An Isochron-Based Solution to Pursuit–Evasion Games of Two Heterogeneous Players

Abstract

In this article, we study a pursuit–evasion game between two players with heterogeneous kinematics, where the pursuer is with damped double-integrator dynamics and the evader is with single-integrator dynamics. The pursuer aims at capturing the evader as soon as possible, while the evader wants to avoid or delay the capture. Traditional methods to solve pursuit–evasion games rely on the Hamilton–Jacobi–Isaacs (HJI) equations and retrogressive path equations, which are very complicated and nonintuitive, thus failing to obtain a complete solution. To overcome these challenges, we develop an intuitive isochron-based method to thoroughly analyze all possible situations of the game and a concise geometric approach to calculate the optimal strategies, providing a complete solution to this game. Specifically, the isochron-based method effectively leverages three main factors: the players' motion capability, the pursuer's capture capability, and the players' states. Based on these, we analyze the players' superiority and the geometrical features of their isochrones and the intersections, thus acquiring concise conditions that determine the game's outcome. For the success-capture cases, we propose a new geometric approach to calculate the target points of the players and then obtain the closed-loop state feedback optimal pursuit and evasion strategies. We then get the corresponding value function and provide a validation using the HJI equation. For the success-evasion cases, we exploit the intersection of the players' isochrones to design some effective evasion strategies, which ensure that the evader can always avoid or delay the capture. Finally, some numerical simulations are carried out to validate the effectiveness and applicability of our results.