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A Di Ingegneria Low Thrust Trajectories in Multi Body

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A Di Ingegneria Low Thrust Trajectories in Multi Body Universita` degli Studi di Pisa Facolt`adi Ingegneria Corso di Dottorato in Ingegneria Aerospaziale XXII Ciclo Tesi di Dottorato Low Thrust Trajectories in Multi Body Regimes Tutore Candidato Prof. Mariano Andrenucci Pierpaolo Pergola Direttore del Corso di Dottorato Prof. Maria Vittoria Salvetti November 2010 A Maria Abstract More and more stringent and unique mission requirements motivate to ex- ploring solutions, already in the preliminary mission analysis phase, going far beyond the classical chemical-Keplerian approach. The present dissertation deals with the analysis and the design of highly non linear orbits arising both from the inclusion of different gravitational sources in the dynamical models, and from the use of electric system for primary propulsion purposes. The equilibrium of different gravitational fields, on one hand, permits unique transfer solutions and operational orbits, on the other hand, the high thrust efficiency, characteristic of an electric device, reduces the propellant mass required to accomplish the transfer. Each of these models, and even better their combination, enables trajectories able to satisfy mission require- ments not otherwise met, first of all to reduce the propellant mass fraction of a given mission. The inclusion of trajectory arcs powered by an electric thruster, providing a low thrust for extended duration, makes essential the use of optimal control theory in order to govern the thrust law and thus design the required transfers so as to minimizing/maximizing specific indexes. The goal is, firstly, to review the possible advantages and the main limits of dynamical models and, afterward, to define methodologies to preliminary design non-Keplerian missions both in interplanetary contexts and in the Earth-Moon system. Special emphasis is given to the study of dynamical systems through which the main features of the Circular Restricted Three Body Model (the first one among the non-Keplerian models) can be identified, implemented and used. Purely ballistic solutions enabled by this model are first indepen- dently explored and after considered as target orbits for electric thrusting phases. Electric powered arcs are used to link ballistic phases arising from the bal- ancing of different gravitational influences. This concept is applied both for the exploration of planetary regions and for interplanetary transfer purposes. Together with low thrust missions to selenocentric orbits designed taking into account both the Earth and the Moon gravity, also transfer solutions toward periodic orbits moving in the Earth-Moon region are presented. These are designed considering electric thrusting arcs and ballistic segments exploring for free specific space regions. In brief, theoretical models deriving from dynamical system theory and from optimal control theory are employed to design non conventional orbits in non linear astrodynamics models. Sommario Requisiti di missione sempre pi`ustringenti e particolari spingono ad esplorare soluzioni, fin dalle fasi preliminari dell'analisi di missione, che vanno ben oltre il classico approccio impulsivo-Kepleriano. Il presente lavoro tratta l'analisi ed il progetto di orbite altamente non lin- eari che emergono sia dall'inclusione di diverse fonti gravitazionali nel modello dinamico, sia dall'impiego di sistemi primari di propulsione elettrici. Da un lato l'equilibrio di pi`ucampi gravitazionali consente soluzioni di trasferimento ed orbite operative uniche nel loro genere, dall'altro l'elevata efficienza propulsiva, tipica di un propulsore elettrico, riduce la massa di propellente necessaria per realizzare il trasferimento. Ognuno di questi due modelli, ed ancor meglio la loro combinazione, definisce traiettorie in grado di rispettare requisiti di missione non altrimenti soddisfabili, primo fra tutti quello di ridurre la massa di propellente richiesta per una specifica missione. L'inclusione di fasi della traiettoria propulse da un sistema elettrico, che fornisce una bassa spinta per una durata prolungata, rende indispensabile la teoria del controllo ottimo per governare la legge di spinta e quindi realizzare il trasferimento minimizzando/massimizzando specifici indici. L'obiettivo `e,in primo luogo, quello di riesaminare i possibili vantaggi e limiti di questi modelli dinamici per poi definire metodologie per la proget- tazione preliminare di missioni non-Kepleriane sia in contesti interplanetari che nel sistema Terra-Luna. Particolare enfasi viene data allo studio dei sistemi dinamici mediante il quale le principali caratteristiche del Modello Ristretto a Tre Corpi (il primo fra i modelli non-Kepleriani) possono essere identificate, implementate e quindi sfruttate. Le soluzioni balistiche che derivano da questo modello sono esplorate dapprima in maniera indipendente e successivamente considerate come orbite obiettivo per fasi propulse. Archi propulsi elettricamente sono utilizzati per connettere fasi di volo balistico che nascono dal bilanciamento di diverse influenze gravitazionali. Tale concetto viene applicato sia per l'esplorazione di regioni planetarie che per trasferimenti interplanetari. Assieme a missioni a bassa spinta verso orbite selenocentriche progettate considerando sia l'attrazione gravitazionale della Terra che della Luna, sono presentate anche soluzioni per trasferimenti verso orbite periodiche che si muovono nel sistema Terra-Luna. Queste ultime sono costruite considerando archi di volo propulso e fasi balistiche che portano naturalmente ad esplorare specifiche regioni dello spazio. In breve, strumenti derivanti dalla teoria dei sistemi dinamici e dalla teoria del controllo ottimo sono impiegati al fine di progettare missioni non convenzionali in modelli astrodinamici non lineari. Acknowledgements Facing this blank page that have to become this acknowledgement section, I worry that my thanks might result in a long trite list of names or in in- adequate mentions of all those people who patiently worked on my side. Nevertheless I'm perfectly aware that I have to thank so many people if I'm at this stage. Of course, the first one I will be always grateful is Prof. M. Andrenucci who gave my incommensurable chances of experiences, that helped me not just from a technical point of view and that is still first close to the person and than to the student. A special thank is for Prof. M. Dellnitz, under whose supervision I had the opportunity to spend the long and extremely fruitful period in the Paderborn applied math department, and for A. Boutonnet and J. Schoenmaekers who gave me the chance to work on their side during my ESOC training. Last but not least, I wish to thank Prof. G. Mengali and A. Quarta for their very useful advices and suggestions. Now I should start a long enumeration of friends, colleagues, mates and mentors among the guys of the Pisa university, of Alta, of the IFIM/PaSco group, of the Astronet project, of the ESOC mission analysis section, and so on. I'm sure that each of you knows how much helped me and how trivial any word on this page could result. Finally, I wish to thank all of those people that make this period (also this one) not just work but fun, relax, experience, life in the complete sense of the word. I received and found so good fellow passenger that can not be just blind Chance. My deepest thanks to all of you guys. Contents 1 Introduction1 1.1 Non-Keplerian Trajectories.................... 1 1.2 Mission Analysis Trends...................... 4 1.3 Goal and Outline......................... 6 2 Dynamical Models7 2.1 The N-body Problem....................... 8 2.2 The Restricted Three Body Problem............... 14 2.2.1 Synodic, Non Dimensional Reference System...... 17 2.2.2 Equations of Motion................... 19 2.2.3 The Integral of the Motion................ 23 2.2.4 Libration Points...................... 26 2.2.5 Equilibrium Regions.................... 31 2.3 Restricted Four Bodies Models.................. 37 2.4 Dynamical System Elements................... 40 3 CR3BP Model Investigation and Exploitation 47 3.1 Periodic Orbits........................... 48 3.1.1 Periodic Orbits around Libration Points......... 48 3.1.2 Global Periodic Orbits................... 52 3.1.3 Stability of Periodic Orbits................ 57 3.2 Manifolds............................. 63 3.2.1 Manifold Globalization Procedure............. 64 3.2.2 Normal Forms to Compute Invariant Manifolds Near and Far the Libration Points.................. 67 3.3 Homoclinic Orbits......................... 76 3.4 Heteroclinic Orbits......................... 77 3.5 Transit Orbits........................... 79 CONTENTS v 3.6 Poincar´eSections......................... 81 3.7 Resonances............................ 84 3.8 Chaos and Lyapunov Exponents................. 90 4 Optimal Control 96 4.1 The Optimal Control Problem.................. 96 4.1.1 Direct Methods...................... 97 4.1.2 Indirect Methods..................... 100 4.1.3 Global Methods...................... 106 4.2 Low Thrust............................ 107 4.3 Two Body Application: Low Thrust Earth-Mars Transfers.... 108 4.4 Three Body Application: Low Thrust Heteroclinic Transfers... 113 4.5 Considerations........................... 118 5 Low Energy, Low Thrust Deep Space Missions 120 5.1 Low Thrust Transition between one dimensional Manifolds... 120 5.1.1 The Moon Tour...................... 124 5.1.2 Earth-Uranus Transfer................... 125 5.2 Low Thrust Halo to Halo Interplanetary Transfers........ 127 5.2.1 Design Strategy...................... 129 5.2.2 Earth-Mars: A Study Case...............
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