Autonomous Spacecraft Guidance for Small-Body Proximity Missions Antonio Canale Delft University of Technology AUTONOMOUS SPACECRAFT GUIDANCEFOR SMALL-BODY PROXIMITY MISSIONS by Antonio Canale 4522958 as part of MSc Thesis in MSc Aerospace Engineering Delft University of Technology Supervisors: Dr. Ir. Erwin Mooij Astrodynamics and Space Missions at TU Delft Prof. Maruthi Akella Aerospace Engineering at UT Austin i To Sergeant Pilot Larry Darrell ii ABSTRACT Low-thrust transfers in the proximity of a celestial body are characterized by a spiral shape, formed by many revolutions, due to the fact that the thrust to mass ratio is usually smaller than the gravity acceleration. From their use for primary purposes on board the Deep Space 1, these propulsion systems have acquired more and more importance. Due to their high specific impulse compared to chemical rockets, resulting in huge propellant mass savings, these systems have been enabling the exploration and the interaction with celestial bodies at increasing distance, up to orders of tens of light minutes. In particular, a special atten- tion has been dedicated to the so-called small-bodies, defined as all the objects in the Solar System not classifiable as planets, dwarf planets or satellites: this interest is not over yet, as confirmed by the future Lucy and Psyche missions by NASA. Although seen as intermedi- ate steps towards interplanetary manned missions, providing incremental capabilities at a low risk, the small-bodies are characterised by environments often not well-known before arrival, tiny standard gravitational parameters and irregularities in their shape, with en- hanced effects, the closer the spacecraft is to their surface. In addition, due to their distance from the Earth, communication delays of several minutes are unavoidable, making neces- sary the shift from the complete ground control of the spacecraft to a more autonomous strategy. This new requirement can be met through the implementation of real-time guid- ance algorithms, characterised by lack of accuracy compared to optimisation methods, but a computational effort low enough to consider them feasible for these operations. In partic- ular, this report proposes the real-time implementable Lyapunov-based simulator by Her- nandez and Akella[2016] with algorithm modifications to account for two of the biggest problems that the two-body problem cannot address when used for small-body proxim- ity operations: the non-impact condition between the spacecraft and the main body, and the violation of the Keplerian assumption on its homogeneous sphericity, limited to the secular effects of the second harmonic in the following. The implementation of these ad- ditional tools as an extension of the core loop of the original simulator, without altering its main architecture, allows not to compromise its strengths on simplicity and real-time im- plementability, while permitting more flexibility. However, before focusing on these mod- ifications, an extensive study of the algorithm capabilities is performed, ranging from its validation with the Q-law and minimum-time transfers to a sensitivity analysis on its main tunable parameters. Advantages are taken of the fact that the simulator proposed by Her- nandez and Akella[2016] requires only the knowledge of the current state of the spacecraft and of the target, allowing the determination of a control law without solving a non-linear system. The innovative nature of the proposed solution is confirmed by the scarcity in the literature of real-time algorithms including the non-sphericity of the main body. iii iv ACKNOWLEDGEMENTS It is right and proper to start this report by saying thanks to those who made this gradua- tion project possible: Prof. Mooij, who first shaped me as a student and then supported me over these months with continuous advice via Skype, and Prof. Akella, who, after accepting me as a member of his research group, made me feel immediately welcomed as an integral part of it. If being supervised by such a relevant academic personality is an honour, being supported by two of them has been amazing. Furthermore, I want to thank my supervi- sors at S[&]T and Airbus Defence & Space for teaching me how to clearly define a problem before trying to solve it and for making me understand how important it is to ask for help when necessary. Finally, I would like to extend my warmest thanks to my Alma Mater, Delft University of Technology, which allowed me to realise my American dream through the prestigious Justus & Louise Van Effen scholarship, and put me in touch with an incredible mentor who helped me to reach some conclusions I would have probably come too late. Over the last few years, I have been so lucky to meet some amazing people coming from all the sides of the world and I am sure I will have a sofa or at least a chair in several cities. Additionally, thanks to friendships that have been able to withstand distances and silences, many of the relationships which have characterised my teen years have remained intact, evolving into something different. Often, looking at pictures while being so far away, I felt the only absent guy in certain circumstances: nevertheless, my friends have been the reason why I can still call my place "home". Unfortunately, misunderstandings made me lose some of the people that most contributed to what it is me right now, and, even if I am extremely sorry for that, I want to thank you for everything. This paragraph is concluded with the awareness that, at the end of this trip, I have been so lucky that the whole universe has been conspiring to make me find what I want and what I have basically always wanted. Finally, I apologise to my family for spending all of my last four birthdays far from home, each one in a different country. I am sorry for obliging them to interact with Skype on daily basis, but, mostly, I am even more disappointed that I have not succeeded yet in explaining to them how to hold their mobile phone in a proper way: it is a discrete failure for an en- gineer, but, even from a strange perspective, I can see how they smile every time we talk. I thank them for allowing a twenty-year-old guy to study two months in England, even if his English skills were incredibly close to zero, and for giving him enough self-esteem to take important decisions. I remember you wanted me to become a lawyer, but I also recall your support when I told you I was dreaming to work as an engineer at NASA. Furthermore, you supported me in Europe and USA, even when I considered changing my career aspirations. I promise it will never happen again, independently from successes and failures, to have me far away for that long: at the end, everything will be fine. Va sempre tutto bene. v vi CONTENTS Glossary ix Acronyms............................................. ix Roman Symbols.........................................x Greek Symbols.......................................... xi Constants............................................. xi 1 Introduction1 1.1 Research Objective....................................2 1.2 Report Structure......................................2 2 Real-Time Guidance Systems5 2.1 Mission Heritage......................................5 2.1.1 Deep Space 1....................................6 2.1.2 Hayabusa......................................7 2.1.3 OSIRIS-REx.....................................9 2.1.4 ARM......................................... 10 2.1.5 Requirements................................... 11 2.2 Integrator Selection.................................... 12 2.3 Lyapunov Algorithms................................... 15 2.3.1 Design Principles................................. 15 2.3.2 Lyapunov Direct Method............................ 16 2.4 Hernandez Algorithm Basics.............................. 17 2.4.1 Levi-Civita Transformation........................... 17 2.4.2 KS Transformation................................ 19 2.4.3 Simulator Architecture.............................. 20 2.5 Conclusions........................................ 20 3 2D Algorithm 23 3.1 Implementation & Verification............................. 23 3.1.1 Matching Semi-Major Axis........................... 24 3.1.2 Matching Eccentricity.............................. 24 3.1.3 Matching Full Orbit................................ 29 3.2 Simulations......................................... 30 3.3 Sensitivity Analysis.................................... 34 3.3.1 Introduction.................................... 34 3.3.2 Control Gains................................... 35 3.3.3 Semi-Major Axis Tolerance........................... 36 3.3.4 Navigation Errors................................. 37 3.4 Conclusions........................................ 42 vii viii CONTENTS 4 3D Algorithm 45 4.1 Implementation & Verification............................. 45 4.1.1 Matching Energy................................. 46 4.1.2 Matching Eccentricity.............................. 47 4.1.3 Matching Inclination............................... 49 4.1.4 Matching Angular Momentum Vector..................... 50 4.1.5 Matching Angular Momentum & Eccentricity Vectors........... 54 4.2 Algorithm Validation................................... 55 4.3 Simulations & Sensitivity Analysis........................... 62 4.4 Conclusions........................................ 65 5 Non-spherical body 67 5.1 Orbit Propagation with J2 ................................ 67 5.2 Linearisation Process................................... 69 5.3 Simulator Architecture.................................. 74