UNIVERSITÁ DIPISA
DOTTORATO DI RICERCA ININGEGNERIA DELL’INFORMAZIONE
R
ELATIVE
M
OTION
D
YNAMICS AND
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IN THE
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ODY
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ROBLEMS
DOCTORALTHESIS
Author
Giovanni Franzini
Tutor
Prof. Mario Innocenti
Reviewers
Prof. Pierluigi Di Lizia Politecnico di Milano Prof. Gianmarco Radice
Singapore Institute of Technology
The Coordinator of the PhD Program
Prof. Marco Luise
Pisa, May 2018 Cycle XXX
Ai miei genitori Antonietta e Vincenzo e a mio fratello Alessandro
Once you fall in love with the ideas, that is so thrilling, there’s not much more to think about except trying to go as deep into the world as you can and being true to those ideas. You kind get lost. And getting lost is beautiful! – David Lynch
Tieniti la Terra uomo, io voglio la Luna! – Michele Salvemini
Ringraziamenti
Acknoledgments
This Thesis is the result of a journey that began near the end of my Master’s Degree, when I had to decide what would be the next step in my life. I decided to start this adventure at the dawn of a day in August 2014. That day I had to wake up early in the morning, since I was leaving for a vacation in Croatia with my friends Matteo and Tommaso. The night before I came home late from a Caparezza gig, and waking up that day was really hard. I submitted the Ph.D. application form just before leaving home at 7 in the morning. At the time, I had a lot of doubts, especially on my future. It turned out to be one of the best decision I have ever made!
In these few pages preceding all the maths and the discussions result of my Ph.D., I would like to acknowledge all the people that supported me during this journey and, most importantly, during my life. The “acade-mic” part of this acknowledgments will be in English, whereas the “non-academic” part will be in Italian.
My deepest and most sincere gratitude goes to my Ph.D. supervisor Ma-rio Innocenti. Thanks to him I learned to do (and I enjoyed doing) research. He allowed me to work autonomously on the topics I most liked, though under is professional and focused guidance. This is how a Ph.D. should be! It was a privilege working with him, and sharing a lot of good time during our travels and the everyday work.
during the 2015 SIDRA PhD Summer School in Bertinoro (FC), Italy).
had the lucky of meeting Pierluigi and Monica. All the talks and the good time spent together were a relief during the first very confused days of my Ph.D. I would like to thank both of them for that, and for the friendship born in that room.
I would also like to, and have to, thank all the people of the controls laboratory at the University of Pisa. I will start from the 5th floor thanking randomly the Underwater Robotics group, better known as the Subaqui: Simone for his idea of preparing bruschette con l’olio using pane cara-sau, Filippo, for introducing the use of glass cups for coffee and for all the discussions on math and control stuffs, Davide, which taught me how controllers design is done in the real world (see Fig. 1), Prof. Riccardo, for teaching us the Italian grammar, Vincenzo, thanks for all the fish riso patate e cozze (and, more in general, the food), and Daniele. I conclude the 5th floor acknowledgments with Michael, which recently joined the laboratory and had the bad luck of sharing the office with the subacqui. Moving to the 6th floor, I would like to thank my office mates Matteo (a.k.a. Robin Hood di Certaldo), Stefano, and Giordana. A special mention goes to my desk neighbor Tommaso (also a subacqueo), my office-lifestyle mentor. It will be a privilege sharing the Thesis defense with you! I would also like to thank the professors of the controls lab: Andrea, Lorenzo and Lucia. We spent a lot of good time all together, and we had the best laboratory dinners! La parte non accademica dei ringraziamenti inizia con un doveroso e più che mai sentito ringraziamento ai miei genitori, Antonietta e Vincenzo, e a mio fratello Alessandro. Grazie di tutto il supporto che mi avete dato in questi anni e che sono sicuro mi darete in futuro. Senza di voi non sarei qui
in questo momento a scrivere queste righe, ma soprattutto non mi troverei dove sono ora! Voglio ringraziare i miei nonni Bruna, Domenico, e Rita, con cui ho condiviso tanti momenti speciali della mia vita. Un grazie anche a nonno Giovanni, che non ho mai conosciuto ma da cui ho ereditato il nome e la passione per la scienza. Un grazie anche ai miei zii e cugini tutti! Tanti sono gli amici che meritano di essere citati, che mi hanno sup-portato durante questi anni e con cui ho trascorso mille momenti stupendi. Grazie a Matteo e Tommaso, con cui ho passato alcune delle vacanze più belle e con cui ho girato un film acclamato dalla critica. Grazie ad Andrea, “compagno d’armi” durante i primi anni di università, Marco per tutte le chiacchierate fatte in questi anni e per il sostegno che mi hai sempre dato, Ciaccio per tutti i cazzotti che ci siamo scambiati (in amicizia eh!), e poi Marcantonio, Lorenzo, Laura, Greta, Alessandro-Frizzales, Federica, Va-lentina, Biffa, fortissimo a calcetto, Jessica, Andrea, Daniele e in generale tutto il gruppo Allerona. Grazie anche a Fabio (nonostante sia juventino), Enrico e Serena, che dai tempi del liceo continuano ad essermi vicini. In particolare, vorrei ringraziare Fabio per i gelati!
Un grazie va anche a tutte le persone che ho incontrato durante la mia vita universitaria a Pisa: a Vincenzo che conosco dalla triennale, la cui barba continua a cresecere negli anni, ai miei coinquilini Emanuele, Sara e Salvatore con cui ho vissuto a stretto contatto per quasi un anno e mezzo, a Lorenzo, il “coinquilino estivo”, e Mattia per una delle migliori estati che abbia mai trascorso a Pisa, e a tutte le altre persone che ho avuto modo di conoscere in 9 anni e mezzo di vita universitaria pisana.
Un grazie enorme va al corso canoa adulti della Canottieri Pisa e a tutti i suoi componenti, con cui ho trascorso allenamenti a dir poco intensi e capaci ogni sera di farmi dimenticare i problemi della giornata. Grazie in particolare a Giacomo, uno dei migliori allenatori che abbia mai conosci-uto, che ha avuto una pazienza enorme nell’insegnarmi ad andare in canoa! Parlando di pazienza, un grazie immenso che non potrà mai essere es-presso con qualche riga va a Giulia. Grazie per sopportarmi e per sup-portarmi ogni giorno. Grazie per essermi accanto ed essere la protagonista di questi momenti di grande cambiamento per la mia vita. Una delle ra-gioni per cui questo dottorato si è rivelato essere la decisione migliore di sempre sei te, perché indirettamente mi ha dato l’occasione di trovarti e di stravolgere la mia vita. Grazie per questo anno e mezzo vissuto insieme e per tutta la gioia che mi hai regalato e continui a regalarmi ogni giorno. Sei la cosa più bella che mi sia capitata!
Summary
The objective of the present Thesis is a detailed study of the relative motion dynamics and control in a two-body and a three-body gravity fields. For the former scenario, a general set of equations for the inclusion of arbitrary or-bital perturbations is derived. The equations are then used for the design of a nonlinear H∞controller based on the state-dependent Riccati equation
technique. A closed-form solution for the H∞control problem for the
rela-tive motion control on elliptic orbits is also presented, based on a linearized time-varying set of equations. Relative motion in the three-body scenario is also studied. In particular, a nonlinear set of equations for relative mo-tion descripmo-tion in the local-vertical local-horizon frame is derived. Star-ting from this set, simplified equations are proposed and their performance compared. The computation of rendezvous maneuvers adopting both im-pulses or continuous thrust is then presented, in order to establish potential feasible trajectories.
Sommario
L’obiettivo della presente Tesi è lo studio dettagliato della dinamica e del controllo del moto relativo nel contesto del problema dei due corpi e del problema ristretto dei tre corpi. Nel primo caso, viene proposto un siste-ma di equazioni generale che permette l’inclusione sistesiste-matica delle per-turbazioni orbitali. Le equazioni ricavate vengono successivamente adot-tate per la progettazione di controllori H∞ non lineari, basati sull’utilizzo
di equazioni di Riccati dipendenti dallo stato del sistema. In aggiunta a questa tecnica viene proposta una soluzione in forma chiusa del problema di controllo H∞ nel caso di orbite ellittiche, risolto tramite l’utilizzo
del-la descrizione lineare tempo variante del moto redel-lativo. Successivamente, l’attenzione viene spostata sullo studio del moto relativo nel contesto del problema dei tre corpi. Viene quindi sviluppata una descrizione esatta del moto relativo, basata sull’utilizzo di un frame a orizzonte locale posto sul veicolo bersaglio. Il sistema di equazioni risultante è non lineare. Vengono quindi proposti vari sistemi di equazioni semplificati, la cui accuratezza è analizzata per mezzo di simulazioni Monte Carlo. Tali modelli semplifica-ti vengono infine impiegasemplifica-ti nel calcolo di manovre in anello aperto di semplifica-tipo impulsivo e a spinta continua.
List of Publications
International Journals
1. G. Franzini, L. Tardioli, L. Pollini, and M. Innocenti, “Visibility Aug-mented Proportional Navigation Guidance”, Journal of Guidance, Con-trol, and Dynamics, vol. 41, no. 4, pp. 987–995, 2018, https: //doi.org/10.2514/1.G002897.
International Conferences/Workshops with Peer Review
1. G. Franzini and M. Innocenti, “Nonlinear H-infinity control of rela-tive motion in space via the state-dependent Riccati equations”, in Proc. 54th IEEE Conference on Decision and Control, Osaka, Ja-pan, Dec. 2015, pp. 3409–3414, https://doi.org/10.1109/ CDC.2015.7402733.
2. G. Franzini, L. Pollini, and M. Innocenti, “H-infinity controller design for spacecraft terminal rendezvous on elliptic orbits using differential game theory”, in Proc. 2016 American Control Conference, Boston, MA, USA, July 2016, pp. 7438–7443, https://doi.org/10. 1109/ACC.2016.7526847.
3. G. Franzini, S. Aringhieri, T. Fabbri, M. Razzanelli, L. Pollini, and M. Innocenti, “Human-machine interface for multi-agent systems
mana-Simulation for Autonomous Systems, Third International Workshop, MESAS 2016, Rome, Italy, June 15-16, 2016, Revised Selected Papers, J. Hodicky, Ed., Springer International Publishing, 2016, pp. 25–39, https://doi.org/10.1007/978-3-319-47605-6_3. 4. M. Razzanelli, S. Aringhieri, G. Franzini, G. Avanzini, F. Giulietti, M.
Innocenti, and L. Pollini, “An haptic display for human and autono-mous system integration”, in Modelling and Simulation for Autono-mous Systems, Third International Workshop, MESAS 2016, Rome, Italy, June 15-16, 2016, Revised Selected Papers, J. Hodicky, Ed., Springer International Publishing, 2016, pp. 64-80, https://doi. org/10.1007/978-3-319-47605-6_6.
5. M. Innocenti, L. Pollini, G. Franzini, and A. Salvetti, “Swarm obstacle and collision avoidance using descriptor functions”, in Proc. 2016 IEEE Conference on Control Applications, Buenos Aires, Argentina, Sep. 2016, pp. 487–492, https://doi.org/10.1109/CCA. 2016.7587877.
6. L. Tardioli, G. Franzini, L. Pollini, and M. Innocenti, “Development of a visibility augmented Proportional Navigation Guidance: a game-theoretic approach”, in Proc. 55th IEEE Conference on Decision and Control, Las Vegas, NV, USA, Dec. 2016, pp. 6135–6140, https: //doi.org/10.1109/CDC.2016.7799212.
7. L. Tardioli, G. Franzini, L. Pollini, and M. Innocenti, “Visibility aug-mentation of the Proportional Navigation Guidance”, in Proc. 2017 AIAA Guidance, Navigation, and Control Conference, Grapevine, TX, USA, Jan. 2017, Paper AIAA 2017-1013, https://doi.org/ 10.2514/6.2017-1013.
8. G. Franzini, M. Tannous, and M. Innocenti, “Spacecraft relative mo-tion control using the state-dependent Riccati equamo-tion technique”, in Proc. 10th International ESA Conference on Guidance, Navigation and Control Systems, Salzburg, Austria, May 2017.
9. G. Franzini and M. Innocenti, “Relative motion equations in the local-vertical local-horizon frame for rendezvous in lunar orbits”, in Proc. 2017 AAS/AIAA Astrodynamics Specialist Conference, Stevenson, WA, USA, Aug. 2017, Paper AAS 17-641.
10. G. Franzini and M. Innocenti, “Effective coverage control for teams of heterogeneous agents”, in Proc. 56th IEEE Conference on Deci-sion and Control, Melbourne, Australia, Dec. 2017, pp. 2836–2841, https://doi.org/10.1109/CDC.2017.8264072.
List of Abbreviations
ARE algebraic Riccati equation. 33
CLERM circular linear equations of relative motion. 101 CNERM circular nonlinear equations of relative motion. 101 CR3BP circular restricted three-body problem. 77
DRE differential Riccati equation. 57 DSG Deep Space Gateway. 2
ECI Earth-centered inertial. 11
ELERM elliptic linear equations of relative motion. 100 ENERM elliptic nonlinear equations of relative motion. 100 ER3BP elliptic restricted three-body problem. 75
GNC guidance, navigation, and control. 1 HJI Hamilton–Jacobi–Isaacs equation. 54
HLEPP Human Lunar Exploration Precursor Program. 90 ISS International Space Station. 2
LERM linear equations of relative motion. 22 LMI linear matrix inequality. 52
LQR linear quadratic regulator. 33 LVLH local-vertical local-horizon. 9
NERM nonlinear equations of relative motion. 20 NRHO near rectilinear halo orbit. 87
ROA region of attraction. 38
SDC state-dependent coefficient. 35
Contents
List of Abbreviations XIII
1 Introduction 1
1.1 Main Contributions . . . 3
1.2 Structure of the Thesis . . . 4
1.3 Notation . . . 5
I Relative Motion in the Two-Body Problem 2 Relative Dynamics in the Two-Body Problem 9 2.1 Coordinate Systems Definitions . . . 11
2.1.1 The Inertial Frame I . . . 11
2.1.2 The Local-Vertical Local-Horizon Frame L . . . 11
2.1.3 The Coordinate Change Matrix Cl i: I → L . . . 12
2.2 Relative Motion Equations . . . 12
2.2.1 LVLH Angular Velocity . . . 13
2.2.2 LVLH Angular Acceleration . . . 16
2.2.3 Nonlinear Equations of Relative Motion . . . 19
2.2.4 Linear Equations of Relative Motion . . . 21
2.2.5 Hill’s Equations . . . 22
2.3 Zonal Harmonics Perturbation . . . 23
2.3.2 Differential Zonal Harmonics Perturbation . . . 26 2.3.3 Time-Derivative Along the H-bar . . . 27 2.4 Atmospheric Drag . . . 28 2.5 Other Perturbations . . . 31 2.6 Conclusions . . . 32 3 Nonlinear H-infinity Control via the State-Dependent Riccati
Equation 33
3.1 State-Dependent Riccati Equation Control . . . 35 3.1.1 Control Problem Statement . . . 35 3.1.2 Extended Linearization Technique . . . 35 3.1.3 SDRE Control Technique . . . 37 3.1.4 Stability of the SDRE Control . . . 37 3.1.5 Optimality of SDRE Control . . . 38 3.1.6 Algebraic Riccati Equation Solution . . . 40 3.1.7 SDRE-H∞Control . . . 41
3.2 SDRE Controllers Design for Relative Motion Control . . . 43 3.3 Simulations Results . . . 46 3.4 Conclusions . . . 51
4 Linear Time-Varying H-infinity Control 53
4.1 Zero-Sum Two-Player Differential Games . . . 55 4.2 H-infinity Control of Relative Motion on Elliptic Orbits . . . 57 4.2.1 The Associated Differential Game . . . 57 4.2.2 Solution of the Differential Riccati Equation . . . 60 4.2.3 H-infinity Norm of The Closed-Loop System . . . 63 4.3 Simulations Results . . . 65 4.4 Conclusions . . . 69
II Relative Motion in the Restricted Three-Body Problem
5 The Restricted Three-Body Problem 73
5.1 Three-Body Rotating Reference Frames S and Mi . . . 74
5.2 The Restricted Three-Body Problem . . . 74 5.3 The Elliptic Restricted Three-Body Problem . . . 76 5.4 The Circular Restricted Three-Body Problem . . . 79 5.4.1 Equilibria of the CR3BP . . . 80 5.4.2 Stability of the Collinear Equilibria . . . 82 5.4.3 Periodic and Quasi-Periodic Orbits in the Linearized
Contents
5.5 Conclusions . . . 90 6 Relative Dynamics in the Restricted Three-Body Problem 91 6.1 Reference Mission Scenario . . . 92 6.2 The Local-Vertical Local-Horizon Frame Li . . . 94
6.3 Nonlinear Equations of Relative Motion . . . 94 6.3.1 Exact Relative Dynamics . . . 94 6.3.2 LVLH Angular Velocity and Acceleration . . . 97 6.4 Simplification of the Equations of Relative Motion . . . 99 6.4.1 CR3BP Assumption . . . 99 6.4.2 Linearization of the Gravitational Acceleration . . . . 100 6.4.3 Relative Motion Equations Sets . . . 101 6.5 Equations Sets Comparison . . . 103 6.5.1 Distance Test Results . . . 105 6.5.2 Speed Test Results . . . 106 6.5.3 Comments on the Applicability of CR3BP at the
Apo-selene . . . 106 6.5.4 Analysis of the Linearization Error . . . 117 6.6 Conclusions . . . 118 7 Rendezvous Maneuvers in the Restricted Three-Body Problem 121 7.1 Literature Review . . . 122 7.2 Adjoint Method Theory . . . 126 7.3 Impulsive Maneuvers . . . 129 7.3.1 Assessing the Maneuver Feasibility . . . 132 7.3.2 Transfer Along V-bar . . . 134 7.3.3 Fly-Around / Transfer to Orbit of Different Altitude . 136 7.3.4 Orbital Plane Correction . . . 139 7.3.5 Impulsive Maneuvers at the Aposelene Using CLERM 139 7.4 Continuous Thrust Maneuvers . . . 141 7.4.1 Station Keeping . . . 145 7.4.2 Straight Line V-bar and R-bar Approaches . . . 146 7.4.3 Continuous Thrust Maneuvers at the Aposelene Using
CLERM . . . 146 7.5 Generalized Multi-Firings Maneuvers . . . 149 7.6 Conclusions . . . 155
8 Conclusions 157
CHAPTER
1
Introduction
Relative motion control is an essential aspect of space missions involving the coordination of multiple spacecraft, or the interception of a cooperative or noncooperative target. Examples include both well-established operati-ons, such as formation flying, rendezvous and docking, as well as future ap-plications like satellite servicing, debris removal, and autonomous in-orbit assembly of space structures.
The first step toward the design of a relative guidance, navigation, and control (GNC) system is the derivation of appropriate models describing the relative motion for the mission scenario considered. Generally, Keple-rian theory is used for characterizing the dynamics of the spacecraft and for deriving the equations sets that describe the relative motion. In this case, the motion is influenced by the primary body about which the spacecraft are orbiting, and the dynamics are the result of a two-body problem. All other forces causing a deviation from the Keplerian model are considered as orbital perturbations. Their effects must be included in the relative dy-namics model if accuracy is critical for mission success. If the analytical characterization of the perturbations is not possible, the controller should counteract their influence on the relative motion in order to meet mission requirements. Robust control techniques must then be adopted to guarantee performance in presence of these disturbances.
pri-mary body, or of a third body, must be carefully handled. As a matter of fact, depending on the mission scenario the assumption of Keplerian mo-tion may be limiting or even inexact when the magnitude of the third body attraction is comparable to that of the primary body. Such situations require the study of the relative motion based on the three-body problem, that is the gravitational attraction of the second primary must be explicitly included into the relative dynamics.
Relative motion in the two-body problem has been studied extensively, in particular due to its importance for missions in Earth orbits. A remarka-ble example of application is the rendezvous and docking with the Interna-tional Space Station (ISS). The ISS case promoted a dramatic technology advancement for relative GNC systems, as well as attention by drawing further the research community on relative motion dynamics and control study. Several relative dynamics models have been proposed in the lite-rature [1, 2], and numerous control techniques have been applied for de-signing relative GNC systems [3–5]. Relative motion in circular and near-circular orbits has been the most studied scenario. In fact, most of the space missions with multiple spacecraft are, or will, operate on these type of or-bits. Furthermore, the relative dynamics in this case is described by set of linear time-invariant models, and linear control theory can be applied. El-liptic orbits have received slightly less attention, even though some future missions will operate on such orbits, e.g. ESA PROBA-3 [6]. In particular, when the target or the formation leader flies an elliptic orbit, the relative dynamics is time-varying, mainly due to the orbital radius evolution along the orbit.
The attention received by the study of relative motion in three-body sce-narios is not comparable to the two-body case. The number of publications dedicated to this subject is significantly smaller compared to the literature devoted to the relative motion in the two-body problem, even though se-veral missions under study are targeting celestial bodies where the third body influence must be explicitly taken into account in the relative dyna-mics. Mission examples are Phobos Sample Return [7], Asteroid Impact and Deflection Assessment[8, 9], and the Human Robotic Lunar Partners-hip[10, 11]. Peculiar non-Keplerian orbits (halo orbits, near rectilinear or-bits, self-stabilized terminator oror-bits, etc.) will be flown in these missions and the dynamics of these orbits influence the relative motion. In particular, the Earth-Moon system is receiving considerable attention, since among the objectives of major national space agencies there is the development of a new space outpost, the Deep Space Gateway (DSG), to be placed in a lunar orbit. The station is conceived as a potential gateway for future exploration
1.1. Main Contributions
missions towards the asteroids and Mars, as well as a staging post to access the lunar surface [10, 12]. The approach to the station and the rendezvous and docking operations will be critical aspects that must be addressed in order to provide a safe access to the station.
In this Thesis the relative motion in the two-body and three-body pro-blem is studied, and new dynamics models and control laws for the deriva-tion of relative GNC systems are proposed.
The first part of the Thesis is dedicated to the relative motion in the two-body problem. In particular, we propose a generalized set of equations designed for including arbitrary perturbations according to the mission sce-nario considered. The derived equation set is used for the development of nonlinear pseudo H∞controllers, for the robust control of relative motion.
A closed-form solution for the H∞control problem for relative motion on
elliptic orbits is also presented.
The second part of the Thesis considers the relative motion in the three-body problem. The relative dynamics is first described in a frame local to the target. In this way, the motion of a chaser vehicle or of a formation follower is characterized from the target or leader perspective. This aspect is particularly important for rendezvous and docking operations, where the trajectories are analyzed from the target perspective in order to monitor the incoming vehicles and assess the trajectory safety. The models derived are used for computing maneuvers that can be adopted for designing a guidance profile aimed at steering the chaser vehicle from a given initial point to the target.
1.1
Main ContributionsA short description of the main contributions of this Thesis is detailed be-low.
• The derivation of a general nonlinear framework for exact description of the relative dynamics in presence of arbitrary perturbations. The set is designed in order to decouple, as much as possible, the orbital perturbations from the Keplerian component of the motion. In this way, arbitrary perturbations can be easily included, avoiding the rede-finition of the whole equation set.
• The nonlinear H∞ control problem of the exact relative dynamics is
solved by means of a sub-optimal technique, the state-dependent Ric-cati equation. The method allows to avoid the direct solution of the
associated Hamilton–Jacobi–Bellman equations, via the solution of algebraic Riccati equationsat each sample time.
• A derivation of a closed-form solution of the linear time-varying H∞
control problem of relative motion in elliptic orbits. Differential game theory is used in order to derive the control law and formally prove the boundedness of the H∞norm of the closed-loop system.
• Nonlinear and linear equation sets for relative motion description in the three-body problem are derived. In particular, the relative dyna-mics are described in the local-vertical local-horizon coordinate sy-stem, centered on the target spacecraft (or leader).
• Maneuvers for rendezvous in three-body scenarios are computed. These include two-consecutive firing, or general multi-firings transfer arcs. The types of firings considered are impulsive and continuous thrust.
1.2
Structure of the ThesisThe Thesis is composed by two main Parts. Part I is devoted to the study of the relative dynamics in the two-body problem, and it is organized as follows:
• Chapter 2 discusses the description of the relative motion in the two-body problem. After introducing the main coordinate systems used in Part I, the Chapter presents the main equation sets for relative motion description. A general framework for the exact nonlinear characteri-zation of the relative motion is provided, and the inclusion of the main orbital perturbations is discussed.
• Chapter 3 presents the state-dependent Riccati equation control techni-que. The Chapter provides an overview of the method and the main theoretical results. The technique is then applied to design guidance systems for relative motion control in presence of orbital perturba-tions. The performance of the controllers obtained are analyzed by means of numerical simulations. A formation flying mission is used as benchmark problem.
• Chapter 4 provides the solution of the H∞ relative motion control
problem. In particular, the linear equations of relative motion, which characterize the relative dynamics on elliptic orbits, are used to design a finite-horizon H∞control law. Differential game theory is used for
1.3. Notation
the closed-loop H∞ norm. The controller is tested on terminal
ren-dezvous missions on elliptic orbits.
Relative motion in the restricted three-body problem analysis constitutes Part II of the Thesis, which is organized as follows:
• Chapter 5 reviews the restricted-three body problem, and the main results related to elliptic and circular problems.
• Chapter 6 is devoted to the description of the relative motion in the restricted three-body scenario. First, the exact dynamics is characteri-zed in the local-vertical local-horizon reference frame, local to a target spacecraft. Then simplified equations sets are derived, based on rather general assumptions. The accuracy of the simplified sets is analyzed by means of extensive Monte Carlo simulations.
• Chapter 7 discusses how rendezvous maneuvers generally adopted in the two-body problem can be replicated in the restricted three-body scenario. In particular, the relative motion equations obtained in Chapter 6 are used for computing impulsive and continuous thrust ma-neuvers in the local-vertical local-horizon frame, which can be com-posed to obtain a rendezvous guidance profile.
Some of the results presented in Chapter 6 and 7 were carried out as part of the ESA Express Procurement EXPRO Simulation tool for rendezvous and docking in high elliptical orbits with third body perturbation (ESA contract no. 4000121575/17/NL/CRS/hh).
The Thesis concludes with Chapter 8, where conclusions are drawn.
1.3
NotationSets, vectors, and matrices
The field of real numbers is denoted with R, and the set of the strictly po-sitive reals is denoted with R+= {x ∈ R : x > 0}. The set of nonnegative reals is instead denoted with R+0 = R+∪ {0}. For any positive integer n,
Rnis the n-dimensional Euclidean space.
Vectors and matrices are denoted using the bold font, e.g. v and A, whereas scalar values with the normal font, e.g. α. Unit vectors are denoted with a hat, e.g. ˆv. Matrix transpose is indicated with the superscript T , e.g. AT. The n × n identity matrix is denoted with I
n, and the n × m zero
matrix with 0n×m. Dimensions are omitted when can be inferred from the
with the notation A > 0 (A < 0) and A ≥ 0 (A ≤ 0), respectively. Matrix inverse, pseudo-inverse, and weighted pseudo-inverse, with W > 0, are indicated with the superscripts −1, ∗, and W ∗, respectively, e.g. A−1, A∗, and AW ∗.
Norms
The Euclidean norm of a vector is denoted with k·k, whereas the operator k·kW, with W ≥ 0, denotes the weighted Euclidean norm, e.g. kvkW = √
vTW v. The L
2 norm of a time-varying vector is denoted with |·|, and
|·|V (t), with V (t) > 0 denotes the weighted version. The H∞ norm of a
system is denoted with |·|∞. Operators
The operators diag {·} and blkdiag {·} denote the diagonal and block-diagonal matrices given by concatenating the arguments of the operators along the diagonal. With hor {·} and ver {·} are denoted the horizontal and vertical concatenation operators. The image operator is denoted with Im {·}, i.e. given f : X → Y , then Im {f } = {y ∈ Y : y = f (x), x ∈ X}. Given a set of vectors v1, . . . , vn, the notation span {v1, . . . , vn} denotes the span
generated by the vectors. Derivatives
A functions is said to be of class Ck(Ω), if it is continuously differentiable k times in Ω, such that C0(Ω) stands for the class of continuous functions
in Ω. For a C1(Rn) f : Rn → R, ∂f(x)/∂x denotes the row of partial
derivatives. The time-derivatives of a vector in a given coordinate system F are denoted enclosing the derivative between · F, e.g. ˙v F represents the first time-derivative of v as seen in the frame F .
Part I
Relative Motion in the Two-Body
Problem
CHAPTER
2
Relative Dynamics in the
Two-Body Problem
Relative motion dynamic models are fundamental for the study of space operations involving coordination of two or more spacecraft, such as forma-tion flying, rendezvous and in-orbit assembly of orbital structures. Accurate prediction and analysis of the evolution of relative position and velocity, es-pecially in the long term, require accurate models that include the orbital perturbations that may influence the spacecraft dynamics.
Many models describing the relative motion in the vertical local-horizon (LVLH) frame under the influence of different perturbations have been proposed in the past. Kechichian presented an exact relative dyna-mics model under J2 and atmospheric drag perturbations, applying
New-tonian mechanics techniques [13]. Xu and Wang adopted the Lagrangian formalism to develop an exact model of relative motion subject to J2
per-turbation [14]. In Reference [15] arbitrary zonal harmonics perper-turbation was considered, and in Reference [16] lunar perturbation in the near Earth relative motion dynamics was introduced. A recent comprehensive survey on relative motion models is presented in [2].
Even if the previously cited works provide exact models, the authors only considered a specific subset of perturbations. Extension of their results
to include other perturbations that may act on the spacecraft and influence the relative motion is not straightforward. To this end a general framework for relative motion analysis in the LVLH frame which can include different perturbations is needed. Casotto proposed a general set of equations for describing the relative dynamics in presence of arbitrary perturbations [17]. However, the influence of the orbital perturbation on the dynamics, and in particular on the precession of the LVLH frame, is not evident in the equa-tions developed, since their contribution is not decoupled from the Keple-rian component of the motion. As a consequence, perturbations influence cannot be properly analyzed. Moreover, since the author considered per-turbations characterized by time-invariant parameters, orbital perper-turbations with parameters varying with time, as in the case of time-varying atmos-phere density or lunar perturbation in low Earth orbits (LEO), cannot be included.
The aim of this Chapter is to provide a general set of differential equa-tions for the exact description of relative motion dynamics in presence of arbitrary orbital perturbations. Unlike the aforementioned works, the pro-posed model offers a structured approach for perturbations inclusion, so that the ones of interest can be readily introduced according to the mission scenario. Such a capability is possible since the perturbations effect on the relative dynamics is separated from the Keplerian component of the motion, as opposed to [17]. The reconfigurability easiness makes the model appea-ling for motion prediction and design of model-based nonlinear controllers and observers. In addition, perturbations effects can be analyzed separa-tely. The equation set proposed was developed using only vector calculus and geometric relations. The inclusion of zonal harmonics perturbation of arbitrary order and atmospheric drag is discussed. General guidelines for inclusion of other perturbations are also given.
The Chapter is organized as follows. In Section 2.1 the main coordi-nate frames for relative motion description are introduced. Section 2.2 dis-cusses the derivation of the equations of relative motion. The different sets proposed in the literature for the relative dynamic description in the LVLH frame are derived, and a general methodology for characterizing the LVLH angular velocity and acceleration, based on [17], is shown. The in-clusion of zonal harmonics perturbations up to an arbitrary order is shown in Section 2.3. Atmospheric drag inclusion in the relative motion models is described in Section 2.4. The inclusion of general perturbations is discus-sed in Section 2.5. Conclusions are drawn in Section 2.6.
2.1. Coordinate Systems Definitions
2.1
Coordinate Systems Definitions2.1.1 The Inertial Frame I
The inertial frame with unit vectors ˆI, ˆJ , and ˆK is denoted as follows: I : nO; ˆI, ˆJ , ˆKo
The frame is centered on the primary body center of mass, and ˆK is aligned to the rotational axis. For orbits about Earth, the Earth-centered inertial (ECI) coordinate system can be chosen as (pseudo-)inertial frame. A possible definition of the ECI frame considers the unit vector ˆI pointing the vernal equinox direction, see [18, Section 9.3].
In this Part, derivation with respect to time in the inertial frame I will be denoted using Leibniz’s notation, i.e. by means of the operator d/dt. 2.1.2 The Local-Vertical Local-Horizon Frame L
Consider the following vectors, denoting the spacecraft state during its flight about the primary body:
• r its position with respect to the primary center of mass; • v = dr/dt its velocity with respect to the primary body; • h = r × v the spacecraft specific angular momentum.
The local-vertical local-horizon (LVLH) coordinate frame L is defined as follows:
L : nr; ˆı, ˆ, ˆko with unit vectors:
ˆı = ˆ × ˆk, = −ˆ h h,
ˆ k = −r
r
where r = krk, and h = khk. The frame in shown in Fig. 2.1. The LVLH frame here considered is defined as in [19, Section 3.1.3]. Due to their definitions, the unit vectors ˆı, ˆ, and ˆk are generally referred to as V-bar, H-bar, and R-bar, respectively1.
In this Part, the time-derivative in the LVLH coordinate frame will be denoted with the Newton’s notation, i.e. by means of upper dots.
Figure 2.1: The local-vertical local-horizon (LVLH) coordinate frame.
2.1.3 The Coordinate Change Matrix Cl
i: I → L
The coordinate change matrix from the inertial frame to the LVLH frame, Cil : I → L can be obtained in two different ways:
• as a function of the spacecraft orbit inclination i, right ascension of the ascending nodeΩ, and true latitude θ;
• as a function of the spacecraft position and velocity vectors.
If the spacecraft orbital elements (i, Ω, θ) are available, then the coordinate change matrix is [13]: Cil(i, Ω, θ) = − cΩsθ− sΩcθci cΩcθci− sΩsθ cθsi − sΩsi cΩsi − ci sΩsθci− cΩcθ − sΩcθ− cΩsθci − sθsi (2.1) where the operators cαand sα denote cos α and sin α, respectively.
The same matrix can be obtained using r, and v: Cil(r, v) =ˆıi ˆi kˆiT
where ˆıi, ˆi, ˆki are the unit vectors of the LVLH frame expressed in the inertial frame, i.e. using the measurements of the spacecraft position and velocity vectors taken in I.
2.2
Relative Motion EquationsConsider a passive spacecraft, in the following referred to as chief, orbiting about a primary body, and a spacecraft which is able to maneuver, denoted
2.2. Relative Motion Equations
as deputy2. The dynamics of the two spacecraft are given by [18]: d2r dt2 = − µ r3r + dc (2.2) d2r d dt2 = − µ r3 d rd+ dd+ u (2.3) where
• µ is the primary body gravitational parameter, given by µ = GM where G is the universal gravitational constant, and M is the primary mass;
• di = di,xˆı + di,y + dˆ i,zk quantifies in terms of acceleration the orbitalˆ
perturbationsacting on the spacecraft;
• u = uxˆı + uy + uˆ zk is the deputy control vector.ˆ
The subscript d denotes deputy’s parameters. When the subscript is dropped, the parameter is referred to the chief, except for the perturbation vectors where the subscripts are always specified.
It is of interest the characterization of the motion of the deputy with respect to the chief, as seen in the LVLH frame. Such a description is useful in order to analyze the trajectories of the deputy as seen from the chief, and to assess the fulfillment of safety requirements, especially during rendezvous and docking operations.
Before proceeding with the equations development, analytical expressi-ons for the angular velocity and acceleration vectors of the LVLH reference frame with respect to the inertial frame must be found.
2.2.1 LVLH Angular Velocity
The analytical expression of the LVLH frame angular velocity with respect to the inertial frame is here derived, following the procedure adopted by Casotto in [17], which is based on purely kinematic relationships.
Consider the time-derivatives of the LVLH frame unit vectors: dˆı dt = ω × ˆı, dˆ dt = ω × ˆ, dˆk dt = ω × ˆk
2Spacecraft nomenclature depends on the mission considered. For formation flying missions, spacecraft are
generally denoted as chief and deputy, or leader and follower. Instead, the terms target and chaser are adopted for rendezvous missions.
The cross product of each of the previous expressions by the relative unit vector yields: ˆ ı × dˆı dt = ˆı × (ω × ˆı) = ω − (ω · ˆı)ˆı ˆ × dˆ dt = ˆ × (ω × ˆ) = ω − (ω · ˆ)ˆ ˆ k × dˆk dt = ˆk × (ω × ˆk) = ω − (ω · ˆk)ˆk
Summing up the previous results we obtain the expression of the LVLH angular velocity as a function of the unit vectors and their derivatives:
ω = 1 2 ˆı × dˆı dt + ˆ × dˆ dt + ˆk × dˆk dt ! (2.4)
The time-derivative of the unit vector ˆk is given by: dˆk
dt = − 1
r(v + ˙r ˆk) (2.5)
Noting that r = −r ˆk and ˙r = − ˙r ˆk, we can write: ˙r = − ˙r · ˆk
= − v · ˆk + (ω × r) · ˆk = − v · ˆk + ω · (r × ˆk)
= − v · ˆk (2.6)
Substitution of Eq. (2.6) into Eq. (2.5) gives: dˆk dt = − 1 r((v · ˆı) ˆı + (v · ˆ) ˆ) = − 1 r (v · ˆı) ˆı (2.7) Note that v·ˆ = 0 since the chief velocity is perpendicular to h by definition of the specific angular momentum.
Derivation with respect to time of ˆ yields: dˆ dt = − 1 h dh dt + ˙hˆk = −1 h dh dt · ˆı ˆı + dh dt · ˆk ˆ k
2.2. Relative Motion Equations = −1 h r × d 2r dt2 · ˆı ˆ ı = −1 h d2r dt2 · (ˆı × r) ˆ ı = −r h d2r dt2 · ˆ ˆ ı (2.8)
where we exploited the following two relationships: ˙h = − ˙hˆ = − dh dt · ˆ ˆ dh dt · ˆk = r × d 2r dt2 · ˆk = d 2r dt2 · (r × ˆk) = 0
The time-derivative of ˆı is given by: dˆı dt = dˆ dt × ˆk + ˆ × dˆk dt = r h d2r dt2 · ˆ ˆ +1 r(v · ˆı) ˆk (2.9) Substitution of Eqs. (2.7), (2.8), and (2.9) into Eq. (2.4), yields:
ω = ωy + ωˆ zk = −ˆ 1 r (v · ˆı) ˆ + r h d2r dt2 · ˆ ˆ k (2.10)
Note that in Eq. (2.10) the component of the angular velocity along the V-bar direction is zero due to the definition of the LVLH frame. The terms ωy and ωz denote the orbital rate and the steering rate of the orbital plane,
respectively.
Eq. (2.10) expresses the LVLH angular velocity as a function of the chief velocity and acceleration vectors projections along the LVLH axes. The equation can be further simplified solving the following dot products:
v · ˆı = 1 hrv · (h × r) = 1 hrh · (r × v) = h r (2.11) d2r dt2 · ˆ = dc· ˆ = dc,y (2.12)
where Eq. (2.2) was used for Eq. (2.12).
Substitution of Eqs. (2.11) and (2.12) yields the simplified expressions for the LVLH angular velocity vector components:
ωy = − h r2 ωz = r hdc,y (2.13a) (2.13b)
Remark 2.1. The angular velocity components of ω in the LVLH frame can also be expressed as a function of the orbital elements derivatives as shown in [13]: ωx = dΩ dt sin i cos θ − di dtsin θ ωy = − dΩ dt cos i − dθ dt ωz = − dΩ dt sin i sin θ − di dtcos θ (2.14a) (2.14b) (2.14c) Gauss’ variational equations provide the expressions for the orbital ele-ments derivatives [18, Section 12.3.5]:
di dt = r cos θ h dc,y dΩ dt = r sin θ h sin idc,y dθ dt = h r2 − r sin θ cos i h sin i dc,y (2.15a) (2.15b) (2.15c) Substitution of Eqs.(2.15) into Eqs. (2.14) gives the same LVLH angular velocity expression of Eqs.(2.13).
2.2.2 LVLH Angular Acceleration
The components of the LVLH angular acceleration ˙ω can be obtained by direct derivation of Eqs. (2.13):
˙ ωy = − 1 r4 ˙hr 2− 2r ˙rh = −1 r ˙h r − 2 ˙rh r2 ! = −1 r ˙h r + 2 ˙rωy ! ˙ ωz = 1 h2
˙rdc,yh + r ˙dc,yh − r ˙hdc,y
Given the time-derivative of the chief specific angular momentum: ˙h = dh dt − ω × h = r × d 2r dt2 − ω × h = r × dc− ω × h = −r ˆk ×dc,xˆı + dc,y + dˆ c,zkˆ +ωy + ωˆ zkˆ × hˆ
2.2. Relative Motion Equations
= −rdc,xˆ
and since ˙h = − ˙hˆ, we have:
˙h = rdc,x (2.16) We conclude that: ˙ ωy = − 1 r(dc,x+ 2 ˙rωy) ˙ ωz = 1 h
˙rdc,y + r ˙dc,y+ rωzdc,y
(2.17a) (2.17b) Remark 2.2. A more general expression for the LVLH angular accelera-tion components can be obtained by derivaaccelera-tion of Eq.(2.10). The angular acceleration along the H-bar is given by:
˙ ωy = − 1 r d2r dt2 · ˆı + v · dˆı dt − ˙r rv · ˆı = −1 r d2r dt2 · ˆı + v · dˆı dt + ˙rωy (2.18) The termv · dˆı/dt can be simplified as follows:
v · dˆı dt = ( ˙r + ω × r) · r h d2r dt2 · ˆ ˆ +1 r (v · ˆı) ˆk =− ˙rˆk − rωyˆı ·ωz − ωˆ ykˆ = ˙rωy
and substituted into Eq.(2.18), obtaining: ˙ ωy = − 1 r d2r dt2 · ˆı + 2 ˙rωy (2.19) The angular acceleration along ˆk is:
˙ ωz = r h ˙r r d2r dt2 · ˆ + d3r dt3 · ˆ + d2r dt2 · dˆ dt − ˙h h d2r dt2 · ˆ ! (2.20) Noting that: d2r dt2 · dˆ dt = d2r dt2 ·ˆı ˆ ı + d 2r dt2· ˆ ˆ + d 2r dt2 · ˆk ˆ k · −r h d2r dt2 · ˆ ˆ ı
= − r h d2r dt2 · ˆı d2r dt2 · ˆ
Eq.(2.20) can be written as follows: ˙ ωz = r h ˙r r d2r dt2 · ˆ + d3r dt3 · ˆ − r h d2r dt2 · ˆı + ˙h r ! d2r dt2 · ˆ ! (2.21) Considering thath = −hˆ, and that
h = r × v
= −r ˆk × ( ˙v + ω × r) = −r ˆk ×− ˙rˆk − rωyˆı
= r2ωyˆ (2.22)
by differentiation of Eq. (2.22), and recalling Eq. (2.19), the derivative of the specific angular momentum norm can be written as follows:
˙h = −2r ˙rωy − r2ω˙y = r
d2r
dt2 · ˆı (2.23)
Introduction of Eq.(2.23) into Eq. (2.21) yields the expression of the angu-lar acceleration along ˆk:
˙ ωz = r h ˙r r d2r dt2 · ˆ + d3r dt3 · ˆ − 2 r h d2r dt2 · ˆı d2r dt2 · ˆ (2.24) The chief jerk in the two-body problem is given by:
d3r dt3 = ∂ ∂t d2r dt2 + ∂ ∂r d2r dt2 v + ∂ ∂v d2r dt2 d2r dt2
where the term∂ (d2r/dt2) /∂t accounts for time variation of the
gravita-tional parameter and of the perturbations’ parameters. In [17], the author considers this term as being equal to zero under the assumption of constant mass distribution and conservative force fields. This results in the following simplified expression forω˙z:
˙ ωz = r h ˙r r d2r dt2 · ˆ − ˆ T ∂ ∂r d2r dt2 v − 2r h d2r dt2 · ˆı d2r dt2 · ˆ
2.2. Relative Motion Equations
Figure 2.2: Relative motion geometry.
Conversely, in our formulation only the gravitational parameter is as-sumed time-invariant. Hence, the formulation here presented can take into account orbital perturbations characterized by time-varying parameters, such as atmospheric drag with time-dependent atmosphere density (see Re-mark 2.9), and the presence of a second primary body (e.g. lunar perturba-tion in LEO, see [16]). In addiperturba-tion, in Eqs.(2.13) and (2.17) perturbations influence is explicit, as opposed to Eqs.(2.10), (2.19), and (2.24).
Note that Eqs. (2.17) can be obtained from Eqs. (2.19) and (2.24), ob-serving that
d3r
dt3 · ˆk = ˙dc,z
2.2.3 Nonlinear Equations of Relative Motion
Relative motion geometry is shown in Fig. 2.2. The relative dynamics are described by introducing the vectors ρ and ˙ρ:
ρ = xˆı + y ˆ + z ˆk, ρ = ˙xˆ˙ ı + ˙y ˆ + ˙z ˆk
denoting deputy relative position and velocity with respect to the chief, as seen from the latter, i.e. in the LVLH frame.
Deputy position and distance from the primary body are: rd = r + ρ = xˆı + y ˆ + (z − r)ˆk
rd= krdk =
p
and its velocity is given by: vd= drd dt = dr dt + dρ dt = dr dt + ˙ρ + ω × ρ (2.25) Derivation of Eq. (2.25) yields the deputy acceleration:
d2r d
dt2 =
d2r
dt2 + ¨ρ + 2ω × ˙ρ + ˙ω × ρ + ω × (ω × ρ) (2.26)
Introducing Eqs. (2.2) and (2.3) into Eq. (2.26), we obtain the nonlinear equations of relative motion (NERM):
¨ ρ + 2ω × ˙ρ + ˙ω × ρ + ω × (ω × ρ) − µ r3r + µ r3 d (r + ρ) = ∆d + u (2.27) where ∆d is the differential perturbations acceleration vector:
∆d = dd− dc= ∆dxˆı + ∆dy + ∆dˆ zkˆ
In terms of LVLH components, Eqs. (2.27) constitutes a set of three nonlinear time-varying second order differential equations:
¨ x = ω2y+ ωz2− µ r3 d x + ˙ωzy − ˙ωyz + 2ωzy − 2ω˙ y˙z + ∆dx+ ux ¨ y = − ˙ωzx + ωz2− µ r3 d y − ωyωzz − 2ωz˙x + ∆dy+ uy ¨ z = ˙ωyx − ωyωzy + ωy2− µ r3 d z + 2ωy˙x − µ 1 r2 − r r3 d + ∆dz+ uz (2.28a) (2.28b) (2.28c) Propagation of Eqs. (2.28) requires the knowledge of chief distance and speed r and ˙r. Chief velocity in the LVLH frame is given by:
v = dr
dt = ˙r + ω × r = −rωyˆı − ˙r ˆk (2.29) Further derivation of Eq. (2.29) with Eqs. (2.13) and (2.17a) gives:
dv dt = ˙v + ω × v = − (r ˙ωy + 2 ˙rωy) ˆı − rωyωz + rωˆ 2y − ¨r ˆ k = dc,xˆı + dc,y + rωˆ 2y− ¨r ˆ k (2.30)
2.2. Relative Motion Equations
Combination of Eqs. (2.2) and (2.30) yields: ¨
r = −µ r2 + rω
2
y − dc,z (2.31)
that can be integrated twice in order to obtain r and ˙r.
Eqs. (2.16), (2.28) and (2.31) are the basis of our general framework for relative motion description. Perturbations are computed separately and then introduced in the equations, obtaining an exact model of the relative motion under the influence of the orbital perturbations of interest.
Remark 2.3. The set composed by Eqs. (2.16), (2.28) and (2.31) con-stitutes a system of 9 first order differential equations. From a system theory point of view, it can be view as a system with state vector x = [x, y, z, ˙x, ˙y, ˙z, r, ˙r, h]T and input vectoru. The disturbances acting on the chiefdc and its time-derivative along ˆ, ˙dc,y, as well as ∆d represent the
system disturbances. The former two affect the LVLH frame orientation, see Eqs. (2.13) and (2.17), as well as chief parameters r, ˙r and h. The latter influences the relative motion directly.
2.2.4 Linear Equations of Relative Motion
Assume the absence of orbital perturbations, i.e. the spacecraft motion is governed only by the gravitational acceleration due to the primary body. Then, Eq. (2.27) simplifies as follows:
¨ ρ + 2ω × ˙ρ + ˙ω × ρ + ω × (ω × ρ) − µ r3r + µ r3 d (r + ρ) = u (2.32) The angular velocity and acceleration vectors, Eqs. (2.13) and (2.17), become equal to:
ω = −h r2 = ˙ˆ f ˆ, ω = −2˙ ˙r r ˙ f ˆ where ˙f denotes the chief true anomaly rate.
The gravitational acceleration on the chief can be linearized by means of a Taylor expansion to first order around the chief position:
µ r3 d rd≈ µ r3r + ∂ ∂r hµ r3r i (rd− r) =µ r3r + µ r3 I − 3rr T r2 ρ (2.33)
Substitution of Eq. (2.33) in Eq. (2.32) yields: ¨ ρ + 2ω × ˙ρ + ˙ω × ρ + ω × (ω × ρ) + µ r3 I − 3rr T r2 ρ = u (2.34) In the literature, Eqs. (2.34) are generally referred to as linear equations of relative motion (LERM) [18]. The equations provide a good description of the relative motion when the distance between the deputy and the chief is sufficiently small with respect to the chief orbital radius (kρk r). According to [19], LERM equations can be used when kρk < 50 km.
From Keplerian orbit theory, we know that h = √µp, where p is the chief orbit semilatus rectum, which can be used to simplify LERM equati-ons as follows: µ r3 = h2 pr3 = ˙f 2r p
In addition, noting that r = −r ˆk, for the LVLH frame we have the following simplification: I − 3rr T r2 = 1 0 0 0 1 0 0 0 2
and Eq. (2.34) writes as follows in terms of LVLH components: ¨ x = ˙f2 1 − r p x − 2 ˙f ˙r rz − ˙z + ux ¨ y = −r p ˙ f2y + uy ¨ z = 2 ˙f ˙r rx − ˙x + ˙f2 1 + 2r p z + uz (2.35a) (2.35b) (2.35c) Eqs. (2.35) constitutes a set of six first order linear time-varying equati-ons. The linearized dynamics is characterized by a decoupling between the in-plane motionalong ˆı and ˆk, from the out-of-plane motion along ˆ. 2.2.5 Hill’s Equations
Eqs. (2.35) further simplify when the chief is on a circular orbit. In this case:
• the LVLH angular velocity is constant and equal to the chief orbit mean motionn = 2π/T , where T is the orbit period;
2.3. Zonal Harmonics Perturbation
• the chief orbital radius is constant and is equal to p. Therefore, LERM equations simplifies as follows:
¨ x = 2n ˙z + ux ¨ y = −n2y + uy ¨ z = −2n ˙x + 3n2z + uz
Eqs. (2.36) are known in the literature as Hill’s equations. The equations were obtained by Hill in [20], for describing the motion of the Moon with respect to the Earth. Sometimes Hill’s equations are also referred to as Clohessy–Wiltshire equations[18], since they were used by Clohessy and Wiltshire in their seminal paper [21] for the design of a terminal rendezvous guidance system, aimed at autonomous assembling of a multi-unit satellite.
2.3
Zonal Harmonics PerturbationThe main assumption at the base of Keplerian theory is the point-mass or spherical modeling of the primary body, and the consequent inverse-squared gravitational field generated. However, satellites experience a gra-vitational attraction besides the point-mass attraction of the primary body. As a matter of fact, primary bodies shape is usually not spheric. For ex-ample, the Earth shape resembles more an oblate ellipsoid, with equatorial radius greater than the polar radius. This flatting is the cause of orbital per-turbations that influence the satellite motion causing deviations from the nominal Keplerian orbit.
Another characteristic that the point-mass/spherical modeling does not capture is the non-homogeneous mass distribution that leads to other devi-ations from the Keplerian model.
Several gravitational potential field models try to include these effects by means of series expansions [18, Chapter 11]. In this Section we consider the spherical harmonic series modeling for deriving the expression of the orbital perturbation associated to the gravitational effects neglected by the point-mass/spherical modeling.
2.3.1 General Analytical Expression
One of the widest adopted model for primary body gravity potential mo-deling is the spherical harmonic series (see [18, Section 11.4]):
U (r, ψ) = −µ r 1 − +∞ X k=2 Req r k JkPk(cos ψ) ! (2.37)
Figure 2.3: Chief orbital parameters.
Table 2.1: Zonal harmonics of Earth gravitational field [18].
Earth zonal harmonics J2 1.0826 × 10−3 J3 −2.5327 × 10−6 J4 −1.6196 × 10−6 J5 −2.2730 × 10−7 J6 5.4068 × 10−7 where:
• Reqis the primary body equatorial radius;
• Jkis the zonal harmonic of order k;
• Pk(ξ) is the Legendre polynomial of order k, defined by the associated
Legendre function: Pk(ξ) = 1 2kk! dk dξk ξ 2− 1k
• ψ is the angle between the primary body polar direction ˆK and the spacecraft position r, see Fig. 2.3.
The first six zonal harmonics for the Earth are given in Table 2.1. Note that the J2is the predominant harmonic.
To include the effect of zonal harmonic perturbations in Eqs. (2.28), the potential of a spherical body, −µ/r, must be first isolated from the
2.3. Zonal Harmonics Perturbation
zonal harmonics potential, since its effect is already included in Eqs. (2.28). Therefore, we write Eqs. (2.37) as follows:
U (r, ψ) = −µ r + UJk(r, ψ) with: UJk(r, ψ) = µ r +∞ X k=2 Req r k JkPk(cos ψ)
Zonal harmonics perturbation acceleration is given by minus the gra-dient of UJk in the spherical coordinates (r, ψ), that is [22]:
dJk = −∇UJk = − ∂UJk ∂r r −ˆ 1 r ∂UJk ∂ψ ˆ ψ (2.38)
Using the geometric relation: ˆ
K = cos ψ ˆr − sin ψ ˆψ and the chain rule, we can write Eq. (2.38) as follows:
dJk = − ∂UJk ∂r r −ˆ 1 r ∂UJk ∂ cos ψ ∂ cos ψ ∂ψ ˆ ψ = −∂UJk ∂r r +ˆ sin ψ r ∂UJk ∂ cos ψ ˆ ψ = −∂UJk ∂r r −ˆ 1 r ∂UJk ∂ cos ψ ˆK − cos ψ ˆr = α(r, ψ) ˆr + β(r, ψ) ˆK − cos ψ ˆr (2.39) For the reader convenience, we defined the following scalar accelerations:
α(r, ψ) = −∂UJk ∂r = µ r2 +∞ X k=2 (k + 1) Req r k JkPk(cos ψ) β(r, ψ) = −1 r ∂UJk ∂ cos ψ = − µ r2 +∞ X k=2 Req r k Jk ∂Pk(cos ψ) ∂ cos ψ where the derivative of the Legendre function is given by:
∂Pk(ξ) ∂ξ = 1 2kk! dk+1 dξk+1 ξ 2 − 1k
Remark 2.4. Note that zonal harmonics perturbation is independent of Ω, since the zonal harmonics gravity potential is axisymmetric and depends only onr and ψ.
Remark 2.5. In Eq. (2.39) it is easy to see that β(r, ψ) is the gravity compo-nent pointing to the equatorial plane due to zonal harmonics perturbations. The perturbation component pointing to the primary body center of mass is given byα(r, ψ) − β(r, ψ) cos ψ instead.
Remark 2.6. If we denote with ri = r
XI + rˆ YJ + rˆ ZK the spacecraftˆ
position in the inertial frame I, we have that cos ψ = rZ/r. We can use
this geometric relation to simplify the expressions above and to avoid the use of the angleψ.
2.3.2 Differential Zonal Harmonics Perturbation
Using Eq. (2.39), we can obtain the expressions in the LVLH frame of the zonal harmonics perturbation for the chief and the deputy.
For the chief, we know that ˆr = −ˆk and
r cos ψ = r sin θ sin i (2.40) where (r, ψ) are chief spherical coordinates (see Fig. 2.3). Thus, Eq. (2.39) for the chief in the LVLH frame simplifies in:
dJk,c = β sin i cos θˆı − β cos iˆ − αˆk
where α = α(r, ψ), and β = β(r, ψ), since, given Eq. (2.1): ˆ
K = sin i cos θˆı − cos iˆ − sin i sin θ ˆk (2.41) For the deputy, Eq. (2.39) writes as follows:
dJk,d = αdrˆd+ βd ˆK − cos ψdrˆd
= (αd− βdcos ψd) ˆrd+ βdKˆ
where αd= α(rd, ψd), βd= β(rd, ψd), and (rd, ψd) are the deputy spherical
coordinates. The unit vector ˆrdis given by:
ˆ rd= 1 rd xˆı + y ˆ + (z − r)ˆk
Therefore, we can write the zonal harmonic perturbations acting on the deputy in the LVLH frame as follows:
dJk,d = (αd− βdcos ψd) x rd + βdsin i cos θ ˆ ı
2.3. Zonal Harmonics Perturbation + (αd− βdcos ψd) y rd − βdcos i ˆ + (αd− βdcos ψd) z − r rd − βdsin i sin θ ˆ k The differential zonal harmonics perturbation is finally given by:
∆dJk =dJk,d− dJk,c = (αd− βdcos ψd) x rd + (βd− β) sin i cos θ ˆ ı + (αd− βdcos ψd) y rd − (βd− β) cos i ˆ + (αd− βdcos ψd) z − r rd + α − βdsin i sin θ ˆ k
2.3.3 Time-Derivative Along the H-bar
To compute the angular acceleration ˙ωz in Eq. (2.17a), we must find the
time-derivative in the LVLH frame of the component along ˆ of dJk,c, i.e.
of the term:
dJk,c· ˆ = −β cos i (2.42)
Derivation with respect to time of Eq. (2.42) yields: ˙ dJk,c· ˆ = − dβ dt cos i − β d cos i dt (2.43)
The time-derivative of β is: dβ dt = − µ r2 +∞ X k=2 Jk Req r k d dt ∂Pk(cψ) ∂ cψ −(k + 2)˙r r ∂Pk(cψ) ∂ cψ (2.44) where cψ = cos ψ, and
d dt ∂Pk(cψ) ∂ cψ = ∂ 2P k(cψ) ∂ c2 ψ d cψ dt (2.45) with ∂2Pk(ξ) ∂ξ2 = 1 2kk! dk+2 dξk+2 ξ 2− 1k
Using Eq. (2.40) along with Eqs. (2.13) and (2.15), we write the time-derivative of cos ψ as follows:
d cos ψ dt = di dtcos i sin θ + dθ dt sin i cos θ
= h
r2 sin i cos θ
= −ωysin i cos θ (2.46)
Introduction of Eqs. (2.45) and (2.46) into Eq. (2.44) yields: dβ dt = µ r2 +∞ X k=2 Jk Req r k ∂2P k(cψ) ∂ c2ψ ωysin i cos θ + (k + 2) ˙r r ∂Pk(cψ) ∂ cψ ! (2.47) Finally, the time-derivative of cos i can be expressed using Eqs. (2.13) and (2.15):
d cos i dt = −
di
dtsin i = −ωzsin i cos θ (2.48) Introducing Eqs. (2.47) and (2.48) in Eq. (2.43), we finally obtain the expression of ˙dJk,c· ˆ.
Remark 2.7. Note that in Eq. (2.48) the angular velocity ωz appears. As
expected, the time-derivative ofcos i is influenced by all the perturbations acting on the chief, not only the zonal harmonics.
Remark 2.8. In order to compute ∆dJk and ˙dJk,c· ˆ, the parameters i, θ,
cos ψ and cos ψdare needed. The former two can be obtained by
propaga-tion of the associate Gauss variapropaga-tional equapropaga-tions in Eqs.(2.15). The term cos ψ can be computed using Eq. (2.40) or as described in Remark 2.6. In the latter case:
rZ = r · ˆK = −r sin i sin θ
For the deputy, using again Eq.(2.40), we have that cos ψd=
rd· ˆK
rd
= 1 rd
(x sin i cos θ − y cos i + (r − z) sin i sin θ) Therefore, to include zonal harmonics perturbation we must propagate two additional first order differential equation fori and θ, i.e. Eqs. (2.15).
2.4
Atmospheric DragAtmospheric drag for chief and deputy can be quantified using the follo-wing formulae [18, Example 12.4]:
da,c = − 1 2ρa(r)C −1 b,cvr,cvr,c (2.49) da,d = − 1 2ρa(rd)C −1 b,dvr,dvr,d
2.4. Atmospheric Drag 0 100 200 300 400 500 600 700 800 900 1000 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 102 Altitude h [km] L o ca l a tm o sp h er e d en si ty ρ [k g / m 3] Beginning of space h= 100 km Sea level ρ= 1.225 kg/m3
International Space Station h= 382.5 km (average)
Hubble Space Telescope h= 559 km
Figure 2.4: U.S. standard atmosphere model 1976.
where:
• ρa(r) is the atmosphere density in r according to the model adopted
(e.g. for the Earth the U.S. standard atmosphere model 1976 [23], shown in Fig. 2.4);
• vris the velocity of the spacecraft relative to the atmosphere;
• Cb = m/(CdA) is the spacecraft ballistic coefficient with m denoting
the spacecraft mass, Cd its drag coefficient, and A its average
trans-versal section area.
If we assume that the atmosphere rotates with the primary body, with constant rotational velocity equal to ωp = ωpK, the spacecraft relativeˆ
velocity vr can be approximated as follows:
vr = v − ωp × r
Let us introduce the vector ∆ω denoting the chief relative angular velo-city with respect to the atmosphere:
∆ω = ω − ωp = ∆ωxˆı + ∆ωy + ∆ωˆ zkˆ
which, using Eq. (2.41), is equal to:
Chief relative velocity and speed with respect to the atmosphere are: vr,c = ˙r + (ω − ωp) × r = −r∆ωyˆı + r∆ωx − ˙r ˆˆ k (2.51) vr,c = q ˙r2+ r2 ∆ω2 x+ ∆ω2y Deputy relative velocity and speed of the deputy are:
vr,d = vd− ωp× rd= vr,c+ ˙ρ + ∆ω × ρ (2.52) vr,d = ( ˙x + (z − r)∆ωy − y∆ωz) 2 + ( ˙y + x∆ωz− (z − r)∆ωx)2 + (y∆ωx− x∆ωy− ˙r) 2 1 2
By means of Eq. (2.52) and defining the coefficients: γc= − 1 2ρa(r)C −1 b,c, γd= − 1 2ρa(rd)C −1 b,d
we can write the differential atmospheric drag perturbation:
∆da= da,d− da,c = (γdvr,d− γcvr,c) vr,c+ γdvr,d( ˙ρ + ∆ω × ρ)
In terms of LVLH components we have:
∆da= (γdvr,d( ˙x + z∆ωy− y∆ωz) − (γdvr,d − γcvr,c) r∆ωy) ˆı
+ (γdvr,d( ˙y + x∆ωz− z∆ωx) + (γdvr,d− γcvr,c) r∆ωx) ˆ
+ (γdvr,d( ˙z + y∆ωx− x∆ωy) − (γdvr,d− γcvr,c) ˙r) ˆk
To compute ˙ωz, Eq. (2.17b), we need the time-derivative of da,c· ˆ in the
LVLH frame. From Eqs. (2.49), (2.50), and (2.51) we have:
da,c· ˆ = γcvr,c∆ωyr = γcωprvr,csin i cos θ (2.53)
Given Eqs. (2.15) and (2.13), the time-derivative of Eq. (2.53) is: ˙
da,c· ˆ = ωpsin i cos θ
dγc
dt rvr,c+ γc( ˙rvr,c + r ˙vr,c)
+ ωpγcvr,cr (ωysin i sin θ − ωzcos i)
with dγc dt = − 1 2C −1 b,c dρa(r) dt = − 1 2C −1 b,c ∂ρa(r) ∂r v (2.54)
Note that in Eq. (2.54) the time-derivative of the ballistic coefficients does not appear, since we implicitly assumed that the chief does not change its mass or shape.
2.5. Other Perturbations
Remark 2.9. The atmosphere density changes also with time, for example due to solar activity cycles. Hence, the coefficientsγcandγdare functions
of time as well as of spacecraft position. If we consider atmosphere time-variability, Eq.(2.54) becomes:
dγc dt = − 1 2C −1 b,c dρa(t, r) dt = − 1 2C −1 b,c ∂ρa(t, r) ∂t + ∂ρa(t, r) ∂r v
However, time-variability of the atmosphere is obviously difficult to esti-mate and may be taken in account only for high fidelity simulations. Remark 2.10. If we consider a static atmosphere model, i.e. the atmosp-here does not rotate with the primary body (as in [24]), we can significantly simplify the previous results. In this case, chief and deputy relative velocity with respect to the atmosphere are, respectively, vr,c = v and vr,d = vd.
The differential perturbation simplifies in:
∆da= (γdvd( ˙x + zωy− yωz) − (γdvd− γcv) rωy) ˆı
+ γdvd( ˙y + xωz) ˆ + (γdvd( ˙z − x∆ωy) − (γdvd− γcv) ˙r) ˆk
In addition, given Eq.(2.29), the atmospheric drag on the chief is equal to: da,c = γcvv = γcv(−rωyˆı − ˙r ˆk)
Therefore, the term ˙da,c· ˆ is equal to zero.
2.5
Other PerturbationsIt is possible to avoid direct calculation of perturbations in the LVLH frame, using the coordinate change matrix Cil, Eq. (2.1). Given the perturbation in the inertial frame, di, the expression in the LVLH frame is dl = Cildi. Consequently, the differential perturbations acceleration vector in I is equal to:
∆dl = Cil did− di c
(2.55) The angular acceleration ˙ωz depends on the time-derivative of dc,z. This
term can be computed using the expression of the disturbance in the inertial frame, dic, by means of the following relation,
˙ dlc= ˙dc,xˆı + ˙dc,y + ˙ˆ dc,zk = Cˆ il ddi c dt − ω l× Cl id i c (2.56)
Eqs. (2.55) and (2.56) may be useful for motion prediction, since one can avoid the analytical computation of perturbations in the LVLH frame. However, they do not provide any physical insight into the relative motion dynamics, unlike the results presented in the previous Sections.
2.6
ConclusionsA general framework for relative motion description in the LVLH frame was developed using simple geometric relations and vector calculus. The equation set proposed can be easily extended to include the perturbations that characterize the mission scenario. We also derived the analytical ex-pression of zonal harmonics perturbation and atmosphere drag for their in-troduction in the proposed equation set. Inclusion of general perturbations was also addressed. Since no approximations were made, all the results presented here are exact and can be used for accurate relative motion ana-lysis.
CHAPTER
3
Nonlinear H-infinity Control
via the State-Dependent
Riccati Equation
The accurate control of relative motion requires the use of those nonlinear equation sets that fully capture the relative dynamics, especially in presence of different sources of perturbations. To deal with the nonlinear dynamics description, nonlinear control techniques must be adopted in order to ex-ploit the equation set accuracy. Different nonlinear techniques have been applied to the relative motion control, such as adaptive control [25, 26], sli-ding mode [27] and computed-torque [28]. However, these solutions do not satisfy any optimality or sub-optimality condition.
The state-dependent Riccati equation (SDRE) control is one of the most promising methods for the systematic design of sub-optimal controllers for nonlinear systems [29]. The method is a generalization of the linear qua-dratic regulator (LQR) for nonlinear systems. In particular, at each sample time the system is linearized and an algebraic Riccati equation (ARE) is solved in order to find the control value according to the LQR control law.
One of the first application of SDRE for relative motion control can be found in [30]. In the literature, SDRE was applied principally for
