Pursuit-Evasion and Differential Games Under Uncertainties
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PURSUIT-EVASION AND DIFFERENTIAL GAMES UNDER UNCERTAINTIES A Thesis Presented to The Academic Faculty by Wei Sun In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Aerospace Engineering Georgia Institute of Technology August 2017 Copyright c 2017 by Wei Sun PURSUIT-EVASION AND DIFFERENTIAL GAMES UNDER UNCERTAINTIES Approved by: Professor Panagiotis Tsiotras Professor Eric N. Johnson Committee Chair, Advisor School of Aerospace Engineering School of Aerospace Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Evangelos A. Theodorou, Professor Anthony J. Yezzi Co-Advisor School of Electrical and Computer School of Aerospace Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Professor Eric Feron Date Approved: 17 May 2017 School of Aerospace Engineering Georgia Institute of Technology To my parents and my sister. iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. P. Tsiotras for his support and guidance throughout my PhD studies. I would also like to extend my appreciation to my co-advisor Dr. E. A. Theodorou and other committee members: Dr. E. M. Feron and Dr. A. Yezzi for their true interest in evaluating my work. I owe the greatest debt of gratitude to my parents and my sister for their never ending support during my graduate studies. This work is dedicated to them as a token of appreciation. Lastly, but by no means least, I wish to thank my labmates Dr. Efstathios Bakolas, Dr. Nuno Filipe, Dr. Oktay Arslan, Mr. Ioannis Exarchos, Mr. Spyridon Zafeiropou- los, Mr. Florian Hauer, Mr. Changxi You among others who have given me many inspirations during my years at Georgia Tech. I would like to extend my thanks to Dr. Andrea L’Afflitto, Dr. Haocao Li, Mr. Tanmay Rajpurohit, Mr. Yunpeng Pan, who have been talented and considerate friends with me. Furthermore, the help in my living and studying by Frank Jiang and his family is gratefully appreciated. iv TABLE OF CONTENTS DEDICATION .................................. iii ACKNOWLEDGEMENTS .......................... iv LIST OF FIGURES .............................. ix SUMMARY .................................... xiii I INTRODUCTION ............................. 1 1.1 Motivation and Goals . .1 1.2 Task Assignment Problem of Spatially Distributed Group of Autonomous Agents and Voronoi Diagram Partitions . .4 1.3 Multiplayer Pursuit-Evasion Problem . .6 1.4 Differential Dynamic Programming . .9 1.5 Comments on the Structure of the Dissertation . 10 1.6 Chapter Description . 12 II A SEQUENTIAL PURSUER-TARGET ASSIGNMENT PROB- LEM UNDER EXTERNAL DISTURBANCES ........... 16 2.1 Introduction . 16 2.2 Preliminaries . 18 2.3 Problem Setup . 20 2.4 Analysis and implementation of the pursuer-target assignment problem 23 2.5 Update algorithm to dynamically generate the ZVD . 30 2.6 Sequential Pursuit of Multiple Targets . 34 2.7 Simulation Results . 37 III PURSUIT-EVASION GAMES IN LINEAR FLOW FIELDS .. 46 3.1 Introduction . 46 3.2 Problem Formulation . 46 3.2.1 Problem Setup . 46 3.2.2 Differential Game Formulation in the Reduced Space . 47 v 3.3 Problem Analysis . 49 3.3.1 Effect of External Field . 49 3.3.2 The Game of Kind . 50 3.3.3 Game of Degree in the Capture Region . 57 3.4 Simulation Results . 59 IV PURSUIT-EVASION GAMES IN GENERAL FLOW FIELDS . 66 4.1 Introduction . 66 4.2 Problem Formulation . 67 4.3 Problem Analysis . 70 4.3.1 Reachable Sets . 70 4.4 Numerical Construction . 75 4.4.1 Level Set Method . 75 4.4.2 Classification of Pursuers . 78 4.4.3 Time-Optimal Paths . 84 4.5 Simulation Results . 86 V PURSUIT-EVASION GAMES UNDER 3-DIMENSIONAL FLOW FIELDS .................................... 95 5.1 Introduction . 95 5.2 Problem Formulation . 97 5.3 Problem Analysis . 99 5.3.1 Pursuer Assignment . 99 5.3.2 Multiple-Pursuers/One-Evader Game . 101 5.4 Numerical Construction . 103 5.4.1 Narrow Band Level Set Method . 103 5.4.2 Pursuer Classification . 107 5.4.3 Time-Optimal Paths . 108 5.5 Simulation Results . 110 VI PURSUIT-EVASION GAMES UNDER STOCHASTIC FLOW FIELDS .................................... 115 vi 6.1 Problem Formulation . 115 6.2 Problem Analysis . 117 6.2.1 Moment Expansion . 117 6.2.2 Reachable Sets . 118 6.3 Numerical Solution . 124 6.3.1 Level Set Method . 124 6.4 Simulation Results . 125 VII GAME THEORETIC CONTINUOUS TIME DIFFERENTIAL DY- NAMIC PROGRAMMING ....................... 127 7.1 Introduction . 127 7.2 Problem Formulation . 128 7.2.1 Derivation of the Backward Propagation of the Value Function 135 7.3 Terminal Condition and the Minimax DDP Algorithm . 136 7.4 Simulation Results . 138 7.4.1 Inverted Pendulum Problem . 138 7.4.2 Inverted Pendulum Problem with Stochastic Disturbances . 138 7.4.3 Two-player Pursuit Evasion Game Under External Flow Field 140 VIIISTOCHASTIC GAME THEORETIC CONTINUOUS TIME DIF- FERENTIAL DYNAMIC PROGRAMMING ............ 147 8.1 Introduction . 147 8.2 Problem Formulation . 148 8.3 Optimal Control Variations . 149 8.4 Backward Propagation of the Value Function . 154 8.5 Simulation Results . 159 IX CONCLUSIONS AND FUTURE WORK .............. 166 9.1 Conclusions . 166 9.2 Projected Reachable Set Approach in Pursuit-Evasion Games . 169 9.3 Differential Games under Dynamical Uncertainties with Learned Dy- namics . 172 vii VITA ........................................ 193 viii LIST OF FIGURES 1 Flip-edge method for generating the Delaunay triangulation. 18 2 Unembeding caused by relocation of a generator. 32 4 3 Zermelo-Voronoi Diagram formed by pursuers at t = 0, XP is the active pursuer . 38 4 Zermelo-Voronoi Diagram formed by pursuers at the first switch time t = 2:6, and trajectories of pursuers and target for t 2 [0; 2:6). 39 5 Zermelo-Voronoi Diagram formed by pursuers at the capture time Tc = 5 5:0, and trajectories of pursuers and target for t 2 [2:6; 5:0], XP is the active pursuer. 40 6 Pursuers and evaders at t = 0. Pursuer 2 is paired with Target 1 and 3. Purser 4 is paired with Target 2. Red curves represents the ZVD at t =0. ................................... 41 7 Trajectories of pursuers and evaders in [0; s1]. Pursuer 2 goes after Target 1 during this period. Target 3 enters the Zermelo-Voronoi cell of Pursuer 5 at this time step and it will be paired with Pursuer 5 henceforth. ZVD at time s1 is depicted in red curves. 42 8 Trajectories of players in [s1; s2]. The target to the left in cyan is captured by Pursuer 2 at time s2. Red curves represents the ZVD at s2. 42 9 Trajectories of players in [s2; s3]. The target to the right in magenta is captured by Pursuer 4 at time s3. Red curves represents the ZVD at s3. 43 10 Trajectories of players in [s3; s4]. The only target left is captured by Pursuer 5 at time s4. Red curves represents the ZVD at the terminal time s4................................... 43 11 Pursuers and evaders at t = 0. Pursuer 2; 4 and 5 are assigned to Target 1; 2 and 3, respectively. Each target and its corresponding active pursuer are depicted by opposite colors for easier recognition, e.g., Target 3 is in yellow and Pursuer 5 is in blue. Red curves represents the ZVD at t =0. ............................ 44 12 Trajectories of players in [0; t1]. The target to the left in cyan is cap- tured by Pursuer 2 at time t1. Red curves represents the ZVD at t1. ....................................... 44 13 Trajectories of players in [t1; t2]. The target to the right in magenta is captured by Pursuer 4 at time t2. Red curves represents the ZVD at t2. 45 ix 14 Trajectories of players in [t2; t3]. The only target left is captured by Pursuer 5 at time t3. Red curves represents the ZVD at the terminal time t3................................... 45 15 The terminal surface C of the game is given by a circle of radius `. The circle is separated by the BUP (4 points on the circle parameterized by s1 through s4) into the usable part (black lines) and the nonusable part (red lines). Every barrier meets the terminal surface at the BUP tangentially. 54 16 Barriers and optmal trajectories of pursuit-evasion problem when A = [0; 10; −5; 0]................................. 60 17 Barriers and optmal trajectories of pursuit-evasion problem when A = [1:4020; −1:0772; 1:4770; 0:7756]. 61 18 Barriers and optmal trajectories of pursuit-evasion problem when A = [0:3188; −0:4336; −1:3077; 0:3426]. 63 19 Velocity components of the system (72) and (73), where the blue and dashed red lines are with respect to the trajectories emanating from R1 and R2, respectively. 63 20 Barriers and optmal trajectories of pursuit-evasion problem when A = [1; 2; 2; 1]. ................................. 64 21 Evolution of the reachability fronts between two pursuers and one evader. 81 22 Level sets of three pursuers and one evader at time T . Here Xf denotes the capture point. 82 23 Level sets of five pursuers and one evader at time T . Pursuers P4 and P5 are each redundant pursuer by definition, but they cannot be removed together. 83 24 Level sets of four pursuers and one evader at time T . Pursuers P1, P2 and P3 are active pursuers, and P4 is a guard. Capture occurs at point Xf ...................................... 84 25 The optimal trajectory and the intermediate reachability front of the evader are shown in dashed blue. The optimal trajectory of the active pursuer is shown in green. 88 26 Optimal trajectories of the three active pursuers in green, red, cyan. Red, cyan and green closed curves represent the reachable fronts of the pursuers at the terminal time. 89 27 Optimal trajectories of the three active pursuers and the reachable fronts of the pursuers at the terminal time.