Modeling of an optimal trajectory at a safe distance in a spacecraft cluster

This paper addresses the problem of modeling an optimal motion trajectory of a spacecraft within a spacecraft cluster consisting of a target and a deputy spacecraft during an undocking maneuver followed by a fly-around at a safe distance. The relative motion of the deputy spacecraft is described in a local orbital reference frame using a linearized dynamical model based on the Hill – Clohessy – Wiltshire equations. An integral performance index is introduced to characterize the tendency of motion near the boundary of the admissible relative position region. The original optimal control problem with an integral cost functional and bounded control inputs proves to be challenging for both analytical and numerical analysis, which limits the direct application of standard optimization techniques. To facilitate the study, a decomposition approach is employed, allowing the problem to be divided into two consecutive stages: transfer to the boundary of the safe zone and subsequent motion along this boundary. The Pontryagin maximum principle is applied to investigate the properties of optimal solutions and to derive analytical expressions for the adjoint variables, making it possible to analyze the structure of optimal control laws. Numerical analysis is carried out using both direct and indirect optimal control methods. A set of quality criteria is introduced to assess the obtained solutions, including the fulfillment of necessary optimality conditions, the Hamiltonian behavior, and the sensitivity of the solution to variations in problem parameters. The comparative analysis shows that direct optimization methods yield more stable and reproducible solutions, while the tangential boundary approach improves the agreement between the two stages of motion at the expense of a moderate increase in the transfer time to the boundary.