End-to-end trajectory design of a mission to the Jovian ......End-to-end trajectory design of a mission to the Jovian Trojan asteroids ... iii. iv. Abstract ... trajectory design of

End-to-end trajectory design of a mission to the JovianTrojan asteroids

Tiago Jorge Assunção de Sousa Bento

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor(s): Paulo Jorge Soares GilRon Noomen

Examination Committee

Chairperson: João Manuel Lage de Miranda LemosSupervisor: Paulo Jorge Soares GilMember of the Committee: João Manuel Gonçalves de Sousa Oliveira

October 2016

ii

Resumo

Inspirado pela diversidade e interesse cientıfico dos asteroides Troianos na orbita de Jupiter, este

trabalho apresenta o projecto de trajectorias de uma missao a este sistema.

Duas missoes diferentes foram consideradas: um rendezvous com o asteroide 624 Hektor, e um

flyby tour de multiplos asteroides. Para os dois cenarios, modelos MGA e MGA-1DSM foram implemen-

tados na optimizacao das trajectorias interplanetarias. A sua implementacao foi adaptada do codigo

do GTOP [1], melhorando a sua fundamentacao teorica, e garantindo melhores resultados. Adicional-

mente, um estudo de diversos algoritmos de optimizacao e apresentado para o modelo MGA, com o

intuito de preencher as lacunas encontradas na literatura.

No caso da flyby tour, um novo problema de optimizacao e desenvolvido e aplicado neste relatorio.

Assim, e conseguida uma optimizacao mais eficiente deste tipo de trajectorias.

A trajectoria final para o 624 Hektor segue uma sequencia MGA Terra-Marte-Jupiter, resultando em

2.4672 km/s de ∆V a bordo do satelite. Para a flyby tour, a sequencia MGA e Terra-Venus-Mercurio-

Venus, e um total de 6 asteroides sao observados. Ao todo, 2.3903 km/s de ∆V a bordo sao necessarios

para tal trajectoria, o que mostra bastantes melhorias face aos designs apresentados por Canalias et

al. [2].

As duas missoes projectadas, e as respectivas melhorias face a literatura, sao sucessos deste

relatorio, e uma consequencia directa dos modelos de trajectoria implementados.

Palavras-chave: Optimizacao de trajectoria, Asteroides Troianos, Flyby tour de asteroides,

Rendezvous de asteroides, Assistencias gravitacionais multiplas

iii

iv

Abstract

Inspired by the diversity and scientific interest of the Jovian Trojan asteroids, this work deals with the

trajectory design of a mission to this system.

Two different types of missions are considered: a rendezvous with asteroid 624 Hektor, and a flyby

tour of different objects. For both of them, MGA and MGA-1DSM models are utilized to optimize the

interplanetary trajectories to the Trojan system. These were adapted from GTOP’s code [1], in order to

increase their theoretical accuracy and assure quality of results. Additionally, an optimization algorithm

survey and tuning study are presented for the MGA implementation, as to fill the gaps found in the

literature.

For the asteroid flyby tour, a new optimization problem is developed and implemented in this report.

This allows for a more efficient optimization of the total flyby tour trajectory.

The designed trajectory to 624 Hektor followed a MGA path of Earth-Mars-Jupiter, resulting in a total

on-board ∆V of 2.4672 km/s. As for the asteroid flyby tour, the MGA route consisted of a Earth-Venus-

Mercury-Venus trajectory, leading to the observation of 6 total asteroids. The on-board ∆V for this result

was 2.3903 km/s, showing improvements over the designs found in Canalias et al. [2].

Both designed missions, and their respective improvements relatively to the literature, are considered

to be achievements of this report, and a direct consequence of the developed trajectory models.

Keywords: Trajectory optimization, Jovian Trojan asteroids, Asteroid flyby tour, Asteroid ren-

dezvous, Multiple gravity assists

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vi

Contents

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Nomenclature xiii

Glossary xv

Acronyms xviii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Similar missions to low-gravity bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Mission design 7

2.1 Trojan cloud characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 High thrust versus low thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Spacecraft properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Mission phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Interplanetary phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Science phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Spacecraft Trajectories 17

3.1 Lambert problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Gravity assists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Unpowered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Powered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Two body versus three body approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Trajectory Optimization 27

4.1 Optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Optimization solution families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Applicable techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

vii

4.2.1 Interplanetary phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.2 Science phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Interplanetary Phase: Implementation and Testing 35

5.1 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.1 MGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1.2 MGA-1DSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Optimization algorithm definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4.1 MGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.4.2 MGA-1DSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Interplanetary Phase: Results 47

6.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1.1 Planetary constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1.2 Ephemeris data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.1.3 Boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 Trajectory to asteroid tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Trajectory to asteroid rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Science Phase: Implementation and Testing 61

7.1 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.3 Optimization algorithm definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 Science Phase: Results 67

8.1 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.2 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.3 Asteroid tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.4 Return DSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

9 Conclusions and Recommendations 77

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

9.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 81

viii

List of Tables

2.1 Characteristics of NEAR-Shoemaker [10], Dawn [13] and Rosetta [11]. . . . . . . . . . . . 11

2.2 Top-level objectives for the optimization of the interplanetary phase of a mission to the

Trojan asteroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Top-level objectives for the optimization of the science phase of a mission to the Trojan

asteroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Computational times of 360 samples of a Lambert problem, solved with Gooding’s [21]

and Izzo’s [1] methods. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Maximum gravitational attraction of every main body in the Solar System, at the Sun-

Jupiter L4. [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1 MGA and MGA-1DSM verification problems’ ∆V specifications. [1] . . . . . . . . . . . . . 38

5.2 Results of the MGA and MGA-1DSM implementations of this thesis for GTOP’s problems,

and comparison to the known solutions. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Results after 100 runs of the selected algorithms for both MGA validation problems, and

comparison to the global minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 List of algorithmic and adaptive scheme variants available in PyGMO’s jDE algorithm. [34] 41

5.5 Best settings and population sizes for the DE algorithm applied to a MGA-1DSM formula-

tion. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1 Planetary constants used for every computation in the interplanetary transfer’s optimiza-

tion. [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Summary of the ephemeris data used, interpolation steps and positional errors attributed

to them, for the Interplanetary phase’s optimization. [23, 31] . . . . . . . . . . . . . . . . . 48

6.3 Boundary values introduced to the interplanetary transfer’s optimization algorithm, in or-

der to define the search space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.4 Notation utilized to denominate planetary sequences. . . . . . . . . . . . . . . . . . . . . 50

6.5 Approximate optimization times, per population, as a function of the number of gravity

assists in each MGA or MGA-1DSM trajectory. The reduction in average computation

time from 2 to 3 gravity assists is due to the change in ephemeris database mentioned in

the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.6 Best solutions derived from the Pareto Front in Fig. 6.3. . . . . . . . . . . . . . . . . . . . 54

ix

6.7 Arrival masses of the spacecraft to the Sun-Jupiter L4 point. Computed from the 7 inter-

planetary trajectory solutions of Table 6.6, and with the Soyuz and Delta IV launchers [37,

38]. The on-board propulsion system utilized was assumed to have a specific impulse of

330 seconds [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.8 Solutions analyzed for the interplanetary transfer to an asteroid rendezvous and their

characteristics. All of the presented trajectories are of the MGA type. . . . . . . . . . . . . 57

7.1 Results of the asteroid tour implementation for three dual Hohmann scenarios, with plan-

ets A, B and C circularly orbiting the Sun at RA, RB and RC , respectively. . . . . . . . . . 63

7.2 Results after 100 runs of the selected algorithms for the first dual Hohmann scenario,

applied to the asteroid tour implementation, and comparison to the global minimum. . . . 64

8.1 Maximum ∆V values (for the whole mission) allowed to be propagated in each of the

tree’s tiers’ solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.2 Approximate optimization time required to propagate a single tier’s solution. . . . . . . . . 70

8.3 Top results for tour sequences with 3 asteroids and their details. The total ∆V presented is

the sum of interplanetary correction with the in-swarm maneuvering. The duration shown

is measured from the first asteroid encounter to the last. Canalias’ [2] top trajectory with

3 asteroids also shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


the sum of interplanetary correction and the in-swarm maneuvering. The duration shown











8.7 Detailed breakdown of encounter epochs and ∆V ’s associated with the best trajectory

obtained for the whole mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.8 Observation times for each individual asteroid in the trajectory of Fig. 8.5, and for multiple

maximum distances to the objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

x

List of Figures

1.1 NEAR-Shoemaker’s orbital path to rendezvous with Eros. [10] . . . . . . . . . . . . . . . . 3

1.2 Rosetta’s orbital path to rendezvous with Comet 67P. [11, 12] . . . . . . . . . . . . . . . . 4

1.3 Dawn’s orbital path to rendezvous with Vesta and Ceres. [13] . . . . . . . . . . . . . . . . 5

2.1 Positions of the L4 and L5 Trojan asteroids, relative to the Sun, on July 31, 2016 [9]. All

coordinates are presented in the ecliptic J2000 frame. . . . . . . . . . . . . . . . . . . . . 8

2.2 Positions of the L4 Trojan asteroids, relative to the stability point, on July 31, 2016 [9]. All

components are with respect to a cylindrical frame, whose description is found in the main

body of text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Example of a low thrust transfer from Earth to a near Earth asteroid (TW229), taking 17

years [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Canalias top asteroid flyby tour trajectory of the Jovian Trojan system. All positions are

presented in an ecliptic J2000 frame. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Traditional Lambert problem representation, with P1 and P2 being the points of departure

and arrival, respectively. F represents the central body. [20] . . . . . . . . . . . . . . . . . 17

3.2 2-D representation of a flyby’s change of direction from the relative incoming to the escape

velocities. [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Gravitational perturbations caused by all planets and solar radiation pressure (with Rosetta

characteristics [11]) from 0 to 6 AU, along a line with all planets aligned. The maximum

accelerations correspond to the value at the border of each planet’s sphere of influence.

[23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Total ∆V of an EMJU trajectory, as a function of two variables. Obtained from the M. Sc.

thesis of Musegaas. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1 Values of n95% for the 18 algorithmic variants and 2 adaptive schemes of PyGMO’s jDE

software. See text for explanation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Results of n95% for a jDE optimization (under optimum settings) of the Cassini 1 problem

as a function of the population size. See text for explanation. . . . . . . . . . . . . . . . . 44

xi

5.3 Distribution of the average and minimum values of n95% for 200 runs of jDE (under opti-

mum settings) as a function of population size. The results are shown as multiples of the

best values obtained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.1 ∆V and travel time results for MGA transfer trajectories, in the case where an asteroid

tour will constitute the science phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 ∆V and travel time results for MGA-1DSM transfer trajectories, in the case where an

asteroid tour will constitute the science phase. . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3 Pareto Front of ∆V and Vrel (relative velocity to L4) for the interplanetary to the Trojan

asteroids, with the intention of having a final asteroid tour in the science phase. Each

marker label indicates its corresponding multiple gravity assist sequence. . . . . . . . . . 53

6.4 Final chosen orbit for an asteroid tour mission, departing from Earth on April 1, 2020. All

ephemeris are presented in an ecliptic J2000 frame. . . . . . . . . . . . . . . . . . . . . . 56

6.5 ∆V and travel time results for MGA transfer trajectories, in the case where a final aster-

oid rendezvous will be had in the science phase. The ’H’ in the end of the sequences

symbolizes the target asteroid 624 Hektor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.6 ∆V and travel time results for MGA-1DSM transfer trajectories, in the case where a final

asteroid rendezvous will be had in the science phase. The ’H’ in the end of the sequences

symbolizes the target asteroid 624 Hektor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.7 Final chosen orbit for a rendezvous with 624 Hektor, departing from Earth on 10/03/2029.

All ephemeris are presented in an ecliptic J2000 frame. . . . . . . . . . . . . . . . . . . . 60

8.1 Positions of the L4 Trojan asteroids, relative to the stability point, in July 31, 2016 [9]. All

coordinates are shown in the ecliptic J2000 reference frame. . . . . . . . . . . . . . . . . 67

8.2 Tree representation of the available combinatorial sequences of Trojan asteroids. The

numbers inside the shapes represent the number of the asteroid. Each tierN corresponds

to N asteroid sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.3 Eccentric anomaly values at the end of flyby tours with 2, 3, 4, 5 and 6 asteroids in the

sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.4 Velocity values of the spacecraft at the last asteroid encounter as a function of the number

of objects observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.5 Final designed orbit for the full mission, departing from Earth on April 1, 2020 and en-

countering 6 asteroids. All ephemeris are presented in an ecliptic J2000 frame. . . . . . . 75

xii

Nomenclature

Greek symbols

α Angular deviation of velocity in a gravity assist [rad]

β Gravity assist’s orbit’s plane orientation [rad]

∆V Impulsive velocity maneuver [m/s]

∆Vα Powered gravity assist’s velocity maneuver for angular correction [m/s]

∆Vp Powered gravity assist’s pericenter velocity maneuver [m/s]

η Fraction of interplanetary leg’s duration before applying a DSM

θ True anomaly [rad]

Ω Right ascension of the ascending node [rad]

ω Argument of pericenter [rad]

ω Inertia weight of Particle Swarm Optimization

Roman symbols

CR Crossover ratio of Differential Evolution

E Eccentric anomaly [rad]

e Orbital eccentricity

F Weighting factor of Differential Evolution

Isp Specific impulse [s]

J Cost function

M Mean anomaly [rad]

n95% Number of function evaluations required to achieve 95% certainty

NP Population size

p Decision vector

xiii

~R Heliocentric position vector [m]

~r Planetocentric position vector [m]

rp Pericenter radius [m]

rp Safe radius of a gravity assist [m]

R Planetary radius [m]

T Travel duration [s]

t Epoch

~V Inertial velocity vector [m/s]

~v Relative velocity vector [m/s]

V∞ Excess velocity from Earth [m/s]

X Cartesian coordinate X [m]

Y Cartesian coordinate Y [m]

Z Cartesian coordinate Z [m]

Subscripts

Sun

⊕ Earth

GA Gravity assist

in Entering the gravity assist’s planet’s sphere of influence

out Exiting the gravity assist’s planet’s sphere of influence

t Gravity assist’s planet

xiv

Glossary

Deep space maneuver

Impulsive maneuver performed outside the sphere of influence of any planet. 3

Dry mass

Mass of a spacecraft with empty propellant tanks. 11

Ecliptic J2000

Pseudo-inertial heliocentric right-hand frame centered on the Sun, with the X axis pointing towards

the vernal equinox, and the Z direction coincident with Earth’s orbital angular momentum The

directions are fixed to those orientations at 12:00 of January 1st, 2000. 8

Ephemeris

Position and velocity of an object at a particular point in time. 8

Eros

Near Earth asteroid targeted by NASA’s NEAR-Shoemaker mission. 2

Excess velocity

Spacecraft’s velocity at an infinite distance from a planet’s center. 14

Flyby tour

Tour mission where no rendezvous is made with the multiple targets. 12

Gravity assist

Act of utilizing a planet’s gravity to change the direction and/or magnitude of the spacecraft’s he-

liocentric velocity. 3

Heliocentric

Relatively to, or centered in, the Sun. 8

Jovian

Of Jupiter. 7

xv

Kernel

File provided by NAIF containing the necessary information to compute the SPICE ephemeris. 48

Kuiper belt

Belt of asteroids located beyond Neptune’s orbit. 2

Launcher

Rocket utilized to launch a spacecraft from the Earth’s surface into space. 11

Main asteroid belt

Belt of asteroids located between the orbits of Mars and Jupiter. 3

MGA

Optimization problem for multiple gravity assist trajectories with (un)powered flybys. 35

MGA-1DSM

Optimization problem for multiple gravity assist trajectories with deep space maneuvers between

each planet. 35

Near Earth asteroid

Asteroid whose rp around the Sun is less than 1.3 AU. 2

Pericenter

Point of minimum distance of an orbit to its central body. 18

Planetocentric

Relatively to, or centered in, a planet. 18

Protoplanet

A large body of matter, orbiting a star, that is thought to be developing into a planet. 3

Rendezvous

A maneuver performed by a spacecraft to insert itself into its target’s orbit. 3

Rendezvous tour

Type of tour mission where rendezvous are made with every target. 13

Solar radiation pressure

Force caused by radiation pressure induced by the Sun. 23

xvi

Sphere of influence

Planetocentric sphere where the planet’s gravitational attraction is significantly larger than the

stellar system’s central star. 18

Subsystem

Component of a spacecraft in charge of a particular task.. 11

Tour mission

Mission whose goal is to observe multiple sequential scientific targets in a short amount of time,

relatively to its full duration. 4

Trojan asteroids

Asteroids located in the vicinity of the L4 and L5 points of the Sun-Jupiter three body system. 1

Wet mass

Mass of a spacecraft including propellant. 11

xvii

Acronyms

ABC

Artificial Bee Colony. 30

AU

Astronomical Unit. 3

DE

Differential Evolution. 30

DSM

Deep Space Maneuver. 15

ESA

European Space Agency. 3

ESOC

European Space Operations Centre. 38

GA

Gravity Assist. 54

GTOP

Global Trajectory Optimization Problems. 19

IAU

International Astronomical Union. 2

jDE

Self adaptive variant of Differential Evolution. 31

JPL

Jet Propulsion Laboratory. 34

xviii

mDE

Self adaptive variant of Differential Evolution. 31

NAIF

Navigation and Ancillary Information Facility. 48

NASA

National Aeronautics and Space Administration. 2

NEAR

Near Earth Asteroid Rendezvous. 2

NLP

Non-Linear Programming. 28

PSO

Particle Swarm Optimization. 30

PyGMO

Python Parallel Global Multiobjective Optimizer. 33

PyKEP

Python Keplerian Toolbox. 33

TU Delft

Delft University of Technology. 39

xix

xx

Chapter 1

Introduction

This work consists of the design of a mission’s trajectory to the Trojan asteroids, which are a collection

of hundreds of thousands of objects in the Sun-Jupiter’s L4 and L5 points. This means that, the complete

trajectory of a spacecraft is defined from the point that it leaves the Earth’s system to the moment its

scientific goal has been achieved.

1.1 Motivation

The Trojan asteroids’ main appeal is that, due to the lack of information about them, there are still

multiple models for their creation, all with interesting consequences. Currently, there are two main

models for the formation/creation of the Jovian Trojan system [3]:

1. Main Trojan asteroids formed from the solar nebula, and were later captured into the L4 and L5

points by Jupiter, thus stabilizing their orbits;

2. Most asteroids formed in other parts of the Solar System, or even outside of it. Due to their orbits,

Jupiter captured them into the stable L4 and L5.

Both of the referred models have implications on the scientific importance of the Trojan asteroids. In

the first model, the asteroids were formed through the same processes as every other object in the Solar

System, therefore a mission to them could reveal more about its formation. Furthermore, since it implies

that the Trojan asteroids formed near Jupiter’s orbit, there is a possibility that beneath their surface lies

a layer of solid water-ice [3]. In an era where the search of life dominates many space missions, this is

extremely valuable from a scientific point of view.

In the second model, the Trojan asteroids are particularly interesting because they provide the pos-

sibility of studying bodies that formed outside the Solar System, in a location closer to Earth. Their

observation could give insight to the composition of bodies originated in those regions, as well to the

formation processes that occurred there.

The Trojan swarm is constantly disrupted by internal collisions [4], which not only leads to the creation

of debris, but also to the possible ejection of material from beneath the asteroids’ surfaces [5]. Such

1

phenomena can not be observed from ground measurements, since the resolution required is too high,

thus requiring close monitoring. However, it has been possible to detect traces of complex organic

molecules in the surface of some Trojan asteroids from ground infrared measurements, suggesting the

presence of the building blocks of life [6]. Although this was possible, not enough information about the

surface composition of the asteroids could be derived, thus deeming the detection inconclusive.

Besides all of this, the Trojan swarm is curious due to its spectral variety, unlike what is commonly

found in other asteroid clusters throughout the Solar System. Two major groups of asteroid families have

been detected in the Sun-Jupiter’s L4 and L5 points [7]:

1. D-type – also called the ”red” group, due to its dark burgundy hue,

2. P and C-types – known as the ”blue” group, because of their grayish blue color.

Previous observations from NASA’s Infrared Telescope Facility have indicated that the Trojan L4

cluster is also composed by primordial bodies, whose composition might be different from that of icy

Kuiper belt objects or main belt asteroids [7]. Such variety in composition and origin is another reason

why these objects are so interesting, and why any mission whose goal is to observe them can yield a

large amount of scientific return.

The importance of providing detailed spectral information of the asteroid ejections and surfaces is

then clear, in order to determine their characteristics and constrain the Solar System’s formation pro-

cesses. In terms of life detection in the universe, the Trojan asteroids are one of the most varied and

closest objects of interest, making them one of the top observational priorities.

In September 2015, NASA awarded 3 M$ to develop concept design studies and analyses for a

mission to the Jovian Trojan asteroids, called LUCY [8]. In case the mission’s conceptual design is

accepted, it will be developed with a 500 M$ budget, starting in September 2016 [8]. The existence of

such a planned mission serves as validation that the Trojan swarm is of high scientific importance, and

should be prioritized in the years to come, in terms of space exploration.

Ephemeris data of the Trojan asteroids indicates that there are nearly twice as much objects in the

L4 than in L5. More specifically, International Astronomical Union (IAU) provides the positions of 4087

Trojans in L4, and only 2201 in L5 [9]. With this in consideration, together with the broader composition

of asteroids in L4, the work developed will focus on that particular cluster. Throughout this thesis report,

any mentions of Trojan asteroids will always be referring to the ones located in L4, unless the L5 cluster

is specifically mentioned as well.

1.2 Similar missions to low-gravity bodies

One of the earliest missions to ever target an asteroid was NASA’s NEAR-Shoemaker, who managed

to be the first ever to orbit and touch down on an asteroid [10]. It was launched in February 1996 with

the intention of targeting Eros, a Near Earth asteroid. The spacecraft was originally designed to simply

orbit this object, and carried several spectrometers, coupled with magnetometers, range-finders and

cameras [10]. Towards the end of its operational lifetime, an attempt at a landing on Eros was performed

2

Figure 1.1: NEAR-Shoemaker’s orbital path to rendezvous with Eros. [10]

to obtain higher resolution pictures and retrieve more data, which ended up being successful. In terms

of mission planning, the full trajectory is shown in Fig. 1.1 [10], where it is visible that several deep

space maneuvers were utilized, as well as gravity assists, which will be explained in Chapter 3. In total,

NEAR-Shoemaker worked for five years.

Another asteroid rendezvous mission that occurred more recently was ESA’s Rosetta, which was

launched in 2004 with the objective of chasing and orbiting Comet 67P. This comet’s orbit spanned

from a perihelion of 1.24 AU (close to the Earth) to an aphelion of 5.68 AU (close to Jupiter) [11].

Naturally, this mission had tougher survivability challenges, relatively to NEAR-Shoemaker, mainly due

to the inefficiency of solar panels at distances from the Sun equal to or larger than Jupiter’s. In a similar

fashion, these are the problems that a mission to the Trojan swarm faces, since they co-orbit with this

planet. Rosetta was composed by an orbiter and a lander, carrying several spectrometers, among other

instruments [11]. On September 30, 2016, the mission ended with a controlled impact between the

orbiter and asteroid [11]. Final observations were performed during the approach before the impact, and

transferred to Earth [11].

The orbital path for Rosetta, shown in Fig. 1.2 [12], was considerably more complicated than that of

NEAR-Shoemaker’s. In total, Rosetta has been operating for around 12 years [11].

A similar mission is NASA’s Dawn spacecraft, whose goal was to target both Vesta and Ceres. These

objects, considered protoplanets at the time, reside in the main asteroid belt, between Mars and Jupiter.

Although they are not officially considered asteroids, they reside in a swarm of objects like these, which

is close to what occurs in the Trojan swarm. Dawn was launched in 2007, utilizing a Martian gravity

3

Figure 1.2: Rosetta’s orbital path to rendezvous with Comet 67P. [11, 12]

assist to then rendezvous and orbit Vesta [13]. Later, it departed from Vesta to rendezvous with Ceres,

being in orbit around it until this day. Its orbital path is defined in Fig. 1.3 [13]. In total, this spacecraft

has been operating for nine years.

Despite having rendezvous with its targets, like NEAR-Shoemaker and Rosetta, Dawn is curious

because it has more than one scientific target (Vesta and Ceres), thus making it an observational tour

mission. From the point of view of a mission to the Trojan asteroids this is interesting, since more than

one object in the swarm may be targeted (due to the wide variety of asteroids in the cluster).

All of these missions can bring something new to this work’s design. First of all, it is noticeable that, in

terms of interplanetary transfers, it is traditional to utilize one or multiple gravity assists, thus reducing the

propellant required. Secondly, it can be seen that the common approach is to target specific asteroids

of interest and rendezvous with them, which can be done for one or several objects. Finally, from Dawn

it is possible to deduce that operating a spacecraft inside an asteroid swarm is feasible, which is good

news since it allows for closer observations inside the Trojan cluster.

4

Figure 1.3: Dawn’s orbital path to rendezvous with Vesta and Ceres. [13]

5

6

Chapter 2

Mission design

In this chapter, the mission developed in this work will be defined. Initially, the asteroid cloud will be

assessed, so that the target system is defined in a more practical way, and analyzed in the next chapters.

Then, a debate between high thrust and low thrust will be presented, together with the definition of limits

to the spacecraft’s properties. Following that, the mission will be structured (i.e. divided in phases) and

individual objectives will be defined, deriving from the general objective of the thesis.

2.1 Trojan cloud characteristics

The Jovian Trojan asteroids are a collection of hundreds of thousands of objects, all co-orbiting the

Sun with Jupiter, 60 ahead and behind it, in the Sun-Jupiter’s L4 and L5 points, respectively. Such a

situation is possible due to gravitational forces of the Sun-Jupiter system, which create those stability

points [14].

In terms of size, it is known that this collection of objects (L4 and L5) is comprised of approximately

1.6 × 105 asteroids with radii between one and hundreds of kilometers [4]. However, this system is con-

stantly disrupted by internal collisions [5], which leads to the belief that this system is widely populated

by collisional debris as well. Furthermore, such collisions also contribute to the ejection of material from

the swarm, at an estimated rate of 10−3 objects/year [4]. Naturally, such an attrition rate is insignificant

when compared to the overall number of asteroids estimated to be present in the Trojan system, which

means that it can be disregarded, from an engineering point of view, for the remaining of this work.

Consequently, by not having any attrition rate considered, instability of the system ceases to exist, and

thus collisions can be neglected as well.

Without collisions, considering that the Lagrange points are stable [14], it is safe to assume that

the Sun and Jupiter’s long term influence on each asteroid has stabilized their orbits, otherwise no

stability would exist. Secular perturbations (i.e. non-periodical) are then compensated by the Sun-

Jupiter influence, eliminating them from the equation. This can be observed in the ephemeris data from

June and August [9], which shows no alteration in orbital parameters. Non-secular perturbations are

commonly significantly smaller than any secular perturbation in the Solar System [15], thus these can

7

-8 -6 -4 -2 0 2 4 6X [AU]

-8

-6

-4

-2

0

2

4

6

8

Y [A

U]

Trojan asteroidsJupiterL4L5Sun

Figure 2.1: Positions of the L4 and L5 Trojan asteroids, relative to the Sun, on July 31, 2016 [9]. Allcoordinates are presented in the ecliptic J2000 frame.

be neglected for this work. Therefore, with no perturbations, the orbits of each individual asteroid can

be assumed constant for this work, relatively to the latest ephemeris data. A similar approximation was

performed in Canalias et al. [2], for a mission to the Trojan asteroids.

Using information from the IAU and its ephemeris data for the Trojan asteroids, taken from the latest

batch on the 20th of June, 2016 [9], it is possible to observe the size of the L4 swarm on July 31, 2016,

like it is shown in Fig. 2.1. The reference frame used was the ecliptic J2000, centered in the Sun.

This already indicates that the Trojan swarm is large, but a more correct assessment can be done

with Fig. 2.2, in a heliocentric cylindrical frame, with r being the radial distance to the Sun (projected

onto Jupiter’s orbital plane), θ the angular distance to L4 (also in Jupiter’s orbital plane) and z the normal

distance to said plane.

It can be seen that the Trojan swarm is found between 4.25 and 6.5 AU of the Sun, projected in

Jupiter’s orbital plane. Furthermore, in terms of angular spread, these objects span from -30 to 50 of the

L4 point. This angular spread causes arc lengths of around 7.53 AU. Finally, the swarm’s perpendicular

spread with respect to Jupiter’s orbital plane goes from -3 to 4 AU. Therefore, cylindrically, the Trojan

asteroids’ swarm has overall dimensions of 2.25×7.53×7 AU, if all the objects in Fig. 2.2 are taken into

account. Neglecting the more distant objects, and focusing on the densest part of the cloud, it can be

considered that the Trojan swarm has dimensions of 0.7×4.7×3 AU, approximately.

It should be noted that the ephemeris data analyzed only contains information regarding 4087 Trojan

asteroids, since the observational data only allows those to be located. Naturally, many more objects

are present in the swarm but, due to their size, there is no ephemeris information, so the data presented

must be interpreted as a statistical sample of the real Trojan cloud. From that sample, it is possible to

draw conclusions on the size of the swarm.

8

-30 -20 -10 0 10 20 30 40 50θ [deg]

4

4.5

5

5.5

6

6.5

r [A

U]

Trojan asteroidsL

4

(a) θr plane

-30 -20 -10 0 10 20 30 40 50θ [deg]

-3

-2

-1

0

1

2

3

4

z [A

U]

Trojan asteroidsL

4

(b) θz plane.

Figure 2.2: Positions of the L4 Trojan asteroids, relative to the stability point, on July 31, 2016 [9]. Allcomponents are with respect to a cylindrical frame, whose description is found in the main body of text.

2.2 High thrust versus low thrust

In any space mission there is a possibility of utilizing high or low thrust propulsion systems. Both

options are fundamentally different and have their consequences in the models applicable for solving

the problem of a mission to the Trojan asteroids. Therefore, the discussion on which propulsion type to

use has to take place.

As its name indicates, the high thrust alternatives refer to systems that provide large impulsive forces,

like bipropellant and other chemical propulsion technologies. Consequently, the thrust periods are small,

in the order of minutes, and can be approximated to instantaneous if the satellite’s travel time or orbital

periods are significantly larger. This is commonly the case with interplanetary travel.

On the other hand, the low thrust alternatives can not be approximated to be instantaneous, and

9

generally consist of ion or electrical propulsion. These are commonly functioning during the whole

operational time of a spacecraft, so that the cumulative velocity change is enough to obtain the desired

trajectory.

High thrust systems are characterized for having specific impulses significantly smaller than those of

low thrust alternatives (at least 3 times smaller for the case of electrostatic propulsion [16]), which ba-

sically means that they are fuel inefficient. This inefficiency translates into larger portions of the space-

craft’s wet mass being comprised of propellant, which in turn reduces the mass available for scientific

payloads.

As for low thrust, despite their significantly higher fuel efficiency, they lead to trajectory solutions that

take more time, since the engines need to operate for longer periods to provide the full ∆V requirement

[16]. For example, on a trip from Earth to Mars, a spacecraft can take several years to complete such

a transfer with low propulsion, taking multiple revolutions around the Sun, while a high thrust option

can perform the same trip in less than a year, with half an orbital period [15]. Another example is the

trajectory presented in Fig. 2.3 from Earth to a Near Earth asteroid, obtained by Seabra [17]. With the

intent of providing more perspective to the reader, regarding the expected travel times of high and low

thrust systems to the Trojan swarm, it is known that Rosetta targeted an asteroid closer to Jupiter and

managed to take 10 years of travel time using high propulsion technology.

Figure 2.3: Example of a low thrust transfer from Earth to a near Earth asteroid (TW229), taking 17years [17].

10

The extra time spent traveling with low propulsion has its risk, since, every day that the spacecraft is

traveling, subsystem failures are more likely. This increase in risk, which is not expected to be significant

relative to that of a high thrust system, can be worth taking depending on the mission’s objective, so a

trade-off must be made in the early design stages. Additionally, the increase in risk can be mitigated by

protecting the spacecraft’s subsystems accordingly (e.g. radiation shielding), at the expense of higher

development costs, which are also not expected to be significantly higher.

An important factor to also consider is the fact that high thrust technology has been implemented

since the start of the space industry, while low thrust only started being demonstrated in 1964 [18] as

part of the main propulsion system of a spacecraft. Furthermore, satellite manufacturers began the (now

popular) commercialization of low thrust engines around 2010 [19]. In other words, high thrust is a more

mature technology than low thrust.

With the description of each system, it is clear that both can not be compared for a general case, as

their superiority depends on the situation (e.g. mission’s goal or budget). In the case of a mission to the

Trojan asteroids, none of these propulsion systems’ advantages (or disadvantages) are enough to single

out one of them as better suited. As such, a choice between one of them has to be made, since it is

not possible to complete this thesis, within its time limit, if both options are investigated. The propulsion

system chosen for this work is high thrust, as the author sees more of an advantage in this technology’s

low travel times and higher maturity, which can ultimately lead to a more attractive mission cost-wise.

2.3 Spacecraft properties

With the main type of propulsion defined (i.e. high thrust), and with information about similar mis-

sions, it is possible to deduce physical constraints that can be applied to the spacecraft in the case of

a mission to the Trojan asteroids. Those constraints will then be utilized throughout the work to narrow

down the optimal trajectories.

Of all the missions mentioned in Section 1.2, NEAR-Shoemaker and Rosetta utilized high thrust

propulsion systems, while Dawn used ion engines, which are low thrust. Therefore, it is possible to

establish direct comparisons with at least Rosetta and NEAR-Shoemaker, although their top-level ob-

servational goals differed significantly. Table 2.1 shows the wet mass and dry mass of each spacecraft.

Mission NEAR-Shoemaker Dawn Rosetta

Wet mass [kg] 800 1240 3000Dry mass [kg] 487 747 1330Propellant-to-wet mass ratio 0.39 0.40 0.56Isp [s] N/A 3100 292Launcher Delta II Delta II 7925H Ariane 5G+ V-158

Table 2.1: Characteristics of NEAR-Shoemaker [10], Dawn [13] and Rosetta [11].

It can be seen immediately that the launch (or wet) masses of the missions are very different, which

has to do with the varying observational natures, targets and the diversity in payload. This can also

be related to the chosen launcher at the time, which may have constrained the spacecraft design so

11

that a certain maximum wet mass was not surpassed. It should also be known that a large amount of

variance is expected in terms of launch mass, for any mission, because there is an abundance of design

choices and objectives directly impacting that. Due to the diversity amongst missions, it is not possible

to generalize and draw reasonable conclusions on launch mass for the case of this work. As such, this

quantity will not be used to limit the design of the mission.

In terms of dry mass, the same can be observed, thus complicating the task of setting boundary

values on the spacecraft for a mission to the Trojan swarm. However, a limitation on dry mass was

already unreasonable, largely because the observational natures of the three considered missions were

very different, which has an impact on the payload types. Furthermore, throughout the years, technology

has been developing so there are solutions today that were not applicable at the time.

Instead, a more reliable comparison can be made between the propellant-to-wet mass ratios, which

indicate how much of the spacecraft is comprised of propellant. These values are 0.39, 0.40 and 0.56,

for NEAR-Shoemaker, Dawn and Rosetta, respectively. These figures are directly related to the ∆V

associated with the missions, because they are a result of the classical Tsiolkovsky’s rocket equation.

So, it makes sense that the ratios for NEAR-Shoemaker and Dawn are similar, given that their objectives

are at approximately the same distance to the Sun (both between Mars and Jupiter). Additionally, con-

sidering that Rosetta traveled further, closer to Jupiter’s orbit, it makes sense that its propellant-to-wet

mass ratio is larger.

Naturally, this rule cannot be generally applied to every mission, because the interplanetary routes

can differ significantly, and the maneuvers associated with them as well. However, in the case of the

three considered missions, the paths are similar, and comparable with what is expected in a mission to

the Trojan swarm.

With that in mind, one can establish a direct relation between their propellant-to-wet mass ratios,

and the one of this work’s mission. The better candidate for that is Rosetta, since it performed a ren-

dezvous with its target asteroid closer to Jupiter’s orbit. Therefore, the assumption will be made that the

spacecraft’s propellant-to-wet mass ratio can not surpass 0.56 plus a margin of 15% (1.5× the margins

commonly used in SMAD [16] for all designs).

Using Tsiolkovsky’s equation, and a high thrust propulsion system with an high-end Isp of 330 sec-

onds [16] (high specific impulse is equivalent to lower consumption values), it is possible to deduce that

the spacecraft can not provide more than 3.34 km/s of ∆V. In order to keep the numbers round, the limit

utilized in this report will be 3.5 km/s, effectively increasing the propellant-to-wet mass ratio’s margin to

18%, instead of 15%.

Due to the specific focus of this thesis on trajectory design, no more constraints can be introduced,

since that would involve subsystem and payload definition, which is not related to this work.

2.4 Mission phases

Similarly to what was shown in Section 1.2, the final mission objective can be performing a single

rendezvous with an asteroid. Alternatively, instead of one asteroid rendezvous, a flyby tour of several

12

objects inside the Trojan swarm could be chosen, like it has been suggested by Canalias et al. [2]. This

consists of a spacecraft entering the Trojan swarm and sequentially passing by a number of objects,

altering its trajectory along the way to target different ones, until the cluster is exited.

Additionally, like it was suggested by Stuart et al. [7], there is a third option, that consists of perform-

ing an asteroid rendezvous tour. In this case, instead of passing by multiple asteroids, the spacecraft

rendezvous with several targets inside the swarm. This means that the satellite would jump between

different asteroid orbits, thus maximizing the observational time of each of them.

These three scenarios result in extremely different final missions, both in terms of trajectory and, in

a real design scenario, payload types. A single rendezvous mission would allow for more repeated

observations of a single asteroid, at the expense of expected higher propellant costs. Besides that, this

scenario would not allow the investigation of the diversity within the Trojan swarm.

In an asteroid flyby tour, several objects would be observed, for short periods of time, thus allowing

the study of different types of asteroids. The propellant costs of this mission would be reduced compared

with the single rendezvous case [2]. An example of this type of trajectory can be seen in Fig. 2.4,

depicting Canalias’ final designed tour.

Figure 2.4: Canalias top asteroid flyby tour trajectory of the Jovian Trojan system. All positions arepresented in an ecliptic J2000 frame. [2]

Regarding the asteroid rendezvous tour, it would be possible to study a large amount of objects,

13

and the observational time of each of them would increase significantly relatively to the asteroid flyby

tour. This is due to the spacecraft co-orbiting the Sun with each asteroid at the time it is observing them.

Naturally, as evidenced by Stuart et al. [7], this type of mission would require very high ∆V’s (minimum

of 7.5 km/s), which surpasses the limit set in Section 2.3, making it impossible to perform with high thrust

systems. As such, this option can be immediately disregarded.

However, in the case of the single rendezvous and asteroid flyby tour scenarios, there is insufficient

information to discard any of them. Consequently, these two cases will be considered:

1. Rendezvous trajectory with a specific Trojan asteroid target (to be determined in this report);

2. Asteroid flyby tour of the Trojan system.

Each of those cases has multiple distinct optimization problems within them. For example, in an

asteroid flyby tour, the spacecraft would need to reach a particular asteroid and, from there, target the

right sequence of objects, at the right time. In order to target that first asteroid it is common practice to

perform multiple gravity assists, which means that the satellite could follow the path Earth-Venus-Earth-

Jupiter-Target, or any other combination of planets. Then, the maximum possible sequence of objects

would need to be determined. With a total of 8 Solar System planets and 4087 Trojan asteroids (with

known ephemeris), this task can prove to be a combinatorial nightmare. Especially if the starting asteroid

is unknown, or if the planetary sequence is undetermined from the start. From a practical perspective, it

is then necessary to divide the mission into two general phases, applicable for both the single asteroid

rendezvous and the flyby tour scenarios:

1. Interplanetary phase – path from Earth to the observational targets,

2. Science phase – segment of the mission in which observations are performed.

2.4.1 Interplanetary phase

The goals for this phases’ optimization are presented in Table 2.2. For the asteroid rendezvous it

is simple to understand why the total ∆V must be minimized. Due to the lack of constraints on the

launch mass of the spacecraft, it is not possible to narrow down feasible launcher options. Therefore,

the ∆V to be minimized must also include the Earth’s excess velocity, which is provided by the launcher

during hyperbolic injection. This helps the final optimum trajectory to not only minimize the on-board ∆V

(performed by the spacecraft), but also the launcher’s ∆V, which in turn maximizes launch mass. This

process is expected to compute trajectories that deliver the maximum mass to the Trojan asteroids.

For the asteroid tour case, the minimization of the final relative velocity leads to an increase in total

observation time. This assures a better arrival trajectory to the swarm, and improved results in the

science phase, since the spacecraft is traveling slower, relatively to the average asteroid of the system.

Naturally, the asteroid tour case is a multi-objective optimization problem, and the final solution must

be a trade-off of the quantities mentioned here.

14

Asteroid rendezvous Asteroid flyby tour

Minimize total ∆V Minimize total ∆V— Minimize final relative velocity to L4

Table 2.2: Top-level objectives for the optimization of the interplanetary phase of a mission to the Trojanasteroids.

2.4.2 Science phase

For the case of the science phase, the two problems are completely different. On one hand, the

asteroid rendezvous scenario does not require any extra maneuvering, assuming that the spacecraft is

perfectly following the target’s orbit. A study of the collisional risk of other asteroids could be performed,

or even an analysis of how many asteroids pass close enough to the target so that the spacecraft can

still perform some measurements, but none of these scenarios require enough ∆V to be included in the

final budget. Therefore, these will not be considered in the optimization algorithm, as parts of the cost

function. However, these additional maneuvers can be taken into account as 10% of the interplanetary

phase’s on-board ∆V, in order to keep the design conservatively realistic.

On the other hand, the asteroid tour scenario requires a completely different optimization process.

It consists of two sections: maneuvering inside and outside the Trojan swarm. According to what was

suggested by Canalias et al. [2], the spacecraft would perform maneuvers inside the swarm, after

each asteroid passage, so that it would target different objects. Outside of the Trojan swarm, after the

spacecraft had already performed all the flybys that it could, the goal would be to perform a DSM so

that it would return to the system a second time and do a second observational round. Depending on

the orbit at which the satellite would exit the system after the first asteroid tour, this could be or not be

possible.

Using this structure, the goals defined for these types of scientific phases are the ones presented in

Table 2.3.

Asteroid rendezvous Asteroid flyby tour

Inside swarm Outside swarm

— Minimize maneuvering ∆V Minimize DSM ∆V— Maximize number of asteroids flown by —

Table 2.3: Top-level objectives for the optimization of the science phase of a mission to the Trojanasteroids.

Again, for the in-swarm optimization, there are multiple objectives, so any solution will have to go

through a trade-off process.

In order to patch the interplanetary and science trajectories, the initial conditions of this phase’s

optimization will be determined by the optimum interplanetary path. Since said path will be optimized to

target the L4 point, it is necessary to correct the last interplanetary leg in order to target each individual

asteroid and search for flyby tour sequences. Thus, with ephemeris data for 4087 L4 Trojan asteroids

[9], the final interplanetary leg will lead to the same amount of initial conditions in the science phase.

15

16

Chapter 3

Spacecraft Trajectories

Before discussing the several optimization techniques that exist, an introduction to astrodynamics’

concepts used in this work needs to be performed. In this chapter, a description of the equations

involved in the Lambert problem, and on a powered and unpowered gravity assist will be done, followed

by a choice between the two and three body approximations.

3.1 Lambert problem

The Lambert problem, or targeter, consists of finding the unperturbed orbit that is able to connect

two particular points, in a specified travel time. This type of targeter is only applicable to central body

formulations. Its traditional representation can be seen in Fig. 3.1, where P1 is the point of departure, at

epoch t1, and P2 is the point of arrival, at epoch t2. Naturally, the travel time is t2 − t1.

Figure 3.1: Traditional Lambert problem representation, with P1 and P2 being the points of departureand arrival, respectively. F represents the central body. [20]

17

For each pair of points, P1 and P2, there are always two trajectory solutions: a short (angular distance

θ) and a long one (angular distance 2π − θ), thus resulting in opposite directions of motion. However,

there are no closed form solutions, so iterative methods must be utilized for that purpose. [1, 21]

The main key to solving this problem is to find the variables that satisfy the time of flight equation.

The iterative methods and input parameters of the problem depend largely on the solver chosen, of

which there are many. The 2 most common Lambert solvers were developed by Gooding [21] and Izzo

[1]. Both methods are very similar, with the main difference that Izzo [1] substitutes a variable into the

time of flight equation, that is proportional to t2 − t1. The purpose of that substitution is to reduce the

number of iterations needed to solve the problem. Furthermore, Izzo’s algorithm [1] also utilizes the

secant method to avoid time consuming derivative evaluations.

In order to preserve space in this report, and given that the Lambert problem is not the focus of this

work, the algorithms and equations for Gooding [21] and Izzo [1] can be found in the bibliography, where

the reader is referred to.

In the work of Musegaas [20], 360 samples of a Lambert problem were solved using Gooding’s and

Izzo’s algorithms. The computational times were recorded, and are here shown in Table 3.1.

Method Maximum [µs] RMS [µs] σ [µs]

Gooding 18 11.35 1.40Izzo 35 6.48 1.09

Table 3.1: Computational times of 360 samples of a Lambert problem, solved with Gooding’s [21] andIzzo’s [1] methods. [20]

It can be seen that, although the maximum computational time experienced in Izzo’s formulation was

larger, the average computational time was approximately 57% of Gooding’s results. Due to that, and

the lower standard deviation of Izzo’s algorithm, this solver will be utilized throughout this work, for any

Lambert problem.

3.2 Gravity assists

When a spacecraft enters the sphere of influence of a planet, with heliocentric inertial velocity ~Vin,

its relative velocity to the target is given by:

~vin = ~Vin − ~Vt, (3.1)

where ~Vt represents the heliocentric inertial velocity of the planet. If said relative velocity is not zero,

then it is known that the planetocentric orbit is hyperbolic, which means that if the spacecraft continues

its trajectory, will pass close to the planet in its pericenter, and escape it with a relative velocity that has

a different direction, but the same magnitude.

The heliocentric inertial velocity of the spacecraft when escaping the planetary system is given by:

~Vout = ~vout + ~Vt. (3.2)

18

The norm of this inertial velocity can be larger than the norm of ~Vin. In practice, performing a gravity

assist (or flyby) means that an interplanetary ∆V is applied. This can be done with or without the use of

propellant, in the case of powered or unpowered gravity assists, respectively.

In general, the spheres of influence of each planet can be considered as significantly smaller than

the Sun’s sphere of influence, which allows to approximate them as infinitesimal points [15]. Such

approximation, leads to the assumption that the flybys occur instantaneously. This approach, commonly

called of patched conics, is something commonly used, even in ESA’s GTOP database, containing

optimization problems for spacecraft trajectories with multiple gravity assists [1].

3.2.1 Unpowered

In the case of an unpowered gravity assist, there is no ∆V applied in the planetocentric orbit, which

means the norm of the relative excess velocity ~vout is the same as for ~vin, but its direction will be

changed. This is shown in Fig. 3.2 [15], with a 2-D representation, where V∞t is equivalent to the norm

of ~vin and ~vout.

Figure 3.2: 2-D representation of a flyby’s change of direction from the relative incoming to the escapevelocities. [14]

With the incoming relative velocity ~vin known, as well as the plane orientation β, together with the

19

pericenter radius rp, it is possible to compute the relative excess velocity as [22]:

e = 1 +rpµt~vin · ~vin,

δ = 2 arcsin (1/e) ,

~i =~vin‖~vin‖

, ~j =~i× ~Vt

‖~i× ~Vt‖, ~k =~i×~j,

~vout = ‖~vin‖(

cos δ~i+ cosβ sin δ~j + sinβ sin δ~k).

(3.3)

3.2.2 Powered

When it comes to a powered gravity assist, ∆V maneuvers are performed during the planetocentric

hyperbolic orbit. These are done to match the departing velocity with the desired direction and intensity.

With the incoming relative velocity ~vin and the desired excess velocity ~vout known, it is possible to

compute the necessary angular change, α. From there, the required pericenter radius, rp, for a gravity

assist ∆V can be computed from [22]:

α = arcsinain

ain + rp+ arcsin

aoutaout + rp

,

ain =µt

~vin · ~vin,

aout =µt

~vout · ~vout,

(3.4)

where µt represents the central planet’s gravitational parameter.

Since Eq. (3.4) is not analytically invertible, a simple Newton-Raphson scheme can be used to ob-

tain the pericenter radius from the necessary incoming and departing velocities. The scheme can be

described by:

F(rip) = arcsinain

ain + rip+ arcsin

aoutaout + rip

− α,

ri+1p = rip −

F(rip)

F’(rip).

(3.5)

This iteration method should converge to the true solution (within < 1 km) in less than 30 iterations, for

any gravity assist in the Solar System [1]. If there are no constraints on the gravity assist’s pericenter

radius, then the necessary impulse at that point, ∆Vp, for the powered flyby, is given by [22]:

∆Vp (~vin, ~vout, rp) =

∣∣∣∣∣√~vin · ~vin +

2µtrp−

√~vout · ~vout +

2µtrp

∣∣∣∣∣ . (3.6)

with rp being the result of iterations from Eq. (3.5). However, generally there are safe radii for gravity

assists associated with each of Solar System’s bodies, due to their atmosphere, satellites or radiation

environment. In those situations, the gravity assist’s pericenter radius must be equal to or higher than

the planet’s safe radius rp. If the pericenter distance obtained from Eq. (3.5) is lower than rp, then ∆Vp

will have to be applied at the limit height, leading to the correct excess velocity value, but to the wrong

20

direction. Therefore, an extra ∆Vα will have to be applied to correct it. This is formally described as:

∆V =

∆Vp, if rp ≥ rp

∆Vp + ∆Vα, if rp < rp

, (3.7)

where rp is the pericenter radius resulting from the Newton-Raphson scheme of Eq. (3.5). The extra

∆Vα can be done either when the satellite enters the planet’s sphere of influence, or when it is escaping

the system.

Entering the sphere of influence

When performed as soon as the satellite enters the sphere of influence, the required ∆Vα is done

in the velocity direction, as to alter the instantaneous vector so that the remaining impulse can be per-

formed at rp. The maneuver’s ∆Vα is computed as shown in [14, 15]:

∆Vα =∣∣∣√~vin · ~vin − vnewin

∣∣∣ ,vnewin =

√µtanewin

,

anewin =sin f(α)

1− sin f(α)rp,

f(α) = α− arcsinaout

aout + rp.

(3.8)

Following the notation introduced in Eq. (3.6), the pericenter ∆Vp will be:

∆Vp = ∆Vp

(vnewin

~vin√~vin · ~vin

, ~vout, rp

). (3.9)

Exiting the sphere of influence

In case the extra correctional maneuver is performed at the point of escape from the planet’s sphere

of influence, then a simple direction change is required. This is because the previously applied pericenter

maneuver already matched the escape velocity’s norm with the desired value.

Therefore, the ∆Vα can be computed as [14, 15]:

∆Vα = 2√~vout · ~vout sin

(α− αmax

2

), (3.10)

where αmax corresponds to the value of α that is obtained with Eq. (3.4) and the pericenter radius set to

rp. Naturally, the pericenter maneuver can be described as:

∆Vp = ∆Vp (~vin, ~vout, rp) . (3.11)

21

3.3 Two body versus three body approximations

In the particular case of a mission to the Trojan asteroids, there are two dynamic models that can be

argued for, since they have their individual advantages and disadvantages. The choice of any of these

severely impacts the optimization methods and techniques that can and will be used from this point on,

so it is very important to assess which one is better. The three main criteria utilized to decide on the

best option are:

1. Accuracy - How precise are the calculations of the model?

2. Computational effort - How much time does it take to obtain a solution?

3. Versatility - How easy is it to study different variants of the same problem?

Accuracy

When the accuracy of a model is analyzed, it is important to determine how closely it approximates

the truth. From a theoretical standpoint, in an interplanetary situation, the third body problem is more

realistic, since it considers more than one attraction force in its equations. However, its equations are

based on the fact that both main bodies circularly orbit each other, which is not true for any planet in the

Solar System. For instance, it is known that Jupiter’s orbit around the Sun has an eccentricity of 0.048,

which is the third highest value for the Solar System [23]. On the other hand, the two-body formulation

disregards extra forces in its derivations, which also is not physically accurate.

0 1 2 3 4 5 6

Distance to Sun [AU]

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

Gravitational attraction [m/s

2]

Sun

Mercury

Venus

Earth

Mars

Solar radiation pressure

Jupiter

Saturn

Uranus

Neptune

Figure 3.3: Gravitational perturbations caused by all planets and solar radiation pressure (with Rosettacharacteristics [11]) from 0 to 6 AU, along a line with all planets aligned. The maximum accelerationscorrespond to the value at the border of each planet’s sphere of influence. [23]

22

Therefore, in order to really assess the accuracy of both models, one should look into the magnitude of

all involved attracting forces, and to the precision of the solution obtaining methods.

In terms of attracting forces, it is possible to deduce the gravitational accelerations of each planet

as a function of the distance to the Sun. This was done in Fig. 3.3. The planets were assumed to

be situated all along the same line, where the gravitational attraction was evaluated. The maximum

acceleration for each planet corresponds to the value on the border of their spheres of influence [14],

hence the observed plateaux. Due to the size of Jupiter’s plateaux, it is possible to see the significantly

larger size of its sphere of influence.

It can be seen that, beyond Mars, Sun and Jupiter’s influence dominate, with the Sun being the main

attractor at all times. At its maximum value, Jupiter’s influence is still less than 24% of the local solar

attraction, and is lower than the maximum attractions of the rocky planets.

More specifically, at the L4 point of the Sun-Jupiter system, where the Trojan asteroids reside, the

gravitational attractions are the ones listed in Table 3.2. These show that Jupiter’s attraction is less

than 0.1% of the Sun’s, thus making it negligible at the L4 point. At the closest point of the swarm to

Jupiter, determined in Fig. 2.2, the planet’s influence is still less than 1%, which is negligible from an

engineering point of view. Furthermore, it is seen from Fig. 3.3 and Table 3.2 that the solar radiation

pressure acceleration is of the order of the maximum Jovian attraction at the Sun-Jupiter L4, thus also

making it insignificant.

Gravitational attraction [10-9 m/s2]Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

219,048.76 0.04 0.72 1.01 0.14 209.10 90.15 1.33 0.49

Table 3.2: Maximum gravitational attraction of every main body in the Solar System, at the Sun-JupiterL4. [23]

Similarly, outside of Jupiter’s sphere of influence, its attraction quickly decays to less than 1% of

the Sun’s. This makes it negligible for a spacecraft’s interplanetary trajectory, given that it maintains a

certain distance from the gas giant. Such a scenario is easily manageable, since the aim of this mission

is the Jovian Trojan system, that is 5.203 AU [23] away from Jupiter, approximately.

Therefore, from a gravitational attraction standpoint, the difference between the two and three body

formulation is negligible for the present mission. This means that both the two and three body models

can be considered equally correct, from an engineering point of view.

In terms of solution obtaining methods, it is clear that, due to being an analytically closed problem,

the two body formulation is more advantageous, as every trajectory can be broken down into a series of

Kepler orbits. Therefore, any quantity that is computed using a two-body model (i.e. orbital parameters,

∆V’s, etc) has no process errors attached to it. On the other hand, each time one needs to perform

a computation using three body equations, there is a need to numerically integrate, thus attributing a

process error to the final solution. This error is a function of the integration method and step size, which

can be optimized, by choosing the most efficient method and analyzing the convergence variation with

the integration step.

23

Furthermore, all of the Trojan asteroids’ ephemeris are given in the ecliptic J2000 frame, which

means that, in order to implement a three body model, all of their positions would need to be converted

to a three body frame, and then propagated. This process is expected to significantly increase the

computational time, as for each integration step, a total of 4087 L4 Trojan asteroids’ ephemeris [9] would

need to be propagated.

Given all this information, it is concluded that the best model, in terms of accuracy, is the two-body

formulation, due to the relative insignificance of third body perturbations around the Trojan system, and

to its computational precision, whose achievement is only possible due to the ability of describing every

trajectory as a series of Kepler orbits.

Computational effort

When it comes to computational effort of the two or three body model, it is important to be aware of

which formulation takes longer to calculate results, since this work must be completed in a finite amount

of time. For this section, let us look at a ∆V minimization problem, in a trajectory from Earth to Jupiter.

In the case of a two-body formulation, if one knows the current spacecraft’s position and velocity

vectors, as well as the heliocentric angular separation to the target (Jupiter), obtaining the ∆V necessary

to reach it is a matter of solving a Lambert problem.

Due to its nature, the three-body problem does not allow such analytical computations. Instead, by

knowing the current position and velocity vectors, one can generate several maneuvers and numerically

propagate those solutions to check if the target is reached. Naturally, this leads to infinite possibilities,

and will require more computational effort than the two-body problem.

Although the examples mentioned here are very simple and straightforward, in reality there are sev-

eral optimization techniques for both models that allow the computation of the timing and magnitude of

the minimum ∆V required for a maneuver [22]. Consequently, the computational effort gap is not as

large as the simple example shown might suggest. However, due to the nature of the physical formu-

lation, which does not allow the accurate prediction of a future position and velocity without numerical

integration, the three body techniques require a much larger computational effort.

In conclusion, in terms of computational effort, the two body model is the most efficient, due to having

the possibility of computing analytical solutions, which consume significantly less time.

Versatility

The versatility of a model concerns its ability to allow the inclusion of additional components in a

simple way. In other words, a versatile model can allow the addition of other forces or maneuver types

(e.g. gravity assists), without requiring a large effort from the programmer.

Extra maneuver types are crucial to a trajectory optimization problem, because they can significantly

reduce a mission’s ∆V. More specifically, in interplanetary trajectories it is common to utilize multiple

gravity assists to reduce the amount of propellant on board of a spacecraft. Excluding spacecraft that go

to Venus, or Mars, which are relatively close to Earth, all interplanetary missions utilize this technique,

24

because it is more efficient.

In the two-body problem, the inclusion of these maneuvers is relatively simple, since their whole

theoretical implementation was derived from a central body formulation and a patched conics approxi-

mation. Furthermore, because of the closed analytical equations available, the determination of other

planets’ positions can be done for any epoch, without the need of integration. The consequent ∆V’s of

a full trajectory can be quickly computed without the need of any iteration to reach the final solution.

On the other hand, with a three body model, the task is severely complicated, since all other planets

need to be constantly tracked and integrated, so that their positions can be determined at a particular

epoch. This task, which is already computationally demanding, can be become even more difficult if a

large number of planets are taken into consideration.

Additionally, besides the large computational effort needed to simply be able to compute a planet’s

position at any time, the three body model requires optimization problems to be solved for each inter-

planetary leg. These optimizations iterate initial guesses of ∆V maneuvers and integrate the consequent

trajectories, in order to arrive at the final solution. Overall, if multiple interplanetary legs are studied, the

iterations required exponentially increase.

With that information it is possible to conclude that, in terms of versatility, the two-body formulation is

superior, mostly due to its simple and efficient methods.

Final conclusion

Taking the accuracy, computational effort and versatility of the two models into account, it can be

concluded that the best option for the Trojan mission optimization is the two-body formulation. Despite

its (slightly) lower realism, since it only includes one main attracting force, this model is more compu-

tationally efficient, and allows the study of more complex trajectories (i.e. multiple gravity assists), thus

allowing for a more versatile and in-depth trajectory design.

25

26

Chapter 4

Trajectory Optimization

In this chapter, a description of the different types of optimization techniques will be performed, in

order to familiarize the reader with the various methodologies available. Following that, the discussion

will focus more on specific techniques that are normally utilized in the optimization of interplanetary

trajectories, with special attention to the ones that are applicable to the problem of designing a mission to

the Trojan asteroids. An additional section will be dedicated to the choice of software for implementation

of the required optimization algorithms.

4.1 Optimal control problem

A dynamical system can be described as [24]:

x(t) = f [x(t), u(t), t], t0 ≤ t ≤ tf , (4.1)

with x(t) being the state vector and u(t) the control vector.

The optimal control problem can then be stated as the determination of the control vector u(t), that

minimizes a cost function J , expressed as [24]:

J = φ[x(tf ), tf ] +

∫ tf

t0

L[x(t), u(t), t]dt. (4.2)

This minimization must occur for a time ranging from t0 to tf . Inequality (or equality) constraints can

also be taken into account, and they are stated as [25]:

S[x(t), u(t), t] ≤ 0, t0 ≤ t ≤ tf . (4.3)

4.1.1 Optimization solution families

Optimization techniques that solve the optimal control problem, presented in Section 4.1, can be

divided into three separate categories: direct, indirect and evolutionary methods [22].

27

These constitute the three major approaches in the world of optimization, although not every single

algorithm falls neatly into one category or another. With that being said, a fourth category, containing

hybrid algorithms, would have to be considered as well, but for the sake of simplicity, only the three main

types will be discussed.

Direct methods

Simply put, direct methods transcribe a continuous optimal control problem into a parameterized

optimization one [22]. This is performed by expressing the control variables as a combination of N

functions, for example. With that parameterization, it is possible to utilize Non-Linear Programming

(NLP) to determine the combination that best optimizes the problem at each particular discrete point

[25].

The advantage of this methodology is that it does not require dealing with perfectly analytical solu-

tions and their complications. However, direct methods are known to be less accurate than their indirect

counterparts, by about 1% [25], which results from the discretization and parameterization of a continu-

ous problem. Furthermore, it is common for direct optimization algorithms to get stuck on local optima

of a problem, instead of converging to the global optimum, since they still rely on gradient methods to

find a solution [25].

Indirect methods

Contrary to the direct methods, the indirect ones rely on the analytically necessary conditions that

solve the optimization problem in a continuous state [22]. In other words, they formally solve the optimal

control problem, determining the control vector that minimizes the Hamiltonian of the system, thus glob-

ally optimizing it. This can either be performed analytically or numerically, through the use of Newton’s

method, for example. [25]

This solution requires the addition of the adjoint variables of the problem, thus duplicating the dynam-

ical system’s size, which complicates it [22]. In a multi-variable optimization problem, like determining

an optimum spacecraft trajectory, this is especially important, because it quickly ramps up the compu-

tational effort necessary for each task. For the sake of simplicity, the mathematical equations involved

in indirect methods will be omitted from this report, as these are not generally used for interplanetary

trajectory optimizations.

However, like it was mentioned before, this methodology is more accurate than the direct methods

since it does not constrain the control variables. [25]

Evolutionary methods

At the other end of the spectrum are evolutionary methods, or metaheuristics, which do not require

understanding of the dynamical system, or of the optimal control problem. Instead, in its most basic

implementation, random control vectors are generated and the solutions continuously computed. After

N iterations the optimal solution is considered to be the best of the randomly generated samples.

28

This can only be done if the cost function J can be entirely described by a set of parameters. This

set of parameters is normally called decision vector, and will be treated in this work with the letter p.

Each evolutionary algorithm has a different way of generating the optimal solution. For example,

in a genetic algorithm, the biological evolutionary process is simulated: an initial batch of samples (or

population) is generated, and its best solutions (or individuals) are then mixed together and mutated in

order to evolve the population. This process is iterated until the global solution stops converging. [22]

Since these types of algorithms do not require the understanding of the optimal control problem, they

are simpler to implement and understand. Furthermore, they are more likely to determine the global

minima in complex solution spaces, since they are not constrained by gradient methods, which may lead

to local optima. [22]

4.2 Applicable techniques

As was discussed in Section 2.4, this work divides the mission to the Trojan asteroids in two stages:

the interplanetary and the science phases. In the case of an asteroid tour scenario, the science phase

is divided into a stage that occurs while the spacecraft is inside the swarm, and outside of it. Therefore,

in the worst case scenario, three different types of optimization algorithms (one for each stage) could be

expected. In this section, for each particular phase of the mission, algorithms commonly encountered

in the literature will be addressed, together with their main characteristics. The final choice of algorithm

will be left for Chapters 5 and 7, since further testing will be necessary to make a decision.

4.2.1 Interplanetary phase

When performing interplanetary transfers to Mercury or any gaseous planet, the Hohmann trajecto-

ries cease to be optimal due to their high ∆V . In order to achieve reduced propellant costs, multiple

gravity assist trajectories are commonly utilized for these scenarios. Since this work deals with an in-

terplanetary transfer to the Trojan asteroids, which co-orbit the Sun along with Jupiter, multiple gravity

assists will have to be analyzed, in order to achieve realistic ∆V values.

In Canalias et al. [2], a multiple gravity assist trajectory was utilized as well, with the flyby sequence

being set to an EVEE – Earth-Venus-Earth-Earth – transfer a priori. However, it is the goal of this work to

determine the optimal flyby sequence, which means that, in the worst case scenario, every combination

of N planets in the Solar System would need to be analyzed.

Due to the vast number of possible flyby sequences, the search space will be large. Additionally, with

the non-co-planar, non-circular, orbits of the Solar System’s planets, the solution space will be vastly

complex, with a large number of strong local minima, as can be seen in some sample problems of ESA’s

GTOP [1]. This can also be observed in Fig. 4.1, where the total ∆V (cost function) was plotted for

an EMJU – Earth-Mars-Jupiter-Uranus – transfer. This figure represents the complexity of the solution

space when only two variables and two gravity assists are taken into account.

These characteristics, together with the large computational effort associated with these types of

29

Figure 4.1: Total ∆V of an EMJU trajectory, as a function of two variables. Obtained from the M. Sc.thesis of Musegaas. [20]

trajectories, are the reason why direct and indirect methods are rarely utilized to optimize them.

Instead, global optimizers of the evolutionary type are commonly implemented [1, 22]. If desired,

direct or indirect methods can be used a posteriori to optimize the evolutionary algorithm’s optimum

solution. However, Musegaas [20] has proven that there is no need to do so, due to the high reliability

and accuracy of evolutionary implementations.

Tests ran by Conway [22] and Musegaas [20] have shown that there are specific algorithms within the

evolutionary branch, that consistently perform better with these types of problems. Those are Differential

Evolution (DE), Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC). Both sources have

indicated DE as the best choice for their types of problems, but the models and validation situations

utilized differ, as do the machines where the algorithms were implemented. This does not allow a safe

inference for the case of a mission to the Trojan asteroids.

In broad terms, these three optimization algorithm’s mode of operation will be described, as to avoid

treating them as black boxes, and helping the reader understand a bit of the philosophies behind them.

Differential Evolution

Differential Evolution, as any other evolutionary algorithm, works with a population of size NP , which

is randomly generated in its first generation. This population contains NP decision vectors (or individu-

als) of the problem, thus corresponding to a total of NP cost function values. [26]

30

For each individual of each generation, the algorithm randomly picks three additional ones, which can

then be combined together into a donor vector. This combination is done with a particular equation,

whose classic formulation is achieved by adding one of the three vectors to the weighted difference of

the other two. That weight is described by the weighting factor F , having any value between 0 and 1.

[26]

However, there are many other equations that describe this operation, and each one corresponds to

a different algorithmic variant. All of the variants still utilize the weighting factor.

So, with every individual of the population’s current generation possessing a donor vector, a trial

vector can be defined for each. In order to do that, a crossover ratio CR, between 0 and 1, is used.

This corresponds to a probability of each individual’s trial vector being equal to their respective donor

vector. The remaining trial vectors remain similar to the individual. [26]

Finally, the individuals are evolved one by one into the next generation. Their newer generation’s

value corresponds either to their last decision vector or to the previously computed trial vector, depending

on which has the minimum cost function value. [26]

This is then repeated for as many generations as the user desires, or until a certain tolerance in each

population’s best solution has been reached in the past generations.

Since DE constitutes a relatively simple algorithm, and is commonly used in several industries and

studies, new adaptations of it have appeared, namely jDE and mDE. These are self-adaptive versions of

DE, where the weighting factor F and crossover ratio CR also mutate throughout the generations. [27]

Particle Swarm Optimization

Particle Swarm Optimization is an algorithm that is based on the behavior of flocks of birds searching

for food.

Again, as in DE, there are generations, each containing a population of sizeNP . The main difference

relatively to DE is that each of the individuals has memory, so their best vector (the vector that lead to

their best cost function value in the previous generations) is stored, also known as pbest. Naturally, this

will lead to the existence of a globally best vector, called gbest. [28]

Then, at each generation, the velocity of each vector is computed, by adding their previous velocity,

multiplied by the inertia weight (ω), to the weighted difference of their distance to pbest, and to the

weighted difference of their distance to gbest. These weights are randomly generated numbers be-

tween 0 and η1 for pbest, and 0 and η2 for gbest. Those variables represent the cognitive and social

components, and can be between 0 and 4. [28]

After each individual’s velocity vector has been determined, their next generation’s decision vector is

computed by adding the previous one to the velocity. After a certain number of generations has been

analyzed, the best solution corresponds to gbest.

31

Artificial Bee Colony

This last algorithm was created based on the behavior or bee colonies when being assigned to outer

food sources.

As such, an initial population of sizeNP is randomly generated and its elements’ cost function values

are determined. These cost function values are similar to food sources, so the less these values are,

the more nectar that food source possesses.

Initially, every individual (or worker bee) has its own food source. From that food source, the bee

is able to visually observe other nearby locations. In order to simulate this, a random decision vector

is generated for every worker bee, and evaluated. If that new decision vector leads to a reduction of

cost function for that particular bee, then it will abandon its current food source and fly to the randomly

generated one. [29]

Following that, there are the onlooker bees, which look at the original worker bees’ food sources, and

choose which of those to occupy, based on their probability (derived from those solution’s fitness value).

[29]

If any one of the original worker bees’ food sources has been vacated, and is not currently occupied

by either the worker or onlooker bees, a random one will be generated by a scout bee. The population’s

decision vectors are maintained in the next generation, only substituting the ones that were replaced by

the scout bees. [29]

This is repeated in every generation, and the solution will be the best food source achieved so far.

4.2.2 Science phase

In the case of Jovian Trojan asteroid tours, two major studies have been published: by Canalias et al.

[2], and more recently Stuart et al. [7]. Both of these deal with trajectories inside the asteroid cloud that

take into account the gravitational attraction of the Sun and Jupiter. More specifically, circular restricted

three body approximations are utilized to analyze the spacecraft’s trajectory, and calculate the asteroids’

position and velocity at any given epoch.

However, as was discussed in Table 3.2 of Section 3.3, in the L4 point of the Sun-Jupiter system,

the Jovian gravitational attraction is less than 0.1% of the local Solar pull. With such a low relative

gravitational attraction, Jupiter is not expected to significantly influence the trajectories, in the period that

the spacecraft is traveling through the asteroid cloud. Plus, 0.1% does not seem to justify the significant

increase in model complexity and computational time, versus the two body approximation.

Consistence-wise, it would not be coherent to adopt a two-body and patched conics approximation for

the interplanetary phase, and a three-body formulation for the periods where the spacecraft is navigating

through the Trojan asteroid cloud. Such is done in Canalias’ paper [2], for example, which is the only of

the two dealing with high thrust trajectories.

With that in mind, for this work, a two-body formulation will be utilized for the science phase, with

the Sun as a central body, similarly to what is going to be implemented in the interplanetary phase.

Consequently, the choice of optimization algorithms can not be based on any published study, validation

32

problem or known common usage.

However, it is possible to establish comparisons with the multiple gravity assist trajectories of the

interplanetary phase. First, both deal with two-body approximations, and since no public information

regarding every asteroid’s mass and diameter is available, these will have to be approximated as mass-

less points, for simplicity. This is, in essence, the same as a patched conics approximation, but with

no spheres of influence. Consequently, all of the necessary heliocentric calculations and mechanics

will remain the same as they were in the interplanetary phase. Second, the asteroid tour optimiza-

tion’s objective is to determine the largest sequence available, with the least ∆V inside the swarm (see

Section 2.4), which is almost the same as in the interplanetary phase. So, several sequences can be an-

alyzed, and their ∆V cost deduced in similar ways to the interplanetary optimization. Furthermore, the

correspondence in physical models and calculations will most likely lead to equivalent solution spaces.

These reasons lead to the belief that the same algorithms of Section 4.2.1 should be applicable to the

science phase, when the spacecraft is inside the Trojan swarm.

In the case of the spacecraft being outside the Trojan swarm, there is a possibility of utilizing direct or

indirect methods, due to only having to optimize one trajectory leg. However, Musegaas [20] has proven

that, with evolutionary algorithms, the solutions can be found safely within 50 m/s of the global optimum,

which is an excellent margin in interplanetary missions, since their normal ∆V is typically anywhere

between 0 and 10 km/s. Due to that, and for the sake of simplicity and consistency, genetic algorithms,

identical to the ones of the interplanetary phase, will be applied.

4.3 Software

With the optimization algorithms narrowed down, a choice needs to be made regarding what software

to utilize in their implementation.

Given that all of the listed algorithms in Section 4.2 are global optimizers, the chosen software should

optimally have a good support for those. Furthermore, the chosen software should also allow for the uti-

lization of off the shelf optimization problems, or custom user made implementations, since that flexibility

is important in spacecraft trajectory optimization, especially in the case of a mission to the Trojan as-

teroids, which is a rather specific/niche scenario. Additionally, if a software were to facilitate the use

of single and multi-objective optimization problems, that would be an added advantage, as its flexibility

would be increased.

Several packages are publicly available, free of charge or with private licensing, but only one matches

all of these criteria: ESA’s Python Parallel Global Multiobjective Optimizer (PyGMO) [27] package, which

is free to download and pair with the Python language. It possesses all of the listed global optimization

algorithms, and some more, together with several local optimizers, both compatible with single and/or

multi-objective problems. It also is built with functions that allow multi-core computations, which can

drastically improve the computational times.

Since the Python language does not feature astrodynamical constants and equations, ESA has also

publicly shared their Python Keplerian Toolbox (PyKEP) [30] package. This allows for the quick computa-

33

tion of ephemeris, from several databases (like JPL [31] or SPICE [32]), and determination of constants.

PyKEP also contains built-in functions for Keplerian propagation of orbits and solving Lambert problems,

which is an advantage.

Both PyGMO and PyKEP are based on C++ implementations, containing wrappers for Python, which

significantly improves their computational time, since C++ is a compiled language, and Python is not.

These packages are also utilized by GTOP [1] and were the basis on which Musegaas [20] did his

work on optimization algorithm surveying for multiple gravity assist trajectories.

34

Chapter 5

Interplanetary Phase: Implementation

and Testing

In the first stage of the mission, it makes sense to look at the interplanetary transfer to the Trojan

swarm, since it helps constrain the search space and maximum allowed ∆V for the observational stage,

which deals with a significantly larger number of possible sequences, due to the thousands of asteroid

ephemeris that are available.

In this chapter, the focus will be given to the implementation of the optimization problems, together

with their optimization algorithms’ validation and tuning. Optimization results will be presented in Chap-

ter 6.

5.1 Optimization problems

For high thrust, there are two main implementations used for multiple gravity assist trajectories [22],

which will be utilized in this thesis’ work:

1. MGA – Multiple powered gravity assists with potential velocity impulses around every planet;

2. MGA-1DSM – Multiple unpowered gravity assists with one deep space maneuver in every inter-

planetary leg.

All the solutions of both formulations can be described by a single decision vector (analogous to a

DNA string) that, when decoded and processed, allows the determination of the corresponding ∆V re-

quired for the total trajectory. The equations for that process have already been presented in Sections 3.1

and 3.2, and will not be repeated here.

However, the decision vectors still need to be presented, and their variables introduced. Additionally,

the decoding process of a decision vector into the final trajectory ∆V will be described.

35

5.1.1 MGA

Formally, this problem can be stated as [22]:

Optimize : J(p),

Subject to : rp(ti) ≥ rip,(5.1)

where J(p) is the cost function, p is the decision vector, rp(ti) is the pericenter radius at the flyby epoch

ti, and rip the constraint on said pericenter radius. The decision vector p is described as [22]:

p = [t0, T1, ..., TN ] , (5.2)

where t0 represents the departing epoch off of Earth, and Ti the travel time of the ith interplanetary

leg. Naturally, N represents the total number of planets in the sequence, not taking into account the

departing one. This decision vector has user-set lower and upper bounds for every entry [22].

With t0 being given, the departing heliocentric position can be determined based on ephemeris data.

Furthermore, with every leg’s duration it is possible to calculate the flyby epochs, thus extracting the

planets’ positions along the trajectory with ephemeris information. These represent each leg’s start and

end point, which along with the duration of said leg (provided in the decision vector), form a Lambert

problem. Its solution allows the computation of the velocities at the beginning and end of an interplane-

tary leg.

Having the incoming and departing heliocentric velocity at a certain flyby planet it is possible to

determine the respective gravity assist ∆V following the equations in Section 3.2.2.

5.1.2 MGA-1DSM

In this particular instance of the multiple gravity assist problem, the formal way of stating it is evi-

denced as [22]:

Optimize: J(p) (5.3)

where the nomenclature is similar to the one presented in Eq. (5.1). In the MGA-1DSM formulation, the

decision vector is slightly different from the MGA case, and can be described as [22]:

p =[t0, V∞, u, v, T1, ..., TN , η1, ..., ηN , r

1p, ..., r

N−1p , β1, ..., βN−1

](5.4)

All its entries are again constrained by user-set upper and lower bounds. The t0 and Ti terms are

analogous to the MGA formulation. Similarly, the rip entries represent the pericenter radius on the ith

flyby planet, an independent variable of the problem, and not a dependent one like in MGA trajectories.

Following that, ηi represents the fraction of Ti at which a DSM is applied. The V∞, u and v terms allow

36

the determination of the departure velocity vector relative to the Earth as [22]:

~v∞ =V∞

(cos θ cosφ~i+ sin θ cosφ~j + sinφ~k

),

with : θ = 2πu,

φ = arccos (2v − 1)− π

2,

~i =~V⊕(t0)

‖~V⊕(t0)‖,

~k =~R⊕(t0)× ~V⊕(t0)

‖~R⊕(t0)× ~V⊕(t0)‖,

~j = ~k ×~i,

(5.5)

where ~R⊕ and ~V⊕ represent the heliocentric Earth position and velocity vectors, respectively.

The departing relative velocity of the spacecraft at every gravity assist, as well as its heliocentric

velocity at the same instant, can be computed from the equations in Section 3.2.1

With all these elements together it is possible to integrate a leg from its start point until ηiTi, where

a Lambert problem is then solved to determine the necessary instantaneous velocity to reach the end

of the leg. The DSM’s ∆V is then computed as the difference of the required Lambert velocity and the

spacecraft’s current velocity.

5.2 Verification

Following the formulations presented in Section 5.1, it is necessary to verify their computational

implementation utilizing validation problems. Such problems are found in the GTOP database [1] and

can be used for both MGA and MGA-1DSM approaches. The ones that have a known solution are:

1. MGA

(a) Cassini 1 — Minimum ∆V of 4.9307 km/s,

(b) GTOC 1 — Maximum cost function of 1,581,950 kg km2/s2.

2. MGA-1DSM

(a) Messenger (Full) — Minimum ∆V of 1.959 km/s,

(b) Messenger (Reduced) — Minimum ∆V of 8.630 km/s,

(c) Cassini 2 — Minimum ∆V of 8.383 km/s,

(d) Rosetta — Minimum ∆V of 1.343 km/s,

(e) SAGAS — Minimum mission duration of 18.19 years.

Depending on the validation problem, the cost function varies, together with what is considered as

the mission’s ∆V. All take into account the powered gravity assist’s or DSM’s ∆V, differing only in the

accountability of the departure and arrival orbital impulse. These can be analyzed in Table 5.1, where

37

Vlau represents the launcher’s excess velocity. Note than when a heliocentric rendezvous is investigated,

the arrival ∆V is the incoming velocity relative to the arrival planet, while in an orbital insertion the ∆V is

performed at a certain arrival pericenter radius and eccentricity.

∆V Departure Arrival

Problem [km/s] Incl. Parameters Incl. Parameters

MGACassini 1 4.9307 Yes V∞ Yes rp = 108950 km, e = 0.98GTOC 1 3.6594 Yes V∞ − Vlau No —

MGA-1DSMMessenger (Full) 1.959 No — Yes rp = 2640 km, e = 0.704Messenger (Reduced) 8.630 Yes V∞ Yes Heliocentric rendezvousCassini 2 8.383 Yes V∞ Yes Heliocentric rendezvousRosetta 1.343 No — Yes Heliocentric rendezvousSAGAS 6.782 Yes V∞ No —

Table 5.1: MGA and MGA-1DSM verification problems’ ∆V specifications. [1]

In GTOP’s solutions, special attention should be paid to the algorithm implementations, which are

provided in ESA’s website [1], since the gravitational parameters do not have the same precision as

the ones implemented in PyKEP, therefore causing any integration/calculation to deviate from the op-

timal solutions. Furthermore, for the MGA problems, it is important to notice that the calculations of

the powered gravity assists’ ∆V do not include the corrections of Eqs. (3.8) and (3.10). Instead, the

pericenter radius is not set to its minimum constraint value, and a penalty is then added to the ∆V for

every kilometer below the threshold. This is theoretically inaccurate, and consequently the advertised

optimal solution is not the true global minimum, but an artificial one. Lastly, the ephemeris calculations

performed for GTOP’s problems utilize analytical polynomials taken from the European Space Opera-

tions Centre (ESOC) library [1], which is advertised in the code as being less accurate than JPL’s data,

but computationally faster.

Solution’s ∆V [km/s]

Problem GTOP Implementation

MGACassini 1 4.9307 4.93065GTOC 1 3.6594 3.65940

MGA-1DSMMessenger (Full) 1.959 1.9588Messenger(Reduced) 8.631 8.6308Cassini 2 8.385 8.3852Rosetta 1.343 1.3428SAGAS 6.782 6.7820

Table 5.2: Results of the MGA and MGA-1DSM implementations of this thesis for GTOP’s problems,and comparison to the known solutions. [1]

Due to these disparities, model implementation verification requires the algorithm to be altered, so

that all the calculations are performed according to GTOP’s implementations. Doing that leads to the

results in Table 5.2 using this thesis’ algorithms – the decision vectors utilized were taken directly from

38

GTOP’s website [1].

All the results obtained with the implemented algorithm are within rounding error of the true optimal

solution, and so we conclude that the computational implementation was performed correctly.

5.3 Optimization algorithm definition

Due to the vast search space associated with both multiple gravity assist trajectory formulations,

metaheuristics are generally utilized for a global optimization of both the flyby sequence and the flyby

parameters. Several algorithms are available in the PyGMO library for the solution of these problems,

but the most relevant are DE, PSO and ABC, as mentioned in Section 4.2.1, along with any variants of

these models, since they are normally utilized for spacecraft trajectory optimization (especially for the

MGA-1DSM formulation), and showing very good global results. [20, 22, 33]

Naturally, every problem is different, and so is the ability of every algorithm to adapt to the solution

space. Some algorithms excel in solution spaces with one or two strong basins of attraction, and then

poorly when a large number of strong local solutions are available. In theory, whenever choosing an opti-

mization algorithm, several should be analyzed for their adaptability to different scenarios (i.e. capability

of providing consistently good results for different problems).

Since the use of these global optimizers is not new in spacecraft trajectory optimization, many of

these studies have already been performed by other investigators. Some of them have even been pub-

lished together with details regarding the tuning of the best optimizer’s settings. This is easily obtained

publicly for the MGA-1DSM formulation (e.g. Conway [22], Musegaas [20]), due to its common use in

real life missions and consequent popularity. One of these studies in particular is even referenced in

GTOP’s webpage [1], and is found as a TU Delft’s M. Sc. thesis performed by Paul Musegaas [20].

In that thesis report, it is concluded that DE is the best optimizer for the MGA-1DSM formulation,

mainly due to its lower overall number of function evaluations required to achieve 95% certainty in the

final solution. This means that less computational time is spent optimizing the problem, if DE is utilized.

Such conclusions are in agreement with what is concluded from other studies, like the one presented

in Conway’s trajectory optimization book [22]. Instead of performing another analysis of optimization

algorithms for this formulation, Musegaas’ results [20] are used for this thesis, and Differential Evolution

(DE) will be implemented with PyGMO’s library.

In the case of the MGA problem, there is only one source of data, which comes from Conway [22].

This study is performed for a set number of function evaluations (80,000 for Cassini 1 and 1,200,000

for GTOC 1) and 100 different runs. However, after all computations are performed, only Cassini 1’s

solution is found within 50 m/s of the global minimum, and only for DE. In the context of interplanetary

travel, a difference of 50 m/s to the optimal solution is not significant [20]. Therefore, it is not clear

from the available data which algorithm is best suited for this case, and a separate study will have to be

performed.

With 100 different runs and the same number of evaluations suggested by Conway [22], 6 optimizers

are analyzed:

39

1. DE — same settings as in Conway (F = 0.7, CR = 0.7) [22],

2. jDE — a self-adaptive variant of DE implemented in PyGMO [34],

3. mDE — a different self-adaptive variant of DE in PyGMO [35],

4. PSO — same settings as in Conway (ω = 0.65, η1 = η2 = 2) [22],

5. ABC — version implemented in PyGMO.

The results of the 100 runs are shown in Table 5.3, where the best results of both problems are

marked in bold. For the Cassini 1 problem only jDE and mDE were able to compute the solution accu-

rately, while for the GTOC 1 problem DE and jDE were the top optimizers. Each run generated a new

population, which was then fed into each of the optimizers, so the results in Table 5.3 show that jDE is

more efficient and versatile: it computes the global solution more accurately and faster.

Problem Result DE jDE mDE PSO ABC

Cassini 1km / s

Solution 4.9307

Mean 8.1993 9.3134 12.4063 12.8089 7.5247Std 3.3626 3.4531 3.5589 3.8381 1.4038Min 5.3034 4.9309 4.9307 5.2547 5.7157Max 16.7116 16.7242 17.2673 27.1471 13.1323

GTOC 1kg km2 / s2

Solution -1,581,950

Mean -1,157,156 -1,170,525 -706,150 -610,642 -594,622Std 142,169 165,856 332,954 296,944 128,619Min -1,576,136 -1,581,928 -1,387,410 -1,210,840 -997,893Max -810,930 -470,813 -44,358 -41,245 -337,085

Table 5.3: Results after 100 runs of the selected algorithms for both MGA validation problems, andcomparison to the global minimum.

In the case of Cassini 1, jDE is not the best algorithm, since it misses the global solution by 0.2 m/s

(significantly lower than the total ∆V). However, in the GTOC 1 problem, where many strong local basins

of attraction are present [22], it almost reaches the global optimum, missing by about 22 kg km2/s2,

considerably smaller than the total solution.

In terms of average solution, it is possible to observe that for the Cassini 1 problem, the ABC al-

gorithm is closest to the global optimum, but due to its small standard deviation, its results are very

concentrated away from the true optimum. In GTOC 1, jDE presents the best mean result, with a rela-

tively small standard deviation. However, the mean and standard deviations of an algorithm only serve

to show how robust its solutions are, having a direct impact on the number of function evaluations re-

quired to have 95% certainty, since a more concentrated swarm of solutions results in better precision.

Nonetheless, a more precise optimizer is not the best option, since it may not achieve the true optimal

result (low accuracy).

40

Overall, due to its better accuracy in very different optimization problems, thus showing its versatility,

the best algorithm for MGA is concluded to be the jDE. Consequently, it will be utilized in this thesis’

implementation.

5.4 Tuning

As with any off the shelf metaheuristic optimization algorithm, the user is required to analyze the

impact of its settings (i.e. tune the software) on its performance. Naturally, this is different for each

problem, and the optimal settings for an MGA model might be different than for an MGA-1DSM situation,

for example.

5.4.1 MGA

Starting with the jDE algorithm, there are only two parameters that can be changed in order to tune

the software, which are the algorithmic and adaptive scheme variants. The first one indicates how a

mutation is computed, and has 18 different settings. The latter describes how the base DE’s parameters

(F and CR) mutate throughout the optimization, having two settings in total. All the options of these two

parameters are presented in Table 5.4.

Variant Algorithmic Adaptive scheme

1 best/1/exp rand2 rand/1/exp rand/33 rand-to-best/exp -4 best/2/exp -5 rand/2/exp -6 best/1/bin -7 rand/1/bin -8 rand-to-best/1/bin -9 best/2/bin -10 rand/2/bin -11 best/3/exp -12 best/3/bin -13 rand/3/exp -14 rand/3/bin -15 rand-to-current/2/exp -16 rand-to-current/2/bin -17 rand-to-best-and-current/2/exp -18 rand-to-best-and-current/2/bin -

Table 5.4: List of algorithmic and adaptive scheme variants available in PyGMO’s jDE algorithm. [34]

To evaluate the performance of jDE, under all these settings, GTOP’s Cassini 1 problem will be

utilized, since it consists of a ∆V minimization, taking into account both the arrival and departure contri-

butions, similarly to what will be performed for the Trojan mission’s interplanetary trajectory. The GTOC 1

validation problem will not be used, since its objective function is different than what will be sought for

the Trojan mission, and because it takes approximately 15 times more function evaluations to obtain a

(close to) optimum solution [22].

41

As a performance index, n95% will be utilized, which represents the number of function evaluations

needed until 95% certainty, that the solution is within 50 m/s of the global optimum, is achieved. This is

similar to what has been initially implemented by Olds et al. [36], and later utilized by Musegaas [20].

This quantity can be computed by analyzing the success rate of several algorithmic runs, psuccess, for a

specific number of evaluations, neval, in the following way [20, 36]:

n95% = nevallog (1− 0.95)

log (1− psuccess)(5.6)

For each algorithmic and adaptive scheme variants, the pseudo-code describing the process taken

to evaluate n95% can be found in Algorithm 1. Each algorithm is set to run for a number of populations,

mruns, and, for each of those, it performs a maximum number of function evaluations, nmax.

Algorithm 1 Computes n95% for all jDE’s settings (A1 through Al) and for a set number of runs, mruns.1: Generate mruns populations with size nind;2: for Ai = A1 : Al do3: Initialize algorithm Ai;4: for j = 1 : mruns do5: for k = 0 : nind : nmax do6: Evolve population j using Ai, for 1 generation;7: Store best fitness in Fj,k;8: end for9: end for

10: for k = 0 : nind : nmax do11: Calculate success rate psuccess from Fj,k;12: Compute n95%,k from psuccess;13: end for14: Store n95%,k vector to performance archive Pi;15: end for

For the case of Cassini 1, 100 runs were performed for every variant, and a maximum of 80,000

function evaluations were done. The results are shown in Fig. 5.1 for the various settings of jDE. Note

that, the settings for which no data is presented represent situations where no success was found even

after 80,000 function evaluations.

Regarding the plot, the red lines symbolize the mean value of the data, with the box delimiting the

25th and 75th percentiles. The upper whisker extends 1.5 times the distance between those percentiles.

If the maximum value of the data is within that range, then it is used instead. Similar logic is applied for

the lower whisker.

The data in Fig. 5.1 show that the best performance setting is to consider the second algorithmic

variant (rand/1/exp), and the first adaptive scheme (random), since its mean n95% is the minimum ob-

tained (2 × 106 function evaluations), and it has a small standard deviation. Better minimum n95% values

were obtained for the first adaptive scheme, but the mean results (red line in the plots) for these cases

were worse.

Having determined the settings for the jDE software, the only thing left to completely tune the algo-

rithm is to determine the optimum population sizes [20], which is done similarly to the process taken

previously. The pseudo-code describing this process is shown in Algorithm 2.

42

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Algorithmic variant

0.0

0.5

1.0

1.5

2.0

2.5

n95%

1e7

(a) First adaptive scheme.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Algorithmic variant

0.0

0.5

1.0

1.5

2.0

2.5

n95%

1e7

(b) Second adaptive scheme.

Figure 5.1: Values of n95% for the 18 algorithmic variants and 2 adaptive schemes of PyGMO’s jDEsoftware. See text for explanation.

The results of such process are shown in Fig. 5.2, where a total of 200 runs were performed, with a

maximum of 80,000 function evaluations. Furthermore, a distribution of the average and minimum values

of n95% is shown as multiples of the best values obtained, in Fig. 5.3, as a function of the population

size. For the latter, nbest represents the best results for the minimum and average n95%, which were

approximately 9×105 and 2×106 function evaluations, respectively. Values that do not belong to the

range described previously are considered outliers and represented with a black box in the plot.

It is noticeable in the data that the optimal population size is 20 for this case, both in terms of minimum

and average n95%. This will be the value utilized for the following calculations.

43

Algorithm 2 Computes n95% for several jDE population sizes (nind,1 through nind,l) and for a set numberof runs, mruns. jDE’s settings are the optimum ones determined (2nd algorithmic variant and 1st adaptivescheme).

1: for ni = nind,1 : Aind,l do2: Initialize algorithm A with optimum settings;3: for j = 1 : mruns do4: Generate population j with size ni;5: for k = 0 : ni : nmax do6: Evolve j using A for 1 generation;7: Store best fitness in Fj,k;8: end for9: end for

10: for k = 0 : ni : nmax do11: Calculate success rate psuccess from Fj,k;12: Compute n95%,k from psuccess;13: end for14: Store n95%,k vector to performance archive Pi;15: end for

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Population size

0.0

0.2

0.4

0.6

0.8

1.0

1.2

n95%

1e7

Figure 5.2: Results of n95% for a jDE optimization (under optimum settings) of the Cassini 1 problem asa function of the population size. See text for explanation.

According to what has been performed by Musegaas [20], there is an accurate proportionality rela-

tion between the optimal population size and the number of variables in the decision vector. With that

information, and knowing that the number of variables nvar in the Cassini 1 problem is 6, it is possible to

deduce an equation for the optimal population size NPopt, represented as:

NPopt =10

3nvar. (5.7)

The equation is linear following the trend that Musegaas [20] identified in his study.

44

15 20 25 30 35

Population size

1.0

1.5

2.0

2.5

3.0

3.5

4.0

n95% [n

best]

Minimum Mean

Figure 5.3: Distribution of the average and minimum values of n95% for 200 runs of jDE (under optimumsettings) as a function of population size. The results are shown as multiples of the best values obtained.

5.4.2 MGA-1DSM

For the case of the MGA-1DSM formulation, as stated previously, the optimum algorithm for it has

been picked based on the results of Musegaas’ study [20]. Consequently, the optimum DE settings and

population sizes are also taken from it.

In DE’s software, there are three parameters that can be tuned: F , CR and algorithmic variant,

with the latter having 10 different settings. All of these parameters have already been described in

Section 4.2.1.

The best settings for the PyGMO’s DE software are the ones presented in Table 5.5. These have

yielded n95% values of 5 × 106 for the Cassini 2 validation problem [20], which has 22 variables in its

decision vector [1].

F CR Mutation strategy Ideal population size

0.7 0.9 1 (best/1/exp) NPopt = 4.5nvar

Table 5.5: Best settings and population sizes for the DE algorithm applied to a MGA-1DSM formulation.[20]

45

46

Chapter 6

Interplanetary Phase: Results

With the algorithms defined in Chapter 5 it is possible to optimize the interplanetary trajectories to

the Trojan swarm. In this chapter, the optimization inputs are presented, followed by the results obtained

and a discussion about them. Finally, a choice of transfer orbits will be done and utilized for the science

phase in the next chapters.

6.1 Input

There are three types of input for the optimization software: planetary constants, ephemeris data

and boundary values. Planetary constants are used in every calculation/integration, ephemeris data

changes the way the planets’ positions are computed, and boundary values set the range where any of

the decision vector’s parameters are generated.

6.1.1 Planetary constants

The constants used in this work were all obtained from PyKEP’s database [30], and include each

planet’s gravitational parameter, radius and safe radius, as well as the Sun’s gravitational parameter. All

of these constants are presented in Table 6.1.

µ R rpObject [m3/s2] [m] [m]

Sun 1.32712440018×1020 – –Mercury 2.2032×1013 2,440,000 2,684,000Venus 3.24859×1014 6,052,000 6,657,200Earth 3.986004418×1014 6,378,000 7,015,800Mars 4.2828×1013 3,397,000 3,736,700Jupiter 1.26686534×1017 71,492,000 643,428,000

Table 6.1: Planetary constants used for every computation in the interplanetary transfer’s optimization.[30]

Note that no information is presented for objects beyond Jupiter, because it is not practical to spend

∆V going to them and then having the spacecraft travel back to the Trojan asteroids.

47

6.1.2 Ephemeris data

For trajectories with two or less gravity assists, the set of ephemeris data used in this work was

obtained from the SPICE toolkit developed by NASA [32], which was already implemented in PyKEP,

and was the most precise method available. This toolkit uses kernels available in the Navigation and

Ancillary Information Facility (NAIF) database [32] to determine the planetary positions and velocities

at a given epoch. The kernel utilized for the computed calculations was de430.bsp, which includes the

positions and velocities of every planet between Mercury and Earth, and of the barycenters of Mars and

Jupiter, from 1549 to 2650.

For any planetary sequence with more than two gravity assists, the ephemeris data was derived

from JPL-Low-Precision [31], being the most precise database available in PyKEP, besides SPICE.

The reason behind this change was the high computational time required for the optimization of these

trajectories with SPICE’s ephemeris, which was one hour per population, or more, depending on the

sequence. With the limited time available for this thesis, that ceased to be feasible, since there were 75

sequences that needed to be optimized, with multiple populations. For example, with 5 simple popula-

tions per sequence, the optimization all those interplanetary routes would take 15.6 days total. And this

would be simply for either MGA or MGA-1DSM, thus the complete computation time would actually be

close to 30 days.

Interpolation of SPICE’s data had to be implemented, due to a software complication with accessing

kernels while optimizing populations in multiple threads. With that in mind, several positions and veloci-

ties throughout the search space were imported from the kernel to a matrix, before initiating the overall

optimization process. Then, any required position/velocity at a given epoch was linearly interpolated,

in order to allow fast computation. In order to achieve better precision with the interpolation of SPICE

data, the step associated with it was set to 0.1 days, leading to interpolation errors of less than half of

JPL-Low-Precision’s errors (see Table 6.2).

The characteristics of the chosen ephemeris databases used, together with their positional precisions

(i.e. error between the true planetary positions and the ones of the ephemeris models), is shown in

Table 6.2.

Flyby sequences with

≤ 2 GA’s > 2 GA’s

Ephemeris database SPICE JPL-Low-PrecisionRange [years] 1549–2650 1800–2050Interpolation Yes NoInterpolation step 0.1 days -

Positional errors [km]Mercury 369.3 1,000Venus 105.8 8,000Earth 55.3 15,000Mars 23.8 30,000Jupiter 2.0 1,000,000

Table 6.2: Summary of the ephemeris data used, interpolation steps and positional errors attributed tothem, for the Interplanetary phase’s optimization. [23, 31]

48

6.1.3 Boundary values

Boundary values need to be set for every parameter in the decision vectors of the optimization

problem. This means that limits are required for launch epochs and leg durations, for both MGA and

MGA-1DSM. Furthermore, for MGA-1DSM, extra boundaries must be defined for V∞, u, v, η, rp and β,

following the representations in Section 5.1.2.

In terms of launch epochs, it was defined that they should happen between the 1st of January 2020,

and 2030. This allows 4 to 14 years of mission planning time, if the development would start this year.

Regarding leg durations, and following GTOP’s sample problems’ limits [1], they were set to be

between 30 and 3000 days. However, for sequences with three or more gravity assists, the ephemeris

data ends in 2050, which forces the maximum leg duration to be equal to 1826 days.

Similarly to GTOP’s sample problems [1] as well, u and v were set to variate between 0 and 1, and η

between 0.01 and 0.99. Besides that, β was defined to go from −π to π radians.

In the case of V∞, this was set to vary from 0.1 to 2.5 km/s, based on GTOP’s validation problems

[1].

Finally, regarding the pericenter radius rp, the values from PyKEP’s database were used, forcing it

to be between each planet’s safe radius and 6 times the planet’s radius. An exception to that rule was

opened for Jupiter, since its safe radius does not allow such range to be set, and therefore the maximum

rp was determined to be 291 times the planet’s radius [1]. The values of each planet’s radius and safe

radius are the ones presented in Section 6.1.1.

All of the boundary values described in this subsection are summarized in Table 6.3.

Flyby sequences with

≤ 2 GA’s > 2 GA’s

t0 [mjd2000] [7305, 10958] [7305, 10958]V∞ [km/s] [0.1, 2.5] [0.1, 2.5]u, v [0, 1] [0, 1]T [days] [30, 3000] [30, 1826]η [0.01, 0.99] [0.01, 0.99]β [rad] [−π, π] [−π, π]

rp

Rocky planets [rp, 6R] [rp, 6R]Jupiter [rp, 291R] [rp, 291R]

Table 6.3: Boundary values introduced to the interplanetary transfer’s optimization algorithm, in order todefine the search space.

6.2 Trajectory to asteroid tour

This section deals with the results of the interplanetary phase’s trajectories, for the mission where an

asteroid tour will occur in the science phase.

As mentioned in Section 2.4, this problem consists of a multi-objective optimization, whose goal was

49

to minimize both the total transfer ∆V and the final relative velocity, Vrel, to the L4 point of the Sun-Jupiter

system. Due to the relation between these two parameters not being confidently known, the use of a

cost function that correlates them will be avoided. Instead, the problem will be optimized for the total ∆V

(escape + gravity assists), as:

J(p) = V∞ +∑

∆VGA, (6.1)

which is commonly used to optimize interplanetary trajectories. In the end, a Pareto Front correlating

both variables will be analyzed for each computed solution.

In order to have as much variety as possible in the final Pareto Front, several populations were

optimized for each multiple gravity assist sequence, so that different solution sets were obtained. The

number of populations per sequence varied between 1 and 20, in order to guarantee that no more than

one hour was spent optimizing a particular sequence. Such decisions had to be made due to the limited

time available for this thesis.

The optimization was performed until flyby sequences with three gravity assists were evaluated,

since, due to the limited time available for this work, it was not possible to compute any more. As a

way to also reduce the computational effort, flyby sequences with two gravity assists, whose first GA

occurred at Jupiter were ignored. Besides that, flyby sequences with three gravity assists, whose first

flyby occurred either at Mercury or Jupiter were also disregarded, due to the impractical ∆V’s that the

first maneuver would require [14].

It should also be noted that, in order for it to be possible to compute trajectories with three gravity

assists, the population sizes had to be capped at 40 individuals. For the case of MGA-1DSM trajec-

tories this allowed the reduction of computational time per population from a little over 1 hour to 5-10

minutes. This significant reduction is thought to be explained by Python’s poor memory management in

the machine used for the calculations. With high usage of memory Python showed significant losses of

performance, in terms of computational times. That lead to two memory dumps during this thesis, which

occurred during the interplanetary optimizations. A preliminary analysis indicated that these dumps oc-

curred due to Python requiring too much memory, and that is the reason why a population cap had to be

introduced.

The results obtained for this type of trajectory, both for the case of MGA and MGA-1DSM trajectories,

are presented in Figs. 6.1 and 6.2, respectively. The flyby sequences are shown in descending minimum

∆V order, so they do not match on both plots.

The sequences’ names represent each planet through its first letter, as described in Table 6.4.

Name Representation

Mercury MerVenus VEarth EMars MJupiter JSun-Jupiter L4 L

Table 6.4: Notation utilized to denominate planetary sequences.

50

EJLEL

EMerJLEMerLEVJJL

EMerMerLEMerMLEMerELEMerVL

EVJMerLEVVL

EVJELEVJL

EVJVLEVJML

EVLEVVJLEVVVLEMJJL

EMMMerLEEJJL

EMJMerLEMJELEVMJL

EMMerJLEMJMLEMJVL

EMEMerLEMMerML

EMEMLEVMVL

EMMerMerLEVVML

EMMerLEMMerELEVMMerLEMVMerLEMMerVLEEJMerL

EMVELEMVVLEVVEL

EMJLEVML

EMMMLEVMerMerL

EVMELEML

EEJVLEMEL

EVMerLEMEELEMVMLEMEVLEMEJL

EVVMerLEMVLEMML

EVMMLEEJELEMVJL

EMMVLEVEVLEEJML

EVEMerLEEL

EEJLEMMJLEEEJLEEEL

EEEELEVEJL

EVMerJLEVMerMLEVMerVL

EVEELEMMEL

EVMerELEVEL

EEMerJLEEEMerL

EVEMLEEMerL

EEMerMerLEEMerMLEEMerVLEEMerEL

EEVVLEEVL

EEEVLEEVJL

EEMMerLEEMJLEEVML

EEVMerLEEMVLEEEMLEEMEL

EEMLEEVEL

EEMML

Flyb

y se

quen

ce

0 5 10 15 20 25 30

∆V [km/s]

05

1015

2025

Travel time [years]

Figure 6.1: ∆V and travel time results for MGA transfer trajectories, in the case where an asteroid tourwill constitute the science phase.

51

EEJMerLEVJJL

EMJMerLEEJML

EVMerELEMJJL

EVMerMerLEMJVLEMJMLEEJJL

EEMerELEMerMerLEMMerJL

EMerJLEMerL

EVMerVLEVMerJLEMerML

EEJVLEVJMerL

EVJMLEVJELEMJELEEJEL

EMerELEMerVL

EMMerMerLEVJVL

EMMerMLEMMerL

EVMerMLEMMerEL

EVMerLEMMMerLEMMerVLEEMerJL

EEMerMerLEEEMerLEEVMerLEEMerVLEMVMerLEEMerMLEVMMerL

EEEELEJLEL

EEMerLEEEJL

EMEMerLEEMMerLEVVMerL

EMVMLEVMJLEMVLEVJL

EMVJLEEEVL

EVLEMMVL

EMJLEEVMLEVMVL

EVMLEML

EMVVLEVMMLEEMJL

EVEMerLEMEVLEMMJLEMVELEMMEL

EEJLEEVL

EEMVLEEL

EEVJLEMEMLEVVJL

EEEMLEMMLEVVL

EVVMLEEMLEMEL

EVEJLEEVELEVVVLEVMEL

EEELEEMEL

EMMMLEEVVL

EVELEVEVLEVEMLEMEJL

EEMMLEMEELEVVELEVEEL

Flyb

y se

quen

ce

0 5 10 15 20 25 30

∆V [km/s]

05

1015

2025

Travel time [years]

Figure 6.2: ∆V and travel time results for MGA-1DSM transfer trajectories, in the case where an asteroidtour will constitute the science phase.

52

In terms of computational effort, the approximate optimization times per population can be seen

in Table 6.5, as a function of the number of gravity assists. Note the dip in computational effort for

trajectories with three gravity assists, which was caused by not only changing the ephemeris database

(see Section 6.1.2), but also by reducing the population size.

Optimization time [s]

Number of GA’s 0 1 2 3

MGA 45 60 120 120MGA-1DSM 60 240 960 600

Table 6.5: Approximate optimization times, per population, as a function of the number of gravity assistsin each MGA or MGA-1DSM trajectory. The reduction in average computation time from 2 to 3 gravityassists is due to the change in ephemeris database mentioned in the text.

The first thing that is observable from Figs. 6.1 and 6.2 is that the MGA-1DSM trajectories are much

more expensive (in terms of ∆V), than MGA alternatives. This may be related to the inherent physics

behind both models, because each of MGA-1DSM’s leg’s ∆V must be larger than (or equal to) the

respective Hohmann transfer [14], due to all of the maneuvers being performed with the same central

body. In the case of MGA trajectories, since the maneuvers are applied with each of the GA’s planets

as central bodies, the leg’s ∆V can achieve even lower values, as it is not constrained by the Hohmann

solution.

Additionally, from the figure it is possible to observe that, in general, MGA-1DSM trajectories take

longer than their MGA counterparts, indicating that powered gravity assists allow for less expensive (in

terms of ∆V) and faster Lambert arcs.

0 1 2 3 4 5 6 7 8 9 10V

rel [km/s]

0

1

2

3

4

5

6

7

8

9

10

∆V

[km

/s]

EMJEL

EVMerVL

EEMerVL

EEVVL

EEMEL EEVELEEMEL

EEVELEEMML

EEMML

EMerL

EMMerEL

EEVEL0

5

10

15

Tra

vel t

ime

[yea

rs]

Figure 6.3: Pareto Front of ∆V and Vrel (relative velocity to L4) for the interplanetary to the Trojanasteroids, with the intention of having a final asteroid tour in the science phase. Each marker labelindicates its corresponding multiple gravity assist sequence.

However, the final solutions will need to be analyzed in terms of ∆V and Vrel, so all the data previ-

ously presented was plotted into a Pareto Front, which is shown in Fig. 6.3. The only data that are not

53

plotted correspond to the solutions that either took more than 15 years of travel time, or that consisted

of 2/3 gravity assist sequences, whose last flyby occurred on Mercury. This is due to the fact that said

sequences always resulted in trajectories whose last leg would come closer to the Sun than Mercury,

thus imposing extreme radiation environments to the spacecraft and making it physically impossible to

execute them.

All the solutions in the plot consist of MGA and MGA-1DSM trajectories. However, all of the solutions

in the Pareto front are MGA, which makes sense, given that significant ∆V reductions could already

be observed in Figs. 6.1 and 6.2. It is possible to notice that the lowest ∆V alternatives happen for

three gravity assist sequences, whose first flyby occurs on Earth. This was also previously observable

in Figs. 6.1 and 6.2, where most of the lowest ∆V sequences possessed the same characteristic.

Taking into account the fact that the Vrel can be artificially corrected with a ∆V impulse, the Pareto

Front shown in Fig. 6.3 could be further refined. For instance, looking at two consecutive hypothetical

Pareto Front solutions —(∆V I , V Irel

)and

(∆V II , V IIrel

):

• If ∆V I ≤ ∆V II + (V IIrel − V Irel), then the first solution is better, since it consumes less propellant

and achieves a smaller final relative velocity,

• If ∆V I > ∆V II + (V IIrel − V Irel), then the second solution is superior, since it could achieve the

same final relative velocity with the addition of an extra velocity impulse, and still consume less

propellant.

With that in mind, and applying it to the previous Pareto Front, allows the determination of the best

trajectories. Effectively, if the additional ∆V is applied to reduce the final Vrel, then all of these have the

same final relative velocity. These solutions’ detailed information is shown in Table 6.6.

∆V [km/s]

Sol. Sequence Departure date [UTC] Duration [years] Excess GA’s Vrel [km/s]

1 E-Mer-L 29/10/2023 2.7 9.6309 ∼ 10−8 4.19692 E-M-Mer-E-L 03/06/2020 10.2 7.4471 ∼ 10−8 4.34293 E-M-J-E-L 06/04/2027 8.7 4.2570 0.0022 4.69744 E-V-Mer-V-L 01/04/2020 4.9 3.0318 ∼ 10−10 5.09615 E-E-Mer-V-L 28/08/2024 4.7 0.0002 2.3139 5.51736 E-E-V-V-L 28/12/2021 5.8 0.0002 1.4392 6.18357 E-E-M-E-L 22/02/2022 5.1 0.0001 0.9098 6.6738

Table 6.6: Best solutions derived from the Pareto Front in Fig. 6.3.

It is quite clear that the trajectories whose first gravity assist occurs at Earth tend to have relatively

insignificant excess velocities, with most of their ∆V being spent on gravity assists. On the other hand,

trajectories who do not have the same type of flyby sequences are the complete opposite, with almost

100% of their ∆V budget being required solely for the escape from Earth.

The latter are superior, because the excess velocity given by the spacecraft can be mitigated, or

even zero’ed, by having the launcher provide it. For the 3rd to 7th solutions, it is known that the Soyuz

launcher, for example, is able to completely provide the necessary excess velocity [37]. In the case of the

2nd solution, Delta IV is known to have the capabilities to deliver such excess velocities [38]. However,

54

for the 1st solution, no launcher was able to be found, with publicly available information and able to

perform the necessary interplanetary injection.

Looking at the 4th trajectory (EVMerVL), in particular, if Soyuz or Delta IV were to completely provide

the excess velocity, the spacecraft launched would need to carry ∼ 10−10 km/s of ∆V for the interplan-

etary trajectory. In practical terms, this is a negligible amount of propellant. The low ∆V for the gravity

assists of this trajectory is thought to be caused by a high sensitivity of the GA’s excess velocity vector

(magnitude and direction) to pericenter maneuvers.

The only disadvantage of having a launcher provide the full excess velocity is that the maximum

launch mass is capped at a lower value. With Tsiolkovsky’s ideal rocket equation, and launcher manuals,

one could investigate the resulting arrival mass at L4. This is done in Table 6.7, for the Soyuz and Delta

IV launchers [37, 38], and an on-board propulsion system with a specific impulse of 330 seconds [16].

Other launchers’ manuals do not provide enough information regarding the relation between launch

mass and Earth’s excess velocity and were not taken into consideration. Nevertheless, the considered

launchers are some of the most common candidates and are enough for this preliminary analysis.

Arrival mass [kg]

Sol. Soyuz Delta IV

1 - -2 - 1650.03 949.4 3797.44 1250.0 4600.05 782.7 2641.66 1025.6 3461.47 1207.9 4076.6

Table 6.7: Arrival masses of the spacecraft to the Sun-Jupiter L4 point. Computed from the 7 interplane-tary trajectory solutions of Table 6.6, and with the Soyuz and Delta IV launchers [37, 38]. The on-boardpropulsion system utilized was assumed to have a specific impulse of 330 seconds [16].

It is possible to see that solution number 4 is better for both considered launchers, since it can deliver

more mass to the Trojan swarm. In the case of Soyuz, this represents an additional 50 kg of mass, and

for Delta IV it is around 600 kg. The latter is much more significant than the former due to the higher

capacity of the rocket. Note that this is not equivalent to carrying 50 or 600 kg more of payload, since

there are multiple subsystems on-board of the spacecraft, and each of their masses is dependent on the

others’. However, it is safe to say that, if more overall mass is delivered to the target, then more payload

mass can be carried as well.

The only downside of this solution is the existence of other trajectories that achieved less Vrel. How-

ever, solution 4 has a Vrel of 5.0961 km/s, which is only 0.9 km/s worse than the best solution’s, and

0.42–1.58 km/s better than the following results of Table 6.6.

Thus, by having a minimum amount of on-board ∆V (almost non-existent), more mass at arrival to the

swarm, and a low Vrel, the 4th solution is best for the interplanetary trajectory to the asteroid tour. The

fact that it practically requires no propellant for the gravity assists increases the amount of maneuvering

that is going to be allowed in the science phase, which attributes a little more flexibility to the design,

and increases this solution’s quality. In terms of risk, due to the negligible ∆V required for the transfer,

55

−6 −4 −2 0 2 4 6X [AU]

−6

−4

−2

0

2

4

6

Y [AU]

ear h 2020-Apr-01

venus 2020-Sep-21

mercury 2020-Nov-27

venus 2022-Jul-11

L4 2025-Feb-16

(a) Full trajectory.

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5X [AU]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Y [AU]

earth 2020-Apr-01

venu 2020-Sep-21

mercury 2020-Nov-27

venu 2022-Jul-11

L4 2025-Feb-16

(b) Close-up of rocky planets.

Figure 6.4: Final chosen orbit for an asteroid tour mission, departing from Earth on April 1, 2020. Allephemeris are presented in an ecliptic J2000 frame.

the maneuvers are practically non-existing, which means that there will be less risk of failure overall.

A plot of the corresponding final orbit is shown in Fig. 6.4, in an ecliptic J2000 reference frame. An

additional close-up of the legs occurring around the rocky planets is also presented, for better analysis.

56

6.3 Trajectory to asteroid rendezvous

Similarly to what was performed in Section 6.2, the optimization algorithms were ran for the desired

asteroid rendezvous target. This target was defined based on the scientific scores for each Trojan

asteroid presented by Stuart et al. [7], where 624 Hektor was shown as the most interesting target,

mostly due to it being the only contact binary in the Trojan swarm. Its ephemeris were obtained from

IAU [9].

This problem consists then of a single-objective optimization of the total ∆V in the interplanetary

trajectory, this time including a heliocentric rendezvous maneuver with the asteroid in question. Conse-

quently, no Pareto Fronts are required. The defined cost function is:

J(p) = V∞ +∑

∆VGA + ∆VArrival. (6.2)

During the optimization, various multiple flyby sequences were analyzed, containing up to three

gravity assists, for the same computational reasons shown previously. The populations used for this

process were the same as for Section 6.2, and the optimizations were performed parallel to the asteroid

tour’s optimizations.

The results are shown in Figs. 6.5 and 6.6, where it is possible to observe the same tendencies as

before, where the MGA trajectories severely outperform the MGA-1DSM results, both in terms of travel

time and ∆V. The sequences all end in the letter H, which represents the chosen target asteroid: 624

Hektor.

It is also possible to again observe that, in terms of MGA trajectories, the best results in terms of ∆V

correspond to flyby sequences whose first gravity assist occurs on Earth. This indicates that the excess

velocity is being diluted into the first gravity assist, similarly to what was observed in Table 6.6.

Thus, in order to fully assess what are the best options, the best solutions of the two best sequences

will be broken down, as well of the four best flyby sequences whose first gravity assist is not at Earth.

Naturally, from what can already be seen in Figs. 6.5 and 6.6, none of those solutions correspond

to MGA-1DSM trajectories. The breakdown of these trajectories’ most important characteristics is in

Table 6.8.

∆V [km/s]

Sol. Sequence Departure date [UTC] Duration [years] Excess GA’s Arrival

1 E-E-M-J-H 29/08/2028 10.0 0.0001 1.0754 3.35372 E-E-V-M-H 21/09/2026 5.8 0.0002 0.8476 4.30413 E-V-E-J-H 26/10/2029 9.3 2.8234 0.4553 3.22004 E-M-J-H 18/01/2025 13.4 4.0388 1.7307 0.73655 E-M-E-J-H 10/03/2029 9.9 3.4225 0.2553 3.25146 E-M-V-J-H 08/02/2029 10.0 3.2045 0.6989 3.2278

Table 6.8: Solutions analyzed for the interplanetary transfer to an asteroid rendezvous and their charac-teristics. All of the presented trajectories are of the MGA type.

57

EMerHEMerMerH

EHEVJMerH

EMMMerHEVMerHEMerVH

EMVMerHEVH

EMMerMerHEVJVH

EMerEHEMEMerHEMJMerHEVMMerH

EVVVHEVMerMerH

EVEMerHEVVH

EMEVHEEJMerHEVVMerHEMerMHEMMerH

EVJEHEMerJH

EEEMerHEMJVHEVJJH

EMMVHEEMerH

EVMerVHEJH

EMVVHEEJVH

EMMerVHEMJEH

EEMerMerHEMEEHEVMVH

EMVHEMMEHEVMEH

EMMerEHEVEVHEVVEHEMVEHEVJMH

EMMerMHEVVJHEVEEHEEEEH

EMMMHEVMerEH

EEJEHEEVH

EMJMHEMMerJHEEVMerHEEMMerH

EEVVHEEEVH

EVMMHEVEHEMEH

EEHEEEH

EVMerMHEMH

EEMerVHEVMH

EEMerJHEMJJHEMMH

EEMerEHEMEMHEMVMHEMMJH

EVMerJHEEJMHEVVMHEEJJH

EVEMHEVMJH

EVJHEEMVHEMVJHEEEJHEMEJH

EEMerMHEMJH

EVEJHEEVEHEEMEHEEMMHEEEMHEEVJHEEMHEEJH

EEVMHEEMJH

Flyb

y se

quen

ce

0 5 10 15 20 25 30

∆V [km/s]

05

1015

2025

Travel time [years]

Figure 6.5: ∆V and travel time results for MGA transfer trajectories, in the case where a final asteroidrendezvous will be had in the science phase. The ’H’ in the end of the sequences symbolizes the targetasteroid 624 Hektor.

58

EMJMerHEVMerVHEVJMerHEEJMerH

EVMerMerHEVMerMH

EVJVHEMJVHEVJJH

EMerMHEMerMerH

EMMerMerHEMerH

EEMerMerHEMJJH

EEMerEHEEJVH

EMMerMHEEVMerH

EMerVHEVJEH

EMMerVHEMJEH

EMMerEHEEJJH

EVJMHEVMerEH

EMerEHEVVMerHEEMerMHEMEMerHEVMerJH

EMVMerHEMerJH

EMMerHEMJMHEVMerH

EMMMerHEVMMerHEEMerVHEEMMerH

EEJEHEEMerH

EMMerJHEVEMerH

EEJMHEEVMH

EEEMerHEVMVHEMEVHEMMVH

EEMerJHEVH

EMVMHEMVVH

EHEMVH

EVEMHEEEEHEEMVHEVVMH

EVMHEVEVHEEVVH

EEVHEVMMHEMVEHEVVVHEVMJHEVMEH

EVVHEEEVH

EMHEMMEH

EJHEMEEHEMEMHEEVJH

EEEMHEEH

EEEHEMVJHEEMEH

EVJHEVVEHEEVEHEMMHEEEJHEVVJHEMEH

EVEJHEEMHEVEH

EVEEHEMMJH

EMJHEMMMHEEMMHEEMJH

EEJHEMEJH

Flyb

y se

quen

ce

0 5 10 15 20 25 30

∆V [km/s]

05

1015

2025

Travel time [years]

Figure 6.6: ∆V and travel time results for MGA-1DSM transfer trajectories, in the case where a finalasteroid rendezvous will be had in the science phase. The ’H’ in the end of the sequences symbolizesthe target asteroid 624 Hektor.

59

It is clear that, for the case of flyby sequences with a first gravity assist on Earth, the excess velocity

is extremely low, not allowing for significant propellant savings on-board of the spacecraft by relying on

direct launcher ejection. Thus, solutions 3 to 6 in Table 6.8 allow for the least on-board ∆V if a launcher

like Soyuz, for example, is used to escape the Earth’s sphere of influence.

Of the six solutions presented, only one does not violate the ∆V limitation of 3.5 km/s set for the

spacecraft in Section 2.3, thus making it the natural choice of trajectory for this mission. Nevertheless,

it is clear that, in practical terms, this mission is the least optimal, as it requires a travel time larger than

what the average spacecraft is able to survive, thus meaning the risks associated with it must be carefully

assessed in later stages of mission design. In a real mission design scenario, with information regarding

the payloads’ characteristics, and accurate estimates of the spacecraft’s dry mass, it would be possible

to set a more realistic on-board ∆V limit. With that, perhaps solutions 3, 5 or 6 of Table 6.8 would prove

to be feasible. As of right now, with the available information, the only feasible trajectory corresponds

to solution 4. With a 10% margin on on-board ∆V, conservatively accounting for the maneuvering

necessary for the spacecraft to be captured by 624 Hektor, solution 4 leads to a propellant-to-wet mass

ratio of 0.57, which is 1.8% larger than Rosetta’s [11].

A plot of the final trajectory is shown in Fig. 6.7, in the ecliptic J2000 reference frame.

Figure 6.7: Final chosen orbit for a rendezvous with 624 Hektor, departing from Earth on 10/03/2029.All ephemeris are presented in an ecliptic J2000 frame.

60

Chapter 7

Science Phase: Implementation and

Testing

With the best interplanetary trajectories designed in the previous chapters, it is possible to expand

on them in the science phase, and investigate the resulting asteroid sequences available.

In this chapter, the discussion will be focused on the implementation of the necessary optimization

problems and algorithms, together withs their testing, similarly to what was done in Chapter 5 for the

interplanetary phase.

7.1 Optimization problems

The trajectory model for the spacecraft inside the Trojan asteroid cloud is based on a two-body

approximation (Sun-spacecraft), similarly to what was done for the interplanetary phase. This goes

against what was developed by Canalias et al. [2] and Stuart et al. [7], since their studies considered

Jupiter’s gravitational attraction when the spacecraft navigated through the asteroid cloud.

This means that, in the case of this work, between every pair of consecutive asteroids, a Lambert

problem can be solved to determine the leg’s required ∆V for the spacecraft to encounter both objects,

similarly to the routine that was implemented in-between consecutive planets in the MGA trajectories

(see Section 5.1.1). However, there is a slight difference, because no sphere of influence is considered

for the asteroids (i.e. no gravitational influence), leading to the ∆V being directly computed from the

required heliocentric change of velocity vector. As such a trajectory model can be defined for a particular

asteroid sequence with N legs (N + 1 asteroids), utilizing a decision vector notation similar to that of the

MGA optimization problem:

Optimize : J(p). (7.1)

The decision vector p is described as:

p = [T1, ..., TN ] , (7.2)

61

where Ti is the travel time of the ith asteroid leg. Naturally, for this work, with the optimization goals

previously established in Section 2.4, the decision vector will have to be translated into a total ∆V for the

maneuvering inside the asteroid swarm.

With the starting epoch being defined from the interplanetary trajectory’s solution, the velocity of the

spacecraft at the first asteroid is already known. This means that, using each leg’s travel durations, and

the ephemeris of every asteroid in the sequence, Lambert problems can be solved identically to what

was performed in Section 5.1.1, in order to have each leg’s ∆V computed. In the end, the total cost

function of a particular asteroid sequence will be defined by the sum of all those components.

Similarly to Canalias’ study [2], this approach approximates each asteroid as an infinitesimal point,

and so the spacecraft trajectory will aim to intercept each asteroid’s position at the encounter epoch

defined by the decision vector. Such an approximation was done, because not all of the asteroids,

whose ephemeris were known (4087 in total), had available information regarding their diameter and

similar assumptions had already been made in the interplanetary phase (i.e. patched conics).

The problem, presented in Eqs. (7.1) and (7.2), defines the input given to the optimization algorithm,

and will be applied from now on to the optimization of the science phase. However, it should be noted

that it significantly differs from the methodologies described by Canalias [2], whose work is the only

other known study of Trojan asteroid flyby tours. Not only does it differ in terms of physical modeling,

as mentioned previously, but it also differs in the definition of the optimization problem. In this work’s

implementation, there is one decision vector for the total trajectory inside the Trojan swarm, meaning

that the optimization algorithms will find the p that minimizes the total cost function of the sequence.

However, Canalias divided this as a succession of N problems, effectively treating each leg as its own

one. In simpler words, let us imagine a sequence of asteroids A–B–C–D, where the goal is to minimize

the total ∆V inside the asteroid cloud. This example’s cost function is:

∆V = ∆VAB + ∆VBC + ∆VCD. (7.3)

In the approach shown in Eqs. (7.1) and (7.2), the optimization algorithm would find the decision vector

that lead to:

min (∆VAB + ∆VBC + ∆VCD) ≡ min ∆V, (7.4)

effectively minimizing the total cost function. However, Canalias’ [2] method would partition this problem

into:

min ∆VAB + min ∆VBC + min ∆VCD 6= min ∆V, (7.5)

which does not minimize the total cost function, because each leg would be constrained by the previous

one’s solution. Canalias’ [2] method would thus optimize the first leg, leading to min ∆VAB , and from

that solution compute the legs that succeeded it. Such methodology would only function if each leg was

independent of the others, and a singular solution could be found for each of them, without constraining

the remaining.

The choice to depart from Canalias’ approach [2] and develop a new one, more similar to the method-

62

ologies of MGA trajectories (see Section 5.1.1), was done to guarantee a more direct relation between

the decision vector and the global solution of the problem.

7.2 Model validation

The model discussed in Section 7.1 is, in many ways, similar to the MGA trajectory model imple-

mented in Section 5.1.1. In fact, all of the necessary algorithms for the calculation of each asteroid

tour’s ∆V were already a part of the MGA trajectories’ code, which was tested and verified in Sec-

tion 5.2. The difference between both routines is that, after having propagated all of the heliocentric

Lambert arcs from object to object, the MGA algorithm would zoom in on each system’s sphere of influ-

ence and compute the required velocity change to match the incoming and outgoing velocities with their

desired values.

In the case of the asteroid tour, that zone of influence is non-existent, because the asteroids are

considered to have no mass. Therefore, the incoming and departure points coincide, spatially and in

time. Thus, by suppressing the parts of the MGA code that operated inside the spheres of influence, the

asteroid tour is completely implemented.

As such, all of the utilized routines have already been validated in Section 5.2, and there is no need

to perform an in-depth validation in this section as well. Since this implementation of the asteroid tour

model is being published for the first time in this thesis, no known validation problems for this particular

scenario are available.

Although no in-depth validation of the routines is strictly necessary, the code should still be tested

for bad implementations, like the wrongfully suppression of parts of the MGA code, or bad variable

assignments, for example. In order to do that, a dual Hohmann transfer scenario was utilized. For that

scenario, we consider three planets — A, B and C — orbiting the Sun in circular co-planar orbits, with

the spacecraft initially at the first planet. From there, the spacecraft should perform a Hohmann transfer

from planet A to B, and following that, from B to C. Naturally, since this is a Hohmann scenario, each of

those transfers is done in half of its transfer orbit’s period, and the ∆V impulses required will be done in

the first transfer orbit’s pericenter and apocenter [14].

Solution’s ∆V [km/s]

Case RA [AU] RB [AU] RC [AU] Theory Implementation

I 1 1.5 2 7.0894 7.0894II 1 1.5 2.5 8.2807 8.2807III 1 1 2.5 5.8148 5.8148

Table 7.1: Results of the asteroid tour implementation for three dual Hohmann scenarios, with planetsA, B and C circularly orbiting the Sun at RA, RB and RC , respectively.

With elementary astrodynamics, the expected ∆V ’s are easily computed by hand, together with the

transfer durations, which will form the decision vector. This decision vector will then be applied to the

implemented code, outputting the implementation’s solution. Three overall scenarios were studied, with

63

three orbital radii — RA, RB and RC — corresponding to the planets A, B and C, respectively. These

parameters, together with the expected and computed trajectory’s ∆V , are listed in Table 7.1.

Of all the tested cases, particular attention should be paid to the third one, because planets A and B

are coincident, which means that the problem is reduced to a single Hohmann transfer from A to C. The

validation results prove to be satisfactory, since they verify that the code was adapted correctly from the

MGA implementation, and is working as expected.

7.3 Optimization algorithm definition

As mentioned in Chapter 4, due to the fact of having similar force models in the interplanetary and

science phases, the optimization algorithms that can be used for the spacecraft trajectory inside the

Trojan swarm are the same as for the MGA and MGA-1DSM problems. This means that, in order to

optimize each asteroid sequence, algorithms that rely on metaheuristics and genetic based routines will

be more effective than direct or indirect methods. This is, again, due to the vast diversity of the solution

space, which can contain various local minima, with some being strong enough to render gradient-

based optimization methods useless. As such, the ability to search for global solutions without being

constrained by any of those gradients is a significant advantage that metaheuristics possess.

Since the optimization problem defined in Section 7.1 is being published for the first time in this

thesis, and for a specific problem of an asteroid flyby tour, there is no point of comparison available.

Consequently, no commonly used optimization algorithms are known for this type of implementation.

Furthermore, since the only two other available studies on Trojan asteroid tours (i.e. Stuart [7] and

Canalias [2]) deal with three body approximations, no parallels can be established, or validation cases

deduced.

As an alternative, the cases studied in Section 7.2 will be used to deduce what algorithm is bet-

ter suited for the asteroid tour problem, because it is known, from basic astrodynamics, that the dual

Hohmann solution is the global minimum [15]. More specifically, only the first validation problem will be

utilized, since all three are extremely similar in solution space, with one strong basin of attraction in the

solutions presented in Table 7.1. As such, with the same algorithms and settings that were presented in

Section 5.3, the results of 100 runs were obtained and listed in Table 7.2.

Problem Result DE jDE mDE PSO ABC

Dual Hohmann1 - 1.5 - 2 AU

km / sSolution 7.0894

Mean 7.6653 7.1799 7.7648 7.7648 7.6134Std 0.8843 0.3260 0.9021 0.9021 0.6151Min 7.0894 7.0894 7.0965 7.0965 7.0894Max 12.4874 8.9677 12.4874 12.4874 9.5290

Table 7.2: Results after 100 runs of the selected algorithms for the first dual Hohmann scenario, appliedto the asteroid tour implementation, and comparison to the global minimum.

64

It can be seen that, despite every algorithm’s minimum solution being within 50 m/s of the expected

global minimum, DE, jDE and Bee Colony came closer. Of those three, jDE was the only one to also ac-

curately optimize the MGA validation problems in Table 5.3, which had more complex solution spaces.

Based on that comparison, and the low minimum solutions and low standard deviations listed in Ta-

ble 7.2, it can be concluded that jDE is the better optimization algorithm for the asteroid tour imple-

mentation. Such conclusion was already expected, due to the close similarities between the MGA and

asteroid tour implementations, which would necessarily lead to similar solution spaces. Naturally, be-

cause the tuning of jDE that was already performed in Section 5.4 for the MGA formulation, a parallel

can be established with the asteroid tour model. This comparison leads to the practical conclusion that

the tuned jDE, utilized for the MGA model, can also be used for the asteroid tour implementation, with

little to no expected negative impact on the overall performance of the optimizer.

65

66

Chapter 8

Science Phase: Results

With the implemented algorithms in Chapter 7 it is possible to compute the Trojan asteroid sequences

of the Science phase. The initial conditions are defined by the decision vector of the trajectory designed

in Section 6.2, where the last interplanetary leg’s powered gravity assist is adjusted to target several

different starting asteroids, instead of L4.

In this chapter, the pruning criteria for the calculation of the asteroid tours will be presented, together

with the results. These will then be discussed and the best solutions will be highlighted.

8.1 Pruning

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5X [AU]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Y [A

U]

Trojan asteroidsL

4

Figure 8.1: Positions of the L4 Trojan asteroids, relative to the stability point, in July 31, 2016 [9]. Allcoordinates are shown in the ecliptic J2000 reference frame.

Due to having ephemeris for a total of 4087 L4 Trojan asteroids [9], there is a large number of possible

sequences to be evaluated. If one would investigate every possible sequence of e.g. five asteroids, there

would be more than 1018 cases and, as was shown in Chapter 6, the computational time could quickly

67

get out of hand. To provide perspective of the dimension of this problem, Fig. 8.1 shows a top view of

the Trojan asteroids’ positions, relatively to L4, on July 31, 2016, in the ecliptic J2000 frame [9].

It is clear that pruning criteria need to be defined for the particular case of a mission to the Jovian

Trojan asteroids, so that the available computational time is feasible and spent effectively. The number

of possible asteroid sequences can be described by a combinatorial tree, similar to the one shown in

Fig. 8.2, where the numbers inside the shapes represent the asteroid indexes.

Figure 8.2: Tree representation of the available combinatorial sequences of Trojan asteroids. The num-bers inside the shapes represent the number of the asteroid. Each tier N corresponds to N asteroidsequences.

Tier N is composed by all the available sequences with N asteroids. In the represented tree, re-

peating asteroids are also shown, although in the case of a mission to the Trojan swarm that is not be

desired, since it does not maximize the variety of observations. Therefore, the first pruning criteria is to

not analyze sequences with repeating asteroids.

However, this is not enough since it only prunes an insignificant amount of branches. For that reason,

it is necessary to prune the remaining based on different indicators. Canalias [2], whose work also dealt

with a tree search of possible Trojan asteroid flyby sequences, utilized criteria like each trajectory leg’s

∆V or travel duration. As mentioned in Chapter 7, Canalias’ optimized individual legs, so from tier 1 to

tier 2 several transfers would be obtained. If, for example, trajectories from asteroid 1 to 2 violated ∆V

or travel duration constraints, that branch would not be propagated to tier 3. Consequently, trajectories

from asteroids 1 to 2 to any of the remaining 4085 objects would not be evaluated.

Although in this work the optimization is done for the whole sequence, and not for individual legs,

it is possible to implement similar pruning criteria. For instance, in the same scenario, all two-asteroid

sequences, departing from the first asteroid, can initially be optimized. If the sequence from asteroid

1 to 2 proves to have a minimum ∆V cost of 10 km/s, for example, then no three-asteroid sequences

going from object 1 to 2, and on to any other body, should be analyzed in tier 3. This is because 10

km/s is already too large of a ∆V for any trajectory. The same could be applied when deciding which

68

asteroids to be analyzed in tier 1, since it is possible to determine the ∆V required to adjust the last

interplanetary leg and target each of the 4087 asteroids.

Given that this is a boundary value optimization problem, and the decision vector variables are al-

ready travel durations for each individual leg, those can not be used as a pruning criteria, as they are

already taken into account by the optimization algorithm. Therefore, considering the main objective of

this part of the mission (see Section 2.4), a certain tier’s solution will only be propagated into the next

tier if it is below a previously defined ∆V ceiling. The ∆V ceiling, or ∆VTotal, is applied for the total

on-board ∆V of the mission (in-swarm maneuvering and interplanetary gravity assists, including the last

leg’s correction to target the first asteroid).

The values for the tolerable ∆V thresholds of each tier will be based on Canalias’ [2] optimal so-

lutions, and on the mission’s propellant constraints. This significantly reduces the computational time

spent in computing solutions worse than Canalias’ [2], which scientifically do not have much interest.

It also allows the pruning criteria to be a bit more strict, since there is already an idea of where the

optimal sequences should reside in the solution space. The values defined in this work are listed in

Table 8.1. Each tier’s value means that its solutions will only be propagated if they respect the defined

ceiling. For instance, a two asteroid tour will only be propagated into a three asteroid tour, if its total ∆V

is smaller than, or equal to, 0.7 km/s. It should be noted that tier 1’s ∆V threshold is applied to the last

interplanetary leg’s correction, which is calculated to target each of that tier’s asteroids.

Tier 1 2 3 4 ≥ 5

∆VTotal [km/s] 0.5 0.7 1.2 1.5 3.5

Table 8.1: Maximum ∆V values (for the whole mission) allowed to be propagated in each of the tree’stiers’ solutions.

8.2 Input

In this particular problem, the only optimization input that should be discussed is the minimum and

maximum asteroid-to-asteroid travel time. These were set to 1 day and 2 years (730.5 days), since they

encompass all of the solutions presented by Canalias [2], conservatively.

In terms of ephemeris data, as mentioned previously in Chapter 2, all of the orbital parameters of

4087 Trojan asteroids were obtained from IAU’s database [9], whose last update was performed in the

20th of June of 2016.

Regarding the astrodynamic constants utilized, only the solar gravitational parameter was required.

The value used was already given in Section 6.1.1.

69

8.3 Asteroid tour

After having defined all procedures and optimization inputs, the results were obtained, considering

the ∆V inside the asteroid cloud as the cost function. This is described as:

J(p) =∑

∆VTour, (8.1)

where ∆VTour represents the individual maneuvers inside the asteroid cloud. The reader should be

reminded that, although the in-swarm maneuvering costs are utilized as the optimization function, the

pruning criteria presented in Table 8.1 refer to the sum of Eq. (8.1) with the ∆V necessary to correct the

last interplanetary leg.

More than 109 sequences were evaluated, of which only approximately 44,000 were acceptable

within the pruning boundaries. The computational efforts associated with this task, more specifically

with the propagation of a tier’s sequence to the next one, can be seen in Table 8.2.

Tier 1 2 3 4 5

Optimization time [min] 3.5 10 30 90 120

Table 8.2: Approximate optimization time required to propagate a single tier’s solution.

Naturally, not every result will be presented in this report, but only the ones that are better. Since the

only point of reference available for this study is Canalias’ paper on Trojan asteroid flyby tours [2], it will

be used to define which results to include in this work. More specifically, trajectories with asteroid tour

sequences of N asteroids will appear in this work, if their total mission’s ∆V is less than Canalias’ top

result [2] for the same number of asteroids observed. For trajectories with more than five asteroids, no

solutions were obtained by Canalias [2], so they will all be presented.

Asteroids in sequence Duration ∆V

I II III [years] [km/s]

2003KM12 2009SW159 2008RE3 0.6 0.32362000QD225 2000AV121 2002GO29 1.1 0.35862000QD225 2000AV121 2012QV34 0.9 0.39432000QD225 2000AV121 2008KT37 0.7 0.4262

2638T-2 2009SG193 2010VB136 0.9 0.43402000QD225 2008KT37 2002GO29 1.1 0.4590

2007QR5 2009SO141 2009SO137 0.8 0.48302000RD88 2010AJ107 2008RZ119 0.6 0.4993

2000QD225 2008KT37 2012QV34 0.9 0.5092

Canalias 0.5380

Table 8.3: Top results for tour sequences with 3 asteroids and their details. The total ∆V presented isthe sum of interplanetary correction with the in-swarm maneuvering. The duration shown is measuredfrom the first asteroid encounter to the last. Canalias’ [2] top trajectory with 3 asteroids also shown forcomparison.

70


I II III IV [years] [km/s]

2000QD225 2008KT37 2012QV34 2002GO29 1.1 0.58742000QD225 2000AV121 2012QV34 2002GO29 1.1 0.68662000QD225 2000AV121 2008KT37 2002GO29 1.1 0.73592000QD225 2000AV121 2008KT37 2012QV34 0.9 0.85582000QD225 2010UQ91 2012QV34 2002GO29 1.1 0.8868

2007QR5 2009SO141 2009SO137 2013AB41 1.1 0.9807

Canalias 1.0290

Table 8.4: Top results for tour sequences with 4 asteroids and their details. The total ∆V presented isthe sum of interplanetary correction and the in-swarm maneuvering. The duration shown is measuredfrom the first asteroid encounter to the last. Canalias’ [2] top trajectory with 4 asteroids also shown forcomparison.


I II III IV V [years] [km/s]

2000QD225 2000AV121 2008KT37 2012QV34 2002GO29 1.1 0.93932000QD225 2010UQ91 2008KT37 2012QV34 2002GO29 1.1 1.2590

Canalias 1.3325



I II III IV V VI [years] [km/s]

2000QD225 2010UQ91 2008KT37 2012QV34 1997UL16 2002GO29 1.1 2.3903

Canalias N/A


The solutions obtained can be seen in Tables 8.3 to 8.6, together with the respective top trajectory

of Canalias [2], for comparison. It is observable that a total of 18 sequences obtained were better than

Canalias’ top results [2], which is a direct consequence of the better interplanetary trajectory and the

improved optimization model used for the asteroid tour.

It can also be seen that the results obtained with the algorithms in Chapter 7 represent maximum

savings of 0.4416 km/s relatively to Canalias’ solutions [2]. This represents a significant amount of fuel

that can be spared from the spacecraft and invested on payload, thus maximizing the scientific return of

the mission, relatively to Canalias’ designs [2].

From Tables 8.3 to 8.6 it is also possible to observe that the duration of the science phase ranges

from 7 to 14 months, approximately. In terms of average asteroid-to-asteroid travel times, this means

71

that the spacecraft takes 2.2 to 3.5 months, approximately. Thus, it can be concluded that the boundary

values, set to the optimization software, were much more conservative than they should have been. With

the narrowing of the search space, significant reductions of the computational effort could be achieved.

Due to the optimization goals of this phase of the mission, which were introduced in Table 2.3, the

best solution must be chosen taking into account the ∆V associated with it, and the number of asteroids

that are observed. These are two very different objectives, that could lead to a Pareto front analysis,

similar to the one performed in Section 6.2. However, in practical terms that is not necessary, since

the main objective of this phase, and the whole mission, is to observe as many asteroids as possible.

With that in mind, it can be immediately deduced that the trajectory of Table 8.6 is superior to all others.

This is due to it being the only trajectory capable of observing 6 asteroids and having a total mission

∆V within the limitations of the spacecraft. The ∆V breakdown of this trajectory, and its associated

interplanetary route, is shown in detail in Table 8.7.

Epoch [UTC] ∆V [km/s]

Interplanetary phase

Earth 01/04/2020 –Venus 21/09/2020 ∼ 10−10

Mercury 27/11/2020 ∼ 10−10

Venus 11/07/2022 0.0300

Science phase

2000QD225 16/02/2025 0.41752010UQ91 24/08/2025 0.50972008KT37 26/11/2025 0.41022012QV34 19/01/2026 0.43571997UL16 28/01/2026 0.58722002GO29 30/03/2026 –

Table 8.7: Detailed breakdown of encounter epochs and ∆V ’s associated with the best trajectory ob-tained for the whole mission.

As curiosity, the observation times of each individual asteroid in the sequence are shown in Table 8.8.

These represent the amount of time the spacecraft spent within a certain distance from each asteroid.

Five of those distances were considered, since no information about the asteroids’ individual masses

and radii was available. The durations listed are, relatively to the overall length of the mission, almost in-

significant, but this was already expected due to the nature of the observation strategy. To provide some

perspective, in Stuart’s work [7], which dealt with an asteroid rendezvous tour (i.e. the spacecraft would

”follow” an asteroid’s orbit until it could hop onto another one’s, and so on), the total observation times

were, at minimum, 450 days. This occurred for sequences of 5 asteroids. However, as mentioned in

Section 2.4, it was not possible to implement a similar observation strategy, since the required ∆V would

largely exceed the reasonable limit set for this mission by 0.5 km/s or more, if high thrust propulsion

systems were to be utilized.

72

Maximum distance [km] 100 200 300 400 500

Asteroids Observation time [s]

2000QD225 34 74 119 159 2042010UQ91 37 67 97 132 1622008KT37 27 57 82 112 1422012QV34 24 49 79 99 1291997UL16 26 46 76 96 1262002GO29 23 43 68 93 113

Total 171 336 521 691 876

Table 8.8: Observation times for each individual asteroid in the trajectory of Fig. 8.5, and for multiplemaximum distances to the objects.

8.4 Return DSM

2 3 4 5 6Number of asteroids in sequence

170

180

190

200

210

220

230

240

250

260

E [deg]

Figure 8.3: Eccentric anomaly values at the end of flyby tours with 2, 3, 4, 5 and 6 asteroids in thesequence.

In this section, the possibility of studying a DSM that would take the spacecraft back to the asteroid

cloud will be discussed. It was this thesis initial goal to study that type of maneuver after the spacecraft

had performed its first asteroid tour. The intent was to possibly return to the asteroid cloud and perform

a second flyby tour, thus maximizing the number of objects observed. Such would be feasible to perform

taking into account the low durations of the trajectories inside the swarm. However, by plotting the best

trajectories of Tables 8.3 to 8.6, plus several other solutions that were not fit enough to be presented

in Section 8.3, it was clear that such a DSM might not be possible. This is due to the fact that all of

the probed solutions’ trajectories observed their last asteroid after having passed the aphelion, which

means that the spacecraft would be heading towards the Sun, with no means of returning to the cloud

73

without spending a large ∆V.

Continuing with the investigation, all of the trajectories whose total ∆V was less than 3.5 km/s were

analyzed regarding their eccentric anomaly, E, at the end of their respective asteroid tours. The results

are shown in Fig. 8.3 for asteroid tour sequences of 2, 3, 4, 5 and 6 objects. It can be seen that the

eccentric anomaly of the obtained solutions is always above 180 , which means that all of the tours that

resulted from the optimization algorithm ended after the heliocentric apocenter. In other words, after

finishing its first asteroid tour of the Trojan swarm, the spacecraft is always heading towards the Sun.

One could argue that, with the trajectories whose final eccentric anomaly is close to 180 it would

be possible to perform such a maneuver, since the spacecraft ends its tour very close to its apocenter.

However, if the spacecraft is ending its tour near the aphelion, then this point is inside the swarm (or

very close to it), which means that it is not possible to return to the Trojan system after the asteroid tours

of Section 8.3. And if it was feasible to target an additional object with a DSM, the optimization algorithm

used in Section 8.3 would have already calculated it, since its boundary values (i.e. asteroid-to-asteroid

travel duration) were very conservatively set.

1 2 3 4 5 6 7Number of asteroids in sequence

2

4

6

8

10

12

14

Vel

ocity

at t

he la

st e

ncou

nter

[km

/s]

Figure 8.4: Velocity values of the spacecraft at the last asteroid encounter as a function of the numberof objects observed.

Given the information of Fig. 8.3, it is then clear that the only way of returning to the asteroid cloud

is for the spacecraft to turn (about) 180 . The necessary ∆V for such maneuver would be double

the spacecraft’s instantaneous velocity at the point of the last encounter, since this occurred after the

aphelion. The velocity values at the spacecraft’s last encounter are presented in Fig. 8.4, as a function

of the number of asteroids seen in the first tour. With a minimum end velocity of 3.7868 km/s, that 180

DSM would require at least 7.5736 km/s of ∆V, which is unfeasible to achieve with the high thrust system

described in Section 2.3. This was already expected, seeing as this type of maneuver, which essentially

inverts the orbital angular momentum of the spacecraft, is never utilized in any circumstance. However,

for the sake of completeness, it was presented in this report.

It is thus concluded that no return DSM is feasible for the asteroid tours achieved. The tendency

74

observed in Fig. 8.3, in terms of end-of-tour eccentric anomaly, is thought to be a direct consequence

of the choice of interplanetary transfer. In Fig. 6.4 it could be seen that the spacecraft would encounter

the L4 point after passing its apocenter. Although L4 and the aphelion were not significantly distant,

at the time, despite this fact, it was thought that, when investigating asteroid tours, there would be

some solutions that would observe their last asteroid before passing the apocenter. Unfortunately, such

assumption was wrong and consequently only one asteroid tour was possible to be performed.

Concluding, the final designed trajectory, with only one asteroid flyby tour of 6 objects, can be seen

in Fig. 8.5, in the ecliptic J2000 reference frame.

−6 −4 −2 0 2 4 6X [AU]

−6

−4

−2

0

2

4

6

Y [AU]

earth 2020-Apr-01

venu 2020-Sep-21

mercury 2020-Nov-27

venu 2022-Jul-11

jupiter 2025-Feb-16

(a) Full trajectory.

−5.0 −4.5 −4.0 −3.5 −3.0X [AU]

2.0

2.5

3.0

3.5

4.0

Y [AU]

earth 2020-A r-01

venus 2020-Se -21

mercury 2020-Nov-27

venus 2022-Jul-11

ju iter 2025-Feb-16

(b) Close-up inside the swarm.

Figure 8.5: Final designed orbit for the full mission, departing from Earth on April 1, 2020 and encoun-tering 6 asteroids. All ephemeris are presented in an ecliptic J2000 frame.

75

76

Chapter 9

Conclusions and Recommendations

9.1 Conclusions

This thesis’ goal was to design the trajectories for a mission to the Trojan asteroids. Two different

observational strategies (i.e. single asteroid rendezvous and flyby tour) were defined in Section 2.4, by

comparing literature studies with the high thrust propulsion system. These two strategies resulted in two

different designs, with two very distinct trajectories.

For the single asteroid rendezvous mission, the spacecraft targeted 624 Hektor, inserting itself into

its orbit, similarly to what ESA’s Rosetta did with Comet 67P [11]. Optimization of multiple gravity assist

trajectories, of the MGA and MGA-1DSM types, lead to a final MGA solution with a EMJ planetary

sequence, requiring 2.4672 km/s of on-board ∆V and 13.4 years of travel time. With a 10% margin for

on-board ∆V, this trajectory resulted in a propellant-to-wet mass ratio 1.8% larger than Rosetta’s [11].

In the case of an asteroid flyby tour, the spacecraft’s trajectory was designed to encounter several

different Trojan asteroids along its way. As a point of reference for this type of mission, the results found

in Canalias et al. [2] were used. Due to the high number of possible planetary and asteroid sequences,

the mission (and its optimization) was divided into an interplanetary phase and a science phase. The first

dealt with all trajectories to the Trojan swarm, using optimization models and algorithms identical to the

ones of the single rendezvous mission. Naturally, the science phase dealt with all scientific observation

trajectories. This phase’s optimization model was adapted from the MGA formulation, thus allowing

the implementation of similar optimization algorithms and processes, reducing the complexity of the

problem. After the optimization took place, the final designed trajectory consisted of an interplanetary

MGA path, with an EVMerV sequence, leading to the observation of six asteroids (one more asteroid

than Canalias’ top result [2]). In total, this solution required 2.3903 km/s of on-board ∆V, representing a

propellant-to-wet mass ratio 6.8% smaller than Rosetta’s [11].

Continuing the comparisons with Canalias et al. [2], the extensive survey of planetary sequences

in the interplanetary phase lead to trajectories with 200–400 m/s less total ∆V, and to asteroid tours

that save 400 m/s or more of on-board ∆V. Although these improvements were not an explicit goal of

this work, they validate the optimization processes used, and the better efficiency of the asteroid tour’s

77

science phase’s model, developed by this work’s author.

Additionally, although it was not an objective of this thesis, the survey of MGA optimization algorithms

(and their tuning) is also considered to be an important achievement of this work, since it serves as an

important bibliographical source for future optimization projects. Such was not existent in the literature

previously to this work, so its formalization and publication with this thesis is important to fill that gap.

In conclusion, the feasible trajectories designed for both missions (i.e. single rendezvous with 624

Hektor and flyby tour), together with their improvements with respect to Canalias’ results [2], are consid-

ered to be important achievements of this work. Considering that the top-level objective of this thesis was

achieved, and that the solutions are feasible to be applied to real space missions, the work developed is

regarded as a success.

9.2 Recommendations

In this section, recommendations for future work based on, or continuing, the progress done in this

report, are presented. These will be divided in topics and areas, for easier consultation.

Asteroid flyby tour

• The impact of launcher performance should be accounted for in the optimizations of interplanetary

trajectories, in a more flexible way. For example, with the a priori choice of a specific launcher

vehicle, it is possible to parameterize the Earth’s excess velocity as a function of the launch mass,

allowing for the computation of the arrival mass to L4. This mass could substitute the ∆V consid-

ered in Section 6.2, resulting in better optimal solutions.

• The number of asteroids could be improved if, instead of targeting the L4 (roughly half point of the

swarm) in the interplanetary phase, the trajectory aimed at the half (or third) of the Trojan swarm

that is closer to Jupiter. This would increase the chances of having tours that started in that area,

and traveled throughout the whole swarm, from end to end.

• If two asteroid tours are desired, then it is recommended that the interplanetary transfer’s opti-

mization takes into account the true anomaly of arrival to the L4 point. This quantity determines

then the direction at which the spacecraft is more likely to find asteroid sequences the first time

it passes through the swarm. In this work, that was not taken into account, so the final optimal

asteroid sequences all ended with the satellite past its aphelion.

• Better total ∆V’s could be achieved if the DSM’s position, between two consecutive asteroids, is

also incorporated into the optimization model. Initial tests (not included in this report for simplicity

sake) suggested that considering it did not have an impact. However, for other interplanetary

trajectories or approach conditions to the Trojan swarm, this may not be the case, and so this

recommendation is done here.

78

• Low thrust trajectories should also be investigated in the future, as they may allow better approach

conditions to the Trojan swarm and asteroid tours. In order to keep the interplanetary travel time

short, the use of low thrust could be limited to the science phase, increasing the complexity of the

spacecraft.

Single asteroid rendezvous

• Given the final ∆V results of Section 6.3, where various solutions violated the defined limit for the

mission, it is suggested to choose of low thrust for the rendezvous of 624 Hektor. The use of a low

thrust system would increase the maximum allowed

• Instead of opting for low thrust, like suggested in the previous bullet, an alternative would be to get

an accurate estimate of the spacecraft’s dry mass, before starting the optimizations. This could

lead to a higher ∆V limit, allowing for other solutions to be now valid.

• With high thrust, the optimization could be done for the on-board ∆V, with the excess velocity from

Earth as a secondary objective. This could show improvements, since the optimization algorithm

would focus on minimizing the propellant ratio on the spacecraft, allowing a Pareto front to be

studied in the end. Instead of a Pareto front, multi-objective optimization could be used.

79

80

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