Trajectory Planning for Spacecraft Formation Reconfiguration Using Saturation Function and Difference-of-Convex Decomposition

The trajectory planning for spacecraft formation reconfiguration (SFR) presents significant technical challenges due to its time-optimal performance index, highly nonlinear terminal formation constraints, potential large-scale spacecraft deputies, and significant requirements for robustness and efficiency in planning. This issue is addressed within the framework of sequential convex programming (SCP) due to its rapid computational capabilities, coupled with two key techniques to enhance SCP's hard-to-ensure convergence. First, to effectively utilize the concave-convergent characteristic of employing SCP to handle pure convex or concave functions, the problem is transformed into an equivalent difference of convex (DC) form. This results in a problem where all components are either convex or concave. A semidefinite problem is constructed to optimize the DC decomposition procedure, thereby achieving fast, reliable, and generalized transformation. Second, saturation functions are then employed to expand the feasible region of the DC problem, overcoming the artificial infeasibility commonly encountered in traditional SCP. A series of bijective mappings are used to connect the saturation function with dissatisfaction across all constraints. By penalizing the saturation function, the SCP procedure can be directed toward optimal solutions. Through rigorous theoretical derivations and sufficient numerical verifications, it can be confirmed that the combination of DC decomposition and saturation function performs exceptionally well in ensuring the convergence of SCP, contributing to the rapid and reliable generation of time-optimal SFR trajectories.