Locating and Capturing an Evader in a Polygonal Environment

This paper contains two main results: First, we revisit the well-known visibility based pursuit-evasion problem and show that, in contrast to deterministic strategies, a single pursuer can locate an unpredictable evader in any simply-connected polygonal environment using a randomized strategy. The evader can be arbitrarily faster than the pursuer and it may know the position of the pursuer at all times but it does not have prior knowledge of the random decisions made by the pursuer. Second, using the randomized algorithm together with the solution of a known lion and man problem [1] as subroutines, we present a strategy for two pursuers (one of which is at least as fast as the evader) to quickly capture an evader in a simply-connected polygonal environment. We show how this strategy can be extended to obtain a strategy for (i) capturing the evader in a polygonal room with a door, (ii) two pursuers who have only line-of-sight communication, and (iii) a single pursuer (at the expense of increased capture time).

[1]  Edward M. Reingold,et al.  "Lion and Man": Upper and Lower Bounds , 1992, INFORMS J. Comput..

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

[3]  J. C. Burkill,et al.  A mathematician's miscellany , 1954 .

[4]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[5]  George J. Pappas,et al.  Greedy control for hybrid pursuit games , 2001, 2001 European Control Conference (ECC).

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

[7]  Hyoun-jin Kim.  Multiple-Agent Probabilistic Pursuit-Evasion Games 1 , 1999 .

[8]  J. Mark Keil,et al.  Decomposing a Polygon into Simpler Components , 1985, SIAM J. Comput..

[9]  Leonidas J. Guibas,et al.  Finding an unpredictable target in a workspace with obstacles , 1997, Proceedings of International Conference on Robotics and Automation.

[10]  Alejandro Sarmiento,et al.  An efficient strategy for rapidly finding an object in a polygonal world , 2003, Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453).

[11]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[12]  Steven M. LaValle,et al.  Optimal navigation and object finding without geometric maps or localization , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[13]  George J. Pappas,et al.  Greedy Control for Hybrid Pursuit-Evasion GamesJo , 2001 .

[14]  Steven M. LaValle,et al.  Visibility-based pursuit-evasion: the case of curved environments , 2001, IEEE Trans. Robotics Autom..

[15]  Kyung-Yong Chwa,et al.  Visibility-Based Pursuit-Evasion in a Polygonal Region by a Searcher , 2001, ICALP.

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

[17]  Kyung-Yong Chwa,et al.  Searching a Polygonal Room with One Door by a 1-Searcher , 2000, Int. J. Comput. Geom. Appl..

[18]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[19]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

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

[21]  J.P. Hespanha,et al.  Multiple-agent probabilistic pursuit-evasion games , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[22]  Jirí Sgall.  Solution of David Gale's lion and man problem , 2001, Theor. Comput. Sci..