Approximate Analytical Winning Regions of the Spacecraft in the Coplanar Pursuit–Evasion–Defense Game

In a pursuit–evasion game, when the evader is unable to escape from the pursuer, the existence of a defender who can defend the evader may change the game outcome. Then, under what conditions (e.g., initial positions, velocities) can the defender guarantee a successful protection? This becomes a significant problem. This study explored this pursuit–evasion–defense game among three coplanar spacecraft, aimed to find the initial orbit conditions that can guarantee the defender–evader (DE) team wins, called winning regions. First, a limit displacement circle method is employed to locate the regions where the defender can arrive earlier than the pursuer, called interception-dominant regions. Based on that, an explicit strategy approach is proposed to transform the solution of the winning region into a cubic or quartic equation regarding the initial state (initial orbital radius ratio, thrust acceleration ratio, and energy constraint). Finally, the approximate analytical expression of the winning region of the DE team was obtained by solving the cubic or quartic equations. Simulation results demonstrate that the computational efficiency of this solution is three orders of magnitude greater than that of numerical methods.