ORBITAL RENDEZVOUS AND SPACECRAFT LOITERING IN THE EARTH-MOON SYSTEM A Thesis Submitted to the Faculty of Purdue University by Fouad Khoury In Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautics and Astronautics December 2020 Purdue University West Lafayette, Indiana ii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF THESIS APPROVAL Dr. Kathleen Howell, Chair School of Aeronautics and Astronautics Dr. Carolin Frueh School of Aeronautics and Astronautics Dr. David Spencer School of Aeronautics and Astronautics Approved by: Dr. Gregory Blaisdell Associate Head of the Graduate School of Aeronautics & Astronautics iii To my parents, Saeb & Lama, and my siblings, Omar & Karmah iv ACKNOWLEDGMENTS "The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land." - T. H. Huxley This work would not be possible without the support of many of my colleagues and mentors. I am grateful for the experiences and interactions I have had (and hopefully continue to have) with each of them. First, I would like to express my gratitude to my adviser Professor Kathleen Howell for her guidance, patience, and encouragement. It has been my honor to serve as her student, teaching assistant, and researcher. I fur- thermore express my gratitude to my fellow researchers in the Multibody Dynamics Research Group, both past and present. Thank you to Andrew C, Robert, Collin, Brian, Emily, RJ, David, Beom, Ricardo, Rohith, Vivek, Juan, Maaninee, Andrew M, Stephen, Kenza, Bonnie, Nick, Paige, Yuki, and Kenji for your technical advice and feedback. Your friendships have enabled me to not just finish my journey in graduate school but to enjoy and learn from it as well. Moreover, I would like to thank the professors in my committee, Professors Carolin Frueh and David Spencer, for reviewing my work and providing valuable insights. I would also like to thank my mentors Dr. Alan Lovell, from the Air Force Re- search Lab (AFRL), Dr. Diane Davis, from a.i. solutions, and the many engineers at the NASA Johnson Space Center. Their collaboration and feedback have been instrumental in establishing the foundations for this work. Moreover, I would like to express my appreciation to the engineers at John Hopkins University Applied Physics Laboratory for their discussions and insight. I look forward to joining their ranks and contributing to the advancement of space technologies and missions. Finally, I thank my family both in the United States and abroad. Your sacrifices v and support have enabled me to pursue my passions and develop a love for learning. To my father, I thank you for instilling in me perseverance and a strong work ethic. To my mother, thank you for teaching me the value of sacrifice and patience. To my brother, thank you for teaching me to recognize beauty in the natural world and humankind's placement in it. Finally, to my sister, thank you for teaching me the importance of displaying kindness and respect to everyone. vi TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::::: viii LIST OF FIGURES ::::::::::::::::::::::::::::::::: ix ABSTRACT ::::::::::::::::::::::::::::::::::::: xiv 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 1.1 Problem Definition ::::::::::::::::::::::::::::: 2 1.2 Previous Contributions ::::::::::::::::::::::::::: 3 1.3 Document Outline ::::::::::::::::::::::::::::: 4 2 BACKGROUND ::::::::::::::::::::::::::::::::: 7 2.1 Keplerian Dynamics :::::::::::::::::::::::::::: 7 2.2 Elliptical Restricted 3-Body Problem ::::::::::::::::::: 12 2.2.1 Pseudo-potential Functions :::::::::::::::::::: 21 2.3 Circular Restricted 3-Body Problem :::::::::::::::::::: 22 2.3.1 Pseudo-potential Functions :::::::::::::::::::: 24 2.3.2 Jacobi Constant ::::::::::::::::::::::::::: 25 2.3.3 Equilibrium Solutions :::::::::::::::::::::::: 26 2.3.4 Zero Velocity Surfaces ::::::::::::::::::::::: 29 2.3.5 Symmetry :::::::::::::::::::::::::::::: 32 2.4 Differential Corrections ::::::::::::::::::::::::::: 33 2.4.1 State Transition Matrix :::::::::::::::::::::: 33 2.4.2 Monodromy Matrix ::::::::::::::::::::::::: 36 2.4.3 Shooting Algorithms :::::::::::::::::::::::: 37 2.4.4 Periodic Family Continuation Techniques ::::::::::::: 49 2.4.5 Periodic Orbit Families ::::::::::::::::::::::: 52 2.5 Selection of Reference Orbits :::::::::::::::::::::::: 64 vii Page 2.5.1 Transitioning Solutions from CR3BP to ER3BP ::::::::: 66 3 RELATIVE MOTION MODELS :::::::::::::::::::::::: 70 3.1 LVLH Frame Definition ::::::::::::::::::::::::::: 71 3.2 Keplerian-Based Relative Motion ::::::::::::::::::::: 72 3.2.1 Nonlinear Equations of Relative Motion ::::::::::::: 78 3.2.2 Linear Equations of Relative Motion ::::::::::::::: 81 3.2.3 Euler-Hill (HCW) Equations :::::::::::::::::::: 83 3.3 Relative Motion in the Restricted Three-Body Problem ::::::::: 84 3.3.1 Nonlinear Equations of Relative Motion ::::::::::::: 85 3.3.2 Linearized Equations of Relative Motion ::::::::::::: 91 3.3.3 Summary of Relative Motion Sets ::::::::::::::::: 92 3.4 Verification and Validation ::::::::::::::::::::::::: 96 3.5 Shooting Algorithms ::::::::::::::::::::::::::: 104 4 APPLICATIONS :::::::::::::::::::::::::::::::: 107 4.1 Orbital Rendezvous :::::::::::::::::::::::::::: 108 4.2 Spacecraft Loitering ::::::::::::::::::::::::::: 131 5 SUMMARY AND RECOMMENDATIONS :::::::::::::::::: 150 5.1 Summary of the Present Work :::::::::::::::::::::: 150 5.2 Recommendations ::::::::::::::::::::::::::::: 152 5.3 Future Work :::::::::::::::::::::::::::::::: 153 REFERENCES ::::::::::::::::::::::::::::::::::: 155 viii LIST OF TABLES Table Page 2.1 Conic classification in the two-body problem ::::::::::::::::: 12 2.2 Characteristic Quantities for Nondimensionalization ::::::::::::: 21 2.3 Libration Point Positions for Different Systems ::::::::::::::: 28 2.4 Libration Point Jacobi Constants for Different Systems ::::::::::: 29 2.5 Monodromy Matrix Eigenvalue Decomposition & Stability Characteristics 37 4.1 9:2 NRHO Rendezvous Maneuver Results ::::::::::::::::: 116 4.2 Small DRO Rendezvous Maneuver Results ::::::::::::::::: 123 4.3 Large DRO Rendezvous Maneuver Results ::::::::::::::::: 130 4.4 Forced Loitering Case Descriptions with ∆v magnitudes ::::::::: 148 ix LIST OF FIGURES Figure Page 2.1 Diagram of the two-body model. ::::::::::::::::::::::: 7 2.2 Diagram of the two-body model in synodic frame S. ::::::::::::: 9 2.3 Diagram of the three-body model in synodic frame S. :::::::::::: 12 2.4 Diagram of the three-body model in synodic frame S. :::::::::::: 14 2.5 Diagram of the ER3BP in synodic frames S (left) and M (right). ::::: 17 2.6 Diagram of the CR3BP in synodic frames S (left) and M (right). ::::: 24 2.7 Positions of the Libration Points in Frame S. :::::::::::::::: 28 2.8 ZVS boundaries (purple) and ZVC (black) case in the Earth-Moon system 30 2.9 Zero-Velocity Curves Progression for the Earth-Moon System ::::::: 31 2.10 Single Shooting Algorithm Diagram :::::::::::::::::::::: 41 2.11 Multiple Shooting Algorithm Diagram :::::::::::::::::::: 42 2.12 Corrected Multiple Shooting Algorithm Diagram :::::::::::::: 43 2.13 Multiple Thrust Firing Schematic ::::::::::::::::::::::: 46 2.14 Planar Periodic Orbit Computation :::::::::::::::::::::: 49 2.15 L1 Lyapunovs in Earth-Moon System ::::::::::::::::::::: 53 2.16 L2 Lyapunovs in Earth-Moon System ::::::::::::::::::::: 54 2.17 L3 Lyapunovs in Earth-Moon System ::::::::::::::::::::: 54 2.18 L1 Northern (top) and Southern (bottom) Halo Families in Earth-Moon System :::::::::::::::::::::::::::::::::::::: 56 2.19 L2 Northern (top) and Southern (bottom) Halo Families in Earth-Moon System :::::::::::::::::::::::::::::::::::::: 57 2.20 Stability index plot for Earth-Moon L2 Southern Halos (left) and identifi- cation of NRHOs in purple (right) ::::::::::::::::::::::: 57 2.21 L3 Northern (top) and Southern (bottom) Halo Families in Earth-Moon System :::::::::::::::::::::::::::::::::::::: 58 x Figure Page 2.22 L1 Axials in Earth-Moon System ::::::::::::::::::::::: 59 2.23 L2 Axials in Earth-Moon System ::::::::::::::::::::::: 59 2.24 L3 Axials in Earth-Moon System ::::::::::::::::::::::: 60 2.25 L1 Verticals in Earth-Moon System :::::::::::::::::::::: 61 2.26 L2 Verticals in Earth-Moon System :::::::::::::::::::::: 61 2.27 L3 Verticals in Earth-Moon System :::::::::::::::::::::: 62 2.28 DROs in Earth-Moon System ::::::::::::::::::::::::: 63 2.29 Northern L2 Butterfly Family in Earth-Moon System :::::::::::: 64 2.30 L2 Southern Halo Family with NRHOs in purple (left) and 9:2 NRHO (right)65 2.31 DRO Family (left) and a small DRO (right) ::::::::::::::::: 65 2.32 DRO Family (left) and a large DRO (right) ::::::::::::::::: 66 2.33 9:2 NRHO in the CR3BP with the selected patchpoint (left) and its ER3BP equivalent (right) ::::::::::::::::::::::::::: 68 2.34 The small DRO in the CR3BP with the selected patchpoint (left) and its ER3BP equivalent (right) ::::::::::::::::::::::::::: 68 2.35 The large DRO in the CR3BP with the selected patchpoints (left) and its ER3BP equivalent (right) ::::::::::::::::::::::::::: 69 3.1 The LVLH frame attached to a target spacecraft with its associated directions72 3.2 Validation scheme along 9:2 NRHO :::::::::::::::::::::: 97 3.3 Position and velocity error results after 12 hrs. for 2B-HCW (top), 2B- LERM (middle), and 3B-LERM (bottom) for the 9:2 NRHO in the CR3BP 98 3.4 Position and velocity error results after 12 hrs. for 2B-HCW (top), 2B- LERM (middle), and 3B-LERM (bottom)