Approximate Analytical Approach for Spacecraft Pursuit–Evasion Game With Reachability Analysis

The problem of spacecraft pursuit–evasion game is typically represented as a two-player zero-sum differential game. The two-point boundary value problem derived from the Pontryagin minimum principle is usually solved by indirect heuristic or nonlinear programming methods to obtain the saddle point solution. However, these methods are computationally expensive and unsuitable for spacecraft on-orbit implementation. This article proposes an approximate analytical approach by combining optimal control and forward reachable set to solve the pursuit–evasion game efficiently. First, a two-sided optimal control problem is converted into an equivalent one-sided minimum-time problem. Then, an explicit and analytical ellipsoid boundary of the approximate reachable set is proposed to determine the optimal terminal time. The approximation strategy can be obtained simultaneously with the proposed terminal geometric conditions. The numerical simulations demonstrate the effectiveness and efficiency of the proposed real-time approach and at least three orders of magnitude faster compared to existing methods. In addition, a Monte Carlo simulation is performed to assess the numerical robustness and reliability of the method, especially its superiority for the short-term game.