In this work, we analyze approximations of capture sets [1] for a visibility based pursuit-evasion game. In contrast to the capture problem, the pursuer tries to maintain a line-of-sight with the evader in free space in our problem. We extend the concept of U set initially proposed in [2] for holonomic players to the scenario in which the pursuer is holonomic. The problem of computing the U set is reduced to that of computing time-optimal paths for the non-holonomic vehicles to an arbitrary line. We characterize the primitives for time-optimal paths for the Dubin’s vehicle, Reed-shepps car and a Differential Drive robot. Based on these primitives, we construct the optimal paths and provide an algorithm to compute the U set.

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