Geodesic-based free terminal time energy optimization for spacecraft clusters trajectory planning in conflict zone

This paper explores the free terminal time two-point boundary value problem for two spacecraft clusters in a conflict zone by introducing a trajectory planning method based on geodesics. Within this context, the clusters are subject to configuration, collision avoidance, communication, and motion constraints. First, the cluster state space under complex multiple constraints is constructed as a manifold. The free terminal time two-point boundary value problem is further transformed into a geodesic solution problem in manifold space and characterized by a partial differential equation. Addressing the multi-peaked nature of this multi-constrained optimization challenge, an initial guess, founded on the Clohessy-Wiltshire equation and employing homotopy principles, is devised. This facilitates algorithm convergence towards a superior peak, even from a bad initial guess. Finally, the Christoffel symbol simplified computation and the fast construction method of cluster boundary are proposed to realize the balance between optimal solution search and algorithm performance. Simulation results show that the proposed algorithm can solve the clusters conflict problem and can converge quickly under the random configuration to achieve cluster energy optimization in the multi-peak problem.