Collision-Free and Guaranteed Capture Winning Strategies for Reach-Avoid Games with Two Heterogeneous Pursuers and One Evader

This paper investigates reach-avoid differential games involving two pursuers and one evader. The pursuers with different capture radii collaborate to protect a target region against the evader whose objective is to reach the target region while avoiding being captured by the pursuers. Previous studies have shown that the cooperation between pursuers can improve their capability of defending the target region against the evader. However, these studies did not account for the collision avoidance between the pursuers when designing pursuit strategies. In this work, the pursuers have different safety radii and must avoid collisions with each other. A feedback capture strategy is proposed to ensure that from a set of states, the pursuers can win the game with collision-free trajectories. This strategy relies on solving an optimization problem induced by the relaxed control barrier function (R-CBF) and a new geometric concept called the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex>-evasion space (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex>-ES). Sufficient conditions to guarantee both the pursuit winning and collision avoidance are presented. Numerical simulations are provided to show the collision-free trajectories of the pursuers under the proposed capture strategy.