Optimization of the Control of a Satellite Formation in a Near-Geostationary Orbit by Means of Solar Sail Propulsion

Optimization of the control of; a satellite formation in a near- geostationary orbit by means of solar sail propulsion

Julien Pelamatti

Optimization of the control of a satellite formation in a near-geostationary orbit by means of solar sail propulsion

by Julien Pelamatti

in partial fulﬁlment of the requirements for the degree of Master of Science in Aerospace Engineering at the Delft University of Technology, to be defended publicly on March 27, 2017

Student number: 4415132 Project duration: May 16, 2016 – March 27, 2017 Thesis committee: Dr. J. (Jian) Guo, TU Delft Dr.ir. M.J. (Jeannette) Heiligers, TU Delft, supervisor Ir. R. (Ron) Noomen, TU Delft Prof.dr.ir. P.N.A.M. (Pieter) Visser, TU Delft

An electronic version of this thesis is available at http://repository.tudelft.nl/. 1

1Cover credit: NASA, https://www.nasa.gov/press-release/nasa-s-cubesat-initiative-aids-in-testing-of-technology-for-solar-sails- in-space

Acknowledgements

For the sake of the poor readers, I will try to avoid clichés and be as concise as possible. The report that follows is the result of what has been, without any doubt, the most challenging and demanding period of my life. I’ve worked harder than I ever have, but even that was barely enough. Very surprisingly, given my legendary laziness, my motivation and focus rarely faded and I like to think that this was only possible thanks to a number of people. A ﬁrst and obvious mention goes to my parents and my sister whose constant and unconditional support, even from a large distance, has been crucial for me to get through the ups and downs of this past year and not give up to become a bartender in Cancún. Subsequently, I would of course like to thank my supervisor, Dr. Heiligers, for her patience, her guidance and her help without which this thesis would have probably resulted in something barely readable and even less interesting than it is. Finally, I would also like to mention Dr. Dirkx for his help during my endless and very frustrating coding related issues.

The list of remaining people I would like to mention in these acknowledgements is very, very long, but I would rather not turn this report into a phone book, not to mention that I’m extremely bad at expressing emotions. For these reasons, I will just say that in the past two years I’ve met some pretty awesome friends who made this stay in Delft a life-changing experience which, if I had the chance, I would do all over again, kind of. Thanks for everything.

"Ladies and Gentlemen, this is Mambo number 5:"

iii

Summary

Space debris, composed of man-made objects orbiting around Earth in an uncontrolled fashion, represents a critical issue for long-term sustainability of outer space activities and for space safety. A reliable and accurate mapping and monitoring of space debris is therefore a critical necessity for modern space industry. For relatively low-altitude orbits, most debris can be accurately tracked by means of ground-based observations. However, for higher-altitude orbits, this debris detection method becomes less precise and smaller debris cannot be tracked anymore. The solution to this problem that will be analyzed in this thesis is the possibility to triangulate the position of the debris with the use of a formation of two or three satellites orbiting near the area of interest. In particular, the possibility of monitoring debris in the geostationary ring will be investigated.

Due to the presence of perturbations, the satellites of the formation will deviate from their nominal orbit and by consequence from the optimal debris triangulation conditions. It will therefore be necessary to maneuver them in order to maintain a relative distance and position between spacecraft that results in a accurate and constant tracking of the geostationary debris. These maneuvers require a form of propulsion in order to provide the necessary thrust. Among the various propulsion technologies available for spacecraft control, this thesis will explore the possibility of using solar sailing. Solar sailing is a propulsion method theorized 100 years ago, but only developed in recent years, that uses the pressure exerted by electromagnetic radiation on a large and lightweight surface attached to the satellite in order to generate thrust. In this thesis, the possibility of maneuvering the satellites by means of solar sail propulsion in order to maintain the formation in an optimal position and shape for the triangulation of debris will be analyzed. This concept will be studied by developing a simulation model describing the orbital dynamics of the spacecraft under the inﬂuence of the Earth’s gravitational attraction, the thrust generated by the sail and a number of perturbations. The control of the formation will be parametrized and subsequently optimized with a self-adaptive differential evolution algorithm. This optimizer will work towards maintaining each satellite of the formation on an optimal orbit, characterized by a number of orbital requirements such as minimum and maximum inclination and inter-satellite distance.

Among the simulations that were performed with different parametrizations of the formation control and with different optimization set-ups, only one complied with all the defined requirements. In fact, it is noticeable that maintaining the inter-satellite distance within the required range for time spans of a year or longer is very challenging and the developed optimization formulation in most cases cannot determine the necessary control law. It can also be noticed that the perturbations that tend to make the formation drift to larger orbital inclinations are more difficult to counteract and that a particular attention needs to be directed to this phenomenon in case long duration simulations are considered. Furthermore, it was found that better results are obtained if the decision variables representing the solar sail attitude are optimized simultaneously for the entire simulation duration rather than separately over limited time periods. However, the results obtained also show that the developed concept tends to control the satellites towards an improvement in the objective function and it is the opinion of the author that a better definition of the control parametrization coupled with a larger computational effort and eventually more suitable optimization algorithms could yield better results and comply with the concept requirements.

Contents

Acknowledgements iii Summary v Nomenclature xi List of abbreviations xv 1 Introduction 1 1.1 Space debris...... 1 1.1.1 Spacecraft removal from low-Earth orbits...... 2 1.1.2 Spacecraft removal from geostationary orbit...... 3 1.2 Problem deﬁnition...... 3 1.2.1 Solution concept...... 4 1.2.2 Propulsion...... 5 1.2.3 Literature heritage...... 5 1.3 Research question and thesis objective...... 5 1.4 Thesis structure...... 6 2 Solar sailing 9 2.1 Physics of solar radiation pressure...... 9 2.2 Concept of solar sailing...... 10 2.2.1 Mission heritage...... 11 2.2.2 Strengths, capabilities and limitations of solar sailing...... 12 2.3 Force models...... 14 2.3.1 Model geometry...... 14 2.3.2 Ideal sail force model...... 15 2.3.3 Optical sail force model...... 16 3 Mission statement and requirements 19 3.1 Mission statement...... 19 3.2 Orbital requirements...... 19 3.3 Nominal orbits and initial conditions...... 20 3.4 Spacecraft characteristics...... 22 4 Orbital dynamics 25 4.1 References frames...... 25 4.1.1 Earth-ﬁxed inertial reference frame (J2000)...... 25 4.1.2 Heliocentric Orbital Frame (HOF) and Heliocentric Inertial Frame (HIF)...... 25 4.2 Reference frame transformations...... 26 4.2.1 Transformation from Heliocentric orbital to heliocentric inertial reference frame.. 27 4.2.2 Transformation from Heliocentric inertial to J2000 frame...... 27 4.3 Two-body motion...... 27 4.4 Perturbed motion...... 29 4.4.1 Spherical harmonics...... 30 4.4.2 Solar radiation pressure acceleration...... 31 4.4.3 Third body perturbations...... 33

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5 Optimization problem 35 5.1 Optimization problem definition...... 35 5.2 Objective function definitions...... 35 5.3 Decision variables...... 37 5.3.1 Constant value representation...... 37 5.3.2 Patched polynomial representation...... 37 5.3.3 Patched sinusoidal representation...... 39 5.3.4 Patched square wave representation...... 40 5.4 Constraints...... 40 5.5 Optimization logic...... 40 6 Numerical methods 43 6.1 Numerical integration...... 43 6.2 Optimization algorithm...... 44 6.2.1 Self-adaptation...... 45 6.2.2 Strengths and weaknesses of the SaDE algorithm...... 46 6.2.3 Seed...... 46 6.3 Application overview...... 47 6.3.1 Tudat libraries...... 47 6.3.2 PaGMO libraries...... 47 6.3.3 Celestial bodies ephemerides...... 47 6.4 Application block diagram...... 47 6.4.1 Orbital mechanics propagation block diagram...... 48 6.4.2 Optimization block diagram...... 48 7 Code verication and validation 53 7.1 Verification of the orbital mechanics propagation block...... 53 7.1.1 Solar System bodies ephemerides...... 53 7.1.2 SRP model...... 54 7.1.3 Spacecraft dynamics propagation...... 54 7.1.4 Cone and clock angle profiles...... 56 7.2 Verification of the optimization...... 57 7.2.1 Seed setting...... 57 7.2.2 Optimization routine...... 57 8 Preliminary results 61 8.1 Solar sail perturbation counteracting capabilities...... 61 8.2 Assessment of low order polynomial representation...... 64 8.3 Sinusoidal representation optimal control parameters...... 64 8.3.1 Optimal frequency...... 65 8.3.2 Optimal sail size...... 66 8.3.3 Optimal population size...... 66 8.4 Constant and square wave representations optimal control parameters...... 67 9 Results 69 9.1 ’Global’ optimization logic...... 69 9.1.1 ’Inclination and distance’ objective function...... 69 9.1.2 ’Observation angle’ objective function...... 70 9.2 ’Revolution-by-revolution’ optimization logic...... 72 9.2.1 ’Inclination and distance’ objective function...... 72 9.2.2 ’Observation angle’ objective function...... 73 9.3 Results comparison...... 75 9.4 Additional constraints or requirements...... 77 9.4.1 Three satellite formation...... 77 9.4.2 Cone angle limitation...... 78 9.4.3 Long duration optimization...... 79 9.4.4 Control effort penalty function...... 80 Contents ix

10 Conclusions and recommendation 87 10.1 Conclusions...... 87 10.2 Recommendations...... 89 Bibliography 91

Nomenclature

Symbol Meaning Unit of measure a Orbital semi-major axis km ak Cone angle polynomial coefficient of degree k - a∗ Absorption coefficient - ~a Acceleration vector m/s2 A Spacecraft/solar sail surface m2 A∗ Spacecraft/solar sail projected area m2 Aα Cone angle amplitude rad Aδ Clock angle amplitude rad A˜ Occulted area m2 Aeq Linear equality constraint matrix - b Linear inequality constraint vector - ˜b Cone angle patched polynomial time delay parameter - beq Linear equality constraint vector - B Lambertian coefficient - B˜ Linear inequality constraint matrix - Bα Cone angle offset rad Bδ Clock angle offset rad c Speed of light m/s c˜ Non-linear inequality constraint function - c∗ Number of profile changes - ck Clock angle polynomial coefficients of degree k - ceq Non-linear equality constraint function - C Generic constant - Cn,m Spherical harmonics model parameter - CR Spacecraft solar radiation pressure coefficient - CR Crossover probability - d Distance between two objects km d˜ Clock angle patched polynomial time delay parameter - d~ Decision variables vector - e Orbital eccentricity - e˜ Local truncation error - E Photon energy J ∆E Energy exchange J f Force intensity N f˜ Objective function - f~ Force vector N F Differential weight - g Penalty function - G Gravitational constant km3/kg·s2 h Planck constant Js h˜ Integration time-step s h∗ Sum of the objective and the penalty functions - HI Heliocentric inertial reference frame - HO Heliocentric orbital reference frame - ∆H Altitude change km i Orbital inclination rad

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Symbol Meaning Unit of measure J Inertial J2000 reference frame - J2,J2,2 Spherical harmonics coefficients - k Sinusoidal frequency multiplier - ~k Runge-Kutta evaluation vector - K Constant attitude angle value - lb Objective function argument lower bound - Ls Sun luminosity W L1 First Lagrange point - m Spacecraft mass kg ~m SRP force direction vector - m0 Resting mass of a particle kg MSun Mass of the Sun kg n Polynomial order - nf Number of mutating vectors passed on to the following generation - ns Number of discarded mutating vectors - n∗ Positive integer - n˜ Population size individuals ~n Sail normal vector - N Number of satellites in the formation - N˜ Search space dimension - o Shadowing function - O Number of parameters to be optimized - p˜ Probability value - ~p Photon momentum vector kg·m/s p~∗ Reflected photon momentum vector kg·m/s ∆~p Momentum exchange kg·m/s P Solar radiation pressure N/m2 P ∗ Shadowed solar radiation pressure N/m2 Pn,m Associated Legendre function - ~q Clock angle reference direction vector - r Orbital radius km r˜ Reflection coefficient - ~r Position vector km rand, randb, rnbr Randomly generated values - R Mean equatorial radius km RJ Rotation matrix - Ro Occulting body radius km RS Occulted body radius km RP Sum of the perturbing forces potential J Rõ Occulting body apparent radius rad R˜S Occulted body apparent radius rad s Fraction of specularly reflected photons - ~s Direction of specularly reflected photons - Sn,m Spherical harmonics model parameter - t Time coordinate s t0 Initial time s T Orbital period s ~t Sail tangential vector - ~u Sun-line vector - ∆T Time interval s ub Objective function argument upper bound - U Potential energy J ~vk Optimizer mutating vector - Contents xiii

Notation Meaning Unit of measure ∆V Change in velocity km/s w Objective function weight - W Energy flux W/m2 x Objective function argument - ~xk Optimizer individual vector - ~xSC Spacecraft position vector km - ~xSun Sun position vector km X First axis of the reference frame - ~y Spacecraft state vector km - km/s ~y0 Initial spacecraft state vector km - km/s Y Second axis of the reference frame - ~z Optimizer trial vector - z Distance from the equatorial plane km Z Third axis of the reference frame - α Sail cone angle rad α˜ Apparent separation between occulting and occulted bodies rad β Sail lightness number - γ Heliocentric orbital frame to J2000 frame rotation angle rad γ∗ Angle characterizing the projection of the debris on the equatorial plane rad δ Sail clock angle rad ∆ Attitude angle variation rad ε Emissivity coefficient - ζ Heliocentric orbital frame to J2000 frame rotation angle rad ~η State vector estimate - θ True anomaly rad θ∗ Angle between the incoming radiation and the SRP force rad Λ Geographic longitude rad µ Gravitational parameter km3/s2 ν Electromagnetic wave frequency Hz ρ Orbital radius projection on the ecliptic plane km σ Sail loading parameter kg/m2 σ∗ Critical sail loading parameter kg/m2 τ Solar sail transmission coefficient - τn Differential evolution optimization control parameter - φ Angle between the sail normal and the SRP force rad φα Cone angle sinusoidal phase rad φδ Clock angle sinusoidal phase rad Φ Geographic latitude rad Φ~ RK4 incremental function - ω Argument of periapsis rad ωα Cone angle sinusoidal frequency rad/s ωδ Clock angle sinusoidal frequency rad/s ω˜ Orbit angular velocity rad/s ω∗ Displaced orbit angular velocity rad/s Ω Longitude of the ascending node rad xiv Contents

Subscript Meaning a Photon absorption A Satellite Athos b Back of the sail best Best individual at a given generation d Space debris e Photon emission Earth Earth end End of the simulation f Front of the sail fc Formation center g Penalty function g˜ Generation number g˜ GEO Geostationary orbit h Spherical harmonics H Halo orbit HIF Heliocentric inertial frame HOF Heliocentric orbital frame i, j Orbiting bodies I Optimization parameter JJ2000 reference frame k Individual number k l Vector element number l m Iteration number m max Maximum value of a given variable o Occulted body obs Observer opt Optimal value of a given variable P Satellite Porthos rand Randomly generated values rs Specular reﬂection of photons ru Non-specular reﬂection of photons s Satellite S Occulted body Sun Sun SC Spacecraft TB Third body perturbations u Optimization parameter 0 Beginning of the integration step 1 Initial transfer orbit 2 Final transfer orbit α Cone angle δ Clock angle List of abbreviations

AU Astronomical Unit

DE Differential Evolution

DoD Department of Defence

ESA European Space Agency

FASTSAT Fast, Affordable, Science and Technology SATellite

GEO Geostationary Earth Orbit

HIF Heliocentric Inertial Frame

HOF Heliocentric Orbital Frame

IADC Inter-Agency space Debris Coordination Committee

IKAROS Interplalentary Kite-craft Accelerated by Radiations Of the Sun

JAXA Japan Aerospace eXploration Agency

LEO Low-Earth Orbit

NASA National Aeronautics and Space Administration

NKO Non-Keplerian Orbits

PaGMO Parallel Global Multiobjective Optimizer

PRNG Pseudo-Random Number Generation

RK Runge-Kutta

SaDE Selft-adaptive Differential Evolution

SBSS Space-Based Space Surveillance

SOF Sailcraft Orbital Frame

SPICE Spacecraft ephemeris, Planet ephemerides, Instrument description kernel, C-matrix, Events kernel

SRP Solar Radiation Pressure

SSA Space Situational Awareness

SSB Solar System Barycenter

TLE Two-Lines Elements

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Tudat TU Delft Astrodynamics Toolbox 1

Introduction

This thesis analyzes the possibility of using a formation of satellites in a near-geostationary orbit propelled by solar sails in order to monitor and track the position of space debris in the geostationary belt. In this chapter, the context in which the space debris problem exists is presented and the need for a way of monitoring small-sized debris, which the studied concept aims to provide, is explained. The problem deﬁnition is followed by a short analysis of the concept of solar sailing. Subsequently, a brief overview of the existing literature treating the possible applications of solar sailing combined with satellite formations is provided. From the previously treated information, the research question and sub-questions as well as the research objective will be deﬁned and analyzed. Finally, a short description of the thesis structure will be provided.

1.1. Space debris

The Inter-Agency space Debris Coordination Committee (IADC) deﬁnes space debris as "all man made objects including fragments and elements thereof, in Earth orbit or re-entering the atmosphere, that are non functional"[1]. Space debris presents a serious danger to active satellites that orbit at similar altitudes, as the relative velocities between the two can reach very high values and therefore a single impact can render the spacecraft partially or completely unusable [2]. Furthermore, if the collision causes the spacecraft to break, new pieces of debris are created. A notable example of this possibility is the collision between the U.S. satellite Iridium 33 and the Russian satellite Kosmos 2251 in 2009 that is estimated to have generated more than 1000 medium-sized debris [3][4]. The ensuing repetition of collisions could cause the amount of space debris to increase exponentially and it is theorized that the uncontrolled growth of the number of objects in Low-Earth Orbit (LEO) could lead to a chain of collisions that would render any type of space-based activity infeasible for hundreds of years. This scenario is known as the Kessler Syndrome [5].

In order to contain the problem, international regulations have been issued: space-system operators are now required to remove their spacecraft from protected regions within 25 years after the end of the operational phase of the mission [1]. These protected regions are LEO (ranging from 160 to 2, 000 km altitude) and the geosynchronous belt (ranging from 35, 586 to 35, 986 km altitude and from −15 to 15 deg of inclination), as illustrated in Figure 1.1.

1 2 1. Introduction

Figure 1.1: Protected zones, according to IADC guidelines [1]

1.1.1. Spacecraft removal from low-Earth orbits The removal of spacecraft from LEO usually requires small amounts of propellant and minor maneuvers are enough to bring the satellite to an altitude where the atmospheric drag completes the de- orbiting within the required time span of 25 years. The average ∆V necessary to comply with the given de-orbiting time is shown in Figure 1.2 as a function of the initial altitude and the type of maneuver. It can be noticed that for particularly low altitudes (< 600 km), a maneuver may not even be required.

Figure 1.2: ∆V required to de-orbit a satellite from LEO within 25 years [6] 1.2. Problem deﬁnition 3

1.1.2. Spacecraft removal from geostationary orbit When considering the spacecraft removal from (near)-geostationary Earth orbits (GEO), a complete de-orbiting becomes very expensive from a propellant consumption point of view. An estimate of the impulse, ∆V , necessary to perform an altitude change maneuver from a geostationary orbit to an altitude where the atmospheric drag completes the process within the required time can be computed as a Hohmann transfer [7]: r r r r √ 1 2r2 1 2r1 4 ∆V = µEarth − 1 + 1 − ' 1.3 · 10 m/s (1.1) r1 r1 + r2 r2 r1 + r2 where

11 3 −2 • µEarth = 3.986004418 · 10 km s is the gravitational parameter of Earth

• r1 = 42, 164 km is the orbital radius of the geostationary orbit

• r2 = 6, 978 km is the orbital radius of the required final circular orbit For this estimate, a final circular orbit characterized by an altitude of 600 km was considered in view of the fact that it is the lower limit for which the de-orbiting of the satellite occurs within 25 years without any maneuver, as shown by Figure 1.2. Apart from the Hohmann impulsive orbit transfer, other possibly more efficient alternatives exist, such as low-thrust transfers or transfers to highly eccentric orbits with apogee at the geosynchronous altitude and perigee at an altitude lower than 600 km. However, these solutions might increase the risk of collisions. Furthermore, a detailed analysis of the optimal de- orbiting strategy for GEO satellites goes beyond the scope of this thesis.

When comparing the results of Eq. 1.1 with the values presented in Figure 1.2, it can be seen that the resulting estimated de-orbiting cost from GEO is one to two orders of magnitude larger than the one from LEO. Performing these maneuvers would therefore substantially increase the propellant budget of GEO missions and by extension their ﬁnancial cost. A more commonly preferred alternative is to perform an altitude change in order to place the spacecraft in the so-called "graveyard orbit", a nearly unused orbit at an altitude 200 − 300 km above the geostationary altitude. More speciﬁcally, as stated by the guidelines, the minimum required change in altitude, ∆H, in order to safely place a satellite in the graveyard orbit is equal to [1]: A ∆H = 235 + C · 103 [km] (1.2) R m where

• CR is the spacecraft solar radiation pressure coefﬁcient • A is the spacecraft cross-sectional area • m is the spacecraft mass However, for nearly two out of three spacecraft launched into GEO, these international recommendations are not followed and the ﬁnal orbits of the satellites are lower than they are supposed to be, which presents a risk to active satellites in the vicinity [8]. Furthermore, it sometimes happens that the on-board systems fail, thus rendering the spacecraft uncontrollable and occasionally also causing the fragmentation of the satellite. A graphical representation of the condition of tracked large-sized objects orbiting at a near-geostationary altitude is shown in Figure 1.3. It can be seen that approximatively 15% of these objects orbit in an uncontrolled manner in GEO, while another 26% drifts with respect to the geostationary orbit (either above or below), thus representing a danger for active satellites.

1.2. Problem deﬁnition From Section 1.1 and in particular Figure 1.3, it is clear that, for the sustainability of the space industry, space exploration and space environment, it is crucial to track and monitor the evolution of the number of space debris and their orbits. For medium and large-sized objects ( ≥ 10−15 cm), this can be easily achieved with ground-based optical observations, which are then used to compute the object’s two-line 4 1. Introduction

Figure 1.3: Condition of tracked large-sized objects orbiting in GEO [8]

element (TLE) set. Instead, for small debris orbiting at high altitudes, these ground-based detection methods prove to be inefﬁcient because of the large distances and because of the limited resolution of the used instruments [2]. Given that small-sized debris can also be fatal to active satellites, an alternative monitoring method is necessary in order to minimize the chances of impacts.

1.2.1. Solution concept

In this thesis, the possibility of monitoring and tracking debris in near-geostationary orbits by using in-situ optical measurements performed by a formation of satellites orbiting near the area of interest will be considered. It is assumed that by greatly reducing the distance between the debris and the observer, it will be possible to detect and track debris of smaller size even with standard instrumentation. Furthermore, a minimum of two satellites will be necessary in order to triangulate the position of the observed debris. However, the possibility of adding a third spacecraft to the formation in order to improve the accuracy of the resulting debris position determination and the robustness of the mission concept will be analyzed.

The only satellites with a similar task currently in orbit around Earth are part of the Space-Based Space Surveillance (SBSS) constellation. This constellation is operated by the United States Department of Defence (DoD) and has the purpose of providing supplementary orbit determination information for GEO satellites to complete the Earth-based measurements and to identify any possible space-based military threat [3]. The SBSS constellation comprises of the Pathfinder satellite orbiting at an altitude of 600 km and two additional satellites in a near-geostationary orbit. However, given the confidential nature of the project, a very limited amount of information is available regarding both the spacecraft specifications and the collected data. Furthermore, the main purpose of the SBSS constellation is to provide orbit determination information for known objects in GEO while identification and charac- terization of new debris is not provided. A mission with the purpose of providing openly available identification and orbit determination information for small-sized debris in GEO could therefore fill a critical gap for what concerns space situational awareness (SSA) and space safety. It is following this rationale that the presented thesis project will be developed. 1.3. Research question and thesis objective5

1.2.2. Propulsion In order to maintain optimal relative positions between the formation satellites and the observed object as well as to avoid collisions, the satellites will need to perform station-keeping and orbital manoeuvres on a regular basis. This is usually achieved by using thrusters. However, the classically used electrical or chemical thrusters consume propellant. The limited on-board storage capability of said propellant puts a severe limit on the mission duration [9]. For this reason, in this thesis the possibility of using solar sailing as a long-lasting and possibly lighter propulsion solution will be analyzed.

In recent years, solar sailing has been developed as a possible alternative to the classical propulsion solutions previously listed. The main idea behind this propulsion method is to exploit the solar radiation pressure (SRP) exerted on a very large, thin and highly reﬂective surface attached to the satellite, called a solar sail, to produce thrust. By its very nature, solar sailing is a very attractive propulsion method as it offers a solution to the problem of ﬁnite propellant.

1.2.3. Literature heritage Although no formation of satellites propelled by solar sails has ever been flown, a number of possible scientific applications of the concept have been proposed and studied in the scientific literature.

A first concept that was analyzed is the possibility of flying a formation of solar sail propelled satellites around the artificial L1 Sun-Earth Lagrange point as a monitoring tool for solar plasma storms [10]. Through a number of simulations, the viability and stability of two different controlled formations for this particular application were demonstrated. Later on, taking inspiration from this work, a similar concept was described and analyzed: using a group of satellites equipped with solar sails flying in formation about the artificial L1 Sun-Earth Lagrange point in order to provide a constant and long- duration monitoring of the polar regions of the Earth through the use of imaging and interferometry [11]. For this particular application, at least two satellites would be needed in order to be able to determine both the optical thickness of the atmosphere and the reflectance of the Earth’s surface.

Based on a concept suggested in Reference [12], another relevant application of a formation of solar sail propelled bodies is the possibility of using the spacecraft as gravity tractors in order to deﬂect the trajectory of potentially harmful asteroids [13]. In this paper, the advantages of using several spacecraft instead of a single one were shown and the great increase of the deﬂection capabilities of the gravity tractor concept was demonstrated. However, it was also shown that, given the complexity of the problem, a highly accurate position and attitude control would be needed in order to avoid collisions between the satellites.

A final relevant mission concept relying on a formation of satellites with solar sails is the GEOSAIL mission concept. Originally proposed by McInnes in Reference [14] to study the geomagnetic tail of the Earth by flying a satellite in a Sun-synchronous displaced orbit, it was then improved and re- analyzed in order to show the advantages of flying a formation of satellites on a similar displaced orbit [15].

From the above it is clear that the application of solar sailing to mission concepts involving satellite formations is relatively new, unexplored and still offers a large amount of possibilities. It is therefore in this context that the thesis project will be developed.

1.3. Research question and thesis objective Combining all of the information given in the previous sections, the main research question that this thesis will aim at answering can be formulated as follows:

Can solar sails be used to maintain a formation of satellites in an optimal shape under the effect of orbital perturbations in order to track and monitor space debris in the geostationary orbit?

The thesis therefore aims at determining whether solar sailing is a viable propulsion method to control a formation of satellites in a near-geostationary orbit. The control is necessary in order to maintain an 6 1. Introduction optimal formation relative geometry allowing to monitor and track the position of space debris.

In addition to this main research question, a number of relevant sub-questions can be formulated:

How does the solar sail size inﬂuence the performance of the formation control and the resulting debris monitoring capabilities in case of a ﬁxed spacecraft mass? The size of a solar sail can vary substantially as a function of the mission application and purpose. During this thesis, an analysis of the relation between the sail size and the formation control performances will be provided.

How does the addition of a third satellite to the formation inﬂuence the control performances and control effort? As previously mentioned, the addition of a third satellite to the formation will yield better space debris tracking capabilities and a more robust mission concept. By producing results both for a two-spacecraft and a three-spacecraft formation, this possibility will be investigated.

How does the performance of the developed control logic vary as a function of the considered orbital propagation duration? The perturbing forces acting on a satellite are usually of a very small intensity. However, the constant presence of a perturbing acceleration could cause the satellite formation to drift more than the sail can compensate for. The evolution of the formation control performance as a function of the orbital propagation duration will therefore be analyzed in this thesis.

What analytical representation of the sail attitude angle proﬁles performs better for formation control? In order to model the control of the formation, the sail attitude angles will be represented by an analytically deﬁned function. During this thesis, various representations will be analyzed. The one yielding the best results will be determined and subsequently used for the further development of the concept.

From the previously described research question and sub-questions, it is possible to deﬁne the objective of this thesis as well:

To control a satellite formation in a near-geostationary perturbed orbit by using solar sails in order to perform the triangulation of space debris in geostationary orbit.

This objective will be achieved by optimizing the steering law of each satellite in the formation so that the perturbing accelerations are counteracted and that the relative position between the satellites and the observed debris results in an accurate triangulation of the latter. The main focus of the thesis will be on the orbital dynamics of the spacecraft necessary to optimize their relative and absolute position. Other mission aspects such as spacecraft instrumentation, power budget, launch, data downlink and ground control are considered to be beyond the scope of this thesis and will therefore not be discussed.

1.4. Thesis structure Following this ﬁrst introductory chapter in which the problem and the suggested concept have been described and brieﬂy analyzed, Chapter 2 will discuss the topic of solar sailing. Starting from a physical description of solar radiation pressure, the possibilities, strengths and limitations of this propulsion concept will be analyzed. A brief heritage study of technologically relevant past space missions will then be presented. Finally, two different models representing the resulting force on an irradiated spacecraft will be described and discussed and the rationale behind the choice of the model to be implemented in the spacecraft dynamics will be explained.

In Chapter 3, the hypothetical mission to which the studied concept is applied will be deﬁned. The mission statement will be determined and discussed. The orbital and geometrical requirements for the formation, derived from the mission statement, will then be provided. Subsequently, the method and rationale used to compute the initial state of each satellite of the formation will be described. Finally, some considerations regarding the spacecraft will be provided. 1.4. Thesis structure 7

In Chapter 4, the orbital dynamics of the satellites in their orbit around Earth will be discussed. This will be done by ﬁrst describing all relevant reference frames, sets of coordinates and coordinate transformations that are used to compute and propagate the satellites dynamics. Subsequently, the equations of motion of a spacecraft orbiting Earth are provided, ﬁrst for a simple Keplerian orbit and then for a more complex perturbed orbit. All relevant perturbing forces are then described and compared in order to determine which ones will be taken into account and which ones can be neglected.

Subsequently, Chapter 5 will discuss the optimization process with which the most suitable control profile for each satellite is determined. First, a number of possible choices for the parameters that define the problem will be presented and the resulting sail attitude control profiles will be discussed. Subsequently, different options for the definition of the problem objective function will be provided and compared.

In Chapter 6, the various functionalities of the application developed for this thesis, as well as the implemented numerical methods and algorithms, will be presented and described. First, the chosen numerical integration method and numerical optimization method will be described and analyzed. Sub- sequently, an overview of the application will be provided and the choices for the used programming language and relevant sets of libraries will be discussed. Finally, a number of block diagrams providing a clear overview of the different functionalities included in the developed application will be shown.

In Chapter 7, the veriﬁcation and validation procedure for the various functionalities developed and used during this thesis project will be described. Every test and relative benchmark will be illustrated separately and the relevant results will be shown and discussed.

In Chapter 8, the preliminary results that were obtained with short duration simulations will be presented and analyzed. The chapter will explain how these optimizations are used to provide an initial overview of the optimizer’s capabilities and to determine the optimal values for some of the simulation control parameters that need to be manually adjusted. The results obtained by varying the sail size, the optimizer population size and when relevant, the sail rotation frequency, will be presented.

In Chapter 9, the results obtained for longer simulation times will be presented and analyzed. The optimizer performance will be compared for various sail attitude representations, various objective function deﬁnitions and for both two- and three-satellite formation cases.

Finally, in Chapter 10, the most relevant results will be highlighted, the conclusions will be drawn and the capabilities and performances of the developed concept will be discussed. Relevant recommendations for future works on the same or similar topic will also be provided.

Solar sailing

The idea that the electromagnetic radiation emitted by the Sun could exert a physical pressure on the body it irradiates was ﬁrst theorized in the middle of the 19th century by J.C. Maxwell in his work "A Dynamical Theory of the Electromagnetic Field"[16]. The development of the space industry led to the analysis and development of this concept as a possible propulsion method, which would allow the spacecraft to accelerate at a constant, low intensity rate without any propellant consumption. As mentioned in Chapter1, this concept will be considered to be the propulsion method used by the satellites of the formation to perform the maneuvers necessary to maintain an optimal formation geometry for the triangulation of space debris.

In this chapter, the concept of solar sailing will be introduced and the physical theory describing the phenomenon of solar radiation pressure will be explained. Subsequently, the actual concept of solar sailing will be discussed and the advantages and disadvantages of this propulsion method will be shown. A mission heritage section is included in order to better show the current capabilities of the propulsion method in actual space missions as well as to illustrate the state of its technological readiness. Finally, two different models describing the force exerted on the sail will be presented and compared. Note that, unless stated otherwise, all information presented in this chapter was derived from Reference [17].

2.1. Physics of solar radiation pressure When photons emitted by the Sun impact a surface at a distance r from the star, an exchange of momentum will occur between the two, thereby exerting a pressure on the surface. In order to derive an expression for this pressure, the energy, E, transported by a photon travelling through vacuum is expressed by combining Planck’s law and Einstein’s mass-energy equivalence: q 2 4 2 2 E = hν = m0c + p c (2.1) where

• h = 6.6261 Js is the Planck constant

• ν is the frequency of the electromagnetic wave

• m0 is the resting mass of the particle • p is the momentum transported by the particle

• c ' 2.9979 · 108 m/s is the speed of light in vacuum

Since photons have a zero resting mass, m0, the transported momentum is equal to: hν E p = = (2.2) c c

9 10 2. Solar sailing

Furthermore, the incoming energy ﬂux is deﬁned as:

L 1 ∆E W = s = (2.3) 4πr2 A ∆T where

26 • Ls = 3.828 · 10 W is the luminosity of the Sun

• A is the surface of the considered body

The energy that crosses the surface A over a time interval ∆T can be written as:

∆E = WA∆T (2.4)

Then, by combining Eq. 2.2 and Eq. 2.4, it is possible de deﬁne the exchanged momentum as:

∆E ∆p = (2.5) c

This results in a pressure, P , acting on the surface equal to:

1 ∆p P = (2.6) A ∆T

Finally, by combining Eq. 2.4, 2.5 and 2.6 it is possible to express the solar radiation pressure as a function of the distance, r, from the radiation source:

W L P = = s (2.7) c 4πr2c 2.2. Concept of solar sailing At a distance of one astronomical unit (AU) from the Sun, the pressure acting on a perfectly reflecting surface perpendicular to the incoming radiation is equal to 9.12 · 10−6 N/m. This value is twice the magnitude of the SRP that would be obtained by using Eq. 2.7 as the exchange of momentum is dou- bled because of the photons’ reflection. When compared to other perturbations acting on the satellite, the magnitude of the resulting SRP force is usually relatively small for normal-sized satellites. This is shown in Figure 2.1, where the intensity of various accelerations acting on a satellite orbiting the Earth as a function of the altitude is provided. It can be seen, for example, that for a satellite in GEO the magnitude of the SRP acceleration is approximatively two orders of magnitude smaller than the one caused by the gravitational attraction of the Moon. Instead, for spacecraft characterized by a large surface, the magnitude of the SRP force can become large enough to be efficiently used as a propulsion method for orbital maintenance, orbital manoeuvres and prolonged deep space propulsion. In order to artificially increase the irradiated surface even further while limiting the increase in mass, a thin and large membrane of highly reflective material, called a solar sail, can be attached to the spacecraft. Given the volume restrictions, it is clearly infeasible to launch a solar sail propelled spacecraft into its orbit with the sail already deployed. The membrane is therefore folded within the satellite structure and needs to be deployed after orbit injection, which represents one of the main technological challenges related to this propulsion concept and puts further restrictions on the choice for the sail material.

For clarity, a typical arrangement of the layers of a solar sail is shown in Figure 2.2. The presented example comprises two sides with different optical characteristics: one with high reﬂective capabilities in order to maximize the thrust that can be obtained and the other one with high emissivity in order to dissipate excess heat. Finally, a middle, polymer layer is included to improve the structural properties of the sail. 2.2. Concept of solar sailing 11

Figure 2.1: Order of magnitude of the perturbing accelerations acting on a satellite orbiting Earth as a function of altitude [18]

2.2.1. Mission heritage In this section, a number of relevant solar sailing related missions or mission concepts is discussed in order to provide the reader with an idea of the propulsion method’s current capabilities and possible applications. It is important to note that solar sailing is still an experimental concept and therefore, all presented mission concepts are either theoretical studies or were technology demonstration missions.

As mentioned at the beginning of the chapter, the concept of solar sailing was theorized over 100 years ago. Since then, several studies have been executed in order to determine the feasibility of certain types of missions by using this propulsion method. A notable example was the study of the possibility of using solar sailing in order to perform a rendez-vous with Comet Halley by the National Aeronautics and Space Administration (NASA) in the 1970s. After thorough analysis, a solar electric propulsion was chosen for the spacecraft. Unfortunately, later on the mission was cancelled due to budgetary issues and the short time available for the development of the probe.

Three notable examples of successful solar sailing technology demonstration missions that have actually ﬂown are IKAROS, NanoSail-D2 and Lightsail-1. Launched as a secondary payload of the Japan Aerospace eXploration Agency (JAXA) Akatsuki spacecraft directed to Venus, IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun) was the ﬁrst spacecraft propelled by solar sails to suc- 12 2. Solar sailing

Figure 2.2: Typical arrangement of solar sail layers [19]

cessfully fly in space1. The spacecraft was launched in 2010 and successfully deployed a 14 × 14 m2 kite-shaped polyamide sail with integrated small solar cells to power the payload. After flying close to Venus and performing a series of programmed operations with testing purposes, IKAROS continued to orbit around the Sun in order to perform additional technology demonstration tasks. It is currently orbiting at a distance of approximatively 0.87 AU from the Sun in hibernation mode. During the mission, the possibility of controlling the attitude of the satellite by using liquid crystal panels on the fringe of the sail was demonstrated. This is achievable due to the fact that the reflectivity of these crystals varies as a function of the applied electric voltage. Therefore, it is possible to obtain different reflectivity coefficients on different parts of the sail, which result in a non-uniformly distributed pressure. This produces a torque that can be used to control the attitude of the sail without the need of propellant or additional dedicated hardware [20].

NanoSail-D2 was a NASA technology demonstration 3-unit cubesat with a mass of approximatively 4 kg and a 10 m2 square-shaped sail. The satellite was launched as a secondary payload of FASTSAT in 2010 and after some initial issues with the separation mechanism, the solar sail was successfully deployed. NanoSail-D2 successfully performed its ﬂight mission, which consisted in demonstrating the possibility of using the sail for an accelerated de-orbiting process2. For this mission, the propulsion aspects of solar sailing technology were neglected and the focus was only oriented towards the deployment mechanism and the de-orbiting capabilities of the solar sail by interaction with the atmosphere. In order to provide the reader with an idea of the sail area to satellite volume ratio of solar sail propelled spacecraft, the picture of NanoSail-D in its fully deployed conﬁguration is shown in Figure 2.3.

LightSail-1 was a solar sail propelled 3-unit cubesat launched in May 2015 as part of a privately funded project of the Planetary Society. Its purpose was to demonstrate and test a number of solar sailing related technologies and to provide scientific data for university-level teaching projects3. The spacecraft successfully deployed its 32 m2 mylar sail in low-Earth orbit and was de-orbited a few days later by atmospheric drag. The follow-up mission, LightSail-2, is planned for launch in the first half of 2017. Its target orbit will be higher than that of LightSail-1 in order to limit the influence of atmospheric drag to a negligible level and thereby being able to properly test the propulsion capabilities of the developed sail. The characteristics of the three previously mentioned spacecraft are summarized in Table 2.1, where the definition of the characteristic acceleration is provided in Section 2.3.2.

For the sake of completeness, it is worth mentioning Sunjammer, a large-sized technology demonstration solar sail propelled spacecraft developed by NASA with the goal of studying the viability of the propulsion method for space-weather monitoring systems4. The spacecraft was scheduled to launch in early 2015, but was concluded in October 2014 due to issues with the contracting companies.

2.2.2. Strengths, capabilities and limitations of solar sailing

1JAXA website, http://www.isas.jaxa.jp/e/forefront/2011/tsuda/index.shtml, accessed: August 2015 2Nanosail-D2 website, http://www.nasa.gov/mission_pages/smallsats/11-148.html, accessed: July 2015 3Lightsail mission description, http://sail.planetary.org/, accessed: August 2015 4Sunjammer website, https://www.nasa.gov/mission_pages/tdm/solarsail/index.html 5Nanosail-D picture, https://www.nasa.gov/mission_pages/smallsats/nsd_deployment_complete.html, accessed: November 2016 2.2. Concept of solar sailing 13

Characteristic Spacecraft Mass [kg] Sail surface [m2] Area to mass ratio [m2/kg] acceleration [m/s2] IKAROS 300 196 0.6533 5.9 · 10−6 Nanosail-D2 4 10 2.5 2.3 · 10−5 Lightsail-1 5 32 6.4 5.8 · 10−5 Sunjammer 32 1444 45.125 4.1 · 10−4

Table 2.1: Relevant characteristics of solar sail propelled spacecraft presented in the heritage study

Figure 2.3: Picture of the fully deployed NanoSail-D during the testing phase5

As mentioned at the beginning of this chapter, solar sail allows the spacecraft to accelerate at a constant, low intensity rate without any propellant consumption. This characteristic makes the propulsion concept particularly suitable for high-energy missions (i.e., missions that require large ∆V ). A first notable example are missions involving an escape from the Solar System, as the cumulated low intensity acceleration over months can result in velocities large enough to reach the Solar System escape velocity. In Reference [21], for example, it is shown that, with a sail surface-to-mass ratio of approximatively 0.29 m2/g, it is possible to exit the Solar System in less than a year by starting from a distance of 1 AU from the Sun. This is achieved by first reducing the distance to the Sun until the orbital radius reaches a value of 0.7 AU. Then, the control law is modified in order to increase the energy of the orbit and escape the Solar System as quickly as possible.

Another category of missions potentially enabled by solar sailing propulsion are the ones requiring the spacecraft to be transferred to a Solar polar orbit (i.e., with an orbital inclination of 90 deg). Assuming the spacecraft to be on a circular orbit with a radius of 1 AU and a zero inclination, an estimate of the ∆V required to maneuver the spacecraft to a Solar polar orbit can be computed as an impulsive thrust equal to [22]: rµ ∆V = 2 Sun sin(π/4) ' 4.2 · 104 m/s (2.8) r where 14 2. Solar sailing

20 3 2 • µSun = GMSun = 1.32712 · 10 m /s is the standard gravitational parameter of the Sun • G = 6.67259 · 10−11 m3/kg/s2 is the Newtonian constant of gravitation [23]

30 • MSun = 1.98855 · 10 kg is the mass of the Sun In Reference [24], however, it is demonstrated that, starting from Earth, it is possible to reach a Solar polar orbit with an orbital radius of 0.48 AU within 5 years from launch by exclusively using SRP.

Finally, a last relevant type of mission for which solar sails are a particularly suitable method of propulsion are the ones involving displaced orbits. Displaced orbits are non-Keplerian orbits (NKO) for which the orbital plane does not intersect the central attracting body. This is achievable, for example, by producing a constant force-component perpendicular to the orbital plane. To create such a constant force is clearly extremely expensive from a propellant consumption perspective if electric or chemical thrusters are employed. The same type of orbit may, however, also be achieved with a properly sized solar sail, inclined at a constant angle with respect to the incoming radiation [17]. A relevant example of a displaced orbit would be an orbit around the Sun lying on a plane parallel to, but not coinciding with, the ecliptic plane.

Solar sailing presents several limitations and technical challenges as well. Most importantly, solar sails can only provide thrust in a direction pointing away from the Sun, which puts strict limitations on the mission possibilities and capabilities in case only this type of propulsion is used. In these cases, solar sailing can be combined with chemical or electric thrusters in order to solve this issue. However, adding a secondary propulsion system results of course in an increased design complexity [25]. Fur- thermore, the presence of a large sail tends to substantially increase the momentum of inertia of the spacecraft, making the steering more problematic and more expensive from an energy perspective. From a technological point of view, this propulsion method presents several challenges as well, such as the need for lightweight, resistant and highly reflective (or emissive) materials that can be safely folded and unfolded and the development of a very reliable sail deployment mechanism, essential to the success of a mission. Although the state of the art of solar sailing related technology is still very experimental and does not allow space agencies to safely consider it as the main propulsion system for expensive spacecraft and/or critical missions, it was shown in the previous section on mission heritage that several technology demonstration missions have been successfully flown and solar sails could therefore be used in the near future, for example, for non-critical low-budget missions with scientific purposes.

2.3. Force models Following the collision of photons with the solar sail, as described in Section 2.1, different phenomena can occur, each one influencing to a certain degree the magnitude and direction of the resulting force acting on the body. Several mathematical models have been developed in order to describe these phenomena with varying accuracy. After a brief description of the problem geometry, the ’ideal sail’ and the ’optical sail’ force models will be presented in this chapter. The first one describes the force exerted on a perfectly reflective sail, while the latter takes into account the absorption, specular reflection, diffuse reflection and emission of the photons by a perfectly rigid and flat sail.

2.3.1. Model geometry The geometry of the problem can be described through the use of the following unit vectors: ~n normal to the sail, ~u aligned with the direction of the incoming radiation, ~s aligned with the direction of the specularly reﬂected radiation and ~t perpendicular to ~n and lying on the ~n-~u plane as shown in Figure 2.4. These vectors are related through the following expressions:

~u = cos α~n + sin α~t (2.9) ~s = − cos α~n + sin α~t (2.10) where α is the angle between −~n and ~u, also known as the cone angle of the sail, which describes the attitude of the sail with respect to the Sun-line ~u. The direction of the resulting acceleration can be 2.3. Force models 15 described with the unit vector ~m, which is inclined at an angle θ∗ with respect to ~u and at an angle φ with respect to ~n. This can be expressed through the following equivalence: α = θ∗ + φ (2.11)

Figure 2.4: Geometry used to deﬁne the SRP induced force, modiﬁed from Reference [17]

In order to uniquely define the attitude of the sail with respect to the incoming solar radiation, the cone angle α is often paired with the clock angle δ, which is defined as the angle between the projection of ~n onto the plane normal to the Sun-line ~u and a reference direction, ~q, lying on said plane, as illustrated in Figure 2.5. The normal can then be defined as: ~n = cos α~u + sin α cos δ~q + sin α sin δ~q × ~u (2.12) In this thesis, the reference direction ~q is defined as the intersection between the plane perpendicular to the Sun-line and the plane defined by the instantaneous Sun-line and the ZHIF -axis of the heliocentric inertial reference frame, defined in Section 4.1.

2.3.2. Ideal sail force model The ideal sail force model assumes the solar sail to be perfectly reflective so that the net produced force is aligned with the sail normal. This can be demonstrated by considering the momentum vector of the incoming photons ~p and the momentum vector of the reflected photons p~∗, which are defined as follows:

~p = p cos α~n + p sin α~t (2.13) p~∗ = −p cos α~n + p sin α~t (2.14) where p is the momentum carried by a photon, deﬁned by Eq. 2.2. The total exchange of momentum is then equal to: ∆~p = p~∗ − ~p = 2p cos α~n (2.15) Furthermore, considering that the projected sail area in the direction of the incoming radiation is equal to: A∗ = A cos α (2.16) the total force exerted on the surface can be expressed as: f~ = 2PA cos2(α)~n (2.17) 16 2. Solar sailing

Figure 2.5: Geometry of the cone and clock angle, modiﬁed from Reference [26]

where P is the SRP described by Eq. 2.7. The acceleration induced by the SRP is then proportional to the area to mass ratio of the satellite and can be written as a function of the lightness number β:

σ∗ L A A β = = S ' · 1.53 · 10−3 (2.18) σ 2πGMSunc m m GM → ~a = β Sun cos2(α)~n (2.19) r2 where

• σ is the mass to surface ratio of the sail, also known as the loading parameter

• σ∗ ' 1.53 · 10−3 kg/m2 is the sail critical loading parameter

• m is the mass of the spacecraft

As stated at the beginning of this section, this model assumes perfect reflection of the incoming radiation. However, commonly used sail materials only have reflective coefficients of approximatively 0.9. The rest of the photons will be absorbed and subsequently re-irradiated by the sail. The ideal sail model is therefore very useful for preliminary analysis of mission concepts and estimates of the produced thrust, but in case a high-fidelity modelling of the acceleration is required, absorption and emission phenomena and other non-ideal effects need to be taken into account.

2.3.3. Optical sail force model The optical sail force model represents a more accurate description of the force produced by the sail than the ideal sail model previously presented. For this reason, the optical sail force model will be used in this thesis, as it provides a fairly accurate description of the force exerted on a solar sail while also being relatively easily implementable and computationally efficient. In this description of the force, apart from the specular reflection of photons, their diffuse reflection, absorption and emission are considered as well. According to this model, the absorption of photons produces a net force that acts along the Sun-line, while both specular and scattered reflection produce a net force that is aligned with the sail normal. Finally, the direction of the force produced by the emission of the radiation depends on the emissive and transmissive properties of the materials from which the various layers of the sail are made of. Nevertheless, by considering the thickness of the sail to be negligible and its temperature to be uniform, it is possible to assume the force caused by the emission of photons to be acting along the sail normal. The total force exerted on the sail due to SRP can then be written as: 2.3. Force models 17

~ f = (frs + fru + fe)~n + fa~u (2.20) where

• fa is the force caused by photon absorption

• fe is the force caused by photon emission

• frs is the force caused by the specular reﬂection of the fraction s of incoming photons

• fru is the force caused by the non-specular diffuse reflection of the fraction 1 − s of incoming photons The expressions for the values of these forces can then be defined as a function of the geometry of the problem and of some physical properties of the surface, such as the reflection coefficient r˜, the absorption coefficient a∗ and the transmission coefficient τ. When assuming that the transmission coefficient is equal to zero on the reflective side of the surface, the remaining parameters are linked by an energy conservation constraint: r˜ + a∗ = 1 (2.21) The magnitude of the force components defined in Eq. 2.20 can then be expressed as:

fa = (1 − r˜)PA cos α (2.22)

frs =rsP ˜ A cos α (2.23)

fru = Bf r˜(1 − s)PA cos α (2.24) εf Bf − εbBb fe = PA(1 − r˜) cos α (2.25) εf + εb where

• Bf and Bb are the coefﬁcients that indicate to which degree the front and back surfaces deviate from the Lambertian condition, respectively. The Lambertian condition is deﬁned as the condition in which the optical properties of an object do not vary with the observing angle.

• εf and εb are the emissivities of the front and back surfaces, respectively.

For the actual implementation of the model in the developed application, it is useful to express the force as a function of its components parallel and perpendicular to the surface, as shown below:

~ 2 εf Bf − εbBb fn = PA{(1 +rs ˜ ) cos α + Bf (1 − s)˜r cos α + (1 − r˜) cos α}~n (2.26) εf + εb ~ ft = PA(1 − rs˜ ) cos α sin α~t (2.27)

The optical sail model force is based on a certain number of assumptions and simplifications that cause the computed acceleration to present a certain inaccuracy. Most importantly, the optical model assumes the sail to be perfectly flat and rigid, while in a more realistic case it will be flexible and present a curvature which will consequently modify both the intensity and the direction of the resulting force [27]. Furthermore, the transmission of heat between the reflective and emissive sides of the surface is also neglected by the optical model, while these two sides are usually at different temperatures and therefore an exchange of energy between the two occurs. Finally, when computing the pressure acting on the sail, only the radiation produced by the Sun was taken into account. A more accurate model would need to consider other sources of radiation such as the Earth and Moon albedo and other sources of pressure such as the charged particles emitted by the outer layers of the Sun (i.e., solar wind). In Figure 2.1 it can be seen that the effect of the Earth albedo induced SRP on a satellite in GEO is approximatively two orders of magnitude smaller than the one induced by direct solar radiation. Similarly, it is computed in Reference [17] that the intensity of the pressure exerted by the solar wind is four orders of magnitude smaller than SRP. Neglecting these phenomena can therefore be considered a reasonable assumption in case high precision values for the estimated spacecraft acceleration are not required. The flat and rigid solar sail assumption, however, might introduce larger errors for both 18 2. Solar sailing the intensity and direction of the resulting force (about 5%)[27]. It is, nevertheless, very complex to model a realistic surface geometry for a solar sail, as several additional geometrical and physical parameters need to be introduced. Furthermore, this would substantially increase the computational weight of the model. For these reasons, it was decided not to include a variable geometry sail model in the developed simulation. 3

Mission statement and requirements

In Section 1.1, the problem caused by the presence of space debris in orbit around Earth was thor- oughly analyzed and the issues related to the determination and tracking of small-sized debris in high- altitude orbits were presented. Using this information, in Section 1.3 the possibility of using a formation of solar sail propelled spacecraft to track the debris was suggested and the thesis research questions and objective were defined. In order to develop the given thesis topic, a theoretical mission concept will be proposed and analyzed. In this chapter, the mission statement will be provided and discussed. From this statement, the necessary mission requirements are then developed and justified. Finally, the nominal orbit and spacecraft characteristics necessary to comply with the defined requirements will be presented. For clarity and synthesis purposes, satellites 1, 2 and 3 will from here on be called Athos, Porthos and Aramis, referencing A. Dumas’ famous novel [28].

3.1. Mission statement A mission statement is necessary in order to properly deﬁne the mission and clarify its rationale. From the mission statement all the relevant mission requirements can then be deduced. For this particular hypothetical application, said statement can be written as:

"Tracking the position of small-sized space debris in the geostationary orbit through an optical triangulation performed by a formation of satellites".

3.2. Orbital requirements As previously stated in Section 1.2, the formation used to track the position of the debris will consist of either two or three spacecraft. In order to properly perform the debris position triangulation, a certain relative position between the satellites and the observed object is required. In this section, the optimal formation geometry and the resulting orbital requirements for the spacecraft will be discussed.

The ﬁeld of view of each spacecraft observation instrument can be represented with a cone with the axis of symmetry aligned with the line of sight of each observation instrument and the aperture determined by the instrument optical resolution. The triangulated position of the debris will then be within the volume of the intersection between the various "observation cones", as illustrated in Figure 3.1 for a two-satellite-planar case. In order to maximize the precision of the debris position determination, it is therefore necessary to minimize the volume of this intersection, which occurs when the lines of sight have a relative orientation of 90 deg [29]. If the formation consists of two spacecraft, this is obtained when the satellites and the observed object form a right triangle, as shown in Figure 3.2. When extend- ing this to the three satellite case, the optimal debris triangulation precision is obtained if the formation and the debris form a tri-rectangular tetrahedron, characterized by having three right face angles at one vertex.

It is also important to note that the formation will not be orbiting at a geostationary altitude. The orbital period of the formation and the observed debris will therefore differ and only a limited time span will be

19 20 3. Mission statement and requirements available for the triangulation process. The same debris will then be observed again when the orbital phase between the satellites and the debris reaches a value of 2π.

Figure 3.1: 2D representation of the observation cone intersection geometry [29]

As mentioned in Section 1.1, the international regulations delimit the protected geostationary belt as the area characterized by an altitude between 35, 586 and 35, 986 km and an orbital inclination of −15 to 15 deg [1], as shown in Figure 1.1. It will be assumed that the formation needs to ﬂy 250 km below the geostationary orbit in order not to interfere or collide with any active satellite. This requirement results in an optimal inter-satellite distance for the debris triangulation of 500 km, as shown in Figure 3.2. The only orbital set-up that would allow for a constant optimal distance between each spacecraft of the formation and the geostationary orbit consists of a set of equatorial circular orbits with a true anomaly offset. This set-up is obviously only valid for a two-satellite case. Furthermore, the presence of perturbing accelerations will cause the satellites to drift apart which would result in a non-optimal tracking of the debris after a relatively short period of time.

To counteract the perturbing forces and maintain an optimal triangulation geometry, the use of solar sailing is therefore proposed. An optimization process is performed in order to ﬁnd the required steering laws of the sails that result in an optimal formation shape and absolute position for debris triangulation purposes over time. It is, of course, unrealistic to maintain the formation in its optimal geometry for the entirety of the mission and the optimizer will therefore allow for solutions that maintain the inter-satellite relative distances within 200 to 800 km. This results in an observation angle varying between 43 and 116 deg, approximatively. In order to provide a constant observation of the geostationary orbit, which is the critical region of interest, the formation is also required to stay at low inclinations. For this reason, the geometrical center of the formation will be bound to a maximum orbital inclination of 1 deg. A summary of the orbital requirements described in this section is provided in Table 3.1.

3.3. Nominal orbits and initial conditions In order to deﬁne the initial conditions of the satellite formation, a set of unperturbed Keplerian orbits that satisﬁes the orbital requirements for the inclination of the satellite formation geometrical center and the inter-satellite distances listed in Table 3.1 along the entire orbital revolution is searched for. To determine whether an orbit complies with the previously listed requirements, it is necessary to express the satellites’ positions with both Keplerian elements, for the inclination requirement, and Cartesian 3.3. Nominal orbits and initial conditions 21

Figure 3.2: Optimal triangulation geometry in the two satellites case

Requirement Value Unit of measure Altitude 35,511 km Orbital radius 41,882 km Minimum inter-satellite distance 200 km Maximum inter-satellite distance 800 km Optimal inter-satellite distance 500 km Minimum formation center orbital inclination -1 deg Maximum formation center orbital inclination 1 deg

Table 3.1: Mission orbital requirements

coordinates, for the absolute position of the formation and the inter-satellite distance requirements. In case unperturbed circular orbits are considered, a direct relationship between the Keplerian elements of the orbit and the resulting Cartesian state vector exists and is illustrated by Fig 3.3. The spacecraft Cartesian coordinates in an Earth-ﬁxed J2000 inertial reference frame (which will be described in detail in Section 4.1) can be computed as follows [22]:

XJ = r[cos(ω + θ) cos Ω − sin(ω + θ) cos i sin Ω] (3.1)

YJ = r[sin(ω + θ) cos i cos Ω + cos(ω + θ) sin Ω] (3.2)

ZJ = r sin(ω + θ) sin i (3.3) where

• r is the orbital radius

• i is the orbital inclination

• Ω is the longitude of the ascending node

• ω is the argument of periapsis

• θ is the true anomaly 22 3. Mission statement and requirements

Figure 3.3: Geometry of the relation between the Earth-centred Cartesian coordinates and the Keplerian elements [22]

The Keplerian elements will be described in more detail in Section 4.3. The Cartesian position of the spacecraft needs to be evaluated along an entire orbit in order to determine the entire profile of the inter-satellite distance. This is achieved by computing it along a dense enough mesh for the true anomaly θ, ranging from 0 to 2π rad. In order to define the initial conditions of the satellites that comply with the mission orbital requirements, sets of Keplerian elements were then randomly generated and assessed. To limit the dimension of the search space, the eccentricity, e, and the argument of the periapsis, ω, were set to 0 deg, while the semi-major axis was set constant to 41, 657 km. Using the same value of the semi-major axis for all the satellites ensures that also the orbital periods will be the same, thus limiting the increase in the inter-satellite distance over time. This leaves only three Keplerian elements to be determined through a Monte Carlo process: inclination i, longitude of the ascending node Ω and true anomaly at the starting epoch of the simulation t0, θ0. No minimum or maximum number of trials was set, by consequence the Monte Carlo search terminates when a set of Keplerian elements that satisfies all requirements is found. A possible set of Keplerian elements obtained with the described process that complies with the mission orbital requirements is provided in Table 3.2. An overview of the nominal unperturbed orbits of the three satellites is provided in Figure 3.4.

Satellite a [km] e i [deg] ω [deg] Ω [deg] θ0 [deg] Athos 41,657 0.0 0.2554 0.0 120.1744 188.8212 Porthos 41,657 0.0 0.2012 0.0 184.7316 123.2081 Aramis 41,657 0.0 0.0564 0.0 350.5606 318.7626

Table 3.2: Unperturbed initial Keplerian elements of the satellite formation

3.4. Spacecraft characteristics As stated in Section 1.3, defining a realistic and comprehensive design for the spacecraft including a detailed analysis of the instrumentation, power budget, launch, data downlink and ground control operations is beyond the scope of the thesis. This is justified by the fact that the simulation of the orbital dynamics of the satellite formation under the influence of the Earth’s central gravitation, the SRP and a number of perturbing forces only requires the position of the acting bodies and the lightness number of the satellites, β, to be computed.

In Table 2.1, the physical characteristics of three successful solar sailing technology demonstration mission were provided. These data can be used in order to deﬁne realistic limits on the sail lightness number. In said table, the area to mass ratios range from 0.6533 to 45.125 m2/kg, which results in lightness numbers ranging from 9.9954 · 10−4 to 6.9 · 10−2. It was arbitrarily decided that, in order not to 3.4. Spacecraft characteristics 23

Athos Porthos 200 Aramis

100

0 z [km] -100

-200 5 5 ×10 4 0 0 ×10 4

y [km] -5 -5 x [km]

Figure 3.4: Nominal unperturbed orbits of Athos, Porthos and Aramis

limit the concept possibilities, the considered lightness numbers for the satellites will range from 5·10−4 to 2.5·10−2. It is important to note that in this thesis it was assumed that the cross sectional-area of the spacecraft bus is negligible when compared to the solar sail area. Therefore, only the sail area will be taken into account when modelling the force resulting from the SRP. Finally, although only the area to mass ratio determines the magnitude of the SRP acceleration, the mass will be considered to be ﬁxed and equal to 3 kg, while the sail area will be allowed to vary. This will allow the readers who are not familiar with solar sailing to have a clearer idea of the actual physical characteristics of the considered satellites.

Orbital dynamics

The motion of the satellite formation in orbit around Earth is determined by the Earth’s central gravitational attraction and a number of perturbing forces. Therefore, the resulting satellite trajectories cannot be calculated analytically and it is necessary to compute them through numerical integration of the spacecraft’s equations of motion. In order to determine the evolution of the satellites’ positions over time, this chapter will give an overview of the dynamics that describe the orbital motion of a solar sail propelled spacecraft. The chapter will start with the definition of the relevant reference frames as well as the used coordinates and their transformations. Subsequently, the simplified and perturbed two- body equations of motion for a spacecraft in orbit around Earth will be provided. Finally, the perturbing forces that will be of influence to the studied concept will be described.

4.1. References frames In astrodynamics, a large number of different coordinates systems and reference frames are used in order to describe the various studied problems in a proper, clear and intuitive way. In this section, the reference frames that are used in this thesis to model the orbital dynamics and the SRP will be presented.

4.1.1. Earth-fixed inertial reference frame (J2000) In order to describe the motion of satellites about Earth, the most commonly used reference frame is the geocentric equatorial and inertial reference frame J(XJ ,YJ ,ZJ ). This reference frame has its origin located in the center of mass of the Earth. The (XJ ,YJ )-plane coincides with the equatorial plane, the XJ -axis points towards the first point of Aries γ (i.e., vernal equinox) while the ZJ -axis is aligned with the Earth’s rotational axis, positive towards the north pole. The direction of the YJ -axis is oriented in such a way that it completes an orthogonal right-handed reference frame. This reference frame is illustrated in Figure 4.1. Due to axial precession phenomena, the first point of Aries cannot be used as a fixed reference over long periods of time. In order to avoid this problem, the XJ -axis of this reference frame is often defined to point toward the first point of Aries measured on the first of January 2000 at 12 : 00 (terrestrial time). This particular reference frame is called J2000.

4.1.2. Heliocentric Orbital Frame (HOF) and Heliocentric Inertial Frame (HIF) The Heliocentric Orbital Frame HO(XHOF , YHOF ,ZHOF ) and the Heliocentric Inertial Frame HI (XHIF ,YHIF ,ZHIF ) are used when modelling the SRP acting on the solar sail. The HIF has its axes XHIF , YHIF and ZHIF oriented in the same way as XJ , YJ and ZJ and is therefore identical to the J2000 reference frame but translated in order to have its origin coinciding with the center of mass of the Sun. The HOF also has its origin coinciding with the Sun. Its XHOF -axis is aligned with the vector pointing towards the spacecraft while the ZHOF -axis is perpendicular to XHOF and lies on the (XHOF ,ZHIF )-plane. Finally the YHOF -axis is oriented in such a way that it completes an orthogonal right-handed reference frame. A graphical illustration of the two reference frames is provided in Figure 4.2.

25 26 4. Orbital dynamics

Figure 4.1: Earth-ﬁxed inertial J2000 reference frame, modiﬁed from Reference [23]

Figure 4.2: Illustration of the HIF and the HOF, modiﬁed from Reference [19]

4.2. Reference frame transformations The relevant direction, position, velocity, acceleration and force vectors considered in the thesis will be expressed in either of the three reference frames of Section 4.1. As operations will be performed between vectors expressed in these three different reference frames, transformations between said frames will be required. In this section, the relevant used transformation will be provided. 4.3. Two-body motion 27

4.2.1. Transformation from Heliocentric orbital to heliocentric inertial reference frame As explained in Section 2.3, the direction of the SRP acting on the satellite is expressed as a function of the solar sail normal. In order to express the SRP and the related force in the heliocentric inertial reference frame, a transformation is therefore necessary. Firstly, the direction of the solar sail normal in the HOF is deﬁned as follows:

 cos α  ~nHOF =  sin α sin δ  (4.1) sin α cos δ

Subsequently, the matrix, RJ represents a rotation from the HOF to the HIF:

 cos ζ cos γ − sin ζ − sin γ cos ζ  RJ =  cos γ sin ζ cos ζ − sin ζ sin γ  (4.2) sin γ 0 cos γ where ζ and γ are rotation angles deﬁned from the instantaneous frame in the following way:

u ζ = arcsin Y (4.3) |~u| u γ = arctan Y (4.4) uX and ~u = [uX , uY , uZ ] is the instantaneous Sun-line, deﬁned as the difference between the coordinates of the spacecraft and the Sun. In the HIF, this corresponds to the position of the spacecraft:

~u = ~xSC (4.5)

It is ﬁnally possible to express the normal vector to the solar sail in the HIF as:

~nJ = RJ~nHOF (4.6)

4.2.2. Transformation from Heliocentric inertial to J2000 frame As mentioned in Section 4.1.2, the heliocentric inertial and the J2000 reference frames have their axes oriented in the same way, with the HIF being centered in the center of mass of the Sun and the J2000 being centered in the center of mass of the Earth. The transformation between the HIF and the J2000 is therefore equivalent to a translation along the Sun-line vector ~u:

~xJ = ~xHIF − ~uHIF (4.7)

4.3. Two-body motion In this section, the theory describing the equations of motion of a satellite in case only the central gravitational attraction of Earth is considered, will be analyzed. Subsequently, the resulting Keplerian orbits and their relevant characteristics will be brieﬂy described.

It is well-known that the gravitational attraction between two bodies i and j, results in an acceleration of the body i equal to [22]: 2 d ~ri mj 2 = G 3 ~rij (4.8) dt rij where

• mj is the mass of the body j

• ~ri is the position vector of the body i

• ~rij = ~rj − ~ri is the relative position between the two bodies 28 4. Orbital dynamics

When considering a closed system composed of n bodies, the total acceleration of the body i due to the gravitational attraction of the other bodies becomes equal to [22]:

2 n d ~ri X mj = G ~r (4.9) dt2 r3 ij j=1,j6=i ij

In a ﬁrst simpliﬁed analysis of the motion of a satellite orbiting a perfectly spherical and homogeneous celestial body (here the Earth), thus equivalent to a point mass, is considered. Any other force that isn’t the gravitational attraction between the satellite and the Earth is neglected. Furthermore, it is assumed that the mass of the satellite is negligible when compared to Earth’s one. The dynamics of the problem can then be described by the central gravitational potential [22]: GM µ U = − Earth = − Earth (4.10) g r r where

• MEarth is the mass of the Earth The acceleration acting on a body due to Earth’s gravitational pull can then be expressed as the negative gradient of the gravitational potential:

d2~r µ = −∇~ U = − Earth ~r (4.11) dt2 g r3 The resulting trajectory that the body follows is a so-called Keplerian orbit, which can be described as a function of the six following parameters, known as Keplerian elements, which will here be deﬁned with respect to the J2000 reference frame as illustrated in Figure 3.3:

• Semi-major axis, a, is calculated as half of the distance between the periapsis and apoapsis of the orbit, which are the points of smallest and largest distance of the satellite from the Earth, respectively.

ra−rp • Eccentricity of the orbit e = , ra and rp being the apoapsis and periapsis of the orbit, ra+rp respectively.

• Inclination, i, deﬁned as the angle between the ZJ -axis and the orbital angular momentum vector of the orbiting body.

• Argument of periapsis, ω, which is the angular distance between the ascending node and the pericenter of the orbit.

• Longitude of the ascending node, Ω, deﬁned as the angular distance between the direction to the point of Aries and the ascending node. The latter corresponds to the point of intersection between the orbit and the equatorial plane during the southern to northern hemisphere crossing of the satellite.

• True anomaly, θ, which is the angular distance between the pericenter of the orbit and the radius vector.

For simplicity purposes, it is common practice to re-write Eq. 4.11 as a set of ﬁrst order differential equations instead of second order ones in order to simplify their integration. This is done by using the state vector ~y of the spacecraft, deﬁned as the vector containing the Cartesian position and Cartesian velocity coordinates:

h iT ~y = XYZ X˙ Y˙ Z˙ (4.12)

The propagation will then require to integrate ~y˙ with the following generic expression [18]:

~y˙ = f(t, ~y) (4.13) 4.4. Perturbed motion 29

In which case, it is possible to combine Eqs. 4.11, 4.12 and 4.13 as: X˙ = X˙ (4.14) Y˙ = Y˙ (4.15) Z˙ = Z˙ (4.16) µ X¨ = − Earth X (4.17) r3 µ Y¨ = − Earth Y (4.18) r3 µ Z¨ = − Earth Z (4.19) r3 (4.20)

4.4. Perturbed motion Keplerian orbits are a representation of the spacecraft motion useful for preliminary mission analysis. In reality, however, forces other than the central gravitational attraction act on a satellite orbiting the Earth and therefore the equations of motion in Eq. 4.11 need to be complemented with perturbing forces [22]:

d2~r µ + Earth ~r = −∇~ R + f~ (4.21) dt2 r3 P where

• RP is the sum of the potentials of the perturbing forces (if applicable) • f~ is the sum of the accelerations that cannot be expressed as a potential In this section, the theory describing the perturbed motion of a satellite will be provided. For a spacecraft orbiting the Earth, the following perturbing forces are of importance (as shown in Figure 2.1). • Earth’s spherical harmonics • Aerodynamic drag • Third body perturbations • Solar radiation pressure • Albedo • Dynamic solid tide The intensities of the resulting accelerations for each one of these forces are shown as a function of the satellite altitude in Figure 2.1. It is important to note that, in the given plot, the SRP and albedo induced accelerations are computed for a standard satellite design with no solar sail and will therefore be several orders of magnitude larger for the analyzed case.

In order to limit the computational load of the simulation, only the Earth central gravitational attraction, the spherical harmonics up to order and degree two (i.e., J2 and J2,2), SRP and the Sun’s and Moon’s gravitational attraction will be included. These forces were chosen because at (near)-geostationary altitudes, their intensity is several orders of magnitude larger than the other perturbations acting on the satellite. In order to give the reader an idea of the time dependency of the perturbing forces and their effect on the spacecraft orbital elements, the dynamics of two uncontrolled satellites (i.e., without solar sails) were propagated from the initial conditions provided in Section 3.3 for a duration of five years. The evolution of the Keplerian elements of Athos (see Table 3.2) is shown in Figure 4.3. The evolution of the inter-satellite distance and the inclination of the formation center is provided in Figure 4.4, while Figure 4.5 illustrates the angle between the lines of sight of the two satellites as a function of time. These figures show that a formation of two uncontrolled satellites violates the requirement on the inclination (of less than one deg) after approximately one year, demonstrating the need for control through the use of solar sails. It can also be seen that the acting perturbations result in a change in the average inter-satellite distance of approximatively 600 km in a time span of five years. 30 4. Orbital dynamics

×10 4 ×10 -3 4.166 1

4.1658

e 0.5

a [km] 4.1656 0 0 500 1000 1500 0 500 1000 1500 Time [revolutions] Time [revolutions] 5 300

0 200 [deg] ω i [deg] 100 -5 0 0 500 1000 1500 0 500 1000 1500 Time [revolutions] Time [revolutions] 150

100 [deg] Ω

50 0 500 1000 1500 Time [revolutions]

Figure 4.3: Evolution of the Keplerian elements of Athos under the inﬂuence of the perturbing forces included in the simulation without control

4.4.1. Spherical harmonics The ﬁrst type of perturbing force to be considered are the spherical harmonics. These perturbations are caused by the non-spherical shape of the Earth and the non-homogeneous density of its interior that result in a non-uniform gravity force ﬁeld. In this case, the gravity potential of Earth at a generic point above the surface can be described by [22]:

" ∞ n n # µEarth X X R U = − 1 + P (sin Φ){C cos mΛ + S sin mΛ} (4.22) h r r n,m n,m n,m n=2 m=0 where • R is the mean equatorial radius • Λ is the geographic longitude • Φ is the geocentric latitude

• Cn,m and Sn,m are model parameters

• Pn,m(sin Φ) is the associated Legendre function of the ﬁrst kind of degree n and order m The associated Legendre function is deﬁned as [22]: 1 dn P (x) = (1 − x2)n (4.23) n (−2)nn! dxn dmP (x) P = (1 − x2)m/2 n (4.24) n,m dxm 4.4. Perturbed motion 31

1 1000

Athos 0 Optimal 500

-1

Inclination [deg] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions] 800 800

600 600 Porthos Optimal 400 400

200 200 Distance [km] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions]

Figure 4.4: Evolution of the inclination of the formation center and of the inter-satellite distance (on the left) and evolution of the debris to satellite distance (on the right) for two uncontrolled satellites under the inﬂuence of the perturbing forces included in the simulation. The red and green lines represent the maximum and minimum allowed values for the represented parameters, respectively Spacecraft-debris distance [km] with x = sin(Φ).

Equtation 4.22 shows that the actual potential is equal to the sum of the main central gravitational potential and a series of correcting terms. The terms with m = 0 are called "zonal harmonics", which represent the influence of the deviations of the mass density in the north-south direction and have no east-west dependency. The terms with m = n 6= 0 are called "sectorial harmonics" and represent the influence of the deviation of the mass density in both the east-west and north-south direction. Finally the terms with m 6= n 6= 0 are called "tesseral harmonics" and represent the influence of all the remaining north-south and east-west mass density deviations. From the previously described force potential, it is possible to calculate the perturbing force due to the spherical harmonics as: µ f~ = −∇(U + Earth ) (4.25) h r Modern Earth gravity field models contain coefficients up to degree and order of more than 2000. How- ever, only a few terms of the perturbed gravitational potential are usually taken into account during preliminary mission analysis work, as the other ones have a negligible influence and lead to a signifi- cant increase of the computational load. In this thesis, only terms up to order and degree two (i.e., J2 and J2,2) will be considered.

4.4.2. Solar radiation pressure acceleration The solar radiation pressure optical model that is used within the dynamics was previously described in Chapter2. As mentioned in said chapter, several methods exist to model this force, each one describing the phenomenon to a different degree of accuracy. The optical sail model was found to be a good compromise between implementation simplicity, computational weight and accuracy.

It is assumed to have one side of the sail with a reflectivity coefficient r˜ of 0.9 and the other side to be characterized by a perfect absorption of the incoming radiation (i.e., r˜ = 0). Furthermore, in order to simplify the implementation of the force model, it is assumed that the radiation due to absorbed photons that are subsequently re-emitted as thermal radiation is the same on the front and back-side of the sail, thus resulting in a null emission related force. Finally, it is assumed that the reflection is specular and that no scattering occurs, this result in a unitary specular reflection factor s = 1. Substituting these assumptions into Eq. 2.26 and 2.27 results in the following expression for the normal and tangential components of the force with respect to the sail:

~ 2 fn = PA(1 +r ˜) cos α~n (4.26) ~ ft = PA(1 − r˜) cos α sin α~t (4.27) 32 4. Orbital dynamics

120 Computed angle 110 Optimal angle

100

60 Debris observation angle [deg] 50

40 0 50 100 150 200 250 300 350 Time [Orbital revolutions]

Figure 4.5: Evolution of the observation angle for two uncontrolled satellites under the inﬂuence of the perturbing forces included in the simulation

When using a solar sail in orbit around the Earth, the possibility of eclipses need to be accounted for. These eclipses occur when a celestial body crosses the instantaneous Sun-line so that the incoming radiation will be partially or completely blocked, thus reducing the magnitude of the pressure exerted on the sail. This phenomenon is taken into account in the computation of the incoming radiation intensity by using the position of the bodies that could potentially be in the line of sight, computed through the use of SPICE ephemerides. In the Tudat repositories, which will be discussed in detail in Section 6.3.1, the reduced SRP due to the partial/total eclipse, P ∗ is computed as:

P ∗ = o · P (4.28) where o is the so-called shadowing function. Both Tudat repositories and SPICE ephemerides will be described in more detail in Section 6.3. In order to compute the value of the shadowing function, ﬁrst, the occulted and occulting bodies apparent radii, R˜S and R˜o respectively, are calculated as follows:

RS R˜S = arcsin (4.29) |~rS − ~rb| Ro R˜o = arcsin (4.30) |~ro − ~rb| (4.31) where

• RS is the actual radius of the occulted body (i.e., Sun)

• Ro is the actual radius of the occulting body (i.e., celestial body)

• ~rS is the position vector of the occulted body

• ~ro is the position vector of the occulting body

• ~rb is the position vector of the irradiated body (i.e., spacecraft) 4.4. Perturbed motion 33

The apparent separation, α˜, between the occulted and occulting body as seen from the irradiated body is also necessary in order to determine the SRP intensity reduction:

(~r − ~r ) · (~r − ~r ) α˜ = arcsin b o b S (4.32) |~rb − ~ro||~rb − ~rS| By using these parameters, it is possible to distinguish three different scenarios: no occultation, partial occultation and complete occultation. The scenario in which no occultation occurs is characterized by an apparent separation which is larger than the sum of the occulted and occulting bodies apparent radii:

if α˜ ≥ R˜S + R˜o (4.33) → o = 1 (4.34)

Partial occultation, instead, occurs when the apparent separation is smaller than the sum of the occulted and occulting bodies apparent radii, but larger than their difference:

if α˜ < R˜S + R˜o &α ˜ > |R˜S − R˜o| (4.35) A˜ → o = 1.0 − (4.36) ˜2 (πRo) where A˜ is the occulted area, computed as follows:

α˜2 + R˜2 − R˜2 A˜ = S o (4.37) 1 2˜α q ˜ ˜2 ˜2 A2 = RS − A1 (4.38) ˜ ˜2 ˜ ˜ ˜2 ˜ ˜ ˜ A = RS arccos(A1/RS) + Ro arccos((˜α − A1)/Ro) − α˜A2 (4.39)

Finally, complete occultation occurs when the apparent separation is smaller than the difference between the occulted and occulting bodies apparent radii:

if α˜ < R˜o − R˜S & R˜S < R˜o (4.40)

or α˜ < R˜S − R˜o & R˜S > R˜o (4.41) → o = 0 (4.42)

In order to limit the computational weight of this functionality, only occultations caused by the Earth and the Moon will be considered.

4.4.3. Third body perturbations The third type of perturbation to be considered is the one due to the gravitational attraction of other celestial bodies. The bodies that exert the largest attraction (besides the Earth) are the Sun and the Moon because of their mass and vicinity. By using the theory of the N-body problem provided in Reference [22] and described in Section 4.3, it is possible to write the perturbing potential caused by the gravitational attraction of a number of external bodies j on the orbit of a body i orbiting body k in an inertial reference frame centred at the position of the body k as: ! X 1 ~ri · ~rj UTB = −G mj − 3 (4.43) rij r k6=k,i j where

• rij is the scalar distance between the satellite and the perturbing body

• ~ri is the position vector of the satellite 34 4. Orbital dynamics

• ~rj is the position vector of the perturbing body The perturbing force is then equal to:

~ fTB = −∇~ UTB (4.44) However, it is important to note that, depending on the position of the bodies, the perturbing forces described in this section might act in the same or in the opposite direction. For this reason, the overall effect of the perturbations can vary considerably along the single orbital revolution as well as along prolonged periods. Long duration simulations are therefore necessary in order to determine the actual effect of the perturbing forces on the orbits of the satellites. 5

Optimization problem

In order for the constellation to comply with the requirements set out in Section 3.2, the attitude of the solar sail of each satellite in the formation needs to be optimal. Because this optimal attitude cannot be known a-priori or calculated analytically, this thesis makes use of numerical optimization. In the first section of this chapter, a formal definition of the problem will be stated. Subsequently, two possible definitions of the objective function that were defined in order to characterize the problem will be described. Subsequently, the various definitions of the decision variables sets that were implemented in the optimization routines will be analyzed. The rationale behind the choice of the solar sail attitude profile representations that the decision variables determine will be discussed. Finally, the two optimization logics that were included in the optimizer will be discussed and their advantages and drawbacks will be highlighted.

5.1. Optimization problem deﬁnition The optimization problem for this application can be formalized as the minimization of an objective function that depends on the state history of the satellites:

min f˜(~y(t, d~), ~y˙(t, d~), d~) (5.1) w.r.t. d~ s.t. c˜(~y(t, d~), ~y˙(t, d~), d~) ≤ 0 ~ ˙ ~ ~ ceq(~y(t, d), ~y(t, d), d) = 0 where

• f˜ is the objective function of the problem.

• d~ is the vector containing the decision variables of the optimization problem. In this case, the decision variables are parameters used to compute the solar sail attitude angles.

• c˜ and ceq are the problem speciﬁc constraint functions.

Please note that in Eq. 5.1 a generic formulation of the objective and the constraint functions was provided. In the actual problem, these functions will not depend on all elements of the state vector, state vector time derivatives and decision variables vector at the same time. In the following sections, a more detailed deﬁnition of the objective functions and the decision variables speciﬁc to the problem at hand will be provided.

5.2. Objective function deﬁnitions As stated in the previous chapters, the goal of the optimization process is to determine the solar sail attitude proﬁles that result in the optimal relative and absolute position of the satellites over time for

35 36 5. Optimization problem debris triangulation purposes. To this end, two possible formulations of the objective function have been deﬁned to highlight different aspects of the problem. For clarity purposes, they will be called ’inclination and distance’ objective function and ’observation angle’ objective function, respectively. In the ﬁrst one, the function value is a weighted sum of the largest orbital inclination reached by the geometrical center of the satellite formation and the differences between the optimal inter-satellite distances and the actual distances, averaged over the entire simulation duration. This results in the following expression:

i=N j=N t=tend ˜ w2 1 X X X f = w1imax + |dij(t) − dopt| (5.2) tend − t0 2 i=1 j=1,j6=i t=t0 where

• imax is the largest orbital inclination reached by the geometrical center of the satellite formation • N is the number of satellites in the considered formation

• dij is the distance between two of the satellites

• dopt is the optimal inter-satellite distance deﬁned in Section 3.2

• w1 and w2 are weights used to rescale the two terms of the objective function The values of the weights need to be tuned in order to better comply with the mission requirements. A ﬁrst estimate was obtained by computing the two terms of the objective function for two uncontrolled satellites over a period of one year. In order to rescale the two terms of the objective function to values −1 −3 −1 of the same order of magnitude, the determined weights were w1 = 60 rad and w2 = 10 km .

The second formulation of the objective function is exclusively based on the relative position between the satellites and the observed debris and does not take into account the absolute position of the formation. Its value is proportional to the deviation of the formation shape from the optimal shape deﬁned in Section 3.2 and is deﬁned as follows:   i=N j=N t=tend i=N t=tend ˜ 1 1 X X X X X f =  |dij(t) − dopt| + |did(t) − dobs| (5.3) tend − t0 2 i=1 j=1,j6=i t=t0 i=1 t=t0 where

• did is the distance between satellite i and the observed debris

• dobs is the optimal distance between the satellite and the observed debris, as deﬁned in Section 3.2 T Here, the debris is assumed to be in a geostationary orbit. Its position vector, ~rd = [Xd,Yd,Zd] , is deﬁned to be aligned with the projection of the radius vector of the formation geometrical center on the equatorial plane and can therefore be computed with the following expressions:

Y γ∗ = arctan fc (5.4) Xfc ∗ Xd = rGEO cos(γ ) (5.5) ∗ Yd = rGEO sin(γ ) (5.6)

Zd = 0 (5.7) where

• Xd, Yd and Zd are the Cartesian coordinates of the observed debris expressed in the J2000 reference frame

• Xfc, Yfc and Zfc are the Cartesian coordinates of the formation geometrical center expressed in the J2000 reference frame

• rGEO is the orbital radius of the geostationary orbit 5.3. Decision variables 37

Penalty function Apart from the previously defined terms of the objective function, it is also possible to define and include a so-called penalty function term, which adds ulterior requirements that the final solution needs to comply with. In the studied case, a penalty function can be added in order to try to minimize the total required formation control effort. Including this additional term in the objective function could allow to produce more realistic solar sail attitude angle profiles. As a first approximation, the control effort necessary to maintain the formation in its optimal shape can by represented by the total control torque. If the perturbing torques acting on the satellites are neglected, minimizing the control torque translates into minimizing the total angular acceleration of each satellite. The penalty function, g, will therefore be equal to the sum of the second time derivatives of the cone and clock angle profiles of each of the satellites averaged over the entire simulation duration:

s=N "t=tend 2 2 # 1 X X d αs(t) d δs(t) g = 2 + 2 (5.8) tend − t0 dt dt s=1 t=t0

In case a penalty function is included in the optimization process, the new objective function, h∗, that will be minimized will be equal to: h∗ = f˜+ g (5.9)

It is important to note that the resulting expressions for f˜, g and h∗ are independent from the length of the time span over which the satellite dynamics are propagated. This allows to analyze and compare the behaviour of the optimizer and the resulting solutions for different simulation durations.

5.3. Decision variables As mentioned in Section 5.1, the purpose of the optimization process is to determine the profile for the cone and clock angles of the sails that result in the lowest value of the problem objective function. These angle profiles are represented with the decision variable vector d~. As it is obviously computationally infeasible to optimize the value of each satellite sail cone and clock angles at every time-step of the dynamics propagation, an alternative representation of the sail attitude profile relying on a limited number of parametersis necessary. The smaller the number of parameters used to describe the problem, the more efficient the optimization process will be. The representations that are considered and analyzed in this thesis are the constant value, the patched polynomial, the patched square wave and the patched sinusoidal.

5.3.1. Constant value representation The ﬁrst and simplest representation for the sail attitude angles is the constant value representation. When this representation is used, the cone and clock angles will assume a constant value, Kα and Kδ, for a given time span:

α(t) = Kα (5.10)

δ(t) = Kδ (5.11)

This representation is extremely simple and therefore easily implementable. Furthermore, no computational processing of the decision variables is required so that no additional calculations occur. However, it is easily understandable that this simple attitude angle representation only allows to characterize a very limited number of solar sail attitude proﬁles and might therefore be too simplistic for the problem at hand.

5.3.2. Patched polynomial representation In the patched polynomial representation the sail attitude angles are described by an nth degree polynomial deﬁned across the time span of an orbit revolution which is then repeated in every subsequent 38 5. Optimization problem orbital revolution. Mathematically this can be expressed as:

k=n X ˜ k α(t) = ak(t − t0 + b) (5.12) k=0 k=n X ˜ k δ(t) = ck(t − t0 + d) (5.13) k=0 where

• t0 is the time at which the satellite crosses the ascending node of the orbit

• ak and ck are the polynomial coefﬁcients of degree k for the cone and clock angles, respectively

• ˜b and d˜are the ’time delay’ parameters for the cone and clock angles, respectively

It is possible to reduce the number of design variables to be optimized by constraining the angle proﬁle to be repeated identical to itself every revolution period T . This results in the angle being the same at the beginning and end of each revolution.

k=n k=n X ˜ k X ˜ k α(t) = α(t + T ) → ak(t − t0 + b) = ak(t + T − t0 + b) (5.14) k=0 k=0 k=n k=n X ˜ k X ˜ k δ(t) = δ(t + T ) → ck(t − t0 + d) = ck(t + T − t0 + d) (5.15) k=0 k=0

Furthermore, in order to consider a more realistic case, the polynomials can be forced to be contin- uously differentiable. This results in smaller physical loads on the sail and a further reduction of the number of design parameters. Two more constraints can be deﬁned:

k=n k=n dα dα X ˜ k−1 X ˜ k−1 = → kak(t − t0 + b) = kak(t + T − t0 + b) (5.16) dt dt t t+T k=1 k=1 k=n k=n dδ dδ X ˜ k−1 X ˜ k−1 = → kck(t − t0 + d) = kck(t + T − t0 + d) (5.17) dt dt t t+T k=1 k=1

These two sets of constraints result in a system of four equations. Any of the polynomial coefﬁcients can then be expressed a function of the others coefﬁcients, and the number of free decision variables is reduced by four.

For long duration simulation (i.e., over hundreds of revolutions), considering a single repeated attitude profile becomes overly simplistic and limits the possibilities of the optimization concept. Therefore the cone and clock angles profiles are allowed to change after a certain number of revolutions. However, every additional profile change results in a larger number of decision variables and a trade-off needs to be made. Different options will therefore be considered and compared.

The previously described continuity and differentiability constraints can also be applied to the switch 5.3. Decision variables 39 points between attitude proﬁles and can be formulated as follows:

k=n k=n X ˜ k X ˜ k αj(t + nT ) = αj+1(t + nT ) → aj,k(t + nT − t0 + bj) = aj+1,k(t + nT − t0 + bj+1) (5.18) k=0 k=0 k=n k=n X ˜ k X ˜ k δj(t + nT ) = δj+1(t + nT ) → cj,k(t + nT − t0 + dj) = cj+1,k(t + nT − t0 + dj+1) (5.19) k=0 k=0 k=n dαj dαj+1 X ˜ k−1 = → kaj,k(t + nT − t0 + bj) = (5.20) dt dt t+nT t+nT k=1 k=n X ˜ k−1 kaj+1,k(t + nT − t0 + bj+1) k=1 k=n dδj dδj+1 X ˜ k−1 = → kcj,k(t + nT − t0 + dj) = (5.21) dt dt t+nT t+nT k=1 k=n X ˜ k−1 kcj+1,k(t + nT − t0 + dj+1) k=1

The number of parameters to be optimized, O, when using an nth degree patched polynomial representation for the sail attitude angles including c∗ proﬁle changes for N satellites is then equal to:

O = N[2(c∗ + 1)(n + 2) − 4(2c∗ + 1)] (5.22)

5.3.3. Patched sinusoidal representation Another possible representation for the sail attitude angles that will be considered is a sinusoidal representation. In this case, the sail cone and clock angles are represented by a sinusoidal function, characterized by an amplitude A, a frequency ω, a phase φ and an offset B, as shown below:

α(t) = Aα cos(ωαt + φα) + Bα (5.23)

δ(t) = Aδ cos(ωδt + φδ) + Bδ (5.24) where the subscripts α and δ indicate the sinusoidal parameters of the cone and clock angles, respectively.

The same continuity constraints described for the previous representation are applied in this case. The sinusoidal functions are patched together so that the values of the attitude angles of the satellites at the beginning and end of every revolution match. Said constraints can be deﬁned as follows:

α(t) = α(t + T ) → Aα cos(ωαt + φα) + Bα = Aα cos(ωα(t + T ) + φα) + Bα (5.25) δ(t) = δ(t + T ) → Aδ cos(ωδt + φδ) + Bδ = Aδ cos(ωδ(t + T ) + φδ) + Bδ (5.26)

By developing these constraint, the frequencies of the sinusoidal functions can be expressed as: 2π ω = k (5.27) α α T 2π ω = k (5.28) δ δ T where kα and kδ are either positive integers or rational fractions. Because kα and kδ can only assume discrete values, they cannot be treated and optimized in the same way as the other continuous decision variables. In order to solve this issue, the optimal values for kα and kδ will be found in an iterative process assuming the same value for both the cone and clock angle of every spacecraft:

k = kα = kδ (5.29) 40 5. Optimization problem

Given the nature of the sinusoidal representation, in this case it was decided not to constrain the attitude angles to having the same ﬁrst time derivative values at the beginning and end of each revolution, as it would considerably limit the formation control possibilities of the optimizer. The number of decision variables to be optimized in this case is then equal to:

O = N[(c∗ + 1) · 6 − 2 · c∗] (5.30)

5.3.4. Patched square wave representation A fourth possible representation for the solar sail attitude that will be considered is the patched square wave. In this case, the value of the cone and clock angles will alternate, with a given frequency, between two values. The square wave function is conceptually similar to the sinusoidal one and can be deﬁned by the same parameters: amplitude A, frequency ω, phase φ and offset B. In case the square wave representation is used, the cone and clock angles values can be expressed as:

α(t) = Aαsgn(cos(ωα · t + φα)) + Bα (5.31)

δ(t) = Aδsgn(cos(ωδ · t + φδ)) + Bδ (5.32) where sgn is a function that returns 1 if its argument is strictly positive, −1 if its argument is strictly negative and 0 if its argument is null. For this reason, the proﬁles of α and δ will present discontinuities. The exact same constraints as for the sinusoidal representation were implemented for the square wave representation as well. The number of decision variables to be optimized in this case are then also equal to: O = N[(c∗ + 1) · 6 − 2 · c∗] (5.33)

5.4. Constraints In the formalization of the optimization problem defined in Eq. 5.1, equality and inequality constraints are included. In the actual implementation, two different constraints are present. First, the decision variables are bounded in order to limit the optimization process search space. In this case, the inequality constraint previously defined is simplified and can be written as:

lb ≤ d~ ≤ ub (5.34) where lb and ub are the lower and upper bounds, respectively. Please note that in this case the bounds are vectors with a dimension equal to the number of decision variables characterizing the problem. The second constraint that is included in the developed application has the purpose of deﬁning the state vectors of the satellites at the initial epoch of the simulation. For each satellite included, this equality constraint can simply be written as: ~y(t0) = ~y0 (5.35) where

• t0 is the initial time of the simulation

• ~y0 is the initial state vector resulting from the preliminary analysis described in Section 3.3

5.5. Optimization logic In order to determine the optimal control profile for each sail, two different optimization logics were considered and implemented. In the first optimization logic, the decision variables describing the solar sail attitude angles for the entire propagation duration are optimized all at once. In this case, the objective function value is computed considering the entire state vectors history obtained at every iteration. This solution puts a severe limit on the number of profile changes that can be performed over the propagation duration, as the number of parameters to be optimized increases linearly with the number of sail attitude modifications. For simplicity reasons this optimization logic will from here on be called ’global’ optimization logic.

An alternative solution is to optimize the sail attitude profiles over short durations, such as one orbital 5.5. Optimization logic 41 revolution. In this case, for example, the objective function value is computed for each single revolution separately so that the decision variables for the given time span are optimized independently from the rest of the simulation. The initial state vectors of the satellites for every revolution are determined by propagating the optimal solutions of the previous revolution. This optimization logic allows to change the attitude profiles very frequently without any loss in the optimization efficiency. However, optimizing the satellites control profiles over limited durations without considering the effect of the resulting choices on the evolution of the orbital conditions over the entire propagation duration might result in orbital changes that yield to poor debris triangulation conditions later on. For simplicity reasons, this optimization logic will be called ’revolution-by-revolution’ optimization logic from here onwards.

Numerical methods

In Chapter4, the dynamics that, when propagated, describe the motion of the formation were deﬁned, while Chapter5 has provided an overview of the optimization problem to be solved. In this chapter, the actual numerical implementation of the dynamics propagation and optimization process will be discussed, together with the relevant aspects of the developed application. First, the implemented algorithms for the integration of the equations of motion and the problem optimization will be described and discussed. Subsequently, a brief overview of the application will be provided and the used language and libraries sets will be described. Finally, an overview of the entire developed application will be provided through the use of block diagrams, in order to provide the reader with a clear idea of its overall functioning.

6.1. Numerical integration Various integration techniques exist, both single and multiple step, each one with a different accuracy and computational weight. During a preliminary literature study [30], several of these methods were analyzed from performance and implementation complexity perspectives. For the propagation of the satellites’ dynamics, it was decided to use a fourth-order Runge-Kutta (RK4) integrator with a ﬁxed step-size. The RK4 integrator is a very commonly used robust integration method that is characterized by a fairly limited computational effort [31].

In order to integrate the dynamics of the satellites, the RK4 method evaluates the equations of motion deﬁned in Eq. 4.13 at four different epochs and/or state vectors at every integration step [18]:

~ k1 = f(t0, ~y0) (6.1) ~ ˜ ˜~ k2 = f(t0 + h/2, ~y0 + hk1/2) (6.2) ~ ˜ ˜~ k3 = f(t0 + h/2, ~y0 + hk2/2) (6.3) ~ ˜ ˜~ k4 = f(t0 + h, ~y0 + hk3) (6.4) (6.5) where

• ~y0 is the state vector at time t0 • h˜ is the integration step-size From these evaluations, the incremental function Φ~ is computed as: 1 Φ~ = (~k + 2~k + 2~k + ~k ) (6.6) 6 1 2 3 4 ˜ and is then used to estimate the value of the state vector at time t = t0 + h:

43 44 6. Numerical methods

˜ ˜ ~η(t0 + h) = ~y0 + hΦ~ (6.7) The local truncation error, e˜, introduced by this numerical integration method at every integration step is equal to: ˜ ˜ e˜ = |~y(t0 + h) − ~η(t0 + h)| (6.8) It can be easily derived that this local truncation error is always smaller than or equal to C · h˜5 (C being a constant). As the number of function evaluations required to propagate ~y(t0) is inversely proportional to the step-size, the total accumulated error will be smaller than or equal to C · h˜4. It can then be deduced that, the smaller the integration step is, the smaller the cumulated truncation error will be. However, a step-size excessively small will also lead to an increase in the accumulated round-off error, numerical error and computational run-time. A trade-off is therefore necessary in order to determine the step-size that works best for the given problem. Some preliminary tests were therefore performed in order to determine a suitable step-size which would result in a good compromise between accuracy and computational efﬁciency. A value of 120 s was ﬁnally selected.

Variable step-size Variations of the Runge-Kutta methods exist that include a routine to automatically adapt the step- size during the integration process. These particular methods can be useful for problems that present large variation of the state vector time derivative over the integration interval, such as problems involving highly eccentric orbits. Variable step-size methods are, however, more demanding in terms of computational effort. Furthermore, a variable step-size would not yield great advantages as the the considered orbits are circular and the perturbations acting on the satellite are relatively small. By consequence, the resulting acceleration components proﬁles are predictably repetitive so that an ap- propriate constant step-size value can be easily determined a priori without a noticeable efﬁciency loss in the integration process.

6.2. Optimization algorithm Given the complexity of the problem that needs to be optimized, classical numerical minimum ﬁnding techniques such as the Newton-Raphson method or computationally inefﬁcient methods such as the Monte Carlo method are not viable options. Instead, this thesis will make use of heuristic optimization techniques. In a preliminary literature study [30], several different heuristic optimization methods were considered, each with their respective strengths and weaknesses. Among the criteria that need to be considered when selecting an optimization method, the most important ones are the type of problem that is to be analyzed, the required solution accuracy and the available computational power. Fur- thermore, the simplicity of the algorithm implementation needs to be taken into account as well, as an easier and shorter implementation could compensate for a longer computation run-time. Directly relating these parameters to a measurable value is a complex task and the choice of the optimization method is therefore often performed by trial and error. Given the large amount of time required to test several different optimization methods, it was decided to choose Self-adaptive Differential Evolution (SaDE), a commonly used and reliable optimization technique that will be described in the following paragraphs.

The differential evolution (DE) optimization technique, originally described by Storn and Price in 1996 [32], belongs to the family of evolutionary algorithms, which are heuristic optimization techniques in- spired by biological evolution mechanisms such as natural selection, random genetic mutation and reproduction. Each set of parameters characterizing a candidate solution is treated as an individual that is part of a "population" which evolves within the search space following the previously listed mechanisms. During this evolution process, the individuals whose set of parameters result in the best values for the objective function "survive" and are passed on to the following generations. The aforementioned population consists of n˜ vectors ~xi containing N˜ values, where N˜ is the dimension of the search space and n˜ is the population size. Every element, l, of each vector of the ﬁrst generation is randomly generated within the bounds deﬁned for the given problem. Subsequently, the objective function value that every individual (vector) yields is evaluated and compared with the others. At each new iteration, a mutating vector ~vk is generated for each vector ~xk of the population, with k = 1, ..., n˜. 6.2. Optimization algorithm 45

The most commonly used deﬁnitions of the mutating vectors are listed below:

~vk = ~xrand1 + F · (~xrand2 − ~xrand3 ) (6.9)

~vk = ~xbest + F · (~xrand1 − ~xrand2 ) (6.10)

~vk = ~xk + F · (~xbest − ~xk) + F · (~xrand1 − ~xrand2 ) (6.11)

~vk = ~xbest + F · (~xrand1 − ~xrand2 ) + F · (~xrand3 − ~xrand4 ) (6.12)

~vk = ~xrand1 + F · (~xrand2 − ~xrand3 ) + F · (~xrand4 − ~xrand5 ) (6.13) where

• rand1, rand2, rand3, rand4, rand5 are randomly generated integers in the interval [1, n˜], k excluded • F ∈ [0, 2] is the so-called "differential weight"

• ~xbest is the individual that yielded the best objective function value at the previous generation

Subsequently, a so-called trial vector, ~zk, is generated as a crossover between the mutating vector ~vk and ~xk. The single element zk,l is chosen in the following way:

zk,l = vk,l if randb(l) ≤ CR or l = rnbr(k) (6.14)

zk,l = xk,l if randb(l) > CR or l 6= rnbr(k) (6.15) where • randb(l) ∈ [0, 1] is a randomly generated real number • rnbr(k) ∈ [0, N˜] is a randomly generated integer that is necessary to make sure that at least one of the elements of ~zk originates from the mutated vector ~vk • CR ∈ [0, 1] is a randomly generated real number called "crossover probability"

Finally, the resulting objective function values computed for every vector ~xk and ~zk are compared and the vector yielding the best objective function value is passed on to the next generation. This process is repeated until the maximum number of generations is reached or until the termination conditions are satisfied. As no reference values for the objective function exist in the available scientific literature, case-specific termination conditions will not be included in the developed application. The optimization process will therefore terminate when a previously set number of generations is reached.

6.2.1. Self-adaptation In order to analytically determine which of the mutating vector definitions provided in Eqs. 6.9-6.13 to use, A.K. Qin and P.N. Suganthan developed an algorithm, called Self-adaptive Differential Evolution (SaDE), that autonomously computes the optimal choice [33]. In its simplest implementation, only Eq. 6.9 and 6.12 are considered, as literature shows that they produce the best results for most applications [34]. The parameters p˜1 and p˜2 represent the probabilities of Eq. 6.9 and 6.12, respectively, to be used at a given iteration. During the first iteration, both mutating vector definitions have the same probability of being used (i.e., p˜1 =p ˜2 = 0.5). Subsequently, during the optimization process, these probabilities will vary in order to render a more optimal search. At every iteration, a vector of dimension n˜ is created and filled with random values ranging from 0 to 1. Every element l of said vector with a value smaller than p˜1 will result in the mutating vector ~vi being defined according to Eq. 6.9, while the remaining ones will be defined according to Eq. 6.12. The values of p˜1 and p˜2 are then updated for the following generation using:

ns1 · (ns2 + nf2) p˜1 = (6.16) ns2 · (ns1 + nf1) + ns1 · (ns2 + nf2) p˜2 = 1 − p˜1 (6.17) where 46 6. Numerical methods

• ns1 and ns2 are the number of mutating vectors ~vk deﬁned in line with Eq. 6.9 and 6.12, respectively, that were passed on to the next generation

• nf1 and nf2 are the number of mutating vectors ~vk deﬁned in line with Eq. 6.9 and 6.12, respectively, that were discarded Furthermore, the parameters F and CR in Eqs. 6.9-6.13 can either be deﬁned as constant, as is done in the classical DE formulation, or determined at each generation g˜ as follows:

Fg˜ = FI + randb1 · Fu if randb2 < τ1 (6.18)

Fg˜ = Fg˜−1 if randb2 > τ1 (6.19)

CRg˜ = randb3 if randb4 < τ2 (6.20)

CRg˜ = CRg˜−1 if randb4 > τ2 (6.21) where

• randb1, randb2, randb3 ∈ [0, 1] are randomly generated real values

• Fu, FI , τ1 and τ2 are constant optimization control parameters In Reference [34] the values for some of the aforementioned control parameters that result in a robust optimization process for most of the functions it was tested on are provided. The SaDE algorithm implemented in the PaGMO libraries (see Section 6.3) adopts these values, which are:

τ1 = 0.1

τ2 = 0.1

FI = 0.1

Fu = 0.9 The remaining control parameters to be tweaked when setting up the optimization process are then the population size n˜ and the number of iterations (generations) to be performed.

6.2.2. Strengths and weaknesses of the SaDE algorithm Compared to the classic minimum finding techniques, evolutionary algorithms present several advantages. The most notable ones are that they do not require any initial guess for the decision variables and they explore the entirety of the search space better than gradient-based algorithms. This results in a considerable robust optimization process and also allows to perform optimizations without determining a realistic first guess beforehand. Furthermore, evolutionary algorithms also put very few to no restrictions on the function to be optimized for what concerns differentiability and continuity. How- ever, no matter how robust the developed algorithm, there is always the possibility for the optimization process to converge to a local minimum without an analytical way to determine whether this indeed occurs. Furthermore, similarly to most heuristic optimization algorithms, the number of decision variables heavily influences the optimizer performances, thus putting a limitation on the number of decision variables that can be used to describe the optimization problem. Finally, when using the SaDE algorithm, a number of control parameters need to be tuned manually before optimal results can be produced, such as the number of generations and population size. This process can be particularly time consuming for computationally heavy problems.

6.2.3. Seed During the optimization process, all values of the parameters that are randomly generated will be computed through a pseudo-random number generation (PRNG) process. The PRNG is an algorithm that generates numbers in a semi-deterministic fashion within the given bounds based on a small set of initial values, called the seed [35]. The probability distribution of the resulting random numbers is similar to the one of truly randomly generated numbers, with the difference that the computational time required to compute them is considerably shorter. Also note that the values of the seed are set manually for every optimization process, which allows the results to be reproducible and comparable. 6.3. Application overview 47

6.3. Application overview The application developed for this thesis has been programmed in C++, a very popular object oriented programming language. The application relies on a number of C++ speciﬁc sets of libraries that allow to greatly reduce the time needed to implement and test most of the included functionalities.

Among the standard sets of libraries that are included, it is worth mentioning the use of Boost1 and Eigen2, that are particularly useful for the efﬁcient handling of memory pointers and linear algebra operations. Instead, for what concerns the specialized applications, the TU Delft Astrodynamics Tool- box (Tudat)3 and Parallel Global Multi-objective Optimizer (PaGMO) libraries4 are used for the orbital dynamics propagation and optimization, respectively.

6.3.1. Tudat libraries The Tudat libraries are used for the orbital dynamics propagation and are a collection of C++ libraries written, maintained and updated by the staff and the students of the Astrodynamics & Space Missions department of the TU Delft Aerospace Engineering faculty3. These libraries provide the user with a large number of ready-to-use functions and functionalities typically necessary when developing orbital mechanics and dynamics applications, such as the automated generation of sets of perturbing forces acting on a body, coordinates and reference frames transformations, handling of the Solar System bodies’ ephemerides and several integration and propagation techniques. The models for all the forces acting on the satellites defined in Section 4.4 can be automatically generated as a function of the spacecraft position and velocity with respect to the other bodies included in the simulation. However, for what concerns the SRP, the only available model in the official repository is the so-called cannon ball radiation pressure model, which computes the force that would act on a partially reflective spherical object with the same cross-sectional area as the considered spacecraft. It was therefore necessary to implement, within the Tudat libraries, the optical sail model described in Section 2.3. Said model is now in the process of being integrated in the official Tudat repository.

6.3.2. PaGMO libraries The optimization block of the developed application makes use of PaGMO, a set of libraries that allow parallel computations of global and local optimisation tasks and are developed and maintained by the European Space Agency (ESA)4. These libraries provide the user with a large number of heuristic optimization techniques for the implementation of constrained and unconstrained functions and several functionalities to interface the optimizer with complex problems. The PaGMO libraries also offer the possibility to easily set-up the optimization process in such a way that it can be run on multiple cores or machines at the same time (i.e., parallel computation).

6.3.3. Celestial bodies ephemerides In order to properly deﬁne the perturbing forces acting on the satellites listed in Section 4.4, the developed application needs the ephemerides of the Earth, the Sun and the Moon. The positions of these bodies at every epoch are obtained through the use of the SPICE ephemerides provided by NASA5, in which they are deﬁned as Chebychev polynomials evaluated at the required time epochs [36]. These ephemerides are then processed through a number of Tudat functionalities. Furthermore, size, mass and other relevant physical characteristics of these bodies used in the simulation are provided by the SPICE tool-kit as well.

6.4. Application block diagram In order to provide the reader with a clearer idea of the functioning of the developed application and its various functionalities, Figures 6.1 and 6.2 show simpliﬁed schematics of the orbital dynamics propagation and the optimization blocks, respectively. It is important to note that each element included in

1Boost libraries website, http://www.boost.org/, accessed: May 2016 2Eigen libraries website, http://eigen.tuxfamily.org/index.php?title=Main_Page, accessed: May 2016 3Tudat libraries website, http://tudat.tudelft.nl/projects/tudat/wiki, accessed: July 2015 4PaGMO libraries website, https://esa.github.io/pagmo/, accessed: September 2016 5SPICE libraries website, https://naif.jpl.nasa.gov/naif/index.html, accessed: June 2016 48 6. Numerical methods the block diagram does not represent a single function or ﬁle, but instead represents a functionality of the application, unless mentioned otherwise in its description.

6.4.1. Orbital mechanics propagation block diagram As described in Chapter4, the orbital mechanics propagation block is used to propagate the dynamics of the satellites for a given time duration under the inﬂuence of a number of different forces. The inputs of this part of the application are the following:

• Decision variables: the decision variables are an input provided by the optimization block, they are used in order to compute the solar sail attitude angles at every time-step of the simulation.

• Sail size: the sail size, together with the spacecraft mass that is set constant to 3 kg within the software, is necessary in order to compute the value of the solar sail lightness number β.

• Initial state: the initial state determines the initial position and velocity of each of the satellites.

• Propagation duration: the propagation duration determines the length of the time span over which the solar sail attitude angles are optimized.

These inputs are then processed in order to determine the state history of the satellites with the following functionalities:

• Spacecraft definition: this function initializes a C++ object for every spacecraft included in the optimization, defining its physical properties such as mass, solar sail lightness number and reflectivity as well as its initial state vector.

• Ephemerides generation: given the list of celestial bodies necessary to compute all forces included in the dynamical model, this functionality loads the necessary SPICE kernels and computes the ephemerides of the celestial bodies at the required epoch.

• Spacecraft dynamics generation: this functionality deﬁnes the equations of motion of all included forces acting on each satellite as a function of their physical properties, their position and the epoch.

• Cone and clock angle proﬁle generation: this function uses the decision variables and the current epoch to compute the value of the solar sail attitude angles. It is called at every time-step of the propagation.

• Propagation of the satellites’ state vectors: this functionality propagates the state vectors of the satellites from a starting value for a speciﬁc time duration by numerically integrating the equations of motion of the spacecraft deﬁned and computed by the previously described functionalities.

• Objective function computation: this function evaluates the value of the objective function deﬁned in Eqs. 5.2 or 5.3 by processing the satellites’ state vectors history. The objective function value is then sent to the optimizer that will use it in order to update the individuals.

6.4.2. Optimization block diagram As described in Chapter5, the objective of the optimizer implemented in the developed software is to compute the parameters that yield solar sail attitude angle proﬁles that result in an optimal control of the formation for debris triangulation purposes. The inputs of the optimization block are the following:

• Seed: integer value used by the PRNG to compute pseudo-random values used in the stochastic optimization process.

• Population size: number of individuals comprising the population.

• Number of generations: number of iterations to be performed by the Self-adaptive differential evolution optimization routine. 6.4. Application block diagram 49

• Number of attitude changes: together with the optimization logic and type of attitude angle representation, the number of attitude changes is necessary in order to determine the number of decision variables to be optimized. As described in Section 5.3, this value represents the number different profiles that will characterize a given attitude angle over the entire simulation duration. • Control representation: type of analytical representation used to parametrize the solar sail attitude angles, as described in Section 5.3. • Objective function definition: definition of the objective function characterizing the problem, as described in Section 5.2.

In order to compute the optimal values of the decision variables, the following functionalities are used within the optimization block:

• Setting bounds: this function provides the optimizer with upper and lower bounds for the decision variables. These bounds are necessary in order to define a precise search space. The more accurately these bounds are defined, the more efficient the optimization process will be. However, overly restrictive bounds could also exclude a possible and feasible optimal solution from the search space. The values of these bounds vary as a function of the attitude angles representation that is used. • Problem definition: this functionality numerically defines the optimization problem as a C++ object with all the relevant fields initialized, such as the definition of the objective function and the value of the decision variables bounds. • Population definition: this functionality numerically defines the population as a C++ object with all the relevant fields initialized. The population is notably characterized by an optimization problem, a number of generations and a population size. • Evolution of the population through self-adaptive differential evolution: this functionality makes the population of candidate solutions evolve in the search space in order to determine the optimal solution to the problem. For every evaluation of the objective function the orbital mechanics propagation block is called. The optimizer provides a set of decision variables and receives the resulting objective function value in return. 50 6. Numerical methods

Figure 6.1: Block diagram of the orbital dynamics propagation block of the developed application 6.4. Application block diagram 51

Figure 6.2: Block diagram of the optimization block of the developed application

Code veriﬁcation and validation

Software developed for scientific applications needs to go through a verification and validation process in order to determine whether all its functionalities were properly implemented and whether it complies with all the project requirements. As explained in Reference [37], validation of a product can be described with the question "are we building the right product?", while verification can be described with the question "are we building the product right?". Given that there are no explicit stakeholder requirements and needs to be taken into account in this thesis project, for simplicity it will be assumed that the software is validated by default. It is, however, still necessary to verify the developed application to make sure that the produced output is correct and matches the given input.

In order to ensure the proper functioning of the written code and developed functionalities, the various blocks comprising the application developed for the project were tested together with their integration. The application testing process is usually performed by having the considered block, function or part of code produce results that can be compared to data present in the scientiﬁc literature or data that is trivially obtainable. As mentioned in Chapter1, the concept that this thesis aims to develop has never been studied before and therefore the literature cannot provide the necessary data to benchmark all the functionalities that need to be tested. For this reason, some of the tested functionalities had to be slightly modiﬁed in order to reproduce models described in reference articles and books. Furthermore, it is important to note that all the functions present in the used libraries were assumed to have been correctly implemented and were therefore not tested.

In this chapter, the rationale behind the verification processes that were performed will be explained, the single tests will be described and finally the relevant results will be analyzed. In the first section, the testing of the most relevant functions and functionalities of the orbital mechanics propagation block described in Section 6.4.1 will be discussed. The second section focuses instead on the testing of the optimization block of Section 6.4.2.

7.1. Veriﬁcation of the orbital mechanics propagation block In this section the veriﬁcation of the relevant dynamics propagation block functions and functionalities is discussed.

7.1.1. Solar System bodies ephemerides As mentioned in Section 4.4, the position of the Sun, the Earth and the Moon are necessary in order to compute some of the forces determining the motion of the satellites. The position of these celestial bodies are provided by the SPICE ephemerides1. In order to verify the proper loading and processing of the aforementioned ephemerides, the resulting position of each celestial body was computed for a duration of one year starting from midnight of the ﬁrst of January 2000. The resulting orbits of the three celestial bodies in the HIF reference frame and the orbit of the Moon in the J2000 reference frame are

1SPICE libraries website, https://naif.jpl.nasa.gov/naif/index.html, accessed: June 2016

53 54 7. Code verification and validation provided in Figures 7.1 and 7.2. As can be seen from said figures, the orbital profiles of the celestial bodies match the expected behaviour: a nearly circular orbit of the Earth around the Sun and a nearly circular orbit of the Moon around the Earth.

Earth Sun Moon

×10 4

8 -2

-1.5 6 -1 4 -0.5 ×10 8 [km]

HIF 2 0 Z 0.5 0 1

-2 1.5 X [km] HIF -2 -1.5 2 -1 -0.5 0 ×10 8 0.5 1 Y [km] 1.5 HIF

Figure 7.1: Orbits of the Sun, the Earth and the Moon over one year in the HIF

7.1.2. SRP model As previously mentioned, the model describing the SRP acting on the solar sails and the produced force was created for this particular application and needs therefore to be verified. A first test that was performed for the SRP model considered the relation between the cone angle and the resulting force. It was tested by computing the resulting normalized SRP induced force for a range of cone angles as shown in Figure 7.3. It can be verified that it matches the plot of the normalized force provided in Reference [17]. A second test performed for the SRP model was the reproduction of the so-called ’force bubble’, which is a plot showing the relation between the components of the force per unit of area parallel and perpendicular to the Sun-line for various values of the solar sail cone angle. Figure 7.4 shows the verification plot (on the left) and the reference plot provided in Reference [38] (on the right). Note that both Figures 7.3 and 7.4 consider the sail to be perfectly reflective and orbiting at a distance of 1 AU from the Sun. In this case as well, it can be seen that the produced results match the reference data and therefore the SRP is assumed to be properly implemented.

7.1.3. Spacecraft dynamics propagation In order to verify the proper computation of the equations of motion of each satellite and the proper integration and interfacing of the SRP model with the rest of the orbital propagation routine, a test was performed. This verification is necessary as the SRP acceleration model was not included in the Tudat libraries and was manually integrated in the dynamics propagation block. In order to perform this verification, it was decided to reproduce a trajectory for which the analytical description is known: a displaced orbit above the Sun, also known as a halo orbit [17]. Displaced orbits are non-Keplerian orbits characterized by an orbital plane that does not intersect with the central attracting body. This type of orbit is achievable by producing a force on the spacecraft with a constant component perpendicular to the orbital plane. It is possible to obtain Sun-centered displaced orbits through the use of a properly sized and properly oriented solar sail, as shown in Figure 7.5. The sail area and sail attitude angles necessary to obtain a displaced orbit characterized by a given radius projected on the HIF (XHIF ,YHIF )-plane, ρ, a given distance from the equatorial plane, z, and a given orbital angular 7.1. Verification of the orbital mechanics propagation block 55

×10 4 4

0 [km] J Z -2 4 2 0 -4 × 5 -2 10 4 2 0 -2 -4 × 5 -4 -6 X [km] 10 Y [km] J J

Figure 7.2: Orbit of the Moon over one year in the J2000 reference frame

velocity, ω∗, can be computed as follows:

{(z/ρ)2 + [1 − (ω∗/ω˜)2]2}3/2 β = [1 + (z/ρ)2]1/2 (7.1) {(z/ρ)2 + [1 − (ω∗/ω˜)2]}2 p 3 ω˜ = µSun/r (7.2) (z/ρ)(ω∗/ω˜)2 α = tan−1 (7.3) (z/ρ)2 + [1 − (ω∗/ω˜)2] δ = 0 (7.4) A = β · m/σ∗ (7.5) where ω˜ is the orbital angular velocity of a circular orbit with radius r. Note that these equations hold for a two-body problem including the Sun and the spacecraft. When creating the test application, it is therefore necessary to manually position the Sun at the origin of the reference frame, as its ephemerides are computed including other celestial bodies. If this is not done, the position of the Sun would itself orbit around the barycenter of the Solar System. Furthermore, the dynamics of the spacecraft were defined to include only the Sun’s central gravitation attraction and the SRP. Finally, it is clear that, if any of these values is not properly defined or if the SRP model is not properly implemented within the dynamics propagation routine, the obtained displaced orbit will not be stable and the resulting trajectory will not resemble a circular orbit if sufficiently long integration times are considered. The target displaced orbit was arbitrarily set to be characterized by the following parameters:

z = 0.5 AU ρ = 0.5 AU ω∗ = 1 rev/year

Substituting these values into Eqs. 7.1-7.5, the following solar sail attitude angles and sail loading are obtained: α = 12.12 deg δ = 0 deg σ = 573.56 m2/kg The trajectory obtained by propagating the spacecraft dynamics with the previously provided parameters for a period of two years is shown in Figure 7.6. As can be seen, the result is the expected circular orbit at a distance of 0.5 AU from the z-axis of the heliocentric inertial reference frame hovering at the same distance from the equatorial plane. The difference between the initial an ﬁnal position due to integration errors was in the order of a kilometer, which was regarded as acceptable. It can therefore be assumed that the SRP model was properly integrated within the dynamics propagation block. 56 7. Code veriﬁcation and validation

0.9

0.8

0.7

0.6

0.5

0.4 Normalized force 0.3

0.2

0.1

0 0 20 40 60 80 Cone angle [deg]

Figure 7.3: Computed normalized SRP induced force (on the left) and reference normalized SRP induced force [17] (on the right) as a function of the cone angle

2 ] 2

N/m 0 µ [ l F -2

-4 0 2 4 6 8 10 F [ µN/m 2 ] s

Figure 7.4: Ideal sail ’force bubble’ computed for a distance of 1 AU from the Sun (on the left) and reference benchmark plot [38] (on the right)

7.1.4. Cone and clock angle profiles As mentioned in Section 5.3, the attitude angles of the sails are computed through the use of an analytical representation. It was therefore necessary to verify that the decision variables were properly processed in order to obtain the correct value for the sail cone or clock angle at every time-step of the simulation. For the sinusoidal and square wave function representations, the verification was performed by computing the resulting sail attitude angles for a duration of five revolutions for a determined set of decision variables and by verifying that the resulting output matched the expected result. The input decision variables are provided in Table 7.1 and the resulting sinusoidal and square wave profiles are shown in Figure 7.7. Instead, the constant angle representation was not tested as in that case the value is passed directly to the SRP model without being processed.

Amplitude [deg] Phase [deg] Offset [deg] 60 45 -20

Table 7.1: Input decision variables for the attitude angle proﬁles generation testing 7.2. Veriﬁcation of the optimization 57

Figure 7.5: Geometry of a halo orbit [17]

0.5

0.4

0.3

0.2 z [AU]

0.1 -0.5

0 0.5 0

0 y [AU] x [AU] -0.5 0.5

Figure 7.6: Halo orbit obtained for the testing of the SRP model and its integration within the dynamics propagation

7.2. Veriﬁcation of the optimization Similarly to the physical models and propagation routines, also the optimization process software needs to be veriﬁed. In this section the tests that were performed in order to ensure that the functionalities of the optimization routine were properly implemented will be discussed.

7.2.1. Seed setting However trivial, it was necessary to ensure that the bound matrix for the decision variables was properly created in order for the optimization to be performed correctly. By outputting the values of the bound vector it was veriﬁed that its number of lines is the same as the number of decision variables that is being considered and that the values it contains are the ones that were stated during the creation of the problem class C++ object.

7.2.2. Optimization routine Testing the optimizer with the same objective functions as the ones defined in Section 5.2 was problematic, as no data for similar cases is available in the existing literature. It was nevertheless possible to test the proper interfacing of the optimizer with the dynamics propagation. For this purpose, two different verification tests were performed. The first verification that was performed is based on a 58 7. Code verification and validation

40 40

20 20

0 0

-20 -20

-40 -40 Angle [deg] Angle [deg]

-60 -60

-80 -80 0 1 2 3 4 5 0 1 2 3 4 5 Time [revolutions] Time [revolutions]

Figure 7.7: Sinusoidal and Square wave function profiles resulting from the verification set-up similar to the one described in Chapters4 and5, including a satellite orbiting in a perturbed near-geostationary orbit with sail attitude angle profiles represented with the use of patched sinusoidal functions, characterized by the amplitude, phase and offset. Each of these parameters is bounded to an interval ranging from 0 to 360 deg. However, instead of the objective function defined in Section 5.2, the objective function now only includes the total required control effort and is defined as follows:

t=tend 2 2 2 2 X d αA d δA d αP d δP f˜ = + + + (7.6) dt2 dt2 dt2 dt2 t=t0 t t t t which, if computed for a large enough number of sinusoidal periods so that the integrated value of the absolute sinusoidal function tends to 1/2, can be approximated as:

f˜ ' A ω2 + A ω2 + A ω2 + A ω2 αA αA δA δA αP αP δP δP where

d2α d2δ • 2 and 2 are the second time derivatives of the sail attitude angles dt dt • The subscripts A and P refer to the satellites Athos and Porthos, respectively

In case the sinusoidal functions representing these angles is set constant for every optimization run, the expected resulting optimal set of parameters is then only characterized by amplitudes equal to 0 ± 2n∗π with n∗ a positive integer. For these optimal (constant) amplitudes, the remaining parameters do not inﬂuence the value of the objective function and therefore will assume random values between 0 and 360 deg. The optimization was run for 100 generations, a population of 40 individuals and a sinusoidal frequency equal to 6π/T for both the cone and clock angle, T being the orbital period of the spacecraft around the Earth. The resulting optimal parameters are shown in Table 7.2 and, as can be seen, the obtained results approximatively match the predicted value. It is fair to assume that more accurate values could be obtained with optimizations including larger populations and more generations.

Athos Porthos Cone angle [deg] Clock Angle [deg] Cone angle [deg] Clock Angle [deg] Amplitude A [deg] 0.6637 0.8459 1.1211 1.1951 Phase φ [deg] 0.9642 132.7253 63.1175 301.7109 Offset B [deg] 94.1603 16.5737 300.6130 157.6065

Table 7.2: Sinusoidal coefficients resulting from the first optimization verification test 7.2. Verification of the optimization 59

A second test was conducted in order to verify the proper interfacing between the optimizer and the orbital dynamics propagation. In this case, a test was set-up in order to obtain the solar sail attitude profile necessary to obtain the same displaced orbit above the Sun that was described in Section 7.1.3 and shown in Figure 7.3. The dynamics propagation set-up is the same as the one presented in Section 7.1.3, except for the sail attitude angles which are not given as constant inputs but are instead represented with decision variables that will be optimized. The optimizer will be set-up to find the set of decision variables that result in an orbit that best fits a given displaced halo orbit. For this purpose, the objective function is therefore defined as follows:

t=tend X p 2 2 f = |ZHIF (t) − zH | + | XHIF (t) + YHIF (t) − ρH | (7.7)

t=t0 where

• XHIF (t), YHIF (t) and ZHIF (t) are the coordinates of the propagated body at time t

• zH , ρH are the geometrical parameters of the target halo orbit This objective function was chosen as it resembles the one described in Section 5.2, in the way that it defines a nominal target orbit that the spacecraft is supposed to follow and penalizes the solution proportionally to how large the deviation from said nominal orbit is along the dynamics propagation. A first verification test was performed by using a patched sinusoidal representation for the sail cone and clock angles. In this case, in order to provide the correct constant values for α and δ, the optimized offsets, Bα and Bδ, are expected to be equal to 12.12 deg and 0 deg, respectively. The expected amplitudes, Aα and Aδ, are expected to be equal to zero, and, finally the optimal phases, φα and φδ, should not influence the resulting orbital motion. The sinusoidal functions frequencies ωα and ωδ are 2π set as a constant input with a value of T , with T the orbital period of the spacecraft around the Sun (equal to a year). The resulting cone and clock angle parameters are shown in Table 7.3. Again, the obtained results seem coherent with the expected values, although they do not precisely match the expected values because of the number of parameters to be optimized and the limited run-time, equal to a year. For thoroughness, the same code verification was run again, this time using the constant representation for the solar sail attitude angles. The results for this second test are provided in Table 7.4 and it can be seen that, by using this simpler angle profile representation, the benchmark values are reproduced with a better accuracy.

Cone angle [deg] Clock Angle [deg] Amplitude A [deg] 2.3363 0.8459 Phase φ [deg] 257.9642 62.7253 Offset B [deg] 14.1603 0.8737

Table 7.3: Sinusoidal coefﬁcients resulting from the second optimization veriﬁcation test

Cone angle [deg] Clock angle [deg] 12.1195 0.0003

Table 7.4: Constant representation cone and clock angles resulting from the second optimization veriﬁcation test

Preliminary results

In Chapters4,5 and6, the application developed for the thesis has been described. In those chapters it was shown that the values of a number of optimization process control parameters cannot be determined beforehand and need to be computed through a trial and error process, common sense and experience in the optimization domain. In this chapter, the simulations that were performed in order to test the capabilities of the developed optimization routine and to determine a number of relevant optimization control parameters will be described. First, the capabilities of the solar sail to counteract the perturbations will be investigated through a point-wise direct optimization of the solar sail attitude. Subsequently, the results obtained for short duration heuristic optimizations by using the different sail attitude angles representations described in Chapter5 will be shown and the efﬁciency of a range of control parameters values will be compared. Finally, a summary of the chosen values for the optimization control variables will be provided.

All results presented in this chapter have been produced by using a value of 123456 for the seed, the ’inclination and distance’ objective function deﬁnition described in Section 5.2 and the ’global’ optimization logic described in Section 5.5, which allows to provide a fair comparison between the various optimization set-ups that will be tested. Please note that this approach is of course overly simplistic, as it would be necessary run simulations with several different seed values and different objective function deﬁnitions and optimization logics in order to properly determine the optimal control parameters. How- ever, due to the large computational cost the optimization runs, only a limited number of preliminary tests could be done.

8.1. Solar sail perturbation counteracting capabilities In order to determine the instantaneous profile of the sail cone and clock angles that best counteract the perturbations acting on the satellite on its nominal Keplerian orbit, the following preliminary analysis was performed: the perturbing forces (SRP excluded) acting on an uncontrolled satellite on the same orbit as Athos (see Section 3.3) were computed for a time span of five years. Through a direct optimization process, the sail attitude angles that minimize the total perturbing acceleration acting on the satellite were computed at every time-step. It is important to note that the perturbation that is countered in this case is the one acting on a satellite orbiting on an unperturbed Keplerian orbit while in reality the residual perturbations will gradually modify the satellite’s orbital elements. Furthermore, this preliminary analysis does not result in an optimal control of the formation for debris triangulation, but only in the optimal steering profile of a sail for perturbation counteracting purposes. The direct optimization was performed using the fmincon.m function of Matlab®, which is a gradient-based non- linear programming optimization tool that finds the minimum of a function f˜(x) subject to the following constraints:

61 62 8. Preliminary results

c˜(x) ≤ 0 ceq(x) = 0 B˜ x ≤ b Aeq x = beq lb ≤x ≤ ub where

• x is a vector of dimension N˜ that represents the argument of the objective function

• c˜(x) and ceq(x) are non-linear constraint functions

• b, beq are linear constraint vectors of dimension equal or larger than N˜

• B˜ and Aeq are the linear constraints matrices characterized by N˜ columns and the same number of rows as b and beq, respectively

• lb and ub are the lower and upper bound for the objective function argument

For this speciﬁc optimization, x is a vector containing the value of the cone and clock angle at the speciﬁc time t. Given that no non-linear constraints are involved, c˜(x), ceq(x) are set constant to 0. Furthermore, no linear equality constraint is involved either and therefore Aeq(x) is set equal to a null matrix:

α(t) x = δ(t) c˜(x) = 0 ceq(x) = 0 0 0 Aeq(x) = 0 0

In order to limit the value of the cone angle between −90 and 90 deg, while keeping the clock angle unbound, B˜ and b are deﬁned as:

−1 0 B˜ = 1 0 π 2 b = π 2 Furthermore, in order to constrain the obtained optimal sail attitude proﬁle to be semi-continuous, the cone and clock angles are allowed to change in value by ∆ = 2 deg between subsequent time-steps. The boundaries on the decision vector at iteration m, lbm and ubm are therefore deﬁned as: αm−1 − ∆ lbm = βm−1 − ∆ αm−1 + ∆ ubm = βm−1 + ∆

Finally, the initial guess at every time-step is deﬁned to be equal to the optimal solution at the previous time-step:

x0,m = xm−1 The initial guess for the ﬁrst time-step was determined through a trial and error process and is equal to: 8.1. Solar sail perturbation counteracting capabilities 63

T x0,m = [10, 30] As an estimate of the optimal sail lightness number β for this particular application is not available, the direct optimization was therefore performed for sail areas of 1, 5, 10, 15 and 20 m2 and a constant satellite mass of 3 kg which corresponds to lightness numbers of 0.51 · 10−4, 2.5 · 10−3, 0.51 · 10−3, 7.6·10−3 and 1.02·10−2. The results of the optimizations are shown in Figure 8.1 and Table 8.1. Figure 8.1 provides the perturbing acceleration (dotted black line) and the residual accelerations for using the aforementioned sail areas. From this ﬁgure it is not easy to determine which is the sail area that best performs for the problem at hand: it can be seen that a sail area of 5 m2 counteracts the perturbing forces at certain epochs of the revolution, while at other times it performs worse than other sail sizes and generates additional perturbations. This is caused by the search space being very limited by the constraint on the allowed change in the cone and clock angles between time-steps and by the fact that gradient-based optimization methods can converge to local minima if the initial guess is not properly set. In order to provide the reader with a clearer idea of the actual performances of each sail area, the residual accelerations averaged over the entire time span are provided in Table 8.1.

×10 -5 1.5 Perturbing acceleration Residual acceleration A = 1 m2 Residual acceleration A = 5 m2 Residual acceleration A = 10 m2 Residual acceleration A = 15 m2 Residual acceleration A = 20 m2

1 /s] 2 acceleration [m

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [days]

Figure 8.1: Residual perturbing accelerations for different sail sizes and a constant satellite mass of 3 kg

Sail size [m2] 1 5 10 15 20 Average 7.5668 · 10−6 6.990 · 10−6 7.708 · 10−6 8.014 · 10−6 8.242 · 10−6 residual acceleration [m/s2]

Table 8.1: Average residual accelerations for different sail sizes and a constant satellite mass of 3 kg

When considering the average residual acceleration rather than the performance at certain epochs, the 5 m2 sail yields the best result in terms of global perturbation mitigation. For this value of the sail area, the resulting sail cone and clock angle values as a function of time are shown in Figure 8.2. In this plot it can be seen that, apart from an initial transition due to the initial guess, the two angle profiles seem to repeat periodically every revolution. While the clock angle alternates smoothly between 50 and 120 deg once per revolution, approximatively, the clock angle presents a minimum value between 5 and 10 deg and two triangle shaped spikes reaching 50 deg every revolution. As mentioned in Section 5.3, 64 8. Preliminary results the sail attitude angle profiles need to be represented by an analytical expression and with a limited number of defining parameters in order to efficiently perform the optimization. Figure 8.2 provides an idea of the potentially optimal shape of the cone and clock angle profiles. From the particular shape of the angle profiles, it can be deduced that no simple analytically defined function will perfectly represent the sail attitude angles. It was therefore decided to focus on the representations characterized by a limited number of parameters such as the constant value, low order polynomial function, square wave function and sinusoidal function, which were previously described in Section 5.3.

250 Cone angle Cock angle 200

150

100 angles [deg] 50

0 0 0.5 1 1.5 2 time [days]

Figure 8.2: Cone and clock angles optimized in order to minimize the perturbing acceleration

8.2. Assessment of low order polynomial representation One of the sail attitude angle representations described in Section 5.3 was the polynomial representation. During initial investigations, the performance of this representation was tested using a third order polynomial. An optimization for a formation of two satellites using a lightness number of 0.0026 (5 m2 sail area) and a population of 20 individuals was run for 100 generations. The obtained optimal polynomial coefficients for the attitude angles were all tending to zero except for the zero and first order coefficients, which represent cone and clock angle profiles linearly increasing at different rates. The optimal coefficients produced by the optimization run that resulted in the lowest value of the objective function are provided in Table 8.2.

Polynomial coefﬁcient order 0 1 2 3 Value 32.7 rad 1.13 · 10−2 rad/s 1.86 · 10−6 rad/s2 −4.26 · 10−7 rad/s3

Table 8.2: Optimal polynomial coefﬁcients

For thoroughness, also the second and fourth degree polynomial representations were tested and both yielded the same kind of linearly increasing profiles as optimal solutions. Although the results are not incoherent nor physically inconsistent, the resulting profiles are very different from the ones expected from the preliminary analysis presented in Section 8.1 and in Figure 8.2. Furthermore, the resulting profiles do not present the periodicity that is to be expected from the solution of the problem at hand (proportional to the satellites revolution period with an integer or rational proportionality coefficient). Finally, it is obvious that the polynomial representation is not optimal as the optimizer would be working on several non necessary parameters characterized by an optimal value equal to zero and would therefore result highly inefficient. For these reasons, it was decided to discard the patched polynomial and to focus on alternative attitude angle representations.

8.3. Sinusoidal representation optimal control parameters In this section, the simulations performed in order to determine optimal values for some of the optimization control parameters will be discussed. The control parameters that will be determined are the 8.3. Sinusoidal representation optimal control parameters 65 sinusoidal function frequency, the sail size and the population size.

8.3.1. Optimal frequency As mentioned in Section 5.3.3, in order to limit the number of parameters to be optimized, the same frequency is used for both the cone and clock angle proﬁles of each satellite of the formation. It was already demonstrated that, by imposing a continuity condition in the patching between two subsequent repetitions of the sinusoidal function, its frequency can be expressed as 2π ω = k (8.1) α α T 2π ω = k (8.2) δ δ T where kα and kδ are rational numbers. For synthesis purposes, from here on the same notation will be used for both rational numbers: kα = kδ = k. In order to determine the optimal value of k, a number of optimizations was performed, each one characterized by 20 individuals, 100 generations and the same seed value, but different values of k. For what concerns the sail lightness numbers, the value yielding to the smallest residual acceleration in Section 8.1 was used: β = 0.0026. The resulting evolution of the objective function values as a function of the generation number and for different sinusoidal frequencies is shown in Figure 8.3. The results obtained for several of the different values of k are not shown in the provided plot, as the resulting objective function values appeared to be much larger than the ones shown in Figure 8.3, thereby rendering the ﬁgure less clear to read. More precisely, the values of k discarded in this plot are 1/10, 1/5, 1/4, 1/2 and 1.

0.55 k = 0 k = 2 k = 3 0.54 k = 4 k = 5 k = 6 k = 7 0.53 k = 1/3

0.52

0.51

0.5

Objective function value 0.49

0.48

0.47

0.46 10 20 30 40 50 60 70 80 90 100 Generation

Figure 8.3: Evolution of the objective function for different frequencies

As can be seen from Figure 8.3, the minimum values of the objective function after 100 generations is obtained for a frequency factor k equal to 5, followed by the ones obtained for k equal to 4. How- ever, depending on the amplitude of the sinusoidal function that the optimization process will yield, a frequency of k = 4 or k = 5 could require a fairly large control effort, which would be expensive from an energy perspective and could put considerable stress on the sail structure as well. The results obtained with k = 1/3 are slightly worse than those for k = 4 and k = 5, but the resulting angular acceleration acting on the satellite would be much smaller. For this reason it was therefore decided to use k = 1/3 for all cases in which the sail attitude angles are represented by a sinusoidal function. The resulting sinusoidal frequencies are therefore equal to: 1 2π ω = ω = ' 2.4752 · 10−5 rad/s α δ 3 T 66 8. Preliminary results

8.3.2. Optimal sail size As mentioned in the introduction, the sail lightness number needs to be calibrated as well in order to determine the value that best ﬁts the given problem. The preliminary analysis presented in Section 3.4 showed that, based on existing and near-term technology, the lightness number of the satellites should be in the range 5 · 10−4 to 2 · 10−2. Assuming a 3 kg spacecraft, for example, this results in a sail size ranging from 1 to 40 m2, approximatively. A number of optimizations was then conducted, each one characterized by 20 individuals, 100 generations, a frequency factor k = 1/3, but different values of the sail area. The results are shown in Figure 8.4 and also in this case, for clarity purposes only the best results are provided, as for sail sizes smaller than 4 m2 and larger than 13 m2, the obtained objective function values are signiﬁcantly larger than the ones presented in Figure 8.4.

0.56 A = 4 m 2 A = 5 m 2 A = 6 m 2 A = 7 m 2 0.55 A = 8 m 2 A = 9 m 2 A = 10 m 2 A = 11 m 2 0.54 A = 12 m 2 A = 13 m 2

0.53

0.52 Objective function value

0.51

0.5

0.49

10 20 30 40 50 60 70 80 90 100 Generation

Figure 8.4: Evolution of the objective function for different sail sizes and a constant satellite mass of 3 kg

It can be seen that, in correspondence with the results described in Section 8.1, the smallest, ﬁnal value of the objective function is obtained with a solar sail surface of 5 m2. This resulting sail area value will therefore be considered to be the optimal one and will be used throughout the remainder of this thesis:

σ∗ 5 β = = σ∗ ' 0.0026 σ 3 · 103

8.3.3. Optimal population size Finally, in order to determine the optimal population size for the problem at hand, a process similar to the ones presented in Sections 8.3.1 and 8.3.2 was used. In this case, each simulation was characterized by the previously determined optimal values for the sinusoidal frequency and sail size. The results for population sizes of 10, 20, 30, 40, and 50 individuals are shown in Figure 8.5.

In order to determine the optimal population size for this speciﬁc problem, it should be ﬁrst taken into account that the computational time increases linearly with the value of the population size and therefore, a population larger than needed will have a negative impact on the optimization process performances. However, in this case, it is shown that it is not the largest population that yields the best results. This is of course due to the random nature of the differential evolution optimization algorithm, but it also shows that a sort of plateau in the search for the optimal parameter set seems to be reached. A population of 40 individuals is thereby chosen as it represents a good compromise between computation run-time and result accuracy. 8.4. Constant and square wave representations optimal control parameters 67

Population size = 10 0.55 Population size = 20 Population size = 30 Population size = 40 0.545 Population size = 50

0.54

0.535

0.53

0.525

0.52 Objective function value

0.515

0.51

0.505

0.5 10 20 30 40 50 60 70 80 90 100 Generation

Figure 8.5: Evolution of the objective function for different population sizes

8.4. Constant and square wave representations optimal control parameters In order to be able to more accurately compare the results obtained by using different attitude proﬁle representations, it was decided to use the same optimal values of the lightness number and population size provided in the previous section for the constant and square wave representations as well. Fur- thermore, given the similarity between the square wave and the sinusoidal representations, it will also be assumed that the optimal frequency is the same for both representations. Therefore in both cases 2π the used frequency will be equal to a third of the orbital frequency: ωα = ωδ = 1/3 T .

An overview of the simulation parameters that follow from the analyses discussed in this chapter are provided in Table 8.3.

Control Parameter Constant representation Sinusoidal and square wave representation 1 2π Sinusoidal frequency N/A 3 T rad/s Sail size 5 m2 Lightness number 0.0026 Population size 40 individuals

Table 8.3: Optimal simulation and optimization control parameters

Results

In Chapter8, optimizations characterized by the possible combinations of optimization logics and sail attitude angle representations were discussed. This allowed to properly estimate the capabilities of both the developed application and the studied concept. It was also possible to determine suitable values for some of the control parameters for the analyzed optimization set-ups. In this chapter, the results produced with these optimized set-ups will be shown and their performance will be discussed and compared. The relevant conclusions will then be drawn. In the ﬁrst section, the results obtained by using the ’global’ optimization logic are presented, while in the second section the results obtained with the ’revolution-by-revolution’ optimization logic will be shown. Subsequently, the various simulations are then compared and the best optimization set-ups are discussed. Finally, a number of additional simulations including further constraints and requirements will be presented and analyzed.

For simplicity and clarity reasons, for each considered set-up only the optimization that yields the best ﬁnal value of the objective function will be discussed. Every optimization run presented in this chapter is characterized by the relevant control parameters that were determined in Chapter8, see Table 8.3.

9.1. ’Global’ optimization logic In this section, the results obtained by using the so-called ’global’ optimization logic are discussed. According to this optimization logic, presented in Chapter5, the decision variables defining the solar sail attitude angle values for the entire simulation duration are optimized simultaneously. The objective function value is therefore evaluated over the entire state vector history. This simulation is performed for one year and considers three attitude angle profile changes. The optimal cases obtained for both definitions of the objective function will be presented.

9.1.1. ’Inclination and distance’ objective function In Figures 9.1 to 9.6 the results obtained by using the ’global’ optimization logic and the ’inclination and distance’ objective function definition are presented. Figures 9.1, 9.3 and 9.5 show the evolution of the instantaneous inclination of the formation center on the top-left plot, the evolution of the inter- satellite distance on the bottom-left plot and the evolution of the distances between the satellites and the observed debris on the plots on the right. Figures 9.2, 9.4 and 9.6 show the evolution of the debris observation angles (left) and the sail attitude angles (right). These plots are obtained for the constant, square wave and sinusoidal representations of the solar sail attitude profiles, respectively. In the presented figures, some recurring phenomena can be noticed, such as the presence of periods in which the inter-satellite distance reaches small values smaller than the minimum allowed value defined in Section 3.2. This obviously also results in small debris observation angles. In Figure 9.1 this can be seen around revolution number 180, while in Figures 9.3 and 9.5 this happens around revolution number 70. It is also interesting to note a particularly low value in the distance between Porthos and the observed debris around revolution number 80, which can be seen in Figure 9.3 and a particularly low inter-satellite distance during the last revolutions of Figure 9.3. Finally, in Figure 9.2 it can be seen that when using the constant representation for the solar sail attitude angles, the back-side of the solar

69 70 9. Results sail is oriented towards the Sun during most of the simulation, which results in a cone angle between 90 to 270 deg. Figures 9.4 and 9.6, instead, show cone angles characterized by varying amplitudes that result in both reﬂective and absorbing sides of the sail being oriented towards the Sun.

For clarity purposes, the plots of Figures 9.3 and 9.4 are presented on a more limited time scales in Figures

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Figure 9.1: Constant attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distance between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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Debris observation angle [deg] 20 0 200 0 200 Porthos cone [deg] 0 100 200 300 Time [Revolutions] Porthos clock [deg] Time [Revolutions] Time [Orbital revolutions]

Figure 9.2: Constant attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and solar sail attitude angles (right).

9.1.2. ’Observation angle’ objective function In Figures 9.9 to 9.14, the results obtained by using the ’global’ optimization logic and ’observation angle’ objective function definition are presented. The results are presented in exactly the same manner as for the ’inclination and distance’ objective function in Figures 9.1 to 9.6. It can seen be from the presented figures that the general trends of the plotted variables are very similar to the ones presented in Section 9.1.1, such as the presence of periods in which the inter-satellite distance reaches values smaller than the minimum allowed value defined in Section 3.2 thus resulting in small debris observation angles. However, a noticeable feature can be seen in Figure 9.13, where it is shown that the amplitude of the inter-satellite distance variation during the revolutions is considerably smaller than the ones shown in the previous plots. However, around revolution number 290 this trend stops and the inter-satellite distance ends up violating the minimum value set by the requirements in the last month 9.1. ’Global’ optimization logic 71

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Figure 9.3: Square wave attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distance between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 400 600 Computed angle Optimal angle 200 400 100 0 200 -200 0 80 -400 -200 0 200 0 200 Athos cone [deg] Athos clock [deg] 60 Time [Revolutions] Time [Revolutions] 600 500 40 400 200 0 20 0 -200 -500 Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.4: Square wave attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. of the simulation. Furthermore, a particularly low inter-satellite distance can be noticed during the last revolutions of Figure 9.9. Finally, similarly to the ’inclination and distance’ objective function deﬁnition, in Figure 9.10 it can be seen that when using the constant representation for the solar sail attitude angles, the back-side of the solar sail is oriented towards the Sun during the entirety of the simulation, which results in a cone angle between 90 to 270 deg. Figures 9.12 and 9.14, instead, show cone angles characterized by varying amplitudes that result in both reﬂective and absorbing sides of the sail being oriented towards the Sun.

By analyzing the results obtained with the ’global’ optimization logic, it can be seen that not all requirements defined in Section 3.2 are met. In fact, the requirement regarding the maximum and minimum allowed inter-satellite distance is not met for any of the attitude angle representations and objective function definitions. As can be expected, in the parts of the orbital evolution in which the inter-satellite distance is smaller than the lower allowed limit, a considerably small debris observation angle is obtained, reaching minimum values of 2 deg. However, the requirements regarding the minimum allowed orbital inclination of the formation center are met with all optimization set-ups presented in this section. It is nevertheless very clear that for slightly longer simulations the orbital perturbations will most probably make the formation drift further away from the equatorial plane so that the requirements regarding the inclination will not be met as well. It can also be seen that the two objective function definitions 72 9. Results

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Figure 9.5: Sinusoidal attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 500 500 Computed angle Optimal angle 100 0 0

80 -500 -500 0 200 0 200 Athos cone [deg] Athos clock [deg] 60 Time [Revolutions] Time [Revolutions] 500 400 40 200 0 20 0 -500 -200 Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.6: Sinusoidal attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. perform similarly for all the considered performance parameters. A noticeable difference is the sinusoidal representation for which the two different objective functions yield fairly different results for what concerns the inter-satellite distance and the debris observation angle.

9.2. ’Revolution-by-revolution’ optimization logic In this section, the results obtained by using the so-called ’revolution-by-revolution’ optimization logic are discussed. According to this optimization logic, presented in Chapter5, the decision variables deﬁning the solar sail attitude angles values are optimized separately for each revolution. The objective function value is therefore evaluated over a single revolution at the time and its optimal result is used as input for the optimization of the subsequent orbital revolution. This simulation is performed for one year. The optimal cases obtained for both deﬁnitions of the objective function will be presented.

9.2.1. ’Inclination and distance’ objective function In Figures 9.15 to 9.20, the results obtained by using the ’revolution-by-revolution’ optimization logic and the ’inclination and distance’ objective function deﬁnition are presented. The results are presented in exactly the same manner as for the ’global’ optimization logic and the ’inclination and distance’ objective function in Figures 9.1 to 9.6. From the presented ﬁgures, it can be seen that the results 9.2. ’Revolution-by-revolution’ optimization logic 73

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Figure 9.7: Square wave attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distance between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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Figure 9.8: Square wave attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle. obtained for this optimization set-up are considerably different from the ones presented in Sections 9.1.1 and 9.1.2. In Figure 9.15 it is shown that, when using the constant representation for the solar sail attitude angles, the inter-satellite distance varies relatively little around its maximum allowed value and slightly exceeds it between revolution numbers 80 and 260. Instead, in Figures 9.17 and 9.19, the inter-satellite distances behave much more poorly and, in both cases, a clear divergence from the target values can be noticed in the last few months of the optimization. In Figure 9.16 it is worth noticing that for the optimization using the constant attitude representation, the debris observation angle never reaches very low values, as was observed for the cases described in Sections 9.1.1 and 9.1.2. Furthermore, when using this optimization logic, both the reﬂective and the absorbing sides of the sail are alternatively oriented towards the Sun, with an alternation frequency depending on the representation that is being used Finally, in Figures 9.17 and 9.19 it can also be noticed that the instantaneous inclination values for these two optimization set-ups exceed the maximum limit values deﬁned by the requirements during the very last revolutions of the simulation.

9.2.2. ’Observation angle’ objective function In Figures 9.21 to 9.26, the results obtained by using the ’revolution-by-revolution’ optimization logic and the ’observation angle’ objective function deﬁnition are presented. The results are presented in exactly the same manner as for the ’global’ optimization logic and the ’inclination and distance’ objective function in Figures 9.1 to 9.6. The results provided for this optimization set-up are considerably 74 9. Results

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Figure 9.9: Constant attitude representation, ’global’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 250 400 Computed angle Optimal angle 200 300 100 150 200 80 100 100 0 200 0 200 Athos cone [deg] Athos clock [deg] 60 Time [Revolutions] Time [Revolutions] 240 40 220 300 200 20 180 200 160 Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.10: Constant attitude representation, ’global’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). similar to the ones presented in Section 9.2.1 and therefore, in order to avoid redundant information, they will not be described in detail. However, it is interesting to discuss the similarity in results between the two objective function definitions. It is the opinion of the author that over a single revolution, the influence of the inter-satellite distance term of the ’inclination and distance’ objective function is larger than the inclination one. By consequence the optimizer tends to find solutions that best control the inter-satellite distance rather than the inclination, yielding therefore results very similar to the ones that are obtained with the ’observation angle’ objective function that focuses solely on inter-satellite and debris to satellite distances.

Similarly to what was described in the previous section, when using the ’revolution-by-revolution’ optimization logic, some of the requirements defined in Section 3.2 are not met. In this case as well, the requirement regarding the maximum and minimum allowed inter-satellite distance is not met for any of the attitude angle representations and objective function definitions that were implemented. In Figures 9.15 and 9.21, it can be interesting to note that, when using the constant attitude representation, the resulting orbital evolution of the formation is very similar to the one that is obtained when solar sailing is not used (see Figure 4.4). For what regards the sinusoidal and square wave function representations, a notable behaviour is the divergence of the measured quantities after the 270th revolution (approximatively), as can be seen in Figures 9.17, 9.19, 9.23 and 9.25. This could be explained by the fact that, during the last part of the simulation, the specific relative position between the Sun, the Earth 9.3. Results comparison 75

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Figure 9.11: Square wave attitude representation, ’global’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 Computed angle 400 600 Optimal angle 200 400 100 200 0 0 80 -200 -200 0 200 0 200 Athos cone [deg] Athos clock [deg] 60 Time [Revolutions] Time [Revolutions] 500 600 40 400 0 200 20 0 -500 -200 Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Time [Revolutions] Porthos clock [deg] Time [Revolutions] Time [Orbital revolutions]

Figure 9.12: Square wave attitude representation, ’global’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. and the satellites makes the counteracting of the perturbations and the optimal control of the formation unattainable by the developed optimizer. More speciﬁcally, as this occurs during the winter period, this could be the result of both the Sun and the satellites being at a large inclinations with respect to the celestial equator, resulting in larger accelerations perpendicular to the orbital plane. This behaviour was not present in the results obtained with the ’global’ optimization logic, as by optimizing the formation control over the entire duration it is possible for the optimizer to preventively compensate for this phenomenon.

9.3. Results comparison In order to compare the performances of the different set-ups used to optimize the formation control, the best resulting objective function values obtained with the various considered sail attitude representations are provided in Table 9.1 and Table 9.2 for the ’global’ and ’revolution-by-revolution’ optimization logics, respectively. It can be seen that the results obtained with different optimization set-ups vary considerably in terms of the obtained objective function values. It is also interesting to note that the best results for the two different considered objective function definitions are obtained with different optimization set-ups. For the ’inclination and distance’ objective function definition, the best results are obtained with the solar sail attitude profiles defined as square waves and changed every three months. 76 9. Results

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Figure 9.13: Sinusoidal attitude representation, ’global’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 Computed angle 500 500 Optimal angle 100 0 0

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0 100 200 300 Porthos clock [deg] Time [Revolutions] Time [Revolutions] Time [Orbital revolutions]

Figure 9.14: Sinusoidal attitude representation, ’global’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values.

Instead, when using the ’observation angle’ objective function deﬁnition, the optimal results are produced by a constant attitude changed at every orbital revolution. When compared to the evolution of the satellites orbits in the case where no control is performed, it is the three months long square wave function optimized with the ’inclination and distance’ objective function deﬁnition that, overall, yields the best results. It is therefore with this optimization set-up that the additional optimizations presented in the following section will be performed.

For thoroughness purposes, the evolution of the objective function value as a function of the generation is provided in Figure 9.27 for the best optimization obtained with the square wave attitude representation, the ’global’ optimization logic and the ’inclination and distance’ objective function. It can be seen that after approximatively 50 generations, a plateau is reached and the objective function stops de- creasing. It is reasonable to speculate that the objective function has reached a local minimum value and that due to the limited number of individuals in the population, the optimizer is not able to search the search space for better solutions before the maximum number of generations is reached. 9.4. Additional constraints or requirements 77

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Figure 9.15: Constant attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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Figure 9.16: Constant attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and solar sail attitude angles (right).

9.4. Additional constraints or requirements The results obtained in Sections 9.1 and 9.2 provided an overview of the capabilities of the developed application with as few constraints as possible (i.e., only the bounds on the decision variables). It is, however, necessary to test the performance of the optimization routine in case some additional constraints or requirements are added, resulting in a more relevant or realistic application. In this section, the results obtained for a number of additional simulations will be discussed.

9.4.1. Three satellite formation As mentioned in Chapter3, the triangulation of space debris can be made more accurate and reliable if the formation comprises three satellites rather than two. A number of optimizations including three spacecraft was therefore performed. The resulting orbital and geometrical parameters are shown in Figures 9.29 and 9.30. As can be expected, the resulting orbital and geometrical behaviours are less optimal than the ones obtained in Sections 9.1 and 9.2. This is of course due to the fact that adding a satellite to the optimization results in an increased number of parameters to be optimized (from 48 to 72 in this speciﬁc case), which results in a less efﬁcient optimization process. Furthermore, the objective function also becomes more complex to satisfy, as ulterior inter-satellite distance terms are added. It is, however, interesting to note that the three triangulation observation angles never tend to very low values at the same time. Adding a third satellite seems therefore promising for what 78 9. Results

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Figure 9.17: Square wave attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

Computed angle 150 600 600 Optimal angle 400 400 200 200 0 0 -200 -200 100 0 200 0 200 Athos cone [deg] Athos clock [deg] Time [Revolutions] Time [Revolutions]

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Figure 9.18: Square attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. concerns the reliability of the concept in providing constant monitoring of the geostationary belt. In order to better show the aforementioned increased robustness, Figure 9.28 shows the best observation angle obtainable among the three available ones at every time-step. As can be seen, the resulting observation angle proﬁle is considerably more constant than the one obtained with only two satellites and is bounded between 20 and 160 deg, thus never reaching critical values.

9.4.2. Cone angle limitation As explained in Section 4.4, the solar sail is modelled in order to have two sides: an absorbing and a reflective one. In the simulations presented up to this point, the attitude profiles were not bounded so that optimal cone angle values larger than 90 deg could be obtained, thus resulting in the absorbing side of the sail being oriented towards the incoming radiation. In reality, however, if this occurs for prolonged periods of time the sail can heat-up and reach very high temperatures, thus compromising its structural integrity and optical properties. For these reasons it was decided to run a number of optimizations in which the cone angle of the sail is limited to the interval [−90, 90] deg. The resulting orbital and geometrical parameters are shown in Figures 9.31 and 9.32. Due to the definition of the square wave function, limiting the maximum and minimum values of the cone angle through a redefinition of the bounds was not easily feasible. For this reason, the cone angle value limitation was implemented by using a value ’symmetrical’ with respect to 90 deg in case it exceeded the given limit 9.4. Additional constraints or requirements 79

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Figure 9.19: Sinusoidal attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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Figure 9.20: Sinusoidal attitude representation, ’revolution-by-revolution’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and satellites attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. value: α = 180 − α deg if α > 90 (9.1) As can be seen from Figure 9.31, limiting the cone angle (and by extension the control possibilities of the solar sails) strongly impacts the formation control capabilities of the optimizer. The resulting orbits very quickly diverge from the desired optimal solutions as the satellites drift apart. Limiting the sail control so that only its reﬂective side is oriented towards the incoming radiation doesn’t seem a feasible solution when using the developed application. A thermal analysis of the sail temperature proﬁle along its orbit could be useful in order to determine whether critical temperatures are actually reached and whether the addition of a heat dissipation hardware could be enough to solve the problem.

9.4.3. Long duration optimization Up to this point, all optimizations that were discussed were focused on the station keeping of the formation for durations of three months to one year. However, given the rationale of the concept, it is of course necessary to assess the performances of the developed application for longer mission lifetimes as well. For this reason, a number of simulations of the control of the formation for time durations of 5 years were performed. The results produced by the best run are shown in Figures 9.34 and 9.33. As could be expected from the results presented up to this point, the very large number of decision variables to be optimized in this case (240) result in a problem clearly too complex for the capabilities 80 9. Results

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Figure 9.21: Constant attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 600 600 Computed angle 400 400 Optimal angle 200 200 100 0 0 -200 -200 0 200 0 200 Athos cone [deg] Athos clock [deg] 80 Time [Revolutions] Time [Revolutions]

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0 100 200 300 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.22: Constant attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). of the developed application and by consequence the resulting orbits diverge considerably from the optimal debris triangulation conditions, as can be seen for example in Figure 9.33. A possible simple solution could be to consider computationally more expensive simulations (i.e., larger populations and larger number of generations). In case this fails as well, a redeﬁnition of the parametrization of the solar sail attitude and of the optimization logic might be necessary.

9.4.4. Control effort penalty function The optimizations discussed in this thesis so far have taken into account the control effort that is necessary in order to control the sail. It is of course understandable that a small control effort results in a smaller power budget and in smaller physical loads acting on the solar sail. It would be therefore convenient to favour optimal solutions that yield the lowest control effort. For this reason, a number of simulations were performed using the objective function described in Section 5.2 and provided in Eq. 5.9 that, apart from the inclination and inter-satellite distance terms, includes a penalty function which tends to limit the resulting control effort. In order to tune the weight of the penalty function, its value was computed for the simulation presented in Figures 9.3 and 9.4. A value for the weight was then determined so that the penalty function value would be equal to half of the value of the objective function. For a more thorough analysis of the penalized optimization, other weight values were also determined in order to obtain penalty function values equal to a third and two thirds of the objective function value. The results produced by the best run are shown in Figures 9.35 and 9.4. Additional constraints or requirements 81

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Figure 9.23: Square wave attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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Figure 9.24: Square wave attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values.

9.36. Differently than what could be logically expected, the results obtained when including the penalty function are considerably good. It is, in fact, the only set-up among all the ones that were analyzed in this thesis, that complies with the inter-satellite distance requirements, as can be seen from Figure 9.35. Compared to the other solar sail attitude angles plots provided in this section, Figure 9.36 presents cone and clock angle plots characterized by a smaller amplitude, which, in case a constant sinusoidal frequency is used, does indeed result in a smaller control effort. However, it must be noted that the other simulations run with the same ’penalized’ set-up but different seed values and/or different penalty function weights do not yield results that comply with the orbital requirements. It is the opinion of the author that this result could therefore be due to either a particularly ’lucky’ run in which the random nature of the SaDE algorithm played a crucial role in determining the optimal control or to the fact that the addition of a penalty term allows the optimizer not to converge too quickly to local optima. Further analysis of the this optimization set-up should be done in order to properly assess whether the addition of the penalty function improves the optimization performance in a constant fashion. 82 9. Results

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Figure 9.25: Sinusoidal attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

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600 600 50 400 400 200 200 0 0 -200 -200

Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.26: Sinusoidal attitude representation, ’revolution-by-revolution’ optimization logic, ’observation angle’ objective function: debris observation angle (left) and solar sail attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values.

Objective function representation Constant Square wave Sinusoidal Uncontrolled ’Inclination and distance’ 1.0502 1.0433 1.0584 1.1451 ’Observation angle’ 2.0306 · 106 2.0885 · 106 2.0835 · 106 2.1236 · 106

Table 9.1: Objective function values for different optimization set-ups and ’global’ optimization logic

Objective function representation Constant Square wave Sinusoidal Uncontrolled ’Inclination and distance’ 1.2873 1.36134 1.33385 1.1451 ’Observation angle’ 2.0050 · 106 2.5714 · 106 3.1930 · 106 2.1236 · 106

Table 9.2: Objective function values for different optimization set-ups and ’revolution-by-revolution’ optimization logic 9.4. Additional constraints or requirements 83

1.2 Optimized objective function 1.18 Uncontrolled objective function

1.16

1.14

1.12

1.1

objective function value 1.08

1.06

1.04 0 20 40 60 80 100 generation

Figure 9.27: Evolution of the objective function value for the best optimization performed with the square wave attitude representation, the ’global’ optimization logic and the ’inclination and distance’ objective function.

160 Best observation angle Optimal angle 140

120

100

Debris observation angle [deg] 40

20 0 50 100 150 200 250 300 350 Time [Orbital revolutions]

Figure 9.28: Proﬁle obtained by plotting the best observation angle among the three available ones at every time-step. 84 9. Results

Athos 1 2000 Optimal 1000 0 0 0 100 200 300 Time [Orbital revolutions] -1 1000 Porthos

Inclination [deg] 0 100 200 300 500 Time [Orbital revolutions] 0 2000 Athos-Porthos 0 100 200 300 Athos-Aramis Time [Orbital revolutions] Porthos-Aramis 1000 1000 Aramis 500 0

0 Spacecraft-debris distance [km]

Distance [km] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions]

Figure 9.29: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distances (bottom-left) and distances between the satellites and the observed debris (right). For the inter-satellites distances, the green and red lines represent the maximum allowed and minimum allowed values.

Athos-Porthos 150 500 500 Athos-Aramis 0 0 Porthos-Aramis -500 -500 Optimal angle 0 200 0 200 Cone [deg] Clock [deg] 100 Time [Revolutions] Time [Revolutions] 500 500 0 0 -500 -500 0 200 0 200 Cone [deg] Clock [deg] 50 Time [Revolutions] Time [Revolutions] 400 500 200 0 0 -500

Debris observation angle [deg] 0 200 0 200 Cone [deg] 0 Clock [deg] 0 100 200 300 Time [Revolutions] Time [Revolutions] Time [Orbital revolutions]

Figure 9.30: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and satellites attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values.

1 5000 Athos 0 Optimal

-1 0

Inclination [deg] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions] 10000 5000 Porthos 5000 Optimal

0 0 Distance [km] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions]

Figure 9.31: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The cone angles of the satellites are bound from −90 to 90 deg. The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km] 9.4. Additional constraints or requirements 85

500 50 150 0 0 -50 -500 0 200 0 200 Athos cone [deg] 100 Athos clock [deg] Time [Revolutions] Time [Revolutions] 500 50 50 Computed angle 0 0 Optimal angle -50 -500 Debris observation angle [deg] 0 0 200 0 200

0 100 200 300 Porthos cone [deg] Time [Revolutions] Porthos clock [deg] Time [Revolutions] Time [Orbital revolutions]

Figure 9.32: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and satellites attitude angles (right). The cone angles of the satellites are bound from −90 to 90 deg. For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values.

5 Athos 5000 Optimal 0

-5 0

Inclination [deg] 0 500 1000 1500 0 500 1000 1500 Time [Orbital revolutions] Time [Orbital revolutions] 15000 Porthos 5000 Optimal 10000 5000 0 0 Distance [km] 0 500 1000 1500 0 500 1000 1500 Time [Orbital revolutions] Time [Orbital revolutions]

Figure 9.33: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

Computed angle 500 500 Optimal angle

150 0 0

-500 -500 0 1000 0 1000 Athos cone [deg] 100 Athos clock [deg] Time [Revolutions] Time [Revolutions] 500 500 50 0 0

-500 -500 Debris observation angle [deg] 0 0 1000 0 1000

0 500 1000 1500 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.34: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function: debris observation angle (left) and satellites attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. 86 9. Results

1 1000 Athos 0 Optimal 500 -1

Inclination [deg] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions] 800 800

600 600 Porthos Optimal 400 400

200 200 Distance [km] 0 100 200 300 0 100 200 300 Time [Orbital revolutions] Time [Orbital revolutions]

Figure 9.35: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function with penalty function: instantaneous inclination of the formation center (top-left), inter-satellite distance (bottom-left) and distances between the satellites and the observed debris (right). The red, green and yellow lines represent the maximum allowed, minimum allowed and optimal values while the blue line represents the computed value. Spacecraft-debris distance [km]

120 600 500 400 Computed angle 100 Optimal angle 200 0 0 -200 80

Athos cone [deg] 0 200 0 200 Athos clock [deg] Time [Revolutions] Time [Revolutions] 60 600 600 400 400 40 200 200 0 0

Debris observation angle [deg] 20 -200 -200 0 100 200 300 0 200 0 200 Porthos cone [deg] Porthos clock [deg] Time [Orbital revolutions] Time [Revolutions] Time [Revolutions]

Figure 9.36: Square attitude representation, ’global’ optimization logic, ’inclination and distance’ objective function with penalty function: debris observation angle (left) and satellites attitude angles (right). For the attitude angles plot, the red, yellow and blue lines represent the maximum, minimum and average computed values. 10

Conclusions and recommendation

In this thesis, the possibility of controlling a formation of two or three satellites by using solar sails was analyzed. The purpose of the formation control was to maintain a relative position between the spacecraft suitable for the triangulation of the position of space debris in the geostationary belt. In Section 1.3, the research question and relative sub-questions of the thesis were deﬁned. In this chapter the answers that were found during this project will be presented, the relevant conclusions will be drawn and a number of recommendations for future works will be provided.

10.1. Conclusions In this section, the answers to the questions that were presented in Section 1.3 will be provided, starting from the main research question and then discussing each sub-question.

Can solar sails be used to maintain a formation of satellites in an optimal shape under the effect of orbital perturbations in order to track and monitor space debris in geostationary orbit?

By analyzing all results presented in Chapter9, a generic answer to this question is hard to provide. Indeed, one of the simulations performed by including the penalty function limiting the control effort did determine a suitable control law for the satellites that would result in an orbital motion complying with the requirements deﬁned in Section 3.2 for an entire year. However, all the remaining simulations that were run during the thesis produced result which did not comply with part or all of the aforementioned orbital requirements. Furthermore, for several of the solar sail attitude angle representations and control logics that were implemented and tested in this report, the obtained orbital motion results in a larger objective function value (i.e., larger orbital inclination, larger differences between the optimal and computed inter-satellite distances and larger differences between the optimal and computed debris to satellite distances) than the one obtained in the case the satellites are not equipped with a solar sail and are not controlled. In conclusion, solar sails can be used to maintain a formation in an optimal shape for debris triangulation purposes, however, the performance of the developed application should be improved before being able to properly assess the actual capabilities of this mission concept.

Among the two optimization logics developed in this thesis, the ’revolution-by-revolution’ optimization logic, which optimizes the decision variables representing the solar sail attitude considering each single revolution separately, seems to be less suitable for the analyzed problem. With this optimization logic, the inter-satellite distance diverges from its nominal and optimal value during the last 60 revolutions. This phenomenon is most likely linked to the fact that, in this case, the solar sail attitude angles are optimized in order to minimize the objective function one revolution at a time. Therefore, it is possible that the computed optimal solutions yield slightly better results in the short term but worse results in the long term. This phenomenon could also be caused by seasonal variations of the position of the Sun with respect to the satellites. The ’global’ optimization logic, on the other hand, which optimizes the decision variables representing the solar sail attitude by considering the entire mission duration, seems to perform better overall. Indeed, as previously mentioned, most of the tested simulation set-ups do

87 88 10. Conclusions and recommendation not seem to work efﬁciently enough to meet all orbital requirements, but the tendency of the optimizer to maintain the orbital parameters within acceptable distance from their nominal value is noticeable for the ’global’ optimization logic.

How does the solar sail size inﬂuence the performance of the formation control and the resulting debris monitoring capabilities in case of a ﬁxed spacecraft mass?

From an analysis of the performances of the optimizer for a range of solar sail sizes (considering a constant satellite mass) for short duration simulations, it followed that the obtained results vary substantially as a function of the sail size. It is also interesting to note that large sail surface-to-mass ratios do not yield the best results. This is due to the fact that, because of the nature of the propulsion system, in certain points of the orbit the thrust that is generated by the sail will have a larger magnitude than necessary, while in other points of the orbit it will be pointing in the same direction as the sum of the perturbations rather than counteracting them. For this reason, a smaller sail seems to be more effective for the required station keeping purposes.

The outcome of the preliminary analysis was that the sail size most suitable for the studied application was 5 m2 for a 3 kg spacecraft, resulting in a lightness number value of 0.0026. Compared to the values of studied concepts described in the literature, this lightness number is fairly small [17], [19]. This is due to the fact that most of the solar sailing related mission concepts are high energy missions and require thereby large sail surface-to-mass ratios to be efﬁcient, whereas for this particular application the required thrust is in the same order of magnitude as the perturbing forces.

How does the addition of a third satellite to the formation inﬂuence the control performances and control effort?

As can be expected, when adding a third satellite to the formation, the optimizer performs slightly worse in terms of maintaining the formation in its optimal shape and geometry. This can be explained by the increase in the number of decision variables that results in a less efﬁcient optimization routine. It is, however, important to note that the addition of a third satellite to the formation does indeed increase the robustness of the mission concept by providing three triangulation measurements at any given time instead of a single one. Due to this, one of the computed ’observation angles’ always has a value close enough to the value yielding optimal debris triangulation. It can therefore be concluded that the addition of third satellite worsens the performance of the formation control but improves the robustness for what concerns the debris triangulation task.

How does the performance of the developed control logic vary as a function of the considered orbital propagation duration?

Most of the simulations that were performed in this thesis involved a short mission duration (three months and one year). However, behind the implementation of solar sailing for a monitoring mission of this type lies of course the idea of a long duration mission. The possibility of maintaining the formation in optimal debris monitoring conditions for a duration of ﬁve years was not demonstrated. Due to the large increase in the number of decision variables, the performance of the optimizer for this sort of long duration propagations is very poor. By consequence, the orbital parameters diverge considerably from their optimal values.

What analytical representation of the sail attitude angle proﬁles performs better for formation control?

Among the solar sail attitude angle representations that were considered during this thesis (i.e., constant, polynomial, sinusoidal and square wave function), the one that performs best for the control of the formation is the square wave function. It is, however, important to remember that this represen- 10.2. Recommendations 89 tation is of course not realistic, as it can imply considerable instantaneous changes in the cone and clock angles depending on the value of their amplitudes. Nevertheless, it does highlight a tendency of the optimal control to be characterized by attitude angles alternating between two values, depending on the relative position between the satellites, the Earth and the Sun. It was also determined that the polynomial representation performs considerably worse that the other representations that were considered, which can be explained by a more indirect relation between the decision variables and the resulting attitude angle proﬁle.

By analyzing it from a more general perspective, the developed application did not yield the results that were expected. It is, however, not clear to the author whether this is simply due to a partially incorrect approach to the optimization of the control of the solar sails, or whether the concept itself is ﬂawed and solar sailing is not suitable for this kind of collaborative Earth-based monitoring missions.

10.2. Recommendations As mentioned at the beginning of this chapter, the developed software could only meet the mission requirements for one of the simulations that were performed. However, some of the results that are shown in Chapter9 show the potential of the studied concept for the control of the satellite formation. For this reason, it can be hoped that with some improvements and some further developments, the proper and efficient functioning of the mission concept could be demonstrated. A first possible direction of improvement is the exploration of other attitude angle representations that might be more suitable for the optimization process that is being performed. An example of a possible interesting representation that was not analyzed due to a lack of time are orthogonal polynomials. A second possible recommendation for future work would be the testing of more complex and efficient optimization algorithms. Possible candidates could, for example, be hybrid optimization algorithms which use evolutionary methods in order to determine the areas of interest in the search space of the problem followed by gradient-based methods in order to find the precise local minimum in said areas. Another addition that could improve the efficiency of the developed software would be the implementation of a machine-learning based dynamical boundary range for the decision variables that are considered at every iteration, based on an analysis of the state vector history of the previous iteration. This would allow to optimize the dimension of the search space that is considered at every iteration, thereby avoid- ing a large amount of unnecessary calculations. It could also be worth handling the different orbital requirements, such as inclination and inter-satellite distance, through a multi-objective optimization process rather than including them in one single objective function. This would allow to make a trade- off between the different objectives and the effort required to optimize them. An alternative to these recommendations could also be to re-think the entire approach and treat this control problem as a shooting problem in order to determine the optimal attitude of the sail at every revolution as part of the initial state of the system that yields the best objective function value.

Further developments of the concept should include a proper rotational dynamics simulation to be coupled with the translational dynamics propagation in order to better study the loads and the perturbing torques acting on the sail. The addition of a rotational dynamics analysis could also be coupled with a feasibility study regarding the possible application of reflectivity modulation technology in order to provide the torque required for the rotational control of the satellites. It would also be necessary to include more complex and accurate solar sail models than the ideal model that was considered in this thesis, in order, for example, to consider the deformation of the sail under the influence of the SRP which results in a non-flat surface. Furthermore, during this thesis it was often assumed that the absorbing side of the sail could be oriented towards the Sun for prolonged periods. This, however, might cause the sail to reach very high temperatures, thus threatening the structural integrity of the sail. It could therefore be interesting to couple the orbital dynamics simulation with a simplified thermal analysis of the satellite in order to determine for how long and how often the absorbing side of the sail can be oriented towards the incoming radiation Finally, a modelling of the actual field of vision of a spacecraft- mounted instrument, including the geometrical limitations due to the presence of the sail, should be included in order to determine the realistic debris position triangulating capabilities of the concept that was developed in this thesis.

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