Numerical Solution of the Three-Dimensional Orbital Pursuit-Evasion Game

Abstract

The problem of interception of an evasive spacecraft by a pursuing spacecraft is formulated as a differential game. Each spacecraft is given a modest capability to maneuver in the three-dimensional space. Interception concludes the game and occurs if the pursuing spacecraft reaches the instantaneous position of the evading spacecraft. The objective of the pursuer is to minimize the time for interception, whereas the evader tries to delay it indefinitely. Saddle-point equilibrium solutions are found using a recently developed direct numerical method that uses the analytical necessary conditions (unlike ordinary direct methods) to find the optimal control for one of the players. The method requires an initial guess of the solution, and this is provided by generating an approximate solution using genetic algorithms. The evolutionary algorithm is employed as a preprocessing technique and is very useful in this context because the trial-and-error selection of first-attempt values for the variables involved is very challenging for the problem at hand. The numerical method is tested on a variety of different starting conditions related to the distinct initial orbits of the two spacecraft. It successfully finds the saddle-point trajectories of the two spacecraft, thus proving its effectiveness and robustness in solving a quite complicated problem such as the three-dimensional orbital game at hand.