APPROACHES FOR MODELING SATELLITE RELATIVE MOTION A Dissertation by KIRK WAYNE JOHNSON Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, Kyle T. Alfriend Co-Chair of Committee, Srinivas R. Vadali Committee Members, John E. Hurtado Igor Zelenko Head of Department, Rodney D. W. Bowersox December 2016 Major Subject: Aerospace Engineering Copyright 2016 Kirk Wayne Johnson ABSTRACT This dissertation explores new approaches for modeling perturbed and unperturbed satellite relative motion. It extends Hoots orbit theory, an analytical averaging-method perturbation solution to the Zonal Problem, to second order. In addition, this study develops a new hybrid numerical/analytical algorithm for converting initial conditions from osculating elements to mean elements, so that a single set of osculating initial conditions may be taken as simulation inputs. Also, this study develops a new version of the Gim- Alfriend State Transition Matrix (GA STM) for linearized perturbed relative motion, in terms of the variables from Hoots theory. These variables, the Hoots elements, are advantageous (although not unique) in that they have no singularities for orbit eccentricity or inclination and they require only one solution of Kepler’s Equation at each time step, even when using the GA STM. The new models are compared by simulation with orbit theories and GA STMs using the so-called nonsingular elements (which are in fact singular for zero inclination) and the equinoctial elements. This study predicts and verifies the order of magnitude of modeling error due to various sources. This study also considers two special applications in satellite relative motion modeling. First, Pro- jected Circular Orbit (PCO) formations, originally defined for unperturbed motion about a circular reference orbit, have important applications and are widely studied. This dissertation removes the singularity for zero inclination by implementing the PCO initial conditions in equinoctial elements, allowing PCO formations to be initialized about equatorial orbits. Furthermore, this study reveals how the choice of variables for writing the PCO initial conditions changes the geometric interpretation of the PCO phase angle parameter α. Sec- ond, this study develops an alternative to the standard methods for mitigating along-track drift in perturbed satellite formations. The new method eliminates all along-track secular motion to first order by sacrificing one degree of freedom in the formation design. ii ACKNOWLEDGMENTS The past three years studying at Texas A&M have been both a tremendous privilege and a great challenge. I’ve had the opportunity to meet and learn from some of the greatest minds in the world in a fascinating area of research. For help in accomplishing my part of the task, I owe a debt of thanks to many. First, as always, ”I thank Christ Jesus our Lord, who has given me strength” (1 Timothy 1:12). I am extremely grateful for the support of my family. To my wife, thank you for all the innumerable ways you’ve helped me throughout this process–I love you, and your encouragement means more to me than I can express! To my son and daughter, thank you for your patience–I’m proud of you both, and I promise we’ll play a family game once this is all over. Thanks also to all those praying for me from a distance, especially Mom and Dad. As for my advisors, Dr. Terry Alfriend and Dr. Rao Vadali, it was an honor to learn from them and work with them. Dr. Vadali’s patient guidance in our weekly meetings and his insightful way of explaining complex topics were critical to what I’ve accomplished. And I have likewise benefited greatly from the depth of Dr. Alfriend’s knowledge and experience. It truly seems to me that they’ve both forgotten more about astrodynamics than I’ll ever know. They have been the best teachers and mentors I could have asked for. I also wish to thank Dr. John Hurtado and Dr. Igor Zelenko for serving on my doctoral committee. And I am grateful for the advice and assistance of those who worked with our research group during my time here: Dr. Kohei Fujimoto, Dr. Hui Yan, Dr. Inkwan Park, and especially Bharat Mahajan, who was always ready to be a cheerful sounding board for ideas. Finally, I am thankful the US Air Force has allowed me to spend a few years studying full-time–it’s been a fantastic opportunity1. 1The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government. iii TABLE OF CONTENTS Page ABSTRACT . ii ACKNOWLEDGMENTS . iii TABLE OF CONTENTS . iv LIST OF FIGURES . vi CHAPTER I INTRODUCTION AND LITERATURE REVIEW . 1 I.A Introduction . 1 I.B Problem Definition . 2 I.C Literature Review . 3 I.C.1 The Zonal Problem . 3 I.C.2 Satellite Relative Motion . 5 I.C.3 Alternative Orbit Elements . 6 I.C.4 Evaluating Models . 8 I.D Outline of Remaining Chapters . 9 CHAPTER II EXTENDED HOOTS THEORY . 10 II.A Introduction . 10 II.A.1 Lie Series . 10 II.B J6 Effects . 11 II.C Second-order Corrections . 13 II.D Method of Simulation . 14 CHAPTER III PROJECTED CIRCULAR ORBITS . 18 III.A Background . 18 III.B Bounded, Linearized, Unperturbed Relative Motion for Circular Chief Orbits . 18 III.C PCO Solutions . 22 III.D Conclusions . 25 CHAPTER IV RELATIVE STATE TRANSITION MATRIX IN TERMS OF HOOTS ELEMENTS 27 IV.A Background . 27 IV.B Sensitivity Matrix D ..................................... 27 ¯ IV.C Mean State Transition Matrix φy¯ .............................. 29 IV.D Relative Transformation Map Σ ............................... 30 CHAPTER V PREVENTING ALONG-TRACK DRIFT . 34 V.A Three-constraint Condition . 34 V.B Two-constraint Conditions . 37 V.C Standard Approach: Approximate Angle Rate Matching . 38 V.D Angle Rate Averaging . 40 V.E Curvilinear Rate Averaging . 42 iv V.F Eliminating the Secular Along-track Motion . 46 V.F.1 Two Constraints . 50 V.F.2 One Constraint . 55 V.G Conclusions . 55 CHAPTER VI SOURCES OF MODELING ERROR . 58 VI.A Numerical Integration Amplifying Input Noise . 58 VI.B Linearized Relative Coordinates . 60 VI.C Approximate Solutions to the Zonal Problem . 68 VI.C.1 Zeroth- and First-order Models . 68 VI.C.2 Second-order Models . 81 VI.D Kepler’s Equation . 86 VI.E Singularities . 88 CHAPTER VII COMPARISON OF MODEL RESULTS . 89 VII.A Comparisons of First- and Second-Order Models . 90 VII.B Comparisons of GA STM Using Hoots, Nonsingular, and Equinoctial Elements . 91 CHAPTER VIII SUMMARY AND CONCLUSIONS . 98 VIII.A Suggestions for Future Work . 99 REFERENCES . 101 APPENDIX A SECOND-ORDER CORRECTION TERMS . 106 A.1 Short-period Correction for Nonsingular Elements . 106 A.2 Short-period Correction for Hoots Elements . 107 A.3 Long-period Correction for Hoots Elements . 113 APPENDIX B ELEMENTS OF D(lp) AND D(sp) . 116 B.1 Long-period . 116 B.2 Short-period . 118 B.3 Polynomial Functions Used in the Sensitivity Matrix . 120 ¯ APPENDIX C ELEMENTS OF φy¯(t; t0) ...............................149 APPENDIX D ELEMENTS OF Σ(t) .................................155 APPENDIX E RELATIVE TRANSFORMATION MAP FOR WHITTAKER VARIABLES . 157 E.1 Background . 157 E.2 Relative Transformation Map Σ ...............................158 E.2.1 Singularity for Equatorial Chief Orbits . 159 E.3 Elements of Σ(t) .......................................160 v LIST OF FIGURES FIGURE Page 2.1 Simulation algorithm flowchart . 17 3.1 HCW relative orbit geometry at t = 0 when αz = αx and ρz = 2ρx . 20 5.1 3D relative trajectory, g = π=2 and α = π=2 ....................... 35 5.2 In-plane relative trajectory, g = 0; π=4; π=2 and α = 0; π=4; π=2 . 36 5.3 In-plane relative trajectory over 1600 orbits, g = π=2 and α = π=2 . 37 5.4 In-plane and 3D relative trajectories, g = π=3 ....................... 38 5.5 Relative orientation of chief (blue) and deputy (red) mean orbit planes . 40 5.6 In-plane contributions to along-track relative motion . 43 5.7 Drift, e = 0:8182 ...................................... 47 5.8 Along-track secular and nonsecular components . 49 5.9 Along-track secular and nonsecular components, two constraints . 51 5.10 1 relative orbit, δa = 0 ................................... 52 5.11 20 relative orbits, δa = 0 .................................. 53 5.12 20 relative orbits, three different drift-prevention strategies . 53 5.13 Along-track secular and nonsecular components for the GTO case study . 54 5.14 Drift in the extrema of y for four cases . 56 6.1 Differences in analytically propagated ECI x-coordinate . 59 6.2 Error: 3 analytical models vs. single numerical solution . 59 6.3 Error with reltol=1e-8 . 61 6.4 Error: each of 3 analytical models vs. its own corresponding numerical solution . 62 6.5 Differences in numerically propagated ECI x-coordinate . 63 6.6 Direct-differencing error: 3 analytical models vs. single numerical solution . 63 6.7 Direct-differencing error: each of 3 analytical models vs. its own corresponding numer- ical solution . 64 6.8 PCO relative error, including STMs . ..