Pursiut-Evasion Games in Presence of Obstacles in Unknown Environments: towards an Optimal Pursuit Strategy

Abstract

In this paper, we will incrementally build a complete pursuit algorithm to deal with a 2-players PEG in presence of a single unknown convex obstacle. We will first provide a sufficient condition to achieve capture without disappearance. Then, we will solve the circular obstacle problem, a particular problem highlighting a necessary trade-off between surveillance and capture. Next, the pole problem, as a generalization, of the convex obstacle problem will be tackled. The solution and the corresponding strategies will be detailed. A quasi-optimal pursuit strategy as regards the time to capture will be provided for the pole problem, and then transposed for the more general convex obstacle problem. For the cases leading to the evader victory in the pole problem, a last strategy allowing a maximal deviation of the line of disappearance will be added to complete to our pursuit algorithm. Finally, our complete pursuit algorithm will be adapted to use a heuristic minimization method instead of the strategy suggested by the resolution of the pole problem for the cases leading to the pursuer victory. Different heuristics, one being an approximation of the solution of the pole problem, will be compared with respect to the size of the capture basin and will highlight thez interest of our pursuit algorithm.

An Introduction to Pursuit-evasion Differential Games

“…Yavin proposed an incomplete information pursuit-evasion differential game by restricting the pursuer's information on bearing and allowed the Evader to have perfect information in the engagement, [35]. Giovannangeli, Heymann and Rivlin tackled the problem of pursuit while avoiding convex obstacles by using Apollonius circles to provide paths inwhich the pursuer's visibility of the evader is guaranteed throughout the engagement, [36]; the geometric concept of Apollonius circle will be formally defined in Section IV-B. In [37], Hexner considered the problem where a parameter is unavailable to only one player at the beginning of the game, and the other has a probability density function describing that parameter was described.…”

Dominance in pursuit-evasion games with uncertainty

“…Reference [11] considers PE in an unknown environment with visibility constraints, and it uses Apollonius circles to identify regions in the game space where capture occurs before loss of visibility. Since visibility constraints are included, the analysis considers only the portion of the standard Apollonius circle that can be reached by both players along straight-line paths, but it does not consider the overall structure of the dominance region or how the dominance regions change due to the presence of the obstacle.…”

Predictive Sampling-Based Robot Motion Planning in Unmodeled Dynamic Environments [Slides]

“…where • is the Euclidean norm [10]. Given a fixed velocity of the evader, the pursuer's optimal strategy consists of moving straight to the point x a on the Apollonius circle that intersects the trajectory of the evader.…”