Orbital Inspection Game Formulation and Epsilon-Nash Equilibrium Solution

This paper studies an orbital inspection game, which involves two spacecraft competing for imaging conditions in an on-orbit inspection mission. First, the main factors affecting the imaging conditions, including the sun angle, sun-angle changing rate, relative distance, and distance changing rate, are analyzed to formulate a realistic multiple-factor inspection game. An approximate switching-type payoff function is specially designed to incorporate all the boundary constraints of those factors into the game model. Then, the analytical necessary conditions for the Nash equilibrium are derived and converted as a two-point boundary value problem (TPBVP). But different from conventional routes to solve the challenging TPBVP, a lighter costate optimization method is proposed, which transforms the TPBVP to a direct optimization problem by employing the conclusion that the optimal thrust directions of both sides are the same and utilizing the theory of the epsilon-Nash equilibrium. The existence of the epsilon-Nash equilibrium is proven, and the necessary conditions for a small epsilon are derived to support the method. Finally, simulations of the GEO inspection missions demonstrated the superiority of the proposed game formulation and the high efficiency and accuracy of the proposed method.