Attitude and orbit control of a solar sail through distributed reflectivity modulation devices

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P

OLITECNICO DI

M

ILANO

M

ASTER

T

HESIS

Attitude and Orbit Control of a Solar Sail

Through Distributed Reflectivity

Modulation Devices

Author: Alessio NEGRI ID: 874935 Supervisor:

Dr. James Douglas BIGGS

A thesis submitted in fulfillment of the requirements for the degree of Master of Science

in

Space Engineering

Department of Aerospace Science and Technology School of Industrial and Information Engineering

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“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.”

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Declaration of Authorship

I, Alessio NEGRI, declare that this thesis titled, “Attitude and Orbit Control of a Solar Sail Through Distributed Reflectivity Modulation Devices” and the work presented in 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 clearly attributed.

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

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POLITECNICO DI MILANO

Abstract

School of Industrial and Information Engineering Department of Aerospace Science and Technology

Master of Science

Attitude and Orbit Control of a Solar Sail Through Distributed Reflectivity Modulation 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 efficient attitude control that is able to precisely and continuously re-point the thrust vector in the required direction for orbit control. One method that has been flight tested on IKAROS is the use of devices capable of changing the reflectivity properties of the sail.

This thesis proposes a new approach to attitude and orbit control of a solar sail with pixelated reflectivity control devices (RCDs) and a logic based on combinations of their state (ON/OFF). A control law that maps an ideal in-plane control torque to the reflectivity property of each pixel is derived, so that ideal controls that guarantee stability can be mimicked. It is shown, as the number of pixels of the sail membrane is increased, the real torque will converge to the ideal one. Different ideal attitude control 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 copying under-actuated and reduced controls, with good performances in terms of attitude angles accuracy, torque requirement, and mimic ability. However, the system is not able to copy the proportional control.

The RCD-based control is extended to incorporate orbit tracking, mimicking an ideal acceleration without introducing torques. To this end, a Linear Quadratic Reg-ulator orbit control is introduced which yields the time history of the attitude angles required 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 the attitude, only the reduced control is analysed. An example station-keeping problem in 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 results show that the sail asymptotically converges to a neighbourhood of the desired loca-tion, where the boundary of this neighbourhood can be decreased by increasing the number of RCD pixels.

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Le vele solari sono una tecnologia che permette di utilizzare i fotoni dal Sole per una propulsione senza carburante. Tuttavia, una delle maggiori sfide è il design di un controllo d’assetto efficiente capace di puntare, precisamente e continuamente, il vettore spinta nella direzione richiesta dal controllo orbitale. Un metodo che è stato testato in volo su IKAROS consiste 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 solare con dispositivi a controllo di riflettività (RCDs) scomposti in pixel e una logica basata sulle combinazioni dei loro stati (ON/OFF). Viene derivata una legge di controllo che mappa una coppia ideale nel piano della vela alle proprietà di ciascun pixel, così che i controlli ideali che garantiscono stabilità possano essere imitati. Viene mostrato che, con il crescere del numero di pixel, la coppia reale converge a quella ideale. Differenti leggi di controllo d’assetto sono comparate, includendo una legge di controllo sotto-attuata, una ridotta e una proporzionale standard. I risultati mostrano che questo sistema riesce a copiare i primi due con buone prestazioni in termini di accuratezza degli angoli d’assetto, richiesta di coppia e abilità di mimica. 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 orbitale basato su un regolatore quadratico lineare viene introdotto, producendo la storia temporale degli 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 ciascun pixel: per l’assetto, solo il controllo ridotto è analizzato. Un problema di stazionamento viene usato come esempio, con lo scopo di stabilizzare la vela su un punto di equilibrio artificiale. I risultati mostrano che la vela converge asintoticamente a un intorno del punto desiderato, i cui limiti si possono ridurre aumentando il numero di pixel.

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Acknowledgements

First, 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 have improved my work, and the extreme cordiality and availability. I am also grateful to Dr. Matteo Ceriotti for having checked and analysed my thesis, since was important for me the opinion of specialists about my work. Thanks also to all the professors that 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 during these years at the university.

The most important acknowledge goes to my mum and dad that gave me the possibility 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 be grateful 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 avermi proposto una tesi sulle vele solari visto che sono sempre stato interessato a questo importante argomento, specialmente oggigiorno. Grazie per i suoi suggerimenti e le sue idee che hanno migliorato il mio lavoro e l’estrema cordialità e disponibilità. Sono grato anche al Dr. Matteo Ceriotti per aver controllato e analizzato la mia tesi, poiché per me era importante l’opinione di specialisti riguardo il mio lavoro. Grazie anche a tutti i professori che nel corso di questi anni mi hanno instillato la passione per lo spazio e l’ingegneria.

Voglio ringraziare mio fratello e tutti i miei amici per avermi incoraggiato durante questi anni 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 moralmente per tutta la mia vita, specialmente in questi ultimi anni. Gli sarò sempre grato per tutti i loro sacrifici che mi hanno permesso di diventare l’uomo che sono oggi.

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List of Figures

1.1 Time-line. . . 3

1.2 Reflectivity control devices typologies. . . 9

2.1 Solar radiation pressure force models. . . 17

2.2 Solar radiation pressure force geometry for a non-perfect, flat Lam-bertian solar sail. . . 18

2.3 Specular reflective + absorptive sail model. . . 19

2.4 Two-body problem geometry. . . 20

2.5 Inertial N and rotating L frames (2BP). . . 23

2.6 Rotating L and body-fixed B frames (2BP). . . 24

2.7 Designed orbit. . . 26

3.1 Euler angles: sequence 1-2-1. . . 32

3.2 Rotations and geometry. . . 35

3.3 Stereographic projection. . . 37

4.1 Reflectivity control device-based attitude control logic scheme. . . 42

4.2 Mesh with regular (•) and un-regular (•) positions (n=4). . . 48

4.3 Multiples of ˜A for regular elements. . . 49

4.4 Multiples of ˜A for un-regular elements. . . 49

4.5 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. . . 52

4.7 Ideal attitude control. . . 53

4.8 Real attitude control. . . 54

4.9 Global surface usage. . . 55

4.10 Local surface usage. . . 56

5.1 Control diagram. . . 59

5.2 Regions of high (D₁) and low (D₂) control torque. . . 64

6.1 Cone and clock angles - ideal (2BP). . . 69

6.2 Angles error - ideal (2BP). . . 70

6.3 Attitude control torque - ideal (2BP). . . 71

6.4 Under-actuated control methods comparison. . . 73

6.5 Angles error - real (2BP). . . 75

6.6 Attitude control torque - real (2BP). . . 76

6.7 Global surface usage (2BP). . . 77

6.8 Local surface usage (2BP). . . 78

6.9 |ζ|. . . 79

7.1 Circular restricted three-body problem geometry. . . 81

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7.3 Euler angles: sequence 1-3-1. . . 85

7.4 Artificial equilibrium points: x−y plane (ρs =0.9). . . 88

7.5 Artificial equilibrium points: x−z plane (ρs=0.9). . . 89

8.1 Reflectivity control device-based orbit control logic scheme. . . 94

8.2 Solar exclusion zone. . . 96

8.3 Designed trajectory. . . 99

8.4 Relative position - ideal (CR3BP). . . 100

8.5 Cone and clock angles - ideal (CR3BP). . . 100

8.6 Angles error - ideal (CR3BP). . . 100

8.7 Time step influence. . . 101

8.8 Surface usage for orbit control. . . 103

8.9 Acceleration error. . . 104

9.1 Trajectory - ideal coupling. . . 107

9.2 Relative position - ideal coupling. . . 107

9.3 Cone and clock angles - ideal coupling. . . 108

9.4 Angles error - ideal coupling. . . 108

9.5 Attitude control torque - ideal coupling. . . 109

9.6 ω1(0)influence. . . 111

9.7 Trajectory - real coupling. . . 113

9.8 Relative position - real coupling. . . 113

9.9 Cone and clock angles - real coupling. . . 114

9.10 Angles error - real coupling. . . 114

9.11 Attitude control torque - real coupling. . . 115

9.12 Surface usage. . . 116

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

A.1 Limb darkening geometry. . . 123

A.2 Deviation from the inverse square law. . . 124

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xiii

List of Tables

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

declared. . . 11

2.1 Solar sail orbit parameters. . . 22

2.2 2BP reference orbit parameters. . . 25

2.3 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). . . 67

6.2 Control gains for the attitude (2BP). . . 67

6.3 Ideal controls solver parameters (2BP). . . 68

6.4 Control gains for different under-actuated controls, with κc = 0.9× 10−3, µc =2 κc, and ρ =1. . . 72

6.5 Ideal controls comparison:= very good,↑= good, ↓= bad,= very bad. . . 80

8.1 CR3BP reference trajectory parameters. . . 97

8.2 CR3BP reference trajectory solver parameters. . . 98

9.1 Initial conditions and control gains for the attitude (CR3BP). . . 105

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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

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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

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xvii

Physical Constants

Astronomical Unit AU=1.495 978 707×108km

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

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

Mass of Earth _M♁ =5.9736×1024kg

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

Proton mass mp =1.672 622×10−27kg

Solar gravitational constant µ₌1.3272×1011km3s−2

Solar luminosity L₌3.832×1026W

Solar mass M₌1.9891×1030_kg

Solar radius R₌696 000 km

Universal gravity constant G=6.672 59×10−20kg km3s−2 Vacuum speed of light c=2.997 924 58×108_{m s}−1

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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 km2s−1

m Mass of the sail-craft kg

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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

˜α Required cone angle rad

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xxi

γ Roll angle rad

δ Clock angle rad

δopt Optimal clock angle rad

˜δ 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

ˆb Body frame

B Euler angles kinematics matrix

c Non-dimensional moment of inertia

C Rotation matrix

C Complex variable

e Principal axis

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H Angular momentum vector kg m2s−1

i Imaginary unit

ˆi 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

ˆu Direction perpendicular to ˆi0₁and ˆb1

w Second parameter

x1, x2, x3 Coordinates of the sphere

z First parameter

ρ Cayley-Rodrigues parameter

σ Modified Cayley-Rodrigues parameter

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υ Stereographic projection

Φ Principal angle rad

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

CHAPTER 4

˜

A 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

nA˜ Multiple of ˜A

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

¯

S Surface area of a quadrant m2

Y Coordinate of the centre of pressure along YB m

¯

Y Y coordinate for a quadrant m

Z Coordinate of the centre of pressure along ZB m

¯

Z Z coordinate for a quadrant m

CHAPTER 5

a Spacecraft bore-sight axis

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D Derivative gain matrix kg m2s−1

D Region in the zrand|wr|2plane

kd Derivative constant gain kg m2s−1

kD Derivative constant gain kg m2s−1

kp Proportional constant gain kg m2s−2

kP Proportional constant gain kg m2s−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 zrand|wr|2

ρ Constant parameter

τ Complex control torque s−2

ωn Closed-loop natural radial frequency rad s−1

˜

ω Angular velocity of the kinematics control rad s−1

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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

ˆi, ˆj, ˆk 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

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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

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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

˜r Front surface reflectivity

s Specular reflective part of ˜r

v Velocity km s−1

e Surface emission coefficient

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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

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n Normal component

N Inertial frame

N0 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

0_,00 _Intermediate

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OPERATORS

¯x Complex conjugate of a complex variable

|x| Absolute value

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

ˆx Unit vector

˙x First time derivative

¨x 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

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1

1 Introduction

1.1 History of Solar Sails

The evolution of solar sails has followed different steps, the most important of which are here listed in chronological order (Figure1.1):

• 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 the first 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 or electricity 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 second law 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 and Hull,1903].

• The first publication, named Extension of Man into Outer Space, by the famous Konstantin 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 sunlight to attain cosmic velocities.”

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

1924.

• 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 Russell Sanders to avoid being recognized.

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

• The first proposal of attitude control of a solar sail was made by R. L. Sohn in

1959for an attitude stabilization problem [Sohn,1959].

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

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

• Colin Robert McInnes extensively studied the dynamics of solar sails from the

1990’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 its famous 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 statically above a planetary pole, called Statite concept, in 1993.

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

• Jason S. Nuss formulated, in 1998, the study of solar sails in the three-body problem 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 the future 2020 human mission to the Moon, prior to a mission to Mars. For a deep analysis of solar sails dynamics in the Earth-Moon system consider [Wawrzy-niak, 2011]. The idea is that multiple spacecraft in Keplerian/non-Keplerian orbits around the Moon can be substituted by a single sail-craft positioned as a relay between a lunar outpost and the ground station on Earth, working as a polesitter. NASA has also proposed a sail-craft for the lunar communication coverage problem.

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

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

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

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1.1. History of Solar Sails 3 1619 J. Kepler 1865 J. Verne 1619 J. C. Maxwell 1865 A. Bartoli 1901 P. Lebedew 1903 E. Nichols G. Hull 1921 K. E. Tsiolkovsky 1924 F. Tsander 1951 C. Wiley 1958 R. L. Garwin 1959 R. L. Sohn 1960 Echo-1 1976 J. L. Wright 1991 R. L. Forward 1990’s C. R. McInnes 1991 J. L. Bell 1993 R. L. Forward 1995 NMP 1998 J. S. Nuss 2004 PVSE ... FUTURE FIGURE1.1: Time-line.

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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 was cancelled for various reasons, but the interest in solar sails increased. From that moment, NASA first, and subsequently the European Space Agency (ESA), start con-sidering the peculiarity of solar sails regarding non-Keplerian orbits, outer planets missions, interstellar missions, ... Initially, solar radiation pressure (SRP) was used to support spacecraft operations:

• The Mariner 10 spacecraft, in 1974−75, used SRP for attitude control around the roll axis, by rotating its solar arrays, since the spacecraft ran low on attitude control gas: this system worked. With commands from the mission controllers, they asymmetrically twisted the solar panels to create the so-called windmill torque 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 single deployable solar panel introduces a solar pressure disturbing torque that was removed by placing on the opposite side a long boom at the end of which a conical-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 on the north side of each satellite to offset the torque resulting from solar pressure on the array. In 2004, the Japanese deployed solar sail materials sub-orbitally from a sounding rocket, which was a key moment in the deployment of gossamer sheets from 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 the ground in vacuum conditions. Their primary objective was to demonstrate success-ful deployment of a lightweight solar sail structure in low Earth orbit. They were the first to deploy a solar sail on Earth, in April 2005 with the first one, and in July of the 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, the mission 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 on L1, while this mission planned to position the sail-craft closer to the Sun in a sub-L1

point, 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 the angular momentum vector within 1 deg of the Sun-line. The Team Encounter Mission considered 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 be placed on a wire to move the centre of mass of the spinning sail-craft and passively stabilize it. The NMP ST7 proposed a flight validation experiment for a sail whose

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1.2. Missions 5 attitude was controlled by shifting its centre of mass with respect to the centre of pressure. However, this system was challenging from the hardware implementation point 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 Kitesail-craft Accelerated by Radiation of the Sun), which was the first in-flight demonstration of solar sailing. It was a 14 m×14 m sail that had a 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. These devices 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 m2NanoSail-D22

(Nano Sail deploy, de-orbit, demonstration, and drag 2): the NanoSail-D1 was lost in a launch vehicle failure aboard a Falcon 1 rocket. Due to malfunctions, the sail-craft was ejected by the main spacecraft (FASTSAT) two months later, and when the sail deployed 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 m2sail, in June 2016. Its main objectives were to demonstrate sail deployment and controlled flight. It was placed in a Sun-synchronous orbit at an altitude of ∼ 820 km. LightSail-1 laid the foundation for the whole LightSail program by demonstrating controlled flight with only the pressure exerted by the solar photons impinging the sail.

1.2.3 Future

Nowadays, the TRL of solar sails has reached the level 6, system/subsystem model or prototype demonstration in a relevant environment (ground or space), but for the next level it requires a system prototype demonstration in a space environment. Solar sails can en-able new mission concepts, as highly non-Keplerian orbits, trajectories beyond the outer planets, hovering along the Sun-Earth line Sun-ward of L1for solar

observa-tion (i.e., Sunjammer mission concept), displacing the L1 point above the ecliptic

to observe the high latitude regions of Earth (i.e., pole-sitter mission concept), long residence time orbits in the Earth’s magnetotail (i.e., GeoSail mission concept), and low-cost multiple rendezvous near-Earth objects missions. However, this requires lighter 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 important aspect 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 sails either 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 mission by JAXA will explore the Jupiter Trojan asteroids with a power solar sail (i.e., a sail equipped with thin-film solar cells). Furthermore, the Planetary Society has planned to launch in the upcoming years LightSail-2 to provide new data and further refine the solar sailing skills of the project. For more details about future mission concepts look at [Vulpetti, Johnson, and Matloff,2015].

1_{https://earth.esa.int/web/eoportal/satellite-missions/i/ikaros}

2_{https://directory.eoportal.org/web/eoportal/satellite-missions/n/nanosail-d2} 3_{https://earth.esa.int/web/eoportal/satellite-missions/l/lightsail-1}

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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 becomes the dominant one and cannot be neglected. In addition, the inclusion of the SRP in the dynamical model creates an infinite set of new libration points, called artificial

equilibrium points(AEPs). All of this depends on the parameters of the sail. She catalogued 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 and computational time. The homotopy method has been used extensively to trajectory optimization 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 one propulsion system to another, different.

TugSat is a CubeSat that adopts a solar sail for de-orbiting of satellites from the geostationary orbit [Kelly et al., 2018]. The idea at the base of this project is the fact that a solar sail can be reused more times to remove a large number of satellites: the maximum capability of the removed satellite is 1000 kg, considering a high-performance sail. The satellites are placed in a retirement orbit, to secure the GEO-ring (Geosynchronous Equatorial Orbit). In this study, a non-linear orbit con-trol is developed, starting from a Lyapunov function (as done typically for attitude control), based on the time derivative of some orbital elements (semi-major axis, ec-centricity, inclination) and the geosynchronous equatorial orbit belt longitude. It is worth noting that SRP is the main cause of unwanted re-enter of satellites in secure graveyard 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 perform close flybys and investigate near Earth asteroids. It is under development by NASA and JPL, and it will be launched in December 2019 as a secondary payload of the Space Launch System’s (SLS) inaugural flight, Exploration Mission 1 (EM-1). It is a 86 m2 sail with a 12 kg main bus of a 6U-CubeSat. Attitude control is performed thanks to reaction wheels, active mass translator, and cold gas thrusters. The ac-tive mass translator is used only for momentum desaturation shifting the centre of mass to the centre of pressure. The system is capable of angular rates of 0.04 deg /s around 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 corrective actions in case of initial trim manoeuvre failure after the separation from EM-1. In this analysis, since only the orbital part is considered, the problem of the tangen-tial component of the SRP acceleration, important in attitude dynamics, is simply underlined but not analysed.

In [Farrés, Heiligers, and Miguel,2018] the problem of transfers from displaced L1and L2Sun-Earth libration points to regions of practical stability around L4 and

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1.3. Recent Studies 7 synodical velocity, it will remain there without any need of station-keeping, with residence 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, since

it is hard to reach orbits around L4and L5for classical spacecraft, a solar sail can do

that in a reasonable amount of time. For example, L5is a good place to observe the

Sun’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 attitude and orbit control of a spinning solar sail based on active shape control of the sail membranes, which becomes a 3D structure. In general a sail is spin-stabilized to counteract disturbances, maintain the correct attitude with respect to the Sun, for long term missions and for large dimensions. They propose a method to control the deformation state of the membranes to change actively the SRP acting on the sail, considering a realistic model for the pressure. They have also introduced the concept of fuel- and geometry-free solar sailing. They manage to control attitude and orbit with the deformation of the sail, and contemporaneously maintain the communication 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 and Earth communication (i.e., force), while the higher orders (i.e., second order static wave) 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 minimum time 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 genetic algorithm (GA). This hybrid method uses SNOPT and GA together in such a way that the disadvantages of the two are counteracted. In fact, GA is used to create a good approximation of the initial point since it is the main problem of SNOPT which shows high dependence on it. Then, GA is stopped and SNOPT continues the optimization process from a certain point on, counteracting the poor conver-gence properties and inability to handle constraints efficiently by GA. They proved that 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 [Stohlman et 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 sail membrane since the remaining consists on tape adhesives. This means that care must 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 a certain method of folding. They have analysed three methods compared to classic hand-folding: 1) folding between folding forms is repeatable, less time-consuming and allows 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 procedure even if some problems of stacking arise. The problem of venting and the decoupling between booms and membrane deployment have been studied.

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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 is applied a certain voltage. The first RCDs where developed from the light control glasses, modified to be radiation tolerant. There are two main types of devices al-ready developed:

• Diffusion RCDs (Figure1.2a and 1.2b) can alternate from diffuse reflection to specular reflection.

• Transmission RCDs (Figure1.2c and 1.2d) can alternate between specular re-flection to transmission. The transition speed of such devices is in the range 10-100 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) put between two polyimide-film with the bottom one aluminium-deposited. One of the main advantage of these devices is that they introduce very small oscillations to the membrane, differently from classical control systems. The characteristics of those devices 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 attitude steering of a spinning solar sail, with a precision in term of Sun angle∼ 10−2deg. One effect that was observed is the introduction of a disturbance torque affecting the roll 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 embedded

within 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 incoming

light, 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 = np

light 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 sail concept based on this type of devices. They declare that the power requirement in the ON state is less than 0.5 mW cm−2. In addition, the change of the fraction of the weighted average momentum transferred to PDLC between the two states is in the order of 0.5, compared to 0.11 of IKAROS devices. The reduction of the PDLC thickness is a way to reduce the power consumption of the device; however, thicker cells provide a larger momentum difference between the two states. The Propellant-less Attitude Control of Solar Sail Technology Utilizing Reflective Control Devices project team 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

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1.4. Reflectivity Control Devices 9 light obliquely (Figure1.2e), introducing two advantages: counteract the windmill effect 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 classic RCD, but the bottom layer has a saw-tooth structure.

(A) Diffusion RCD OFF state [Funase et al.,2010]. (B) Diffusion RCD ON state [Funase et al.,2010]. (C) Transmission RCD OFF state [Munday,2016]. (D) Transmission RCD ON state [Munday,2016].

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

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1.4.2 Applications

In [Borggräfe et al.,2014b] the use of RCDs is analysed in the domain of the two-body problem (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, separated by a line.

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

“When distributing controlled regions of high and low reflectivity across the surface, a wide range 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, in the 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 to the rotation. In addition, they consider a realistic behaviour for the SRP model and all 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 the adoption of RCDs since are able to reduce the propellant required to maintain the spacecraft 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 that some assumptions have been made, even if not stated. They adopt backstepping methods since suitable for controlling "subsystems", as the coupled orbit-attitude case. The torque around the roll axis is neglected, and a reaction wheel is used to control the sailcarft around it; to quote them:

“The torque generated by the solar radiation pressure force transverse to the surface is not considered 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 the

solar 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 this

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1.4. Reflectivity Control Devices 11 In [Gao et al., 2018] is analysed a highly-fidelity model, coupled orbit-attitude propellant efficient station-keeping for a hybrid sail in the Earth-Moon system using RCDs. 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 improves the 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, which is the typical main result and it is always better than a system without them. They considered also the presence of thin-film solar cells on the sail surface. They state that optimal periodic controllers have a convergence speed greater than classical linear quadratic regulator (LQR). They switch the RCD state between specular and absorptive, 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 the SRP is not considered with all the components, quoting them:

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

TABLE1.1: RCD-based papers comparison:X =used,× =not used,

n.d. = not declared.

Orbital model Sail model Pressure Control law ID

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 the orbit, SRP model, components of the force used in the attitude control laws, type of control considered. A characteristic of all those researches is that the surface covered of RCDs is always considered as a continuum, without taking into account the real dimensions of the single device, and the SRP force component lying in the sail plane is always neglected.

To the best of this author’s knowledge, there was only one application where the sail 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 behaving like 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 reflectivity control devices to create that torque, but only in the plane of the sail. The logic is based 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

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

In a recent work, the problem of three-axis attitude control of a solar sail with the adoption of RCDs has been analysed [Theodorou,2016]. He analysed the problem of expanding the control in the three-dimensional domain by rotating at fixed angles the 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 high number of combinations also considering a small number of devices. In fact, in his analyses he considered at maximum 5 RCDs per sides, for a total of 20. The problem was 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 ON and OFF states, creating a polyhedron. In addition, a Sun map is built, which is a graphic representation of the Sun angles where the control requirements are met: it requires 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 ideal attitude control law generated to be supplied to the real control was a proportional integral derivative (PID) control. He studied also a hybrid system consisting of three reaction wheels (one for each axis of the sail), together with the RCDs, that provide a larger torque for a limited time in order to achieve higher slew rates and larger angular 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 high RCDs 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 and attitude 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 simplificradi-ation about its component lying in the sail plane when analysing the generated torque. In all the other works presented before this is not done, but here it is shown that its effect is not always negligible. In this section the sentences contained in the papers underline that the torque along the normal to the sail is given by wrinkles and/or buckles of the membrane, that are not considered. However, the control itself, based on RCDs, introduces in any case a torque along the normal due to that component of the SRP. The analysis is performed both in the two-body problem (attitude alone) and the circular restricted three-body problem (coupled orbit-attitude). Regarding the devices, here they are used to study an alternative approach for the control. For this reason, the existing devices are not considered; instead, a different logic capable of switching between specular and absorptive state is adopted, as done also in other works. Due to the adoption of these devices, a more realistic behaviour for the sail is considered in the coupled control analysis. Finally, the RCDs are distributed on the surface as pixels, so with their own dimensions, and can work independently one to the other. This thesis expands the application of distributed reflectivity control de-vices for a large number of dede-vices, combining combinations logic and an alternative approach.

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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 followed by the description of the 2BP model perturbed by the SRP, and a reference orbit is generated.

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

4 The attitude control based on RCDs is fully described. First, the combinations logic and the mesh is described, with all the steps to follow for its creation. Then, the attitude control logic is described, and the performance analysed considering 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 more detail since slightly different from the one found in the literature.

6 Ideal and real controls are used to track the desired attitude history introduced in 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 classical LQR control for the orbit is developed.

8 The orbit control based on RCDs is described, and the performance evaluated using 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 the union of the two controls developed separately for orbit and attitude: only the reduced (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 and SimulinkR . ToR help the reader, lists of abbreviations, physical constants, and symbols can be found just before this chapter. The symbols are divided among the chapters: symbols al-ready defined in previous chapters are not redefined next, except if they change meaning.

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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)

whereE is the energy transported by a photon of frequency ν and h is the Plank’s constant. Then, from the Einstein’s special relativity theory can be found that, since the 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. From

Equation2.1 and 2.2, the momentum can be derived p= E

c = h ν

c (2.3)

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

W = L 4 π r2 0 r0 r 2 =W0 r0 r 2 (2.4) where L is the solar luminosity, r0 is the mean Sun-Earth distance equivalent to

1 AU (astronomical unit), and W0is the energy flux at 1 AU. By defining the energy

transported across a surface of area A normal to the incident radiation in time∆t

∆E =W A∆t (2.5)

, and adoptingEquation2.3 and 2.4, the expression of the SRP can be finally obtained P= 1 A ∆p ∆t = 1 A ∆E c∆t = W c = W0 c r0 r 2 = P0 r0 r 2 (2.6) Adopting the typical average value of W0 = 1367 W m−2, the value of the SRP at

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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 electromagelectromag-netic wave is given by (E=c B, e0 µ0 =1/c2) U= 1 2 e0 E 2₊ 1 2 µ0 B2 = B 2 µ0 = e0 E2 (2.7)

where e0is the free space permittivity, µ0is the free space permeability, E is the

elec-tric field, and B is the magnetic field. An important quantity in the electromagnetic theory 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-tion2.7 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 atEquation2.5, it can be found that

U = W

c (2.11)

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

pem = S

c2 =⇒ pem =

U

c (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 exerted

on a perfectly absorbing surface coincides with the energy density of the electromag-netic wave

P= F

A =U=

W

c (2.14)

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

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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), and

diffusely reflected (ρd), such that

ρa+ρs+ρd=1 (2.15)

In this model, the transmitted fraction ρtis neglected. The SRP force acting on a flat,

Lambertian1surface is then modelled as

FSRP= P S ρa <s, n>s+2 ρs <s, n>2n+ρd<s, n> s+ 2 3 n = P S<s, n> (ρa+ρd)s+ 2 ρs <s, n> +2 3 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 at Appendix 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 90 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s = 1 d = 1 a = 1

FIGURE2.1: Solar radiation pressure force models.

Attitude and orbit control of a solar sail through distributed reflectivity modulation devices

P

OLITECNICO DI

M

ILANO

M

T

Attitude and Orbit Control of a Solar Sail

Through Distributed Reflectivity

Modulation Devices

Declaration of Authorship

Abstract

Acknowledgements

Contents

List of Figures

List of Tables

List of Abbreviations

Physical Constants

List of Symbols

1 Introduction

1.1

History of Solar Sails

1.2

Missions

1.3

Recent Studies

1.4

Reflectivity Control Devices

1.5

Thesis Content

2 Solar Sail Models and Dynamics

2.1

Solar Radiation Pressure

2.2

Force Models