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Safe Spacecraft Rendezvous and Proximity Operations Via Reachability Analysis Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2021 Safe Spacecraft Rendezvous and Proximity Operations via Reachability Analysis Paul A. Grossi Follow this and additional works at: https://scholar.afit.edu/etd Part of the Aerospace Engineering Commons Recommended Citation Grossi, Paul A., "Safe Spacecraft Rendezvous and Proximity Operations via Reachability Analysis" (2021). Theses and Dissertations. 4977. https://scholar.afit.edu/etd/4977 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Safe Spacecraft Rendezvous and Proximity Operations via Reachability Analysis THESIS Paul A. Grossi, Captain, USSF AFIT-ENY-MS-21-M-302 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio Distribution Statement A. APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED. The 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, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENY-MS-21-M-302 SAFE SPACECRAFT RENDEZVOUS AND PROXIMITY OPERATIONS VIA REACHABILITY ANALYSIS THESIS Presented to the Faculty Department of Aeronautics & Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Master of Aeronautical Engineering Paul A. Grossi, BS, Mechanical Engineering Captain, USSF March 25, 2021 Distribution Statement A. APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED. AFIT-ENY-MS-21-M-302 SAFE SPACECRAFT RENDEZVOUS AND PROXIMITY OPERATIONS VIA REACHABILITY ANALYSIS THESIS Paul A. Grossi, BS, Mechanical Engineering Captain, USSF Committee Membership: Major Costantinos Zagaris, PhD Chair Lieutenant Colonel Kirk Johnson, PhD Member Major Joshuah Hess, PhD Member Adedeji B. Badiru, PhD Dean, Graduate School of Engineering and Management AFIT-ENY-MS-21-M-302 Abstract The rapid expansion of the utilization of space by nations and industry has pre- sented new challenges and opportunities to operate efficiently and responsibly. Many current and future operations require precise and safe rendezvous and proximity oper- ations (RPO) with a guarantee of the feasibility of the rendezvous maneuver. Reach- ability analysis is the process of computing the set of states that can be reached given all admissible controls and can be a valuable component in an autonomous mission planning system if conducted efficiently. In the current research, reachability analysis is used with several relative motion models to show that all ranges of orbits can be computed in milliseconds, and that it is a feasible approach for on-board autonomous mission planning. Reachability analysis is then combined with an Artificial Potential Function (APF) derived guidance control law to conduct safe spacecraft rendezvous between a deputy in a Natural Motion Circumnavigation (NMC) relative orbit around a chief while avoiding obstacles. While the APF employed in this research requires improvements for trajectory computation, this research demonstrates the feasibility of combining reachability analysis with an APF for safe, on-board, autonomous mission planning. iv To my grandparents, whose kindness, generosity, and laughter live on. v Acknowledgements I would foremost like to express my gratitude to my advisor, Major Costantinos Zagaris for your support and guidance throughout the entire process and as my re- search goals evolved considerably along the way. I would also like to thank Major Joshuah Hess for his enthusiastic love for all RPO related topics. Your dedication and willingness to join our video calls at midnight on Saturdays to offer help with our work goes above and beyond. My appreciation extends as well to Lt Col Kirk Johnson for serving on my committee and providing valuable insights to refine this research. I owe much of my affinity for engineering to Mr. K, mentor and friend, who has shown me from a young age how any problem can be tackled with a bit of resource- fulness, ingenuity, duct tape, and perseverance. Thank you to my \quaranteam" who helped keep me relatively sane through a pandemic and Master's program. Of course, thank you to Mom, Dad, and my brothers, whose love and support is endless and energizing. Lastly, to my wife, for always believing in me and never giving up on encouraging me, and to my cats and dog for always keeping me company as I worked late into the night. Paul A. Grossi vi Table of Contents Page Abstract . iv Dedication . .v Acknowledgements . vi List of Figures . ix List of Tables . xii I. Introduction . .1 1.1 Motivation and Background . .1 1.2 Problem Statement . .2 1.2.1 Research Questions . .3 1.2.2 Research Scope and Assumptions . .4 1.3 Organization of the Thesis . .5 II. Literature Review . .7 2.1 Spacecraft Relative Motion Models . .7 2.1.1 Reference Frames . .7 2.1.2 The Keplerian Two-Body Problem . 10 2.1.3 Relative Orbital Dynamics . 11 2.1.4 Linearized Relative Equations of Motion . 14 2.1.5 Clohessy-Wiltshire Relative Motion Model . 15 2.1.6 Schweighart-Sedwick Relative Motion Model . 18 2.1.7 Yamanaka-Ankersen Relative Motion Model . 20 2.1.8 Gim-Alfriend Relative Motion Model . 21 2.1.9 Relative Orbital Elements . 22 2.2 Control Methodologies . 24 2.2.1 Impulsive Maneuvers . 25 2.2.2 Optimal Control . 27 2.2.3 Artificial Potential Functions . 31 2.3 Reachability Analysis . 33 III. Solution Methodology . 40 3.1 State Dynamics . 41 3.2 Reachability Analysis . 42 3.2.1 The State Ellipsoid . 43 3.2.2 The Control Ellipsoid . 43 3.2.3 Computing the Reach Set . 44 vii Page 3.3 Evaluating Reach Sets With Different Relative Motion Models........................................................ 45 3.4 Artificial Potential Function . 46 3.5 Reach/APF Algorithm . 48 IV. Results and Analysis . 50 4.1 Reach Set Variation Between Relative Motion Models . 50 4.1.1 Test Instances . 50 4.1.2 Reach Set Comparison at LEO . 51 4.1.3 Reach Set Comparison at GEO . 62 4.2 Trajectory Planning using Reachability Analysis with APFs......................................................... 73 4.2.1 Test Instances . 73 4.2.2 Case 1: Strong Attractive Potential Function . 75 4.2.3 Case 2: Weak Attractive Potential Function . 80 4.2.4 Case 3: Obstacles moving on an NMC orbit . 86 V. Conclusions and Recommendations . 88 5.1 Conclusions. 88 5.2 Recommendations for Future Work . 89 5.2.1 Tuning via Linear Regression . 89 5.2.2 Potential Functions and Control Laws . 89 5.2.3 Parallelization . 90 Bibliography . 91 viii List of Figures Figure Page 1.1 Flow chart of guidance methodology . .4 2.1 Earth-Centered Inertial (ECI) reference frame with position r and velocity v vectors shown . .9 2.2 Local-Vertical, Local-Horizontal (LVLH) reference frame with position r and velocity v vectors shown . .9 2.3 Parameterizations of chief and deputy inertial and relative orbits [1] . 12 2.4 Planar projection of relative motion trajectory with ROE labeled [2] . 24 2.5 Construction of an Artifical Potential Function . 31 2.6 Notional representations of forward and backward reach sets........................................................... 34 2.7 Illustration of computational complexity of HJB reachability (reproduced from Chen [3]) . 35 3.1 Research methodology flowchart . 40 3.2 Flow chart of guidance methodology . 49 4.1 Reach set with all models overlaid in LEO (e = 0); Top: 1 min interval, Bottom: 10 min interval . 52 4.2 Reach set with all models overlaid in LEO (e = 0:1); Top: 1 min interval, Bottom: 10 min interval . 53 4.3 Reach set with all models overlaid in LEO (e = 0:7); Top: 1 min interval, Bottom: 10 min interval . 54 4.4 Euler angle of reach set ellipsoid semi-major axis (LEO); Top: 1 min interval, Bottom: 10 min interval . 56 4.5 Volume of reach set ellipsoid (LEO); Top: 1 min interval, Bottom: 10 min interval . 57 4.6 Trace of reach set ellipsoid (LEO); Top: 1 min interval, Bottom: 10 min interval . 58 ix Figure Page 4.7 Reach set computation.
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