Abstract: The orbital game between spacecraft has significant importance for space safety. Compared to the classical "one-to-one" orbital game problem, the "many-to-one" orbital game faces challenges such as high-dimensional state parameters, undefined roles of pursuers, and variable terminal conditions, making traditional differential strategy methods difficult to solve. To overcome these challenges, this paper proposes a geometric method for the "many-to-one" orbital game based on the reachable set of spacecraft. Firstly, the reachable set equivalence representation of Nash equilibrium points in the orbital game is established based on the theory of reachable sets of spacecraft. Then, the envelopment of the reachable set of spacecraft at any time is accurately solved using a grid point search method. Subsequently, the relative geometric relationship between the reachable sets of spacecraft in-volved in the game is determined by computing the solid angle of the triangulated closed surface. Finally, the binary search method is used to determine the terminal time of the game, satisfying the geometric conditions of the reachable set of Nash equi-librium points, thereby determining the spatial position coordinates of the spacecraft at the end of the game and completing the solution to the "many-to-one" orbital game problem. Simulation results demonstrate that for a typical "three-to-one" orbital game scenario, the traditional differential strategy method takes over 2 hours on an ordinary personal computer. In contrast, the pro-posed geometric method can provide the same solution within 8 minutes.

Key words: Orbital game, multiple-pursuer/one-evader, reachable set, Nash equilibrium, geometrical method