A DIFFERENTIAL GAMES APPROACH FOR ANALYSIS OF SPACECRAFT POST-DOCKING OPERATIONS By TAKASHI HIRAMATSU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 c 2012 Takashi Hiramatsu 2 I dedicate this to everyone that helped me write this manuscript. 3 ACKNOWLEDGMENTS My biggest appreciation goes to my advisor Dr. Norman G. Fitz-Coy for his great help and support. Every time I talked to him he motivated me with critical responses and encouraged me whenever I was stuck in the middle of my research. I also thank my committee Dr. Warren Dixon, Dr. Gloria Wiens, and Dr. William Hager for their supports. Finally, I thank my colleagues in Space Systems Group and all other friends, who directly or indirectly helped me throughout the years I spent at University of Florida. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS..................................4 LIST OF TABLES......................................8 LIST OF FIGURES.....................................9 ABSTRACT......................................... 11 CHAPTER 1 INTRODUCTION................................... 13 1.1 Spacecraft Rendezvous and Docking..................... 13 1.1.1 Cooperative Scenarios......................... 14 1.1.2 Noncooperative Scenarios....................... 14 1.2 Small Satellites................................. 15 1.3 Game Theoretic Approach........................... 15 2 MATHEMATICAL BACKGROUND FOR THE APPROACH............ 18 2.1 Differential Games and Control Theory.................... 18 2.1.1 Minimax Strategy............................ 20 2.1.2 Nash Strategy.............................. 20 2.1.3 Stackelberg Strategy.......................... 21 2.1.4 Open-Loop Strategies for Two-Person Linear Quadratic Differential Games.................................. 23 2.2 Numerical Methods to Optimal Control Problem............... 24 2.3 Bilevel Programming.............................. 25 3 TECHNICAL DESCRIPTION............................ 27 3.1 Reduction of Stackelberg Differential Games to Optimal Control...... 30 3.2 Conversion of Stackelberg Differential Games to Stackelberg Static Games 32 3.3 Costate Mapping of Stackelberg Differential Games............ 35 3.4 Conclusion................................... 45 4 DYNAMICS OF DOCKED SPACECRAFT..................... 47 4.1 Formulation of Dynamics........................... 47 4.1.1 Relative Motion Dynamics of a Satellite............... 47 4.1.1.1 Translation.......................... 48 4.1.1.2 Rotation............................ 48 4.1.2 Dynamics of Two Docked Satellites.................. 49 4.2 Simulation.................................... 52 4.2.1 Case I: Nonzero Linear Velocity.................... 52 4.2.2 Case II: Nonzero Rotational Velocity................. 58 5 4.2.3 Case III: Nonzero Linear and Rotational Velocities......... 62 4.3 Conclusion................................... 66 5 LINEAR CONTROLLER DESIGN WITH STACKELBERG STRATEGY..... 67 5.1 Post-Docking Study with Linear Quadratic Game.............. 67 5.2 Simulation and Results............................ 69 5.3 Conclusion................................... 75 6 SOLUTIONS TO TWO-PLAYER LINEAR QUADRATIC STACKELBERG GAMES WITH TIME-VARYING STRUCTURE........................ 77 6.1 Game Based on Additive Errors........................ 78 6.1.1 Open-loop Stackelberg Solution.................... 80 6.1.2 Closed-loop Stackelberg Solution................... 85 6.2 Game Based on Multiplicative Errors..................... 92 6.2.1 Open-loop Stackelberg Solution.................... 94 6.2.2 Closed-loop Stackelberg Solution................... 99 6.3 Simulations and Results............................ 99 6.4 Conclusion................................... 106 7 CONCLUSION AND FUTURE WORKS...................... 108 APPENDIX A OPTIMALITY CONDITIONS OF TWO-PERSON STACKELBERG DIFFERENTIAL GAMES........................................ 109 A.1 Fixed Final Time................................ 109 A.1.1 Follower’s Strategy........................... 109 A.1.1.1 Variation of the augmented cost functional........ 110 A.1.1.2 Optimality conditions.................... 112 A.1.2 Leader’s Strategy............................ 112 A.1.2.1 Variation of the augmented cost functional........ 113 A.1.2.2 Optimality conditions.................... 115 A.2 Free Final Time................................. 115 A.2.1 Follower’s Strategy........................... 116 A.2.1.1 Variation of the augmented cost functional........ 116 A.2.1.2 Optimality conditions.................... 118 A.2.2 Leader’s strategy............................ 119 A.2.2.1 Variation of the augmented cost functional........ 119 A.2.2.2 Optimality conditions.................... 122 A.3 Linear-Quadratic Differential Game...................... 122 A.3.1 Fixed Final Time............................ 123 A.3.2 Free Final Time............................. 123 6 B RISE STABILITY ANALYSIS............................ 126 B.1 Rise Feedback Control Development..................... 126 B.2 Stability Analysis................................ 128 C COSTATE ESTIMATION FOR THE TRANSCRIBED STACKELBERG GAMES 132 C.1 Transformed Optimality Conditions...................... 132 C.2 Discretization of Two-person Stackelberg Differential Games....... 134 C.3 KKT Conditions and Costate Mapping.................... 136 REFERENCES....................................... 141 BIOGRAPHICAL SKETCH................................ 147 7 LIST OF TABLES Table page 4-1 The simulation parameters for Case I....................... 55 4-2 The simulation parameters for Case II....................... 59 4-3 The simulation parameters for Case III....................... 63 5-1 The simulation parameters for the linear quadratic game............. 72 6-1 The simulation parameters for the Stackelberg-RISE controller......... 105 8 LIST OF FIGURES Figure page 1-1 A design iteration through satellite post-docking analysis............. 17 3-1 Relationship among optimization problems.................... 29 3-2 Direct and indirect methods............................. 30 4-1 A representation of the position of a satellite with the inertial and the nominal reference frames.................................... 47 4-2 A satellite with a body-fixed reference frame i .................. 48 F 4-3 An exeggerated view of two satellites near the nominal orbit........... 50 4-4 Two satellites initially on the same nominal orbit.................. 53 4-5 Two satellites initially radially aligned........................ 54 4-6 Case I: the interaction forces applied to the SV and the RSO.......... 54 4-7 Case I: the interaction torques applied to the SV and the RSO......... 56 4-8 Case I: the linear motion of the RSO relative to the SV.............. 56 4-9 Case I: the rotational motion of the RSO relative to the SV............ 57 4-10 Case II: the interaction forces applied to the SV and the RSO.......... 58 4-11 Case II: the interaction torques applied to the SV and the RSO......... 60 4-12 Case II: the linear motion of the RSO relative to the SV............. 60 4-13 Case II: the rotational motion of the RSO relative to the SV........... 61 4-14 Case III: the interaction forces applied to the SV and the RSO.......... 62 4-15 Case III: the interaction torques applied to the SV and the RSO......... 64 4-16 Case III: the linear motion of the RSO relative to the SV............. 64 4-17 Case III: the rotational motion of the RSO relative to the SV........... 65 5-1 Two rigid bodies on circular orbits.......................... 67 5-2 The resultant trajectory................................ 73 5-3 The control force inputs................................ 74 5-4 The control torque inputs............................... 75 6-1 The relationship among the current and the desired orientations......... 92 9 6-2 Two docked satellites approximated as two rigid bodies connected via a torsion spring......................................... 100 6-3 f (t) and g(t) as respective weights on the game and an arbitrary disturbances. 104 6-4 The simulation results for the Stackelberg and RISE controller......... 107 10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A DIFFERENTIAL GAMES APPROACH FOR ANALYSIS OF SPACECRAFT POST-DOCKING OPERATIONS By Takashi Hiramatsu August 2012 Chair: Norman G. Fitz-Coy Major: Mechanical Engineering An increase of responsive space assets will contribute to the growing number of spacecraft in space and in turn the growing potential for failures to occur. The number of spacecraft which has past its operational life also keeps increasing. Without proper treatment these satellites become space debris, which could lead to more failure due to collision with other spacecraft. Thus, there will be a need for effective debris abatement (i.e., repair and /or disposal of failed satellites) which will require autonomous service satellites. Such a “tow truck” concept is expected to take a crucial role for sustainable small satellite utilization in the future. Current and past investigation where autonomous docking plays an important role have all considered “cooperative” interactions between satellites. That is, either the target has the same goals as the service vehicle or the target is not actuated and passively follows the lead of the service vehicle. Cooperative scenarios are not always guaranteed, thus it is imperative that we consider