Near-optimal continuous control for spacecraft collision avoidance maneuvers via generating functions

This paper presents a suboptimal continuous control algorithm that would enable an active spacecraft to avoid collision with inactive space objects. To prevent collision, a penalty function whose value soars as a spacecraft is about to collide with other objects is inserted into the cost functional of an optimal control problem. Then, a two-point boundary value problem for a Hamiltonian system is constructed, and is solved with the generating functions. This algorithm can be coded step-by-step to obtain suboptimal feedback continuous control laws as truncated power series with no initial guess or iteration. This advantage over direct optimizations, however, requires moderate efforts to develop higher-order generating functions and update the penalty function parameters heuristically. In the illustrative examples, the above process allows an active satellite to smoothly circumvent other space objects or forbidden regions. The overall process is also useful for obtaining an appropriate initial guess for the direct optimization approaches in case of numerically sensitive problems because of the definiteness in its solution procedure.