Three-body confrontation differential game for an aggressive attacker

In this work, we study the three-body confrontation differential game, in which an Attacker attempts to capture a Target while a Defender guards the latter. We decompose the three-body game into two subgames based on the game of kind theory: the included angle game and the distance ratio game. By integrating the Apollonius circle with these subgames, we define the semipermeable surface and infer the optimal guidance strategies for the three players on this surface. Unlike existing methods, our method effectively addresses the coupling relationship between the target, defender, and attacker, resulting in greater applicability. Based on the derived semipermeable surface, the conditions for the three players to win the game are also presented. These winning conditions are comprehensively considered from both lines of sight, including the angle and the ratio between the Attacker-Target distance and the Attacker-Defender distance, thereby providing a game solution through an analysis of the players' dynamics. From the perspective of viewing angle ratio, our method bridges the gap in the viewing angle game outcomes compared to existing methods. To demonstrate the effectiveness of our method, we provide simulation examples that illustrate the three-body confrontation game in various scenarios.