Autonomous Rendezvous and Docking with Tumbling, Uncooperative, and Fragile Targets under Uncertain Knowledge by Hailee Elida Hettrick B.S., Cornell University (2017) Submitted to the Department of Aeronautics and Astronautics in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 © Massachusetts Institute of Technology 2019. All rights reserved. Author.............................................................. Department of Aeronautics and Astronautics May 10, 2019 Certified by. David W. Miller Professor, Aeronautics and Astronautics Thesis Supervisor Accepted by . Sertac Karaman Chairman, Department Committee on Graduate Theses 2 Autonomous Rendezvous and Docking with Tumbling, Uncooperative, and Fragile Targets under Uncertain Knowledge by Hailee Elida Hettrick Submitted to the Department of Aeronautics and Astronautics on May 10, 2019, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics and Astronautics Abstract As efforts to expand humanity’s presence in space continue to increase, aneedfor spacecraft to autonomously perform in-space close proximity maneuvers without a human operator increases, as well. Such in-space close proximity maneuvers include active debris removal, satellite servicing, and in-space assembly. Active debris removal will facilitate the continued use and access to low Earth orbit, mitigating the expo- nential debris growth occurring due to decrepit satellites and rocket bodies colliding. Satellite servicing will provide the capability to repair and refurbish spacecraft, elon- gating the lifetime of valuable assets both locally orbiting Earth and on routes further out in the solar system. In-space assembly is the means by which large space struc- tures are developed in orbit. Currently, such feats occur with the help of astronauts and robotic arms (i.e. the continued development of the International Space Station). However, for increased benefit, in-space assembly must occur autonomously, without a human in-the-loop, in order to create large structures in locations unideal for hu- mans or with a non-negligible communication latency. These three reference missions need the software enabling autonomous rendezvous and docking to reach a technical readiness level to be employed with confidence. In-space close proximity maneuvers share a standard sequence of events described in this thesis. The focus of this thesis address the terminal approach trajectory to soft docking, the contact dynamics of docking between two spacecraft, the optimization of the detumble procedure to bring the Target to stabilization, and adaptive control techniques to handle uncertainties in spacecraft knowledge. The software developed in support of these subproblems is included in the appendices and is largely based on implementation with the Synchro- nized Position Hold Engage Reorient Experimental Satellites (SPHERES) platform or with the characteristics of SPHERES considered. Thesis Supervisor: David W. Miller Title: Professor, Aeronautics and Astronautics 3 4 Acknowledgments This research was conducted primarily under the support of the Massachusetts Insti- tute of Technology Department of Aeronautics and Astronautics through a research assistantship. The author thanks the department for its support and Dr. David Miller for his guidance through the research. Additional gratitude is expressed to the SPHERES team and former postdoc Danilo Roascio for assistance with using the SPHERES testbed. Lastly, many thanks to the never-ending support and wisdom from family and the friendship and community provided by Jess, Alexa, Cadence, and Golda. 5 6 Contents 1 Introduction 21 1.1 Motivation for Autonomous Rendezvous & Docking . 21 1.1.1 History of Rendezvous and Docking . 21 1.1.2 Reference Missions for Autonomous Rendezvous & Docking . 24 1.2 SPHERES Platform . 30 1.2.1 Status of Testbed . 32 1.3 Relative Operations for Autonomous Maneuvers . 33 1.4 Research Objective . 34 1.5 Thesis Roadmap . 35 2 Literature Review 37 2.1 Trajectory Optimization . 37 2.2 Nonlinear Control . 38 2.2.1 Sliding Mode Control . 39 2.2.2 Adaptive Control . 39 2.3 Attitude Stabilization under Actuator Loss-of-Effectiveness . 43 2.4 Soft Docking & Stabilization of a Tumbling Target . 44 3 Trajectory Optimization 47 3.1 Stage Description . 47 3.2 Synchronous Radial Approach via Second Order Differential Equation 48 3.3 Differential Equation Propagator Architecture . 56 3.3.1 R-Bar Alignment . 57 7 3.3.2 Forward Propagation . 59 3.3.3 Determine Final Position . 59 3.3.4 Backwards Propagation . 60 3.3.5 Radial Shift . 62 4 Adaptive Control for Thrust and Inertia Uncertainties 67 4.1 Thruster Degradation Characterization . 67 4.1.1 Thruster Characterization in Test Session 97 . 68 4.1.2 Thruster Force Characterization . 70 4.1.3 Maintenance Session . 73 4.2 In-Flight Adaptive PID Sliding Mode Position and Attitude Controller 82 4.2.1 Problem Formulation . 83 4.2.2 Software and Hardware Implementation . 88 4.2.3 Results & Discussion . 90 5 Docked Dynamics 101 5.1 Contact Dynamics of Docking . 101 5.2 Modeled Docked Dynamics . 105 5.3 Discrete Solver for Docked Dynamics . 110 5.4 Detumble Optimization . 112 5.4.1 Indirect Optimization on Angular Acceleration Profiles . 113 5.4.2 Direct Optimization Approach . 117 5.4.3 Comparison of Optimization Results . 120 6 Conclusions and Future Work 123 6.1 Summary of Thesis . 123 6.2 Contributions . 126 6.3 Recommendations for Future Work . 127 A Trajectory Optimization 129 A.1 Fuel-Optimal Trajectory Optimization Functions . 129 A.2 Synchronous Radial Approach Propagator . 135 8 B Adaptive Control for Uncertainty 149 B.1 Thruster Characterization Plots . 149 B.2 Thruster Characterization Functions . 150 B.3 Adaptive PID SMC Controller . 165 C Docked Dynamics Functions 171 C.1 Contact Dynamics of Docking Event . 171 C.2 Setting up Target and Chaser . 174 C.3 Discrete Solver . 177 C.4 Direct Optimization Using GPOPS . 184 9 10 List of Figures 1-1 Remote Manipulator System on STS-125, Hubble berthed to Shuttle [8] 23 1-2 Orbital Express Mission: ASTRO and NEXT [13] . 23 1-3 Effective number of LEO objects larger than 10 cm [15] . 25 1-4 Conceptual diagram of potential capturing methods [17] . 26 1-5 Conceptual diagram of potential removal methods [17] . 26 1-6 13-year construction progress of the International Space Station . 27 1-7 Astronauts install set of corrective lenses for flawed mirror in HST on first servicing mission (STS-61) [25] . 28 1-8 Canadarm2 post-releasing Dragon spacecraft [27] . 29 1-9 Two SPHERES equipped with UDPs performing docking maneuver onboard the ISS . 30 1-10 Expanded view of SPHERES . 32 1-11 Spatial representation of the stages of ROAM . 33 2-1 Diagram of a generic adaptive control system configuration from Lan- dau’s textbook [44] . 40 2-2 Diagram of a model reference adaptive control system . 41 2-3 Diagram of a self-tuning control system . 42 2-4 Diagram of a gain scheduling control system . 42 3-1 Reference frames used in discussion of trajectory generation . 48 3-2 Sequence of optimizations and function-fitting that led to a trajectory- generation differential equation from [38] . 49 3-3 Δ푉 of five trajectory optimizations with fixed final time . 50 11 3-4 Euclidean norm of acceleration components as fraction of the Euclidean norm of total acceleration . 51 3-5 Sum of Euclidean norm of radial and tangential acceleration compo- nents as fraction of the Euclidean norm of total acceleration . 52 3-6 Vector elements of acceleration components for each generated trajectory 54 3-7 Chaser’s trajectory from call to ode45 using Listing 3.1 . 56 3-8 Illustrating the necessary initial conditions for the propagator i.e. the Chaser is at the desired radius and the Chaser is along the Target’s R-bar . 58 3-9 Timeline illustrating the Chaser’s intersections with the Target’s R-bar at the desired initial radial length away from the Target . 58 3-10 Forward propagate Target’s orientation and rotation rate for 푛 periods 59 3-11 Generated trajectory yielded from 푡푓 from forward propagation prior to radial shift to align initial position with Chaser’s current position . 63 3-12 Generated trajectory after radial shift with initial position aligned with Chaser’s current position . 64 3-13 Illustration of timeline shift after shifting the radial approach to align with the Chaser’s current position . 65 4-1 Location and indices of thrusters on satellite with noted exhaust direction 68 4-2 Thruster characterization: raw accelerometer and gyroscope data from IMU in test session 97 . 69 4-3 Thruster characterization: 200 ms impulse accelerometer data sepa- rated by axis . 70 4-4 Calculated force magnitude of each thruster on Blue and Orange satellites 73 4-5 Test session 101: calculated force of each thruster and each satellite . 75 4-6 Test 11: closed-loop x-translation telemetry . 77 4-7 Test 12: closed-loop y-translation telemetry . 78 4-8 Test 13: closed-loop z-translation telemetry . 79 4-9 Test 14: closed-loop x-rotation telemetry . 80 12 4-10 Test 15: closed-loop y-rotation telemetry . 80 4-11 Test 16: closed-loop z-rotation telemetry . 81 4-12 Flowchart of the in-flight adaptive PID sliding mode controller asim- plemented in C-code . 91 4-13 A comparison of commanded position and attitude vs. measured teleme- try from simulation . 92 4-14 Adaptive PID SMC time histories corresponding to position and atti- tude from simulation . 93 4-15 Trajectories plotted in the 푥-푦 plane from simulation . 95 4-16 Trajectories plotted in the 푥-푦 plane from hardware tests . 96 4-17 A comparison of commanded position and attitude vs. measured teleme- try from hardware tests . 97 4-18 Adaptive PID SMC time histories corresponding to position and atti- tude from hardware tests . 98 5-1 Chaser and Target during instance before docking occurs denoted with separate position, velocity, and attitude vectors .