Time-Optimal Motion Strategies for Capturing an Omnidirectional Evader Using a Differential Drive Robot

In this paper, we consider the problem of capturing an omnidirectional evader using a differential drive robot in an obstacle-free environment. At the beginning of this game, the evader is at a distance (the capture distance) from the pursuer. The goal of the evader is to keep the pursuer farther than this capture distance for as long as possible. The goal of the pursuer is to capture the evader as soon as possible. In this paper, we make the following contributions. We present closed-form representations of the motion primitives and time-optimal strategies for each player; these strategies are in Nash equilibrium, meaning that any unilateral deviation of each player from these strategies does not provide to such player benefit toward the goal of winning the game. We propose a partition of the playing space into mutually disjoint regions where the strategies of the players are well established. This partition is represented as a graph, which exhibits properties that guarantee global optimality. We also analyze the decision problem of the game and we present the conditions defining the winner.

[1]  Steven M. LaValle,et al.  Visibility-based Pursuit—Evasion with Bounded Speed , 2008, Int. J. Robotics Res..

[2]  Marcelo H. Ang,et al.  Motion Planning for 3-D Target Tracking among Obstacles , 2007, ISRR.

[3]  Rafael Murrieta-Cid,et al.  A homicidal differential drive robot , 2012, 2012 IEEE International Conference on Robotics and Automation.

[4]  Antony W Merz,et al.  The Homicidal Chauffeur - A Differential Game , 1971 .

[5]  T. Başar,et al.  Dynamic Noncooperative Game Theory, 2nd Edition , 1998 .

[6]  Lynne E. Parker,et al.  Distributed Algorithms for Multi-Robot Observation of Multiple Moving Targets , 2002, Auton. Robots.

[7]  Geoffrey A. Hollinger,et al.  Search and pursuit-evasion in mobile robotics , 2011, Auton. Robots.

[8]  Alejandro Sarmiento,et al.  Surveillance Strategies for a Pursuer with Finite Sensor Range , 2007, Int. J. Robotics Res..

[9]  Rafael Murrieta-Cid,et al.  Tracking an omnidirectional evader with a differential drive robot , 2011, Auton. Robots.

[10]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[11]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[12]  Tomás Martínez-Marín,et al.  Optimal motion planning by reinforcement learning in autonomous mobile vehicles , 2011, Robotica.

[13]  Jason M. O'Kane,et al.  On the Value of Ignorance: Balancing Tracking and Privacy Using a Two-Bit Sensor , 2008, WAFR.

[14]  Leonidas J. Guibas,et al.  Visibility-Based Pursuit-Evasion in a Polygonal Environment , 1997, WADS.

[15]  S. Shankar Sastry,et al.  Probabilistic pursuit-evasion games: theory, implementation, and experimental evaluation , 2002, IEEE Trans. Robotics Autom..

[16]  Héctor H. González-Baños,et al.  Real-time combinatorial tracking of a target moving unpredictably among obstacles , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[17]  Tomás Martínez-Marín,et al.  Integration of Cell-Mapping and Reinforcement-Learning Techniques for Motion Planning of Car-Like Robots , 2009, IEEE Transactions on Instrumentation and Measurement.

[18]  Sourabh Bhattacharya,et al.  On the Existence of Nash Equilibrium for a Two-player Pursuit - Evasion Game with Visibility Constraints , 2008, Int. J. Robotics Res..

[19]  Héctor H. González-Baños,et al.  Motion strategies for maintaining visibility of a moving target , 1997, Proceedings of International Conference on Robotics and Automation.

[20]  A. Melikyan.  Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games , 2012 .

[21]  Pushkin Kachroo,et al.  Differential Drive Robot , 2007 .

[22]  Leonidas J. Guibas,et al.  A Visibility-Based Pursuit-Evasion Problem , 1999, Int. J. Comput. Geom. Appl..

[23]  Alon Efrat Héctor H. González-Baños Stephen G. Kobourov Palaniappan.  Optimal Motion Strategies to Track and Capture a Predictable Target , .

[24]  P. Souéres,et al.  Shortest paths synthesis for a car-like robot , 1996, IEEE Trans. Autom. Control..

[25]  Masafumi Yamashita,et al.  Searching for a Mobile Intruder in a Polygonal Region , 1992, SIAM J. Comput..

[26]  Timothy H. Chung.  On Probabilistic Search Decisions under Searcher Motion Constraints , 2008, WAFR.

[27]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[28]  Devin J. Balkcom,et al.  Time Optimal Trajectories for Bounded Velocity Differential Drive Vehicles , 2002, Int. J. Robotics Res..

[29]  Volkan Isler,et al.  Capturing an Evader in a Polygonal Environment with Obstacles , 2011, IJCAI.

[30]  Wei Yang,et al.  Robotic Routers: Algorithms and Implementation , 2009, Int. J. Robotics Res..

[31]  S. Sastry,et al.  Probabilistic pursuit-evasion games: a one-step Nash approach , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[32]  Arik Melikyan,et al.  Generalized characteristics of first order PDEs , 1998 .

[33]  Sampath Kannan,et al.  Randomized pursuit-evasion in a polygonal environment , 2005, IEEE Transactions on Robotics.

[34]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[35]  Huifang Wang,et al.  A Geometric Algorithm to Compute Time-Optimal Trajectories for a Bidirectional Steered Robot , 2009, IEEE Transactions on Robotics.

[36]  Rufus Isaacs,et al.  Differential Games , 1965 .

[37]  Rafael Murrieta-Cid,et al.  A Sampling-Based Motion Planning Approach to Maintain Visibility of Unpredictable Targets , 2005, Auton. Robots.

[38]  Gaurav S. Sukhatme,et al.  Tracking Targets Using Multiple Robots: The Effect of Environment Occlusion , 2002, Auton. Robots.

[39]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.