (which experiences local uncontrollability) can beapproximately studied in terms of its time-invariant approx-imations (which have mean controllability). This empiricalrule for attaining stability is directly related to the physicsunderlying the problem.

Observing the slow actuation guideline presented in [31],the stabilizing desired torque (Tdes) in steady-state is ex-pected to be several times smaller than the ones availableto approximate it with the relay actuation of the LCADCS(depending on the saturation of the relay actuators). In thissense, the use of relay-type actuators can make it hard toattain stability in steady-state using the torque projectionapproach. In simulation, it only was possible to stabilize thenanosatellite with relay-type actuation when the maximummagnetic actuation was limited to 0.01Am2. The resultsfor these simulations are given in the next section. Eventhough the use of a magnetorquer saturation at 0.01Am2

could stabilize the nanosatellite, the time for stabilizationwas exaggeratedly large (about 9 orbits) and the steady-stateactuation was prohibitive in terms of energy when comparedto usual continuous actuation profiles, due to the extensivechattering of the magnetorquers.

It should be noted that it can be found previous cases in

6

the literature using the above approach. In [15] the authoruses the same type of discontinuous relay actuation as in theLCADCS. However, the system disposed of a momentumwheel. Nonetheless, the steady state actuation still was notadequate to the LCADCS requirements. To the best of theauthors knowledge, there are no solutions in the literaturefor 3-axis magnetic stabilization with the use of relay-typeactuation with magnetorquers, such that could be directlyapplied to the LCADCS development. Therefore, it wasnecessary to work out a solution to that project obstacle inthe LCADCS.

Recently, several works have approached the 3-axis magneticstabilization of spacecrafts using Model Predictive Control(MPC) techniques [16], [17], [18]. In principle, if “future”information would be available, the LCADCS could decidethe best instantaneous actuation in order to gain overallstability. However, the computational burden for the MPCcomputation and the discontinuous form of the actuationmake these solutions unsuitable for the LCADCS.

The relay actuated magnetorquers can assume 27 configura-tions by switching their polarity (see Table 1). Due to thereduced number of possible actuations, it is reasonable toconsider a one step ahead prediction before the control actiondecision. Thus, it would be possible to decide among the27 possible actuations for the one that has the most desirablelocal instantaneous effects. Even though, the problem stillsin how to define the desired local instantaneous effects on thesystem to gain overall stability.

In this direction, the control strategy first calculates thepredictions using (16) and (15)

qk+1 (u) = Ω (ωk)qk, (22)

ωk+1 (u) = J−1∆t(

Tk (u)−[

ω×]

Jωk

)

+ ωk, (23)

with u = 1, · · · , 27 corresponding to the 27 possible actua-tions.

Then the variable λ, corresponding to an error vector, isdefined as

λk = Cpqerrk +Cdωerrk ,

qerrk =[

Ξ(

qref−1

)

qref−1

]

qk,

ωerrk = ωk − ωrefk ,

(24)

where Cp and Cd as constant gain matrices to be tunned,qref is the pointing reference, ωref is the angular velocityreference, and the inverse quaternion is defined as

q−1 =

[

−qq4

]

. (25)

Now, an error variable λk+1 (u) can be predicted for eachpossible actuation. This permits the definition of a first errormeasure given simply by the euclidean norm of λ. This error

measure is related to the distance from the actual systemconfiguration from the solution surface λ = 0

λnk= ‖λk‖. (26)

Rigorously, at each instant in time, only the dynamics per-pendicular to the geomagnetic field can be reduced from thesystem. If the actuation is defined seeking for the maximumdecrease from λnk

to λnk+1, this could leave uncontrollable

dynamics in the system that degrade the pointing accuracy.Depending on the actuation power and the orbit period, thiscould cause instability. This is the equivalent to the empiricalrule pointed in [31] in which the system should convergerestricted to slow dynamics.

Thus, it is necessary to define a way to restrict the actuationover the error variable λk to maintain the stability. To doso, a second error measure was defined in terms of the errorprojection in the local geomagnetic field B

λ⊥k=λk ·Bk

‖Bk‖. (27)

The effect of the actuations over this second error measurecan be predicted assuming that the geomagnetic field ofthe Earth is approximately constant in an inertial coordinatesystem for the control time window (1 sec) as

λ⊥k+1= λk+1·Bk+1

‖Bk+1‖,

Bk+1 = DCM (qk+1)DCM (qk)TBk.

(28)

By deciding the actuation observing the predictions for thetwo measures (λ⊥ and λn), it is possible to reduce the localerror through kinematic movements trying to maintain thecontrollability over the remaining error. The u-th actuationis selected from the set of possible actuations as

minu,

λ⊥k+1(u)≤λ⊥k

λnk+1(u)≤λnk

(λnk+1(u)) (29)

If no actuations respect the restriction, the system usesnull actuation by default. This approach allows the choicefor more powerful actuators (with maximum induction of0.1Am2). The evaluation of the proposed system is addressedin the next section.

6. SIMULATIONS

A first simulation for the determination system was done forthe spacecraft in stationary position and considering the sunalways visible when not in eclipse. The results are shown inFigure 5 for the x-axis of the spacecraft. It can be seen that theeffect of the varying magnetic field is reflected in a wave formof the 3σ uncertainty. The more one given axis has magneticcomponents in it, the bigger is the signal-noise ratio. So, thevalleys occur when large amounts of the magnetic field are in

7

a given axis, and the peaks in the uncertainty occur when themodule felt by the axis is null.

The use of relay-type actuators to approximate a continuousdesired torque Tdes, as discussed in Section 5, was evaluatedin several simulations. The stabilization was attained onlywhen the maximum induction in the magnetorquers waslimited to 0.01Am2. Figures 6 and 7 show simulation resultsfor the stabilization at the origin with the nanosatellite startingwith 2 deg /s angular velocity error and 90 deg pointing errorin the y-axis. Due to the small saturation limits in the relaymagnetorquers, the time for stabilization is exaggeratedlylarge, as is clear in Figure 6. The steady-state response andactuation profile is detailed in Figure 7, showing the energyproblems cited in Section 5.

The one step ahead predictive approach for the control strat-egy, proposed in Section 5, was evaluated with the sameinitial conditions used in the simulation illustrated in Figures6 ans 7. The simulation results are shown in Figures 8 and9. In Figure 8, a considerable increase in the performanceis observed. The time for stabilization droped to around4 orbits, given the availability of more powerful relay ac-tuators. Interestingly enough, this control strategy confersmean stability to the system accounting only with one stepahead predictions. In Figure 9, the steady state responseand actuation profile are presented. Despite the fact thatthe actuation is bigger (0.1Am2) than the one used with thetorque projection strategy (0.01Am2), Figures 9 and 7 showthat the proposed strategy has a better steady-state profileconcerning energy costs and maximum errors.

To roughly evaluate the effects of residual magnetic dipoleson the system, small magnetic dipoles based on data fromother nanosatellites [26] were applied equally distributed inthe 3 axes. A steady-state simulated result is shown inFigure 10. In this simulation the errors remained lower than6 deg in each axis, which is an usual pointing precision formost low-cost attitude determination and control systems. Amore detailed evaluation of the proposed strategy with moretrustworthy magnetic data will be done in future works.

Finally, a simulation was performed combining the determi-nation and control parts of the LCADCS to evaluate the pro-posed control behavior in the presence of velocity and point-ing uncertainties from the LCADCS determination part. Theinitial conditions were the same of the previous simulations.The system stood within 3 deg in each axis during the steady-state, and the transient phase was unaffected by the uncer-tainties. The simulation results are shown in Figures 11 and12. The orientation error, estimated angular error (surroundedby the 3σ confidence interval from the estimation), and the3 deg tolerance for each axis is depicted in Figure 11. Figure12 shows the convergence of the overall LCADCS systemand the actuation profile for each magnetorquer. ComparingFigures 9 and 12, it is clear that the actuation profile is moreactive when the uncertainties are present.

7. CONCLUSION

The design and simulation for the LCADCS is presented.The proposed attitude determination part of the system wascapable of delivering attitude and angular velocities estimatesso that the control stability of the system was not compro-mised. For simplicity, in this work only the bias of the gyroswere considered in the filter, but other parameters comingfrom gyros or magnetometers could be as well added to the

state vector. In the control part of the LCADCS, a controlstrategy was developed to maintain the stability, with 3 degpointing accuracy, considering the actuation restrictions, at-titude determination characteristics and orbit profile of thenanosatellite. The simulation results show that the systemstabilization is attained. The proposed controller showed topromote energy savings and better results when comparedto approaches using the direct approximation of continuouscontrollers by relay actuation.

ACKNOWLEDGMENTS

The authors thank the Brazilian National Council for Sci-entific and Technological Development (CNPq), the Dne-propetrovsk National University (DNU), and the BrazilianSpace Agency (AEB) for making this work possible.

REFERENCES

[1] P. Fortescue, G. Swinerd, and J. Stark, Spacecraft Sys-tems Engineering. Wiley, 2011.

[2] J. Bouwmeester and J. Guo, “Survey of worldwide pico-and nanosatellite missions, distributions and subsystemtechnology,” Acta Astronautica, vol. 67, no. 7, pp. 854–862, 2010.

[3] H. Ashida, K. Fujihashi, S. Inagawa, Y. Miura, K. Oma-gari, N. Miyashita, S. Matunaga, T. Toizumi, J. Kataoka,and N. Kawai, “Design of tokyo tech nano-satellitecute-1.7+ apd ii and its operation,” Acta Astronautica,vol. 66, no. 9, pp. 1412–1424, 2010.

[4] T. Xiang, T. Meng, H. Wang, K. Han, and Z. Jin, “De-sign and on-orbit performance of the attitude determina-tion and control system for the zdps-1a pico-satellite,”Acta Astronautica, vol. 77, pp. 182–196, 2012.

[5] M. Ovchinnikov, A. Ilyin, N. Kupriynova, V. Penkov,and A. Selivanov, “Attitude dynamics of the first russiannanosatellite tns-0,” Acta Astronautica, vol. 61, no. 1,pp. 277–285, 2007.

[6] A. Mohammed, M. Benyettou, M. Sweeting, andJ. Cooksley, “Initial attitude acquisition result of thealsat-1 first algerian microsatellite in orbit,” in Net-working, Sensing and Control, 2005. Proceedings. 2005IEEE. IEEE, 2005, pp. 566–571.

[7] M. Ovchinnikov, V. Pen’ko, O. Norberg, andS. Barabash, “Attitude control system for the firstswedish nanosatellite munin,” Acta Astronautica,vol. 46, no. 2, pp. 319–326, 2000.

[8] J. Crassidis and J. Junkins, Optimal estimation of dy-namic systems. Chapman & Hall, 2011, vol. 24.

[9] J. Crassidis, F. Markley, and Y. Cheng, “Survey of non-linear attitude estimation methods,” Journal of guidancecontrol and dynamics, vol. 30, no. 1, p. 12, 2007.

[10] O. Diaz, “Analysis and comparison of extended and un-scented kalman filtering methods for spacecraft attitudedetermination,” Ph.D. dissertation, Monterey, Califor-nia Naval Postgraduate School, 2010.

[11] J. Crassidis and F. Markley, “Unscented filtering forspacecraft attitude estimation,” Journal of GuidanceControl and Dynamics, vol. 26, no. 4, pp. 536–542,2003.

[12] J. Kim, J. Crassidis, S. Vadali, A. Dershowitz, andU. Alliance, “International space station leak localiza-

8

tion using attitude disturbance estimation,” in IEEEAerospace Conference, 2003, pp. 3475–3494.

[13] M. Lovera and A. Astolfi, “Spacecraft attitude controlusing magnetic actuators,” Automatica, vol. 40, no. 8,pp. 1405–1414, 2004.

[14] R. Wisniewski, “Satellite attitude control using onlyelectromagnetic actuation,” Ph.D. dissertation, AalborgUniversity. Department of Control Engineering, 1996.

[15] T. Meng, H. Wang, Z. Jin, and K. Han, “Attitudestabilization of a pico-satellite by momentum wheeland magnetic coils,” Journal of Zhejiang University-SCIENCE A, vol. 10, no. 11, pp. 1617–1623, 2009.

[16] X. Chen and X. Wu, “Model predictive control of cubesatellite with magneto-torquers,” in IEEE InternationalConference on Information and Automation (ICIA),2010. IEEE, 2010, pp. 997–1002.

[17] Y. Cao and W. Chen, “Automatic differentiation basednonlinear model predictive control of satellites usingmagneto-torquers,” in 4th IEEE Conference on Indus-trial Electronics and Applications, 2009. ICIEA 2009.IEEE, 2009, pp. 913–918.

[18] T. Krogstad, J. Gravdahl, and P. Tondel, “Explicit modelpredictive control of a satellite with magnetic torquers,”in Proceedings of the 2005 IEEE International Sym-posium on, Mediterrean Conference on Control andAutomation Intelligent Control, 2005. IEEE, 2005, pp.491–496.

[19] B. Schutz, B. Tapley, and G. Born, Statistical orbitdetermination. Academic Press, 2004.

[20] S. Maus, S. Macmillan, S. McLean, B. Hamilton,A. Thomson, M. Nair, and C. Rollins, “The us/uk worldmagnetic model for 2010-2015,” NOAA Technical Re-port NESDIS/NGDC, Tech. Rep., 2010.

[21] J. van der Ha and M. Shuster, “A tutorial on vectors andattitude [focus on education],” Control Systems, IEEE,vol. 29, no. 2, pp. 94–107, 2009.

[22] F. Markley, “Attitude error representations for kalmanfiltering,” Journal of Guidance, Control, and Dynamics,vol. 26, no. 2, pp. –, 2012.

[23] B. Wie, Space vehicle dynamics and control. AIAA,2008.

[24] D. Vallado, Fundamentals of astrodynamics and appli-cations. Springer, 2001, vol. 12.

[25] J. Wertz, Spacecraft attitude determination and control.Kluwer Academic Pub, 1978, vol. 73.

[26] J. Springmann, J. Cutler, and H. Bahcivan, “Mag-netic sensor calibration and residual dipole characteriza-tion for application to nanosatellites,” Toronto, OntarioCanada, vol. -, pp. –, 2010.

[27] S. Tanygin and M. Shuster, “The many triad algo-rithms,” in AAS/AIAA 17th Space Flight MechanicsMeeting, Sedona, Arizona, 2007, pp. 81–99.

[28] H. Leeghim, Y. Choi, and B. Jaroux, “Uncorrelated un-scented filtering for spacecraft attitude determination,”Acta Astronautica, vol. 67, no. 1, pp. 135–144, 2010.

[29] M. VanDyke, J. Schwartz, and C. Hall, “Unscentedkalman filtering for spacecraft attitude state and pa-rameter estimation,” in Spaceflight mechanics 2004:proceedings of the AAS/AIAA Space Flight MechanicsMeeting held February 8-12, 2004, Maui, Hawaii, vol.119. Amer Astronautical Society, 2005, p. 217.

[30] Y. Cheon and J. Kim, “Unscented filtering in a unitquaternion space for spacecraft attitude estimation,” inIndustrial Electronics, 2007. ISIE 2007. IEEE Interna-tional Symposium on. IEEE, 2007, pp. 66–71.

[31] E. Silani and M. Lovera, “Magnetic spacecraft attitudecontrol: a survey and some new results,” Control Engi-neering Practice, vol. 13, no. 3, pp. 357–371, 2005.

[32] M. Wood and W. Chen, “Attitude control of magneti-cally actuated satellites with an uneven inertia distribu-tion,” Aerospace Science and Technology, vol. -, pp. –,2011.

[33] M. Reyhanoglu and J. Hervas, “Three-axis magneticattitude control algorithms for small satellites,” in Re-cent Advances in Space Technologies (RAST), 2011 5thInternational Conference on. IEEE, 2011, pp. 897–902.

[34] M. Guelman, R. Waller, A. Shiryaev, and M. Psiaki,“Design and testing of magnetic controllers for satellitestabilization,” Acta Astronautica, vol. 56, no. 1, pp.231–239, 2005.

[35] J. Wen and K. Kreutz-Delgado, “The attitude controlproblem,” IEEE Transactions on Automatic Control,vol. 36, no. 10, pp. 1148–1162, 1991.

BIOGRAPHY[

Gabriel F. Oliveira received his B.S.degree in Mechatronics Engineeringfrom the University of Brasilia in 2011.He is currently a Masters Degree Stu-dent at University of Brasilia with fo-cus in Aerospace Engineering workingat the Automation and Robotics Labo-ratory (LARA). During his Masters De-gree program, he spent 9 months in Dne-propetrovsk, Ukraine, involved with the

development of small-spacecraft subsystems.

Anatoliy M. Kulabukhov AnatoliyMikhailovich Kulabukhov received anEngineering Diploma in the area ofautomatic control systems from Dne-propetrovsk National University (at thattime - Dnepropetrovsk State University)in 1972. After 2 years of military servicehe returned to the university and hasbeen working at the Department of Au-tomatic Control Systems since then. In

2002 he obtained a PhD degree in technologies of manufac-turing of flying vehicles. He carried out many research worksfor the Design Office “Yuzhnoye” and the UMZ plant (bothenterprises are involved in the rocket and satellite industry).Currently he is the head of the Department of AutomaticControl Systems at the Faculty of Physics and Technologyof Dnepropetrovsk National University and the coordinator ofan all-Ukrainian project “A Satellite of the Ukrainian Youth”.

9

Vladimir A. Larin Vladimir Alekseye-vich Larin graduated from Taganrog Ra-dio Engineering Institute in 1960 withan engineering diploma in the area ofautomation and remotely controlled sys-tems. For over 10 years he workedfor the Design Office “Yuzhnoye”, anenterprise involved in designing rocketsand satellites, as a designing engineerand a researcher. In 1972 he obtained

his PhD. Since 1971 he has been working in DnepropetrovskNational University. For over 15 years he was the head of theDepartment of Automatic Control Systems of that university.Now he is a professor of that department.

Vladimir V. Belikov Vladimir Vik-torovich Belikov received an Engineer-ing Diploma in the area of automaticcontrol systems from DnepropetrovskNational University (at that time - Dne-propetrovsk State University) in 1984.After 2 years of military service he re-turned to the university and has beenworking at the Department of AutomaticControl Systems since then. He partici-

pated in many researches for the Design Office “Yuzhnoye”and the UMZ plant (both enterprises are involved in the rocketand satellite industry). Currently he is a senior teacher at theDepartment of Automatic Control Systems at the Faculty ofPhysics and Technology of Dnepropetrovsk National Univer-sity.

Joao Y. Ishihara Joao Ishihara hasa Bachelor degree in Electrical Engi-neering from the Polytechnic Univer-sity of Sao Paulo (1990), where he alsoobtained a Master (1996) and a Ph.D.(1998) degree in Electrical Engineer-ing. From 1998 to 2002 he held apost-doctoral position at the Engineer-ing School of Sao Carlos, University ofSao Paulo. Since 2004 is an assistant

professor at the University of Brasilia and a control systemresearcher affiliated with the Laboratory of Automation andRobotics (LARA). He has experience in Electrical Engineer-ing with emphasis on control of electronic process, feedbacksystems, working mainly in the following topics: robust con-trol and filtering, descriptor systems, systems with Markovianjumps. Joo Ishihara is a Fellow Researcher of Productivityof the National Council for Scientific and TechnologicalDevelopment (CNPq).

Renato A. Borges Renato Borges re-ceived a Master (2004) and a Doctoral(2009) degree in Electrical Engineeringfrom the University of Campinas anda PhD degree in Electrical Engineer-ing from the University of New Mexico(2009) (USA). From 2009 to 2011 heheld a post-doctoral fellowship from theState of Sao Paulo Research Foundationworking at the School of Electrical and

Computer Engineering at the University of Campinas. He iscurrently an assistant professor at the Electrical EngineeringDepartment of the University of Brasilia, and a control systemresearcher affiliated with the Laboratory of Automation andRobotics (LARA). His main research interests are Lyapunov

stability theory, stability analysis of uncertain linear andnonlinear systems, linear systems with parameter variationsand finite-time stability.

Henrique Ferreira Henrique Ferreirahas a Bachelor degree in Electrical En-gineering from the Polytechnic Univer-sity of So Paulo (2003), where he alsoobtained a Master (2004) and a Ph.D.(2008) degree in Electrical Engineering.He is currently an associate professorat the University of Brasilia and a con-trol system researcher affiliated with theLaboratory of Automation and Robotics

(LARA). He has experience in Engineering Electric, withemphasis on control of electronic process, feedback systems,working mainly in the following topics: robust control robust,nonlinear control, process control, optimization, magneticlevitation and lamination.

10

0 1 2 3 4 5 6 7 8 9−0.5

0

0.5

deg

Error1

Estimated

3σ

0 1 2 3 4 5 6 7 8 9−2

0

2x 10

−5

rad/s

ω1

0 1 2 3 4 5 6 7 8 9−5

0

5x 10

−6

time (orbits)

rad/s

param1

Figure 5. Estimation results for x-axis. Angle, angular velocity and gyro bias estimation errors surrounded by the 3σconfidence intervals.

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

quate

rnio

n

q1

q2

q3

q4

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

time (orbits)

deg/s

ω1

ω2

ω3

Figure 6. Convergence of the relay-actuated torque projection (Tdes) controller. Maximum induction limited to 0.01Am2.

11

10 12 14 16 18 20 22 24 26 28 30

−202

Err

or

(deg)

10 12 14 16 18 20 22 24 26 28 30−0.01

0

0.01

Am

2

M1

10 12 14 16 18 20 22 24 26 28 30−0.01

0

0.01

Am

2

M2

10 12 14 16 18 20 22 24 26 28 30−0.01

0

0.01

time (orbits)

Am

2

M3

Figure 7. Detailed view of the steady-state response and actuation profile of the relay-actuated torque projection controller.In the upper part the error angles for the 3 axes are shown to be within the 3 deg requirements. The actuation profile of eachmagnetorquer is detailed below.

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

quate

rnio

n

q1

q2

q3

q4

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

time (orbits)

deg/s

ω1

ω2

ω3

Figure 8. Convergence for the proposed one step ahead predictive control strategy. Maximum induction limited to 0.1Am2.

12

6 8 10 12 14 16 18 20−0.5

0

0.5

Err

or

(deg)

6 8 10 12 14 16 18 20−0.1

0

0.1

M1

Am

2

6 8 10 12 14 16 18 20−0.1

0

0.1

M2

Am

2

6 8 10 12 14 16 18 20−0.1

0

0.1

M3

time (orbits)

Am

2

Figure 9. Detailed view of the steady-state response and actuation profile of the proposed control strategy. In the upper partthe error angles for the 3 axes are shown to be within the 3 deg requirements. The actuation profile of each magnetorquer isdetailed below.

6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

deg

6 8 10 12 14 16 18 20−1

−0.5

0

0.5

1x 10

−4

time (orbits)

rad/s

Error1

Error2

Error3

Figure 10. Steady-state response for the application of a residual magnetic dipole of 2.5×10−4Am2. The errors for orientationand angular velocities are shown for the 3 axes.

13

0 1 2 3 4 5 6 7 8 9 10−5

0

5

Error1

deg

0 1 2 3 4 5 6 7 8 9 10−5

0

5

Error2

deg

0 1 2 3 4 5 6 7 8 9 10−5

0

5

Error3

time (orbits)

deg

Figure 11. Determination and control evaluation. A detailed view of the error is presented surrounded by the 3σ (dashedlines) accuracy from the determination part for each axis. The 3 deg tolerance for each axis are also presented in horizontal thindashed lines.

0 1 2 3 4 5 6 7 8 9 10−1

0

1

quate

rnio

n

0 1 2 3 4 5 6 7 8 9 10−0.1

0

0.1

M1

Am

2

0 1 2 3 4 5 6 7 8 9 10−0.1

0

0.1

M2

Am

2

0 1 2 3 4 5 6 7 8 9 10−0.1

0

0.1

M3

time (orbits)

Am

2

Figure 12. Determination and control evaluation. The convergence of the LCADCS can be checked in the upper part of thefigure observing that the components of the vector part of the attitude quaternion tend to zero (origin). The actuation profile foreach magnetorquer is detailed below.

14