Relative-Motion Trajectory Generation and Maintenance for Multi-Spacecraft Swarms
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Relative-Motion Trajectory Generation and Maintenance for Multi-Spacecraft Swarms by Rahul Rughani A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ASTRONAUTICAL ENGINEERING) May 2021 Copyright 2021 Rahul Rughani Acknowledgements I would like to acknowledge and sincerely thank a number of people for making this work pos- sible. First, I thank my research advisor, David Barnhart, who has shown me that a rigorous approach and a steady supply of snacks can solve any problem. Thanks to Dr. Mike Gruntman for your advice in navigating the winding path of academic journal publications. Thank you to Tyler Presser for your assistance in analyzing the potential applications of swarm trajectories for Geosynchronous orbit slot sharing. Thanks to Kyle Clarke for your assistance in characterizing the state, or rather chaos, or low-earth spacecraft density over the next few decades. Thank you to Dell, Linda, Marlyn, and the rest of the Astronautics department for all their logistical and financial support during my time as a Teaching Assistant. Thank you to the USC Space Engi- neering Research Center (SERC) and the Information Sciences Institute (ISI) for all the resources and research opportunities offered throughout this dissertation process. Thank you as well to the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this dissertation (https://carc.usc.edu). I would like to thank my committee members for their support and advice throughout the course of my research, and Missy Rogers for her help with proofreading and editing. I would like to thank all of my friends for helping me de-stress during these hectic times, especially Matt, Jon, Eric, and Rohan for the weekly Civilization VI games where I could vent my frustrations by way of global domination. ii Thank you to my parents, Avnish and Dipti Rughani, and my sister Shalini Rughani for, among many things, teaching me to be inquisitive and to reach for the stars. I would like to thank all of my extended family for their support and encouragement, and to forgive them for all their continual inquisitions of \when are you going to be done with school?" (Sonya, do we really ever stop learning?) And finally, above all, I would like to thank my partner, Becca Rogers, for her constant encouragement and support that has helped me to finally conclude this adventure. iii Table of Contents Acknowledgements ii List of Tables vii List of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Motivations . .1 1.2 Current State of the Art . .6 1.3 Problem Statement . .8 1.4 Solution Approach . .9 1.5 Assumptions and Spacecraft Design . .9 1.6 Challenges . 15 1.7 Overview of Thesis . 16 1.7.1 Chapter 2 { Background . 16 1.7.2 Chapter 3 { Generating Initial Trajectories for the Spacecraft Swarm . 17 1.7.3 Chapter 4 { Trajectory Maintenance for Spacecraft Swarms . 18 1.7.4 Chapter 5 { Behavioral Stresses of the System . 18 1.7.5 Chapter 6 { Swarm Configuration Example Scenario . 19 1.7.6 Chapter 7 { Other Considerations . 20 1.7.7 Chapter 8 { Application to GEO Spacecraft . 20 1.7.8 Chapter 9 { Conclusion . 20 Chapter 2: Background 21 2.1 Rendezvous and Proximity Operations . 21 2.1.1 Mathematical Formulation . 24 2.1.2 Orbit Maintenance . 28 2.1.3 Perturbation Effects . 30 2.1.4 Spherical Harmonic Representation of Earth's Gravitational Field . 33 2.2 Formation Flying . 36 2.3 Ground-Based Analogs . 38 2.3.1 Insect Swarm Comparisons . 40 2.4 Kalman Filtering . 41 2.4.1 Mathematical Formulation . 42 2.4.2 Sensor Fusion Kalman Filter . 46 2.5 Genetic Algorithms . 48 iv Chapter 3: Generating Initial Trajectories for the Spacecraft Swarm 53 3.1 Defining the Swarm Parameters . 53 3.2 Solving for Spacecraft Swarms . 54 3.2.1 Overview of Trajectory Generation . 54 3.2.2 Initial Trajectory Generation . 54 3.2.3 Trajectory Generation for Large Swarms . 58 3.2.4 Trajectory Modification for New Spacecraft Insertion . 60 3.2.5 Considerations for Construction and Aggregation . 62 3.3 Comparing the GA Method to Arbitrary Trajectories . 62 Chapter 4: Trajectory Maintenance for Spacecraft Swarms 66 4.1 Overview . 66 4.2 Kalman Filtering for Real-Time Operations . 66 4.3 Patched RPO using Kalman Filtering . 69 4.4 Stationkeeping Maneuvers . 71 4.5 Considerations for Electric Propulsion . 72 Chapter 5: Behavioral Stresses of the System 74 5.1 Overview . 74 5.2 Unexpected Loss of Vehicle . 77 5.3 Response to Dynamic Construction Environment . 79 5.4 Collision Avoidance Schemes . 87 Chapter 6: Swarm Configuration Example Scenario 90 6.1 Initial Conditions . 91 6.2 Results . 92 6.2.1 Trajectory Generation . 92 6.2.2 Transfer Trajectories to Acquire Spacecraft Sections . 93 6.2.3 Transfer Trajectories to Rendezvous in Assembly Zone . 96 6.2.4 Death of a Spacecraft { Debris Generation . 98 6.2.5 Stationkeeping Maneuvers to Recycle Trajectories . 100 6.2.6 ∆V Usage at Each Stage . 100 Chapter 7: Other Considerations 102 7.1 Orbital Reconnaissance . 102 7.2 Self-Aggregating Swarm . 103 7.3 Computational Distribution . 103 7.4 Light-time Delay for Autonomous Operations . 106 7.5 Foreign Object Threats . 107 7.6 Irregular Keep-Out Zones . 107 7.7 Search for Globally Optimal Solutions . 108 7.8 Implement Advanced Continuous-Thrust Trajectory Generation Techniques . 108 Chapter 8: Application to Geostationary Spacecraft Sharing Slots 109 8.1 Trajectory Generation & Collision Avoidance . 110 8.2 Patched RPO . 112 8.3 Stationkeeping Maneuvers . 113 8.4 Numerical Results from Simulation Trials . 114 8.4.1 Comparison to KOREASAT three satellite swarm . 114 8.4.2 Comparison to Convex Optimization Strategy . 116 8.4.3 Comparison to 4 co-located GEO spacecraft . 119 8.5 Summary of GEO Applicaton Results . 121 v Chapter 9: Conclusions and Ongoing Work 123 9.1 Real-Time Kalman Filtered Simulations . 123 9.2 GEO Swarms Analysis . 125 9.3 Hardware Testing . 125 9.4 Future Work . 126 References 127 Appendix A Computational Processes . 146 A.1 Swarm Generation . 146 A.2 Swarm Modification . 147 A.3 Genetic Algorithms . 148 A.4 Sensor Fusion Kalman Filter . 162 Appendix B Spherical Harmonic Gravity Model Coefficients . 164 vi List of Tables 2.1 Associated Legendre Polynomials. This table gives a few sample expansions for the associated Legendre function, with geocentric latitude used [51]. 34 3.1 Constraints for 10 Spacecraft Swarm Example . 57 3.2 Orbital Elements of Reference Trajectory . 65 8.1 Bi-Weekly ∆V Results for KOREASAT Comparison . 115 8.2 Yearly ∆V Results for DLR Comparison . 118 8.3 Yearly ∆V Results for Real-World Comparison . 121 vii List of Figures 1.1 Current Spatial Density in Orbit (Figure courtesy of Kyle Clarke) . .2 1.2 Predicted Future Spatial Density in Orbit (Figure courtesy of Kyle Clarke) . .3 1.3 NovaWurks HISAT Spacecraft . 14 1.4 HISAT Platform Configuration . 15 2.1 Slightly eccentric orbit allows relative motion [50] . 23 2.2 Swarm of Spacecraft in Relative Motion . 24 2.3 LVLH Coordinate Frame . 25 2.4 Trajectory Offsets for Various Levels of Position Injection Error . 29 2.5 Trajectory Offsets for Large Injection Errors . 30 2.6 Trajectory Drift for different gravity models . 32 2.7 Directed Graph Diagram . 39 2.8 Example Error Ellipse for a Satellite . 46 2.9 Sensor Fusion Diagram . ..