Attitude and Orbit Control of a Solar Sail Through ...€¦ · Attitude and Orbit Control of a...

POLITECNICO DI MILANO

MASTER THESIS

Attitude and Orbit Control of a Solar SailThrough Distributed Reflectivity

Modulation Devices

Author:Alessio NEGRI

ID:874935

Supervisor:Dr. James Douglas BIGGS

A thesis submitted in fulfillment of the requirementsfor the degree of Master of Science

in

Space Engineering

Department of Aerospace Science and Technology

School of Industrial and Information Engineering

Academic Year - 2017/2018

http://www.polimi.ithttp://www.aero.polimi.ithttp://www.aero.polimi.ithttp://www.ingindinf.polimi.it

“Scientific progress is the discovery of a more and more comprehensive simplicity.”

“Il progresso scientifico consiste nella scoperta di una semplicità sempre più esaustiva.”

Georges Lemaître

“I can calculate the motion of heavenly bodies, but not the madness of people.”

“Posso calcolare i movimenti dei corpi celesti ma non la follia della gente.”

Isaac Newton

“I never think of the future: it comes soon enough.”

“Non penso mai al futuro: esso arriva fin troppo presto.”

Albert Einstein

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Declaration of AuthorshipI, Alessio NEGRI, declare that this thesis titled, “Attitude and Orbit Control of a SolarSail Through Distributed Reflectivity Modulation Devices” and the work presentedin it are my own. I confirm that:

• This work was done wholly or mainly while in candidature for a research de-gree at this University.

• Where I have consulted the published work of others, this is always clearlyattributed.

• Where I have quoted from the work of others, the source is always given. Withthe exception of such quotations, this thesis is entirely my own work.

• I have acknowledged all main sources of help.

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POLITECNICO DI MILANO

AbstractSchool of Industrial and Information EngineeringDepartment of Aerospace Science and Technology

Master of Science

Attitude and Orbit Control of a Solar Sail Through Distributed ReflectivityModulation Devices

by Alessio NEGRI

Solar sails are an enabling technology that utilizes photons from the Sun for fuel-free propulsion. However, one of the major challenges is the design of an efficientattitude control that is able to precisely and continuously re-point the thrust vectorin the required direction for orbit control. One method that has been flight tested onIKAROS is the use of devices capable of changing the reflectivity properties of thesail.

This thesis proposes a new approach to attitude and orbit control of a solar sailwith pixelated reflectivity control devices (RCDs) and a logic based on combinationsof their state (ON/OFF). A control law that maps an ideal in-plane control torque tothe reflectivity property of each pixel is derived, so that ideal controls that guaranteestability can be mimicked. It is shown, as the number of pixels of the sail membraneis increased, the real torque will converge to the ideal one. Different ideal attitudecontrol laws are compared, these include an under-actuated, a reduced, and a stan-dard proportional control law. The results show that this system succeeds in copyingunder-actuated and reduced controls, with good performances in terms of attitudeangles accuracy, torque requirement, and mimic ability. However, the system is notable to copy the proportional control.

The RCD-based control is extended to incorporate orbit tracking, mimicking anideal acceleration without introducing torques. To this end, a Linear Quadratic Reg-ulator orbit control is introduced which yields the time history of the attitude anglesrequired for station-keeping on a libration point orbit. The resulting coupled orbit-attitude control is mapped optimally to ON/OFF states to each RCD pixel: for theattitude, only the reduced control is analysed. An example station-keeping problemin the solar sail circular restricted three-body problem is demonstrated in simula-tion with the aim of stabilizing the sail on an artificial equilibrium point. The resultsshow that the sail asymptotically converges to a neighbourhood of the desired loca-tion, where the boundary of this neighbourhood can be decreased by increasing thenumber of RCD pixels.

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Le vele solari sono una tecnologia che permette di utilizzare i fotoni dal Sole per unapropulsione senza carburante. Tuttavia, una delle maggiori sfide è il design di un controllod’assetto efficiente capace di puntare, precisamente e continuamente, il vettore spinta nelladirezione richiesta dal controllo orbitale. Un metodo che è stato testato in volo su IKAROSconsiste nell’utilizzo di congegni capaci di cambiare le proprietà di riflettività della vela.

Questa tesi propone un nuovo approccio al controllo d’assetto e orbitale di una vela solarecon dispositivi a controllo di riflettività (RCDs) scomposti in pixel e una logica basata sullecombinazioni dei loro stati (ON/OFF). Viene derivata una legge di controllo che mappa unacoppia ideale nel piano della vela alle proprietà di ciascun pixel, così che i controlli ideali chegarantiscono stabilità possano essere imitati. Viene mostrato che, con il crescere del numerodi pixel, la coppia reale converge a quella ideale. Differenti leggi di controllo d’assetto sonocomparate, includendo una legge di controllo sotto-attuata, una ridotta e una proporzionalestandard. I risultati mostrano che questo sistema riesce a copiare i primi due con buoneprestazioni in termini di accuratezza degli angoli d’assetto, richiesta di coppia e abilità dimimica. Tuttavia, il sistema non è in grado di copiare il controllo proporzionale.

Il controllo basato sugli RCD è esteso per incorporare il tracciamento dell’orbita, simu-lando un’accelerazione ideale senza introdurre coppie. A questo fine, un controllo orbitalebasato su un regolatore quadratico lineare viene introdotto, producendo la storia temporaledegli angoli d’assetto richiesti per stazionare su un’orbita di un punto di librazione. Il con-trollo accoppiato risultante viene mappato in maniera ottimale agli stati ON/OFF di ciascunpixel: per l’assetto, solo il controllo ridotto è analizzato. Un problema di stazionamento vieneusato come esempio, con lo scopo di stabilizzare la vela su un punto di equilibrio artificiale. Irisultati mostrano che la vela converge asintoticamente a un intorno del punto desiderato, icui limiti si possono ridurre aumentando il numero di pixel.

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AcknowledgementsFirst, I would like to thank my supervisor Dr. James D. Biggs for having pro-

posed me a thesis about solar sails since I have been always interested in this im-portant topic, especially nowadays. Thanks for his suggestions and ideas that haveimproved my work, and the extreme cordiality and availability. I am also grateful toDr. Matteo Ceriotti for having checked and analysed my thesis, since was importantfor me the opinion of specialists about my work. Thanks also to all the professorsthat during these years have instilled in me the passion for space and engineering.

I want to thank my brother and all my friends for having encouraged me duringthese years at the university.

The most important acknowledge goes to my mum and dad that gave me thepossibility to study and becoming an engineer. They have supported me both phys-ically and morally for all my life, especially in these last years. I will always begrateful for all theirs sacrifices that allowed me becoming the man that I am today.

Prima di tutto, vorrei ringraziare il mio supervisore Dr. James D. Biggs per avermiproposto una tesi sulle vele solari visto che sono sempre stato interessato a questo importanteargomento, specialmente oggigiorno. Grazie per i suoi suggerimenti e le sue idee che hannomigliorato il mio lavoro e l’estrema cordialità e disponibilità. Sono grato anche al Dr. MatteoCeriotti per aver controllato e analizzato la mia tesi, poiché per me era importante l’opinionedi specialisti riguardo il mio lavoro. Grazie anche a tutti i professori che nel corso di questianni mi hanno instillato la passione per lo spazio e l’ingegneria.

Voglio ringraziare mio fratello e tutti i miei amici per avermi incoraggiato durante questianni all’università.

Il riconoscimento più importante va a mia madre e mio padre che mi hanno dato la possi-bilità di studiare e diventare ingegnere. Mi hanno supportato sia fisicamente che moralmenteper tutta la mia vita, specialmente in questi ultimi anni. Gli sarò sempre grato per tutti i lorosacrifici che mi hanno permesso di diventare l’uomo che sono oggi.

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Contents

Declaration of Authorship i

Abstract iii

Acknowledgements v

List of Figures xii

List of Tables xiii

List of Abbreviations xv

Physical Constants xvii

List of Symbols xix

1 Introduction 11.1 History of Solar Sails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Recent Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Space Flight Mechanics Meeting 2018 . . . . . . . . . . . . . . . 61.3.2 AIAA SciTech 2019 Forum . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Reflectivity Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Functioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Thesis’s Novelty . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Thesis Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Solar Sail Models and Dynamics 152.1 Solar Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Electromagnetic Theory . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Force Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Equations Of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Orbit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

viii

3 Attitude Dynamics and Kinematics 273.1 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Attitude Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.3 w-z Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Reflectivity Control Device-Based Attitude Control 414.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Ideal Attitude Control Laws 575.1 Proportional Derivative Control . . . . . . . . . . . . . . . . . . . . . . . 575.2 Bore-sight Guidance and Control . . . . . . . . . . . . . . . . . . . . . . 595.3 Under-actuated Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.1 Kinematics Control . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3.2 Dynamics Control . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.3 Alternative Formulation . . . . . . . . . . . . . . . . . . . . . . . 64

6 Simulations of the Attitude Dynamics 676.1 Ideal Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Real Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 Controls Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Sun-Earth-Sail System Dynamics 817.1 Circular Restricted Three-Body Problem . . . . . . . . . . . . . . . . . . 81

7.1.1 Equations Of Motion Without Solar Radiation Pressure . . . . . 817.1.2 Equations Of Motion With Solar Radiation Pressure . . . . . . . 84

7.2 Artificial Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . 877.3 Ideal Orbit Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3.1 Linearised Equations of Motion . . . . . . . . . . . . . . . . . . . 907.3.2 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8 Reflectivity Control Device-Based Orbit Control 938.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.2.1 Trajectory Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.2.2 Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9 Simulations of the Coupled Dynamics 1059.1 Ideal Coupled Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.3 Real Coupled Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

10 Conclusions 117

A Sun & Sail Realistic Models 123A.1 Limb-Darkened Solar Disc . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.2 Finite Solar Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.3 Complete Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.4 Solar Wind Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

ix

B Partial Derivatives of the Pseudo-Potential 127

C Partial Derivatives of the Acceleration in Position 129

D Partial Derivatives of the Acceleration in Orientation 131

Bibliography 133

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List of Figures

1.1 Time-line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Reflectivity control devices typologies. . . . . . . . . . . . . . . . . . . . 9

2.1 Solar radiation pressure force models. . . . . . . . . . . . . . . . . . . . 172.2 Solar radiation pressure force geometry for a non-perfect, flat Lam-

bertian solar sail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Specular reflective + absorptive sail model. . . . . . . . . . . . . . . . . 192.4 Two-body problem geometry. . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Inertial N and rotating L frames (2BP). . . . . . . . . . . . . . . . . . . . 232.6 Rotating L and body-fixed B frames (2BP). . . . . . . . . . . . . . . . . 242.7 Designed orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Euler angles: sequence 1-2-1. . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Rotations and geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Stereographic projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Reflectivity control device-based attitude control logic scheme. . . . . . 424.2 Mesh with regular (•) and un-regular (•) positions (n = 4). . . . . . . . 484.3 Multiples of Ã for regular elements. . . . . . . . . . . . . . . . . . . . . 494.4 Multiples of Ã for un-regular elements. . . . . . . . . . . . . . . . . . . 494.5 Example of attitude control torque generation:

yellow pixels⇒ ON not used, orange pixels⇒ ON for attitude con-trol, grey pixels⇒ OFF for attitude control. . . . . . . . . . . . . . . . . 51

4.6 Available control torques. . . . . . . . . . . . . . . . . . . . . . . . . . . 524.7 Ideal attitude control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.8 Real attitude control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.9 Global surface usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.10 Local surface usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1 Control diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Regions of high (D1) and low (D2) control torque. . . . . . . . . . . . . 64

6.1 Cone and clock angles - ideal (2BP). . . . . . . . . . . . . . . . . . . . . 696.2 Angles error - ideal (2BP). . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3 Attitude control torque - ideal (2BP). . . . . . . . . . . . . . . . . . . . . 716.4 Under-actuated control methods comparison. . . . . . . . . . . . . . . . 736.5 Angles error - real (2BP). . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.6 Attitude control torque - real (2BP). . . . . . . . . . . . . . . . . . . . . . 766.7 Global surface usage (2BP). . . . . . . . . . . . . . . . . . . . . . . . . . 776.8 Local surface usage (2BP). . . . . . . . . . . . . . . . . . . . . . . . . . . 786.9 |ζ|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.1 Circular restricted three-body problem geometry. . . . . . . . . . . . . . 817.2 Rotating L and body-fixed B frames (CR3BP). . . . . . . . . . . . . . . . 84

xii

7.3 Euler angles: sequence 1-3-1. . . . . . . . . . . . . . . . . . . . . . . . . 857.4 Artificial equilibrium points: x− y plane (ρs = 0.9). . . . . . . . . . . . 887.5 Artificial equilibrium points: x− z plane (ρs = 0.9). . . . . . . . . . . . 89

8.1 Reflectivity control device-based orbit control logic scheme. . . . . . . 948.2 Solar exclusion zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.3 Designed trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.4 Relative position - ideal (CR3BP). . . . . . . . . . . . . . . . . . . . . . . 1008.5 Cone and clock angles - ideal (CR3BP). . . . . . . . . . . . . . . . . . . . 1008.6 Angles error - ideal (CR3BP). . . . . . . . . . . . . . . . . . . . . . . . . 1008.7 Time step influence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.8 Surface usage for orbit control. . . . . . . . . . . . . . . . . . . . . . . . 1038.9 Acceleration error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.1 Trajectory - ideal coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.2 Relative position - ideal coupling. . . . . . . . . . . . . . . . . . . . . . . 1079.3 Cone and clock angles - ideal coupling. . . . . . . . . . . . . . . . . . . 1089.4 Angles error - ideal coupling. . . . . . . . . . . . . . . . . . . . . . . . . 1089.5 Attitude control torque - ideal coupling. . . . . . . . . . . . . . . . . . . 1099.6 ω1(0) influence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.7 Trajectory - real coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . 1139.8 Relative position - real coupling. . . . . . . . . . . . . . . . . . . . . . . 1139.9 Cone and clock angles - real coupling. . . . . . . . . . . . . . . . . . . . 1149.10 Angles error - real coupling. . . . . . . . . . . . . . . . . . . . . . . . . . 1149.11 Attitude control torque - real coupling. . . . . . . . . . . . . . . . . . . . 1159.12 Surface usage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

10.1 Torque regions logic: low (green), medium (orange), and high (red). . . 118

A.1 Limb darkening geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.2 Deviation from the inverse square law. . . . . . . . . . . . . . . . . . . . 124A.3 Complete force model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xiii

List of Tables

1.1 RCD-based papers comparison: X = used, × = not used, n.d. = notdeclared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Solar sail orbit parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 2BP reference orbit parameters. . . . . . . . . . . . . . . . . . . . . . . . 252.3 2BP reference orbit solver parameters. . . . . . . . . . . . . . . . . . . . 25

3.1 Solar sail attitude parameters. . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 ON/OFF combinations: in red the ones adopted for attitude control,in blue the ones possibly usable for an orbit control. . . . . . . . . . . . 44

4.2 Mesh precision choice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Initial conditions for the attitude (2BP). . . . . . . . . . . . . . . . . . . 676.2 Control gains for the attitude (2BP). . . . . . . . . . . . . . . . . . . . . 676.3 Ideal controls solver parameters (2BP). . . . . . . . . . . . . . . . . . . . 686.4 Control gains for different under-actuated controls, with κc = 0.9×

10−3, µc = 2 κc, and ρ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5 Ideal controls comparison: � = very good, ↑ = good,

↓ = bad,� = very bad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.1 CR3BP reference trajectory parameters. . . . . . . . . . . . . . . . . . . 978.2 CR3BP reference trajectory solver parameters. . . . . . . . . . . . . . . 98

9.1 Initial conditions and control gains for the attitude (CR3BP). . . . . . . 1059.2 Coupled problem solver parameters (CR3BP). . . . . . . . . . . . . . . 106

xv

List of Abbreviations

2BP Two-Body Problem

3BP Three-Body Problem

ACS3 Advanced Composites-based Solar Sail System

AEP Artificial Equilibrium Point

AIAA American Institute of Aeronautics and Astronautics

A-RCD Advanced-Reflectivity Control Device

COM Centre Of Mass

CR3BP Circular Restricted Three-Body Problem

EM Exploration Mission

ESA European Space Agency

GA Genetic Algorithm

GEO Geosynchronous Equatorial Orbit

HCI HelioCentric Inertial

I.C. Initial Condition

JPL Jet Propulsion Laboratory

LCD Liquid Crystal Display

LQR Linear Quadratic Regulator

MOI Moment Of Inertia

NASA National Aeronautics and Space Administration

NMP New Millennium Program

NOAA National Oceanic and Atmospheric Administration

PD Proportional Derivative

xvi

PDLC Polymer-Dispersed Liquid Crystal

PID Proportional Integral Derivative

PVSE Presidential Vision for Space Exploration

RCD Reflectivity Control Device

SLS Space Launch System

SNOPT Sparse Non-linear OPTimizer

SRP Solar Radiation Pressure

ST Space Technology

SWP Solar Wind Pressure

TRL Technological Readiness Level

xvii

Physical Constants

Astronomical Unit AU = 1.495 978 707× 108 km

Free space permeability µ0 = 4 π 10−7 H m−1 [T m−1 A−1]

Free space permittivity e0 = 8.854 187 817 6× 10−12 F m−1

Mass of Earth M♁ = 5.9736× 1024 kg

Plank’s constant h = 6.626 069 57× 10−34 J s

Proton mass mp = 1.672 622× 10−27 kg

Solar gravitational constant µ� = 1.3272× 1011 km3 s−2

Solar luminosity L� = 3.832× 1026 W

Solar mass M� = 1.9891× 1030 kg

Solar radius R� = 696 000 km

Universal gravity constant G = 6.672 59× 10−20 kg km3 s−2

Vacuum speed of light c = 2.997 924 58× 108 m s−1

xix

List of Symbols

CHAPTER 1

L1,2,3 On-axis Lagrangian points

L4,5 Off-axis Lagrangian points

ne Extraordinary refractive index

no Ordinary refractive index

np Polymer matrix refractive index

CHAPTER 2

a Acceleration vector m s−2

A Surface area normal to the incident radiation m2

A Attitude matrix between two frames

B Magnetic field T [Wb m−2]

E Electric field V m−1

E Photon’s energy J

f Frequency rad d−1

F Force vector N

h Orbital angular momentum vector km2 s−1

m Mass of the sail-craft kg

m Real sail SRP resultant force direction

xx

n Outward sail normal

p Photon’s momentum kg m s−1

pem Electromagnetic momentum vector Pa/(m/s)

P Solar radiation pressure N m−2

r Sun-Earth distance km

r Sun-sail position vector km

r1 Sun position vector km

r2 Sail-craft position vector km

R COM position vector km

s Outward Sun direction

S Solar sail surface area m2

S Poynting vector W m−2

t Time s

t Sail tangent unit vector

ux Direction of propagation of the wave

U Electromagnetic energy density J m−3

v Velocity vector km s−1

W Energy flux W m−2

x, y, z Position coordinates of the sail in N km

X, Y, Z Orthogonal axes of a right-handed set

α Cone angle rad

αopt Optimal cone angle rad

α̃ Required cone angle rad

β Lightness number

xxi

γ Roll angle rad

δ Clock angle rad

δopt Optimal clock angle rad

δ̃ Required clock angle rad

θ Effective cone angle rad

ν Photon’s frequency Hz

ρa Absorptivity coefficient

ρd Diffuse reflectivity coefficient

ρs Specular reflectivity coefficient

ρt Transmittivity coefficient

σ Sail loading parameter g m−2

σ∗ Critical sail loading parameter g m−2

Ψ0 Constant amplitude deg

Ψ1 Variable amplitude deg

ω Angular velocity vector between two frames rad s−1

CHAPTER 3

a, b, c Coefficients of the w− z parametrization

b̂ Body frame

B Euler angles kinematics matrix

c Non-dimensional moment of inertia

C Rotation matrix

C Complex variable

e Principal axis

F Generic frame

xxii

H Angular momentum vector kg m2 s−1

i Imaginary unit

î Inertial frame

I Identity matrix

J Moment of inertia kg m2

J Moment of inertia tensor kg m2

m Mass kg

n Number of masses

q Quaternion

r Position vector m

R Attitude matrix

R1 Rotation matrix for z

R2 Rotation matrix for w

R Real variable

S Inverse of B

S2 Unit sphere

u Torque vector N m

û Direction perpendicular to î′1 and b̂1

w Second parameter

x1, x2, x3 Coordinates of the sphere

z First parameter

ρ Cayley-Rodrigues parameter

σ Modified Cayley-Rodrigues parameter

τ Control torque s−2

xxiii

υ Stereographic projection

Φ Principal angle rad

ω Complex angular velocity in the body frame rad s−1

CHAPTER 4

Ã Area of the RCD/element/pixel m2

CP Centre of pressure vector m

D Depth matrix

f Fraction of usage

L Side length of a quadrant m

n Order of the mesh

nÃ Multiple of Ã

N Number of rows/columns of the mesh

Ne Number of elements/pixels contained in a quadrant

P Solar radiation pressure vector N m−2

S̄ Surface area of a quadrant m2

Y Coordinate of the centre of pressure along YB m

Ȳ Y coordinate for a quadrant m

Z Coordinate of the centre of pressure along ZB m

Z̄ Z coordinate for a quadrant m

CHAPTER 5

a Spacecraft bore-sight axis

b Reference direction

xxiv

D Derivative gain matrix kg m2 s−1

D Region in the zr and |wr|2 plane

kd Derivative constant gain kg m2 s−1

kD Derivative constant gain kg m2 s−1

kp Proportional constant gain kg m2 s−2

kP Proportional constant gain kg m2 s−2

n Unit vector perpendicular to a and b

P Proportional gain matrix kg m2 s−2

ζ Closed-loop dumping

θ Angle between a and b rad

θe Angle error rad

θr Angle of the cone from b rad

κ First kinematics proportional constant gain s−1

κc Constant gain s−1

λ Dynamics proportional constant gain s−1

µ Second kinematics proportional constant gain s−1

µc Constant gain s−1

ξ Ratio between zr and |wr|2

ρ Constant parameter

τ Complex control torque s−2

ωn Closed-loop natural radial frequency rad s−1

ω̃ Angular velocity of the kinematics control rad s−1

ωr Reference angular velocity rad s−1

xxv

CHAPTER 6

e Smoothing parameter s−2

CHAPTER 7

0 Zero matrix

a Non-dimensional acceleration vector

A State matrix

B Matrix of partial derivatives

d1 Non-dimensional Sun position from the barycentre

d2 Non-dimensional Earth position from the barycentre

F Solution of the algebraic Riccati equation

G Control gain matrix

î, ĵ, k̂ Base unit vector in S

J Cost function

L Lagrangian point

LA Artificial equilibrium point

L∗ Characteristic length km

M1 Mass of the Sun kg

M2 Mass of the Earth kg

M3 Mass of the sail kg

M∗ Characteristic mass kg

n Sail normal

P Input matrix

xxvi

Q Weighting matrix of the state

r Non-dimensional value of R0

r1 Non-dimensional value of R1

r2 Non-dimensional value of R2

R Weighting matrix of the control

R0 Position of the sail from the origin km

R1 Position of the sail from the Sun km

R2 Position of the sail from the Earth km

T∗ Characteristic time s

U Pseudo-potential function

x, y, z Non-dimensional coordinates in S

x State vector

θ Control input vector rad

µ Mass ratio

ν Margin of stability

ξ, η, ζ Relative position with respect to an equilibrium point

Ω Angular velocity of the Sun-Earth system rad s−1

CHAPTER 8

a, b, c Coefficients of a polynomial

a Acceleration vector mm s−2

n Number of pixels

xxvii

APPENDIX A

B Non-Lambertian coefficient

f Component of Fn

F Deviation factor

I Specific Intensity W m−2

m Mass kg

n Mean number density cm−3

P Pressure N m−2

P∗ Inverse square law pressure N m−2

r Sun-sail distance km

r̃ Front surface reflectivity

s Specular reflective part of r̃

v Velocity km s−1

e Surface emission coefficient

λ Angle between Sun surface normal and sail position rad

xxviii

SUBSCRIPTS & SUPERSCRIPTS

0 At 1 AU

10 Constant value around XB

1 : 3 Components 1, 2, 3

1, 2, 3 Components of a vector in B

1, 2, 3, 4 Quadrants of the sail

a× b Matrix of a rows and b columns

A Attitude

b Back surface

B Body-fixed frame

c Commanded

C Centre of mass

d Desired

e Error

ext External

f Front surface

id Ideal

i, j, l, k, m, h Indexes

i O From O to i (similarly for the others)

L Orbit frame

Lagrangian or equilibrium point

LN To pass from N to L (similarly for the others)

LQR Using the LQR control

max Maximum value

xxix

n Normal component

N Inertial frame

N′ Inertial frame of support

NU Not-used

O Origin of the N frame

Orbit

OFF Off state

ON On state

p Proton

r Relative

RCD Using the RCD-based control

reg Regular

S Synodic frame

SRP Solar Radiation Pressure

t Tangential component

unreg Un-regular

w Solar wind

x, y, z Components of a vector

Partial derivatives

α Derivative with respect to the cone angle

δ Derivative with respect to the clock angle

′,′′ Intermediate

∗ Evaluated at the equilibrium point

xxx

OPERATORS

x̄ Complex conjugate of a complex variable

|x| Absolute value

||x|| Norm of a vector (alternatively the symbol not in bold)

x̂ Unit vector

ẋ First time derivative

ẍ Second time derivative

< x, y > Scalar product

x∧ y Vector product

XT Transpose

tr(X) Trace of a matrix

[x×] Cross product matrix

min(x, y) Minimum between x and y

max(x, y) Maximum between x and y

∇ Gradient

∂ Partial derivative

∑ Summation

diag(x) Diagonal matrix from a vector

Dedicated to my family

1

1 Introduction

1.1 History of Solar Sails

The evolution of solar sails has followed different steps, the most important ofwhich are here listed in chronological order (Figure 1.1):

• Johannes Kepler postulated in 1619 the idea that sunlight exerts a pressure,and as a result creates the dust tail of comets [Kepler, 1619]: it was one of thefirst evidence of the pressure exerted by the photons coming from the Sun.

• Jules Verne published in 1865 the famous From the Earth to the Moon [Verne,1970], where he wrote

“[...] there will some day appear velocities far greater than these, of which light orelectricity will probably be the mechanical agent [...] we shall one day travel to the

moon, the planets, and the stars.”

• James Clerk Maxwell in 1864 deduced, and mathematically proved, that elec-tromagnetic radiation exerts a pressure [Maxwell, 1865].

• Adolfo Bartoli determined in 1876 the sunlight pressure, but from the secondlaw of thermodynamics [Bartoli, 1884].

• The sunlight pressure was proved experimentally by Pytor Lebedew in 1901[Lebedew, 1902], and Ernest Nichols and Gordon Hull in 1903 [Nichols andHull, 1903].

• The first publication, named Extension of Man into Outer Space, by the famousKonstantin Eduardovitch Tsiolkovsky, dates back to 1921 [Tsiolkovsky, 1921].He introduced the concept of solar sail with the following words

“[...] using tremendous mirrors of very thin sheets to utilize the pressure of sunlightto attain cosmic velocities.”

• Fridrickh Tsander was the first to publish a practical paper on solar sails in1924.

• Carl Wiley’s article in Astounding Science Fiction, 1951, discussed about feasibil-ity and design of solar sailing [Wiley, 1951]: he used the pseudonym of RussellSanders to avoid being recognized.

• Richard Lawrence Garwin’s first technical archival journal publication waspublished in 1958 [Garwin, 1958].

• The first proposal of attitude control of a solar sail was made by R. L. Sohn in1959 for an attitude stabilization problem [Sohn, 1959].

• In 1960, Echo-1 thin-film balloon felt solar pressure effects in orbit.

2 Chapter 1. Introduction

• Jerome L. Wright, from the Jet Propulsion Laboratory (JPL), made a proposal fora rendezvous mission with comet Halley in 1976, based on researches relatedto construction and trajectories of solar sails [Wright and Warmke, 1976]. Itwas the first serious mission study.

• Robert Lull Forward first proposed in 1991 the use of solar sails for stationarypositions (e.g., levitated non-Keplerian trajectories) [Forward, 1991].

• Colin Robert McInnes extensively studied the dynamics of solar sails from the1990’s, with works on the restricted three-body problem and the relative con-trol, and the introduction of real solar pressure models [McInnes and Macpher-son, 1991, McInnes et al., 1994, McInnes, 1998], finally culminating with itsfamous book [McInnes, 1999].

• Julia L. Bell examined the Sun-Earth libration point orbits with the solar radi-ation pressure, in 1991 [Bell, 1991].

• Robert Lull Forward patented a satellite with a solar sail that hovers staticallyabove a planetary pole, called Statite concept, in 1993.

• Solar sail concepts started to be funded by the National Aeronautics and SpaceAdministration (NASA) for the New Millennium Program (NMP) in 1995, withthe aim of increasing its Technological Readiness Level (TRL).

• Jason S. Nuss formulated, in 1998, the study of solar sails in the three-bodyproblem framework [Nuss, 1998].

• More recently, in 2004, the Presidential Vision for Space Exploration (PVSE) mo-tivated the development of solar sails for communications Moon-Earth for thefuture 2020 human mission to the Moon, prior to a mission to Mars. For a deepanalysis of solar sails dynamics in the Earth-Moon system consider [Wawrzy-niak, 2011]. The idea is that multiple spacecraft in Keplerian/non-Keplerianorbits around the Moon can be substituted by a single sail-craft positioned asa relay between a lunar outpost and the ground station on Earth, working asa polesitter. NASA has also proposed a sail-craft for the lunar communicationcoverage problem.

Solar sails apply the same principle of comets’ tail, but on a larger surface, whichis able to reflect the incoming photons. It is an alternative propulsion system thatcan extend the possibilities of moving throughout the solar system. This is basedalso on its peculiarity, which is the absence of propellant: the limits are imposed bythe life of the materials. In addition, trajectory and attitude are hugely coupled for asolar sail.

Solar sails can be either spin-stabilized or three-axis stabilized. Typically, theformer have three advantages on the latter: easier angular momentum management,safer voyage for Sun-racking motion, larger acceleration. However, they requiremuch more fuel to change the attitude and they are not suitable for a continuousattitude changing.

Classical control systems, such as wheels and thrusters, are not practical for asolar sail since they introduce additional mass and the latter use also expendablemass. So, other control mechanisms are available: gimballed control booms, controlvanes, control-mass translation, reflectivity modulation. More details can be foundin [McInnes, 1999]. In addition, a sail is characterized by a flexible structure anduncertainties on the solar pressure model.

1.1. History of Solar Sails 3

1619 J. Kepler

1865J. Verne

1619J. C. Maxwell

1865A. Bartoli

1901P. Lebedew

1903E. Nichols

G. Hull

1921 K. E. Tsiolkovsky

1924F. Tsander

1951 C. Wiley

1958R. L. Garwin

1959R. L. Sohn

1960Echo-1

1976 J. L. Wright

1991R. L. Forward

1990’sC. R. McInnes

1991J. L. Bell

1993R. L. Forward

1995NMP

1998J. S. Nuss

2004PVSE

...FUTURE

FIGURE 1.1: Time-line.


1.2 Missions

1.2.1 Past

The first mission employing a solar sail dates back to the 1970’s, when the JPL pro-posed a rendezvous mission for the comet Halley. Unfortunately, this mission wascancelled for various reasons, but the interest in solar sails increased. From thatmoment, NASA first, and subsequently the European Space Agency (ESA), start con-sidering the peculiarity of solar sails regarding non-Keplerian orbits, outer planetsmissions, interstellar missions, ... Initially, solar radiation pressure (SRP) was usedto support spacecraft operations:

• The Mariner 10 spacecraft, in 1974− 75, used SRP for attitude control aroundthe roll axis, by rotating its solar arrays, since the spacecraft ran low on attitudecontrol gas: this system worked. With commands from the mission controllers,they asymmetrically twisted the solar panels to create the so-called windmilltorque about the roll axis: for more details about this effect look at [Wie, 2008,pp. 743-749].

• The idea of Sohn was then used for geostationary and interplanetary satellites.For example, for INSAT and GOES satellites, an asymmetry given by a singledeployable solar panel introduces a solar pressure disturbing torque that wasremoved by placing on the opposite side a long boom at the end of which aconical-shaped solar sail was located [Markley and Crassidis, 2014, Chapter 1].

In 1993, the Russian Space Agency (ROSCOSMOS) launched a 20 m diameter, spin-ning mirror called Znamya 2, with the aim of irradiating solar power to the ground.Two Indian communications satellites (INSAT 2A in 1992 and INSAT 3A in 2003),powered by a 4-panel solar array on one side, possessed a solar sail mounted onthe north side of each satellite to offset the torque resulting from solar pressure onthe array. In 2004, the Japanese deployed solar sail materials sub-orbitally from asounding rocket, which was a key moment in the deployment of gossamer sheetsfrom spacecraft.

Between 2001 and 2005, NASA developed two different 20 m solar sails (fabri-cated by ATK Space Systems and L’Garde, Inc., respectively) and tested them on theground in vacuum conditions. Their primary objective was to demonstrate success-ful deployment of a lightweight solar sail structure in low Earth orbit. They werethe first to deploy a solar sail on Earth, in April 2005 with the first one, and in July ofthe same year with the other one. One month before that, the Planetary Society devel-oped a 30 m, 105 kg sail-craft, called Cosmos 1; however, due to a rocket failure, themission was lost: it would have been the first spin-stabilized, free-flying solar sail.

The NMP Space Technology 5 (ST5) proposed a mission, called Geostorm Warn-ing Mission, whose aim was to provide real-time monitoring of the solar activity,especially related to the geomagnetic storms (from which the name) caused by coro-nal mass ejections. In fact, actual space weather forecasting satellites are placed onL1, while this mission planned to position the sail-craft closer to the Sun in a sub-L1point, shortening the warning time. For this mission, a square solar sail was selected,of dimensions 76 m× 76 m and spin-stabilized at 0.45 deg /s in order to maintain theangular momentum vector within 1 deg of the Sun-line. The Team Encounter Missionconsidered a sail of the same dimensions as the previous one, with the aim of escap-ing the solar system in less the 5 years. In this project, the payload was intended to beplaced on a wire to move the centre of mass of the spinning sail-craft and passivelystabilize it. The NMP ST7 proposed a flight validation experiment for a sail whose

1.2. Missions 5

attitude was controlled by shifting its centre of mass with respect to the centre ofpressure. However, this system was challenging from the hardware implementationpoint of view, and, unfortunately, the NMP closed in 2009.

1.2.2 Present

In May 2010, the Japanese Space Agency (JAXA) launched the famous IKAROS1 sail-craft (i.e., Interplanetary Kitecraft Accelerated by Radiation of the Sun), which wasthe first in-flight demonstration of solar sailing. It was a 14 m × 14 m sail that hada spin rate of 2 rad min−1. It was also the first to use Liquid Crystal Display (LCD)panels to control the attitude by switching ON-OFF them on opposite sides. Thesedevices are also called reflectivity control devices (RCDs) and were switched be-tween specular and diffuse reflectivity conditions.

Few months later, in November 2010, NASA launched the 10 m2 NanoSail-D22

(Nano Sail deploy, de-orbit, demonstration, and drag 2): the NanoSail-D1 was lost ina launch vehicle failure aboard a Falcon 1 rocket. Due to malfunctions, the sail-craftwas ejected by the main spacecraft (FASTSAT) two months later, and when the saildeployed the battery died and the communications stopped. In any case, it success-fully stowed and deployed the sail, and demonstrated the de-orbit functionality.

The Planetary Society succeeded in launching the LightSail-13, a three-element(3U) CubeSat with a 32 m2 sail, in June 2016. Its main objectives were to demonstratesail deployment and controlled flight. It was placed in a Sun-synchronous orbit atan altitude of ∼ 820 km. LightSail-1 laid the foundation for the whole LightSailprogram by demonstrating controlled flight with only the pressure exerted by thesolar photons impinging the sail.

1.2.3 Future

Nowadays, the TRL of solar sails has reached the level 6, system/subsystem model orprototype demonstration in a relevant environment (ground or space), but for the next levelit requires a system prototype demonstration in a space environment. Solar sails can en-able new mission concepts, as highly non-Keplerian orbits, trajectories beyond theouter planets, hovering along the Sun-Earth line Sun-ward of L1 for solar observa-tion (i.e., Sunjammer mission concept), displacing the L1 point above the eclipticto observe the high latitude regions of Earth (i.e., pole-sitter mission concept), longresidence time orbits in the Earth’s magnetotail (i.e., GeoSail mission concept), andlow-cost multiple rendezvous near-Earth objects missions. However, this requireslighter and larger sails, thus reducing the ratio of vehicle mass to sail area, maximiz-ing the acceleration. The development of smaller spacecraft has been an importantaspect for solar sails, since the dimensions of the sail can be reduced a lot, com-pared to the 800 m × 800 m Halley concept. It is a good choice to propose solar sailseither in hybrid propulsion systems, or to assist the attitude control; this would re-duce risks and difficulties for a future advancement in the TRL. The future missionby JAXA will explore the Jupiter Trojan asteroids with a power solar sail (i.e., a sailequipped with thin-film solar cells). Furthermore, the Planetary Society has plannedto launch in the upcoming years LightSail-2 to provide new data and further refinethe solar sailing skills of the project. For more details about future mission conceptslook at [Vulpetti, Johnson, and Matloff, 2015].

1https://earth.esa.int/web/eoportal/satellite-missions/i/ikaros2https://directory.eoportal.org/web/eoportal/satellite-missions/n/nanosail-d23https://earth.esa.int/web/eoportal/satellite-missions/l/lightsail-1

https://earth.esa.int/web/eoportal/satellite-missions/i/ikaroshttps://directory.eoportal.org/web/eoportal/satellite-missions/n/nanosail-d2https://earth.esa.int/web/eoportal/satellite-missions/l/lightsail-1


1.3 Recent Studies

Farrés has analysed the dynamics of solar sails in the Earth-Sun restricted three-body problem including specular reflection and absorption in the sail model [Farrés,2017]. The main reason is that at the libration points the effect of the SRP becomesthe dominant one and cannot be neglected. In addition, the inclusion of the SRP inthe dynamical model creates an infinite set of new libration points, called artificialequilibrium points (AEPs). All of this depends on the parameters of the sail. Shecatalogued periodic and quasi-periodic motions close to those AEPs.

In [Sullo, Peloni, and Ceriotti, 2017] is described a new method to compute min-imum time 2D trajectories for solar sails starting from a given low-thrust solution.The idea is to use homotopy to link the law-thrust problem with the solar sail op-timal control. The reason is that the first is easier to solve, and its solution is con-verted in the second thanks to numerical continuation. The result is that this alter-native method is better than a classical genetic algorithm in terms of accuracy andcomputational time. The homotopy method has been used extensively to trajectoryoptimization of low thrust propulsion, but not to solar sails, which is far more diffi-cult. The nice thing is that homotopy allows to change the thrust provided by onepropulsion system to another, different.

TugSat is a CubeSat that adopts a solar sail for de-orbiting of satellites fromthe geostationary orbit [Kelly et al., 2018]. The idea at the base of this project isthe fact that a solar sail can be reused more times to remove a large number ofsatellites: the maximum capability of the removed satellite is 1000 kg, consideringa high-performance sail. The satellites are placed in a retirement orbit, to secure theGEO-ring (Geosynchronous Equatorial Orbit). In this study, a non-linear orbit con-trol is developed, starting from a Lyapunov function (as done typically for attitudecontrol), based on the time derivative of some orbital elements (semi-major axis, ec-centricity, inclination) and the geosynchronous equatorial orbit belt longitude. It isworth noting that SRP is the main cause of unwanted re-enter of satellites in securegraveyard orbits after retirement to the GEO-belt, and here is used to secure them.

1.3.1 Space Flight Mechanics Meeting 2018

Near Earth Asteroid (NEA) Scout is a solar sail mission whose aim is to performclose flybys and investigate near Earth asteroids. It is under development by NASAand JPL, and it will be launched in December 2019 as a secondary payload of theSpace Launch System’s (SLS) inaugural flight, Exploration Mission 1 (EM-1). It is a86 m2 sail with a 12 kg main bus of a 6U-CubeSat. Attitude control is performedthanks to reaction wheels, active mass translator, and cold gas thrusters. The ac-tive mass translator is used only for momentum desaturation shifting the centre ofmass to the centre of pressure. The system is capable of angular rates of 0.04 deg /saround pitch and yaw axes, and 0.02 deg /s around roll axis. In [Pezent, Soody,and Heatonz, 2018] high fidelity trajectories are studied for different launch dates.The objective of the study is to design alternative trajectories for initial correctiveactions in case of initial trim manoeuvre failure after the separation from EM-1. Inthis analysis, since only the orbital part is considered, the problem of the tangen-tial component of the SRP acceleration, important in attitude dynamics, is simplyunderlined but not analysed.

In [Farrés, Heiligers, and Miguel, 2018] the problem of transfers from displacedL1 and L2 Sun-Earth libration points to regions of practical stability around L4 andL5 is analysed. The benefit is that, if a sail-craft reaches these regions with zero

1.3. Recent Studies 7

synodical velocity, it will remain there without any need of station-keeping, withresidence times greater than 1000 years. A possible application is for space weather,from a different perspective with respect to L1. The reason for this study is that, sinceit is hard to reach orbits around L4 and L5 for classical spacecraft, a solar sail can dothat in a reasonable amount of time. For example, L5 is a good place to observe theSun’s activity, trying to forecast faster important events like coronal mass ejections,exploiting the counter-clockwise rotation of the Sun.

In [Takao, Mori, and Kawaguchi, 2018] is introduced an new method for attitudeand orbit control of a spinning solar sail based on active shape control of the sailmembranes, which becomes a 3D structure. In general a sail is spin-stabilized tocounteract disturbances, maintain the correct attitude with respect to the Sun, forlong term missions and for large dimensions. They propose a method to controlthe deformation state of the membranes to change actively the SRP acting on thesail, considering a realistic model for the pressure. They have also introduced theconcept of fuel- and geometry-free solar sailing. They manage to control attitudeand orbit with the deformation of the sail, and contemporaneously maintain thecommunication link with the Earth. In practice, the first order deformation mode(i.e., first order static wave) is used to control the orbit for power generation andEarth communication (i.e., force), while the higher orders (i.e., second order staticwave) are used for attitude control (i.e., torque).

1.3.2 AIAA SciTech 2019 Forum

In [Selvaraj and Shankar, 2019] is analysed an alternative optimizer for minimumtime transfer trajectories of solar sails, thus enlarging the range of mission appli-cations typically hard for classical spacecraft. They introduce a hybrid optimizer,compared to classical methods as sparse non-linear optimizer (SNOPT) and geneticalgorithm (GA). This hybrid method uses SNOPT and GA together in such a waythat the disadvantages of the two are counteracted. In fact, GA is used to createa good approximation of the initial point since it is the main problem of SNOPTwhich shows high dependence on it. Then, GA is stopped and SNOPT continuesthe optimization process from a certain point on, counteracting the poor conver-gence properties and inability to handle constraints efficiently by GA. They provedthat the hybrid optimizer performs better than the other two considered singularly.It is worth underlining that they have considered an ideal sail model in the per-turbed two-body domain, without attitude dynamics.

In conclusion, some lessons learned about laboratory work on solar sail mem-branes performed by NASA Langley Research Center are discussed in [Stohlmanet al., 2019]: it is supporting the Advanced Composites-based Solar Sail System(ACS3) project that is a solar sail demonstration concept for a 6U Cubesat. The mem-brane material contributes for about half of the total volume of the assembled sailmembrane since the remaining consists on tape adhesives. This means that caremust be taken in the package of these membranes. There are different fold patterns(e.g., entirely-parallel z-fold, half-parallel z-fold, partial fan fold) and they need acertain method of folding. They have analysed three methods compared to classichand-folding: 1) folding between folding forms is repeatable, less time-consuming andallows complex folding patterns; 2) folding in a channel is not well-suited for com-plex patterns but fast; 3) folding with local stiffness variation speeds up the procedureeven if some problems of stacking arise. The problem of venting and the decouplingbetween booms and membrane deployment have been studied.


1.4 Reflectivity Control Devices

1.4.1 Functioning

A reflectivity control device is a thin film device capable of controlling the orien-tation of liquid crystal components, placed in between two electrodes, at which isapplied a certain voltage. The first RCDs where developed from the light controlglasses, modified to be radiation tolerant. There are two main types of devices al-ready developed:

• Diffusion RCDs (Figure 1.2a and 1.2b) can alternate from diffuse reflection tospecular reflection.

• Transmission RCDs (Figure 1.2c and 1.2d) can alternate between specular re-flection to transmission. The transition speed of such devices is in the range10-100 ms.

Diffusion RCDs have been adopted for the IKAROS mission, as an optional at-titude control system. They consist in a polymer-dispersed liquid crystal (PDLC) putbetween two polyimide-film with the bottom one aluminium-deposited. One of themain advantage of these devices is that they introduce very small oscillations to themembrane, differently from classical control systems. The characteristics of thosedevices and the control logic adopted for IKAROS are discussed in [Funase et al.,2010]. The attitude control experiment consisted in a stable and fuel-free attitudesteering of a spinning solar sail, with a precision in term of Sun angle ∼ 10−2 deg.One effect that was observed is the introduction of a disturbance torque affecting theroll axis when the RCD panels were in use.

A transmission RCD is similar to the previous one but without the need for po-larisers. In this case, the micro-metric liquid crystal droplets are optically birefrin-gent with ordinary (no) and extraordinary (ne) refractive indexes, and embeddedwithin an optically isotropic polymer matrix with refractive index np [Ma, Murray,and Munday, 2017]. It behaves in two ways:

• In the OFF state, the droplets are randomly oriented (refractive indexes differ-ent from np), causing a variation in the refraction index seen by the incominglight, leading to diffuse reflection of it (i.e., scattered light).

• In the ON state, the droplets align with the electric field generated by the volt-age applied, such that the light experiences a refractive index no. If no = nplight does not see any change in the refraction index and so it is transmitted.

This logic that utilizes also the transmission improves the performance of the mo-mentum change of three-/four-times compared to classical devices based on reflec-tivity only. In [Ma, Murray, and Munday, 2017] is presented a steerable solar sailconcept based on this type of devices. They declare that the power requirement inthe ON state is less than 0.5 mW cm−2. In addition, the change of the fraction ofthe weighted average momentum transferred to PDLC between the two states is inthe order of 0.5, compared to 0.11 of IKAROS devices. The reduction of the PDLCthickness is a way to reduce the power consumption of the device; however, thickercells provide a larger momentum difference between the two states. The Propellant-less Attitude Control of Solar Sail Technology Utilizing Reflective Control Devices projectteam at NASA is working on these devices [Munday, 2016].

In [Ishida et al., 2017] a new concept of RCD is proposed, which is called ad-vanced-RCD. A diffusion RCD reflects the light specular in the ON state, while dif-fusively in the OFF state. An A-RCD substitutes the ON behaviour by reflecting the

1.4. Reflectivity Control Devices 9

light obliquely (Figure 1.2e), introducing two advantages: counteract the windmilleffect and introduce an additional degree of freedom for the attitude control. It al-lows generating a torque along the sail normal. Practically it is similar to a classicRCD, but the bottom layer has a saw-tooth structure.

(A) Diffusion RCDOFF state

[Funase et al., 2010].

(B) Diffusion RCDON state

[Funase et al., 2010].

(C) Transmission RCDOFF state

[Munday, 2016].

(D) Transmission RCDON state

[Munday, 2016].

(E) Advanced-RCD [Ishida et al., 2017].

FIGURE 1.2: Reflectivity control devices typologies.


1.4.2 Applications

In [Borggräfe et al., 2014b] the use of RCDs is analysed in the domain of the two-bodyproblem (2BP), considering two different logics:

1. The reflectivity is assumed to have a linear variation across the surface, mod-ulated in a continuous way between diffuse and specular reflection. However,this concept is not available nowadays.

2. Discrete regions of specular and diffuse reflectivity are considered, separatedby a line.

The sail model does not take into account wrinkles, so assumes a perfectly flat sailsurface; quoting them:

“When distributing controlled regions of high and low reflectivity across the surface, a widerange of torques can be generated in the sail plane, however, torques perpendicular to the

surface are not possible.”

In [Mu, Gong, and Li, 2015] is proposed an orbit-attitude control with RCDs, inthe 2BP, comparing two different modulation modes:

• The diffusion mode changes the reflectivity between specular and diffuse.

• The absorption mode switches between specular and absorptive.

They show that the second is better than the first, in terms of attitude control.In [Oguri, Kudo, and Funase, 2016] RCDs are utilized for attitude control, and

the uncertainty factors related to sail deformation and optical property discussed.They consider a spinning sail and so the RCDs have to be switched accordingly tothe rotation. In addition, they consider a realistic behaviour for the SRP model andall the pressure components are considered even if due to the sail deformation.

In [Tamakoshi and Kojima, 2018] the coupled orbit-attitude control law for a hy-brid solar electric propulsion solar sail in the Earth-Moon three-body problem (3BP)system is studied for the communication Earth-Moon’s far side. They consider theadoption of RCDs since are able to reduce the propellant required to maintain thespacecraft into orbit: the ON state reflects all the sunlight, while the OFF state ab-sorbs all of it. It is given a strong importance to the attitude control, since it influ-ences also the orbit control: the attitude equations are written in the relative frame,and not in the absolute frame where are valid in the classical form, meaning thatsome assumptions have been made, even if not stated. They adopt backsteppingmethods since suitable for controlling "subsystems", as the coupled orbit-attitudecase. The torque around the roll axis is neglected, and a reaction wheel is used tocontrol the sailcarft around it; to quote them:

“The torque generated by the solar radiation pressure force transverse to the surface is notconsidered in this paper because the force affecting the local dynamics of the membrane due

to wrinkles and/or buckles is very small, and thus its effect on the rotational dynamics of thesolar sail is also very small. In other words, it is assumed that the solar radiation pressure

normal to the surface, [...], generates the torque to change the attitude of the solar sail in thispaper.”

1.4. Reflectivity Control Devices 11

In [Gao et al., 2018] is analysed a highly-fidelity model, coupled orbit-attitudepropellant efficient station-keeping for a hybrid sail in the Earth-Moon system usingRCDs. The hybrid system consists of solar electric propulsion together with RCDs,the latter used for both orbit and attitude control. The use of these devices improvesthe controllability of a solar sail since they add an additional degree of freedom,whose usage is indicated in highly perturbed systems as the Earth-Moon one. In hy-brid systems, the RCDs are adopted to save propellant in the orbital domain, whichis the typical main result and it is always better than a system without them. Theyconsidered also the presence of thin-film solar cells on the sail surface. They statethat optimal periodic controllers have a convergence speed greater than classicallinear quadratic regulator (LQR). They switch the RCD state between specular andabsorptive, and the tracking is considered reached when the absolute difference be-tween actual and desired attitude angles is less than 0.1 deg. Also in this case theSRP is not considered with all the components, quoting them:

“Ignoring the very small torque generated by the SRP force transverse to the sail surface”

TABLE 1.1: RCD-based papers comparison: X = used, × = not used,n.d. = not declared.

Orbital model Sail model Pressure Control lawID

2BP 3BP Ideal Real x y z Orbit Attitude

1 X × X × X × × × X

2 X × X × X × × X X

3 n.d. n.d. × X X X X × X

4 × X X × X × × X X

5 × X X × X × × X X

All these works are compared in Table 1.1 in terms of: model adopted for theorbit, SRP model, components of the force used in the attitude control laws, type ofcontrol considered. A characteristic of all those researches is that the surface coveredof RCDs is always considered as a continuum, without taking into account the realdimensions of the single device, and the SRP force component lying in the sail planeis always neglected.

To the best of this author’s knowledge, there was only one application where thesail surface has been considered as a collection of RCDs. In [Borggräfe et al., 2014a]the membrane of the sail is modelled considering discrete reflectivity cells behavinglike specular reflecting mirrors that can switch between 0 and 1 as reflectivity value.An ideal attitude control based on quaternions is used as reference for the real con-trol that finds the optimal pattern in terms of number and combination of reflectivitycontrol devices to create that torque, but only in the plane of the sail. The logic isbased on the possible combinations ON/OFF of all the cells considered, which in-crease very fast with the number of cells, even if the possible torques generated are


far less. All the torques are pre-calculated before and stored in a lookup table fromwhich, during the simulation, the system searches for the best torque componentsthat match the desired ones. The precision reached by this system, in terms of pitchangle, is in the order of 0.1 deg, only the attitude dynamics is considered, the sailis not spinning, and no torques are generated about the roll axis since the SRP isconsidered always perpendicular to the sail surface.

In a recent work, the problem of three-axis attitude control of a solar sail with theadoption of RCDs has been analysed [Theodorou, 2016]. He analysed the problemof expanding the control in the three-dimensional domain by rotating at fixed anglesthe devices placed at the edges of the sail. He used the combinations logic to estab-lish the possible torques to generate. Again, the problem was related to the highnumber of combinations also considering a small number of devices. In fact, in hisanalyses he considered at maximum 5 RCDs per sides, for a total of 20. The problemwas the computational time required, and the idea was to simulate with few RCDs,but considering redundancy in their number. The author generated the torque enve-lope with all the possible combinations of the torque generated by the RCDs in ONand OFF states, creating a polyhedron. In addition, a Sun map is built, which is agraphic representation of the Sun angles where the control requirements are met: itrequires the calculation of the control envelope for each Sun angle. As can be noted,for a large number of devices the procedure requires a lot of computational time,with the possibility of exceeding the memory capacity at a certain point. The idealattitude control law generated to be supplied to the real control was a proportionalintegral derivative (PID) control. He studied also a hybrid system consisting of threereaction wheels (one for each axis of the sail), together with the RCDs, that providea larger torque for a limited time in order to achieve higher slew rates and largerangular accelerations. However, this can introduce vibrations in the sail membrane.However, only the attitude part is analysed, and the effect of the usage of a highRCDs number on the orbit is not considered.

1.4.3 Thesis’s Novelty

This thesis analyses the adoption of reflectivity control devices for both orbit andattitude control, with main emphasis on the attitude part. First of all, the solar radi-ation pressure is considered in its vectorial form, without any sort of simplificationabout its component lying in the sail plane when analysing the generated torque. Inall the other works presented before this is not done, but here it is shown that itseffect is not always negligible. In this section the sentences contained in the papersunderline that the torque along the normal to the sail is given by wrinkles and/orbuckles of the membrane, that are not considered. However, the control itself, basedon RCDs, introduces in any case a torque along the normal due to that componentof the SRP. The analysis is performed both in the two-body problem (attitude alone)and the circular restricted three-body problem (coupled orbit-attitude). Regardingthe devices, here they are used to study an alternative approach for the control. Forthis reason, the existing devices are not considered; instead, a different logic capableof switching between specular and absorptive state is adopted, as done also in otherworks. Due to the adoption of these devices, a more realistic behaviour for the sail isconsidered in the coupled control analysis. Finally, the RCDs are distributed on thesurface as pixels, so with their own dimensions, and can work independently one tothe other. This thesis expands the application of distributed reflectivity control de-vices for a large number of devices, combining combinations logic and an alternativeapproach.

1.5. Thesis Content 13

1.5 Thesis Content

The remaining of the thesis is divided into the following chapters:

2 The model of the SRP is introduced, deriving the adopted form of the force, af-ter a preliminary derivation of the expression of the pressure. This is followedby the description of the 2BP model perturbed by the SRP, and a reference orbitis generated.

3 The attitude dynamics is recalled. In particular, the Euler equations are firstderived, with all the simplifications adopted. Then, the kinematics equationsare shown for different representations: Euler angles, quaternions, and thew − z parameters. The latter are a peculiar attitude representation, so all therelations are here derived.

4 The attitude control based on RCDs is fully described. First, the combinationslogic and the mesh is described, with all the steps to follow for its creation.Then, the attitude control logic is described, and the performance analysedconsidering a reference ideal control torque.

5 Proportional derivative, reduced, and under-actuated ideal controls are de-scribed. The last control, based on the w− z parameters, is described in moredetail since slightly different from the one found in the literature.

6 Ideal and real controls are used to track the desired attitude history introducedin Chapter 2, and finally compared. Only the attitude dynamics is considered,no matter of what happens to the orbit that for the moment is considered un-coupled.

7 The circular restricted three-body problem model for a solar sail is described,with particular emphasis on the definition of equilibrium points. A classicalLQR control for the orbit is developed.

8 The orbit control based on RCDs is described, and the performance evaluatedusing as reference acceleration the one generated with the LQR control.

9 The simulations involving the coupled dynamics, with the ideal and real con-trols, are shown. In this work, the control for the coupled dynamics is theunion of the two controls developed separately for orbit and attitude: only thereduced (bore-sight) control is considered.

10 At the end, the main considerations and improvements are discussed, and fu-ture developments proposed.

For all the simulations, the software adopted are Matlab R© and Simulink R©. Tohelp the reader, lists of abbreviations, physical constants, and symbols can be foundjust before this chapter. The symbols are divided among the chapters: symbols al-ready defined in previous chapters are not redefined next, except if they changemeaning.

15

2 Solar Sail Models and Dynamics

2.1 Solar Radiation Pressure

The expression of the solar radiation pressure can be obtained either from the quan-tum theory [McInnes, 1999, pp. 34-36], or from the electromagnetic one [Focardi,Massa, and Uguzzoni, 2010, pp. 151-160]: this is related to the wave-particle dual-ism.

2.1.1 Quantum Mechanics

From quantum mechanics, the radiation pressure is related to the momentum trans-ported by photons, the quantum packets of energy. To obtain the relative expression,first consider the Plank’s law

E = h ν (2.1)

where E is the energy transported by a photon of frequency ν and h is the Plank’sconstant. Then, from the Einstein’s special relativity theory can be found that, sincethe photon has a zero rest mass, its energy is simply given by

E = p c (2.2)

where p is the momentum of the photon and c the vacuum speed of light. FromEquation 2.1 and 2.2, the momentum can be derived

p =Ec=

h νc

(2.3)

It is known that the energy flux at a distance r from the Sun is given by

W =L�

4 π r20

(r0r

)2= W0

(r0r

)2(2.4)

where L� is the solar luminosity, r0 is the mean Sun-Earth distance equivalent to1 AU (astronomical unit), and W0 is the energy flux at 1 AU. By defining the energytransported across a surface of area A normal to the incident radiation in time ∆t

∆E = W A ∆t (2.5)

, and adopting Equation 2.3 and 2.4, the expression of the SRP can be finally obtained

P =1A

(∆p∆t

)=

1A

∆Ec ∆t

=Wc

=W0c

(r0r

)2= P0

(r0r

)2(2.6)

Adopting the typical average value of W0 = 1367 W m−2, the value of the SRP at1 AU is P0 = 4.56× 10−6 N m−2.

16 Chapter 2. Solar Sail Models and Dynamics

2.1.2 Electromagnetic Theory

From the electromagnetic point of view, momentum is transported by electromag-netic waves. It is known that the energy density of an electromagnetic wave is givenby (E = c B, e0 µ0 = 1/c2)

U =12

e0 E2 +1

2 µ0B2 =

B2

µ0= e0 E2 (2.7)

where e0 is the free space permittivity, µ0 is the free space permeability, E is the elec-tric field, and B is the magnetic field. An important quantity in the electromagnetictheory is the Poynting vector, defined as

S =E× B

µ0=⇒ ||S|| = c e0 E2 (2.8)

where have been assumed planar electromagnetic waves (E ⊥ B). Comparing Equa-tion 2.7 and 2.8, the following relation holds

||S|| = c U (2.9)

The energy that passes through a surface A normal to the direction of the elec-tromagnetic wave in time ∆t is

∆E = ||S|| A ∆t = c U A ∆t (2.10)

Looking at Equation 2.5, it can be found that

U =Wc

(2.11)

The electromagnetic radiation transfers not only energy, but also momentum; infact, the impulse density is given by

pem =Sc2

=⇒ pem =Uc

(2.12)

At this point, from the momentum (p) and the impulse theory

∆p = F ∆t

pem A c ∆t = F ∆t ux

U A ux = F ux

(2.13)

, where ux is the direction of propagation of the wave. Finally, the pressure exertedon a perfectly absorbing surface coincides with the energy density of the electromag-netic wave

P =FA

= U =Wc

(2.14)

It is the same result obtained with quantum mechanics and a consequence of theparticle-wave dualism.

2.2. Force Models 17

2.2 Force Models

The real force model considered is taken from [Wie, 2008, pp. 749-752]. The im-pinging photons on a surface are partly absorbed (ρa), specularly reflected (ρs), anddiffusely reflected (ρd), such that

ρa + ρs + ρd = 1 (2.15)

In this model, the transmitted fraction ρt is neglected. The SRP force acting on a flat,Lambertian1 surface is then modelled as

FSRP = P S[

ρa < s, n > s + 2 ρs < s, n >2 n + ρd < s, n >(

s +23

n)]

= P S < s, n >[(ρa + ρd) s +

(2 ρs < s, n > +

23

ρd

)n] (2.16)

where S is the sail surface, s is the Sun direction (pointing outward from the Sun),and n is the sail normal (pointing outward). For a more accurate model look atAppendix A. From this model, three particular cases can be derived:

• Perfect specular reflective sail (ρs = 1)

FSRP = 2 P S < s, n >2 n (2.17)

• Perfect diffuse reflective sail (ρd = 1)

FSRP = P S < s, n > (s + 2/3 n) (2.18)

• Perfect absorptive sail (ρa = 1)

FSRP = P S < s, n > s (2.19)

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

s = 1

d = 1

a = 1

FIGURE 2.1: Solar radiation pressure force models.

1A Lambertian surface appears equally bright when viewed from any aspect angle.


In Figure 2.1 are represented all the three models, projected along n, for differentvalues of α ∈ [0 deg, 90 deg]: α is the angle between the sail normal and the Sundirection, called Sun aspect angle or cone angle. It is worth noting that the magni-tude of the SRP force decreases moving from the perfect reflective to the absorptivesail. The thesis is focused on the adoption of RCDs mainly for attitude control. Thedifference between the specular and the diffuse models changes sign at α = 48.2 deg,complicating the design. Instead, the largest difference among the three models ex-ists for the perfect reflective and the absorptive models, without sign changes. Forthis reason, the devices will switch between these two states, increasing the resultingtorque generated, and using less of them.

n

t

Incoming Photons

s

Specularly Reflected Photons

θ

α− θ

α

α

Fn

Ft

m

FIGURE 2.2: Solar radiation pressure force geometry for a non-perfect, flat Lambertian solar sail.

Now, consider the 2D representation of the sail in the plane (n, s), as shown inFigure 2.2. Following that scheme, and defining t ⊥ n as the direction tangential tothe sail, Equation 2.16 can be rewritten as follows

Fn = P S[(1 + ρs) cos2 α +

23

ρd cos α]

(2.20a)

Ft = P S (ρa + ρd) cos α sin α (2.20b)

θ represents the angle between the Sun direction and the effective direction of theSRP force for a non-perfect, flat Lambertian sail. Since the sail will be equippedwith RCDs, it is obvious that it will not behave like a perfect sail, except if the RCDsare not used. For this reason, a more appropriate model for the sail has to combinespecular reflective and absorptive characteristics (i.e., ρd = 0), and Equation 2.20 canbe simplified using only ρs

Fn = P S (1 + ρs) cos2 α (2.21a)

Ft = P S (1− ρs) cos α sin α (2.21b)

2.2. Force Models 19

The following equation relating θ and α holds

α− θ = arctan(

FtFn

)= arctan

(1− ρs1 + ρs

tan α)

(2.22)

The behaviour of θ for different values of α and ρs is shown in Figure 2.3: for aperfect reflective sail α = θ. It can be noticed from Figure 2.3a that, when the spec-ular reflectivity decreases, the effective maximum angle of the SRP force decreases(θmax): this reduces the performance of the sail and must be taken into account. Fig-ure 2.3b shows that, for a given value of α, decreasing ρs reduces the effective angleθ and this effect is enhanced for high values of α. The selection of the appropriatevalue of ρs will be described in the appropriate section. To conclude, the correspond-ing expression of Equation 2.16 for a specular reflective + absorptive sail becomes

FSRP = 2 P S cos α[

ρs cos α n +12(1− ρs) s

](2.23)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

s = 1.0 (

max = 90.00 deg)

s = 0.9 (

max = 64.16 deg)

s = 0.8 (

max = 53.13 deg)

s = 0.7 (

max = 44.43 deg)

s = 0.6 (

max = 36.87 deg)

max(

s)

(A) ρs fixed.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40 = 0 deg = 10 deg = 20 deg = 30 deg = 40 deg

(B) α fixed.

FIGURE 2.3: Specular reflective + absorptive sail model.


2.3 Two-Body Problem

For the first analysis of the real control, only the attitude dynamics is studied; so, asimple model is considered, which is the two-body problem. Then, to analyse theeffect of the usage of the RCDs, which create a non-perfect sail, the coupling betweenattitude and orbit is discussed and analysed to see how to solve the problem. Forthat aim, the circular restricted three-body problem (CR3BP) model will be consid-ered, also for a possible mission application. For both models, the primary massconsidered is the Sun.

2.3.1 Equations Of Motion

The Newton’s equations of motion are valid in an inertial reference frame N. Theone adopted is centred in the Sun, with XN pointing toward the Ares point (ver-nal equinox, March 21), ZN is normal to the ecliptic plane, while YN completes theright-handed set. This frame is also called heliocentric inertial (HCI) system [Griffinand French, 2004, pp. 121-125]: this choice for XN is always adopted when analysingproblems in the solar system. It is worth noting that the Sun is not fixed, but movesorbiting around the centre of the Milky Road (precisely around a super-massiveblack hole) at a mean velocity of ∼ 250 km s−1. However, this effect is clearly negli-gible in this analysis. For the derivation of the equations of motion, the proceduredescribed in [McInnes, 1999, pp. 113-114] is followed. As described in [Brown, 2002,pp. 46-47], the assumptions at the base of the 2BP are:

• The motion of a spacecraft is governed by the attraction of a single body.

• The mass of the spacecraft is negligible compared to the main body.

• The two bodies are considered as point masses.

• The only forces acting on the system are gravitational and centrifugal, bothalong the connecting line (unperturbed 2BP).

XN′

YN′

ZN′ Sun

r1r2

r = r2 − r1COM

R Solar Sail

n

FIGURE 2.4: Two-body problem geometry.

2.3. Two-Body Problem 21

Start considering an inertial frame N’ (Figure 2.4) in which are considered theSun and the solar sail at distance r1 and r2, respectively, from the origin of that frame:r is the position vector of the sailcraft with respect to the Sun. The position of thecentre of mass (COM) of the Sun-sail system is given by

R =M� r1 + m r2

M� + m(2.24)

where M� = mass of the Sun, m = mass of the sail. Now, adopting the Newton’ssecond law of motion for the two point masses, the following relations hold

M� r̈1 =G M� m

r2r̂ (2.25a)

m r̈2 = −G M� m

r2r̂ + aSRP (2.25b)

, where G is the universal gravity constant. Regarding the SRP acceleration, in thiscase the ideal sail model is adopted (ρs = 1), since the attitude is assumed uncoupledfrom the orbit: the model to use is the one of Equation 2.17.

A design parameter peculiar of solar sails is called sail loading parameter

σ =mS

(2.26)

Another parameter connected to σ is the lightness number β defined as

β =σ∗

σ(2.27)

where σ∗ is called critical sail loading parameter, with the following expression

σ∗ =L�

2 π G M� c=

2 W0 r20µ� c

=2 P0 r20

µ�= 1.53 g m−2 (2.28)

where µ� = G M� is the solar gravitational constant, and Equation 2.4 and 2.6have been adopted.

The final expression for the SRP acceleration is

aSRP = βG M� m

r2< r̂, n >2 n (2.29)

Summing Equation 2.25a and 2.25b, and adopting Equation 2.24, the acceleration ofthe COM is obtained

R̈ = βG M� mM� + m

< r̂, n >2

r2n (2.30)

It can be noted that, differently from the classic 2BP without SRP, the COM acceler-ation is different from zero. However, since m � M� (i.e., M� + m ∼ M�), thisacceleration can be considered negligible. Then, transforming the support N’ frameinto the inertial reference frame N, located at the COM, R becomes equal to zero. Inthis way, the following relation holds (derived from the differentiation of Equation2.24)

r̈ = r̈2 − r̈1 =(

1 +m

M�

)r̈2 (2.31)


Inserting Equation 2.25b into 2.31, and considering that m � M�, the equations ofmotion for a perfectly reflective sail around the Sun are:

r̈ = −µ�r2

r̂ + βµ�r2

< r̂, n >2 n (2.32)

The solar sail parameters considered in this thesis are taken from [Heligers etal., 2014] and reported in Table 2.1. These data are referred to the NASA Technol-ogy Demonstration Mission called Sunjammer23, with the aim of targeting the sub-L1region for advanced solar storm warning, thus almost doubling the warning time:it integrated two sensors for space weather monitoring, a solar wind analyser anda vector magnetometer. This mission was led by industry manufacturer L’GardeInc. with the participation of the National Oceanic and Atmospheric Administration(NOAA), with the aim of increasing the TRL of the L’Garde solar sail from 6 to 9. Itsmain objectives were:

• To demonstrate segmented deployment of a solar sail.

• To demonstrate attitude control plus passive stability and trim using beam-tipvanes.

• To execute a navigation sequence with mission-capable accuracy.

• To fly and maintain position at L1 and pole-sitter positions.

The sail deployed successfully at the L’Garde facility in Tustin, California, in Septem-ber 2013: the test demonstrated the successful coupling and deployment of the sailand deploying beam under the conditions imposed by Earth’s gravity and atmo-sphere. This sail was intended to be launched in January 2015 from the KennedySpace Centre at Cape Canaveral, Florida, aboard a Space Exploration Technologies(SpaceX) Falcon 9 launcher; however, this project stopped in 2014, before the flighttesting, due to integration issues. For more details about the mission analysis look at[Heligers et al., 2014]. This sail has been selected since characterized by a reasonabledimension of the surface, and a realistic value for the sail lightness parameter: themass here considered does not include bus, payload, and disposable support mod-ule. The mass of the reflectivity control devices is neglected in all the analyses hereperformed.

TABLE 2.1: Solar sail orbit parameters.

m S σ β

45 kg 1444 m2 31.16 g m−2 0.049 344

2https://www.nasa.gov/mission_pages/tdm/solarsail/index.html3https://www.aerospace-technology.com/projects/sunjammer-solar-space-mission/

https://www.nasa.gov/mission_pages/tdm/solarsail/index.htmlhttps://www.aerospace-technology.com/projects/sunjammer-solar-space-mission/

2.3. Two-Body Problem 23

2.3.2 Reference Frames

The orbit of the sail can be described using either the couple position and velocity,or the orbital elements (semi-major axis, eccentricity, inclination, right ascension ofthe ascending node, anomaly of the pericentre, true anomaly). There are methods tointegrate the 2BP equations with the SRP perturbation directly in terms of the orbitalparameters (e.g. Lagrange variational equations), but to be more general (also in caseof the adoption of more complicated models where the orbital elements are no moredefined, e.g. non-Keplerian orbits) this is not done.

XN ≡ Ares Point

YN

ZN

Sun

XL ≡ r̂ ≡ sLYL ≡ ĥ∧ r̂

ZL ≡ ĥSolar Sail

FIGURE 2.5: Inertial N and rotating L frames (2BP).

Now, it is possible to build the rotating frame L, which is centred at the Sun androtates with the solar sail in its orbit around the Sun (Figure 2.5): XL is directed asthe position vector from the Sun (r̂), ZL is normal to the orbital plane and alignedwith the orbit angular momentum vector (h = r ∧ v), while YL completes the right-handed set. The rotation matrix to pass from N to L is then given by

ALN =

r̂T(ĥ∧ r̂)TĥT

∈ R3×3 (2.33)From this matrix, it is possible to derive the angular velocity between N and L, ex-pressed in L [Markley and Crassidis, 2014, pp. 68-69]

[ωLN ×L

]= −ȦLN ATLN −→

[ωLN ×L

]=

0 −ωz ωyωz 0 −ωx−ωy ωx 0

(2.34)It is worth noting that the position of the Sun in the L frame is aligned with XL, suchthat the unit vector of the Sun is

sL = r̂ =[1 0 0

]T (2.35)


In the case of solar sails, the typical input for the attitude, coming from the as-trodynamics, is a couple of angles describing the orientation of the sail normal withrespect to the Sun direction. In general, the angles used are the so called cone angleα and clock angle δ [McInnes, 1999, pp. 115-118]. The former describes the anglebetween the sail normal and the Sun direction, while the latter identifies the angleformed by the projection of the normal in the (YL, ZL) plane starting from ZL andmoving clockwise: there is not a unique way to define the clock angle, as will bedescribed in Chapter 7. The two angles can be clearly seen in Figure 2.6. If the rollangle γ, defining the rotation of the sail around XB, is added to the other two, thebody-fixed frame B can be fully defined, with XB aligned with the sail normal nB,and YB and ZB lying in the sail plane and passing through the axes of symmetryof the sail (parallels to the sides). The expression of the sail normal in the rotatingframe can now be derived

nL = cos α r̂ + sin α sin δ ĥ∧ r̂ + sin α cos δ ĥ (2.36)

XL ≡ r̂ ≡ sL

YL ≡ ĥ∧ r̂

ZL ≡ ĥ Sun

δ

XB ≡ nB ≡ 1

YB ≡ 2

ZB ≡ 3

γ

α

FIGURE 2.6: Rotating L and body-fixed B fr

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