Maintaining strong mutual visibility of an evader moving over the reduced visibility graph

In this paper, we address the problem of determining whether a mobile robot, called the pursuer, is able to maintain strong mutual visibility (a visibility notion between regions over a convex partition of the environment) of an antagonist agent, called the evader. We frame the problem as a non cooperative game. We consider the case in which the pursuer and the evader move at bounded speed, traveling in a known polygonal environment with or without holes, and in which there are no restrictions as to the distance that might separate the agents. Unlike our previous efforts (Murrieta-Cid et al. in Int J Robot Res 26:233–253, 2007), we give special attention to the combinatorial problem that arises when searching for a solution through visiting several locations in an environment with obstacles. In this paper A preliminary version of a portion of this work has been presented at the IEEE International Conference on Robotics and Automation (Murrieta-Cid et al. 2008). B Israel Becerra [email protected] Rafael Murrieta-Cid [email protected] Raul Monroy [email protected] Seth Hutchinson [email protected] Jean-Paul Laumond [email protected] 1 Centro de Investigación en Matemáticas, CIMAT, Guanajuato, Mexico 2 Tecnológico de Monterrey, Escuela de Ingeniería y Ciencias, Atizapan, Estado de México, Mexico 3 University of Illinois at Urbana-Champaign, Urbana, IL, USA 4 LAAS-CNRS, University of Toulouse, Toulouse, France we take a step further, namely, we assume an antagonistic evader who moves continuously and unpredictably, but with a constraint over its set of admissible motion policies, as the evader moves in the shortest-path roadmap, also called the reduced visibility graph (RVG). The pursuer does not know which among the possible paths over the RVG the evader will choose, but the pursuer is free to move within all the environment.We provide a constructivemethod to solve the decision problem of determining whether or not the pursuer is able to maintain strong mutual visibility of the evader. This method is based on an algorithm that computes the safe areas (areas that keep evader surveillance) at all times. We prove decidability of this problem, and provide a complexity measure to this evader surveillance game; both contributions hold for any general polygonal environment that might or not contain holes. All our algorithms have been implemented and we show simulation results.