Spacecraft Constant-Thrust Minimum-Time Rendezvous via Reachable Set Theory

This paper presents a geometric reachable-settheory-based solution for spacecraft orbital rendezvous under constant-magnitude continuous thrust, addressing two problems: minimum-time trajectories for given thrust magnitude and minimum thrust-magnitude requirements for specified rendezvous time. The proposed methodologies are developed for close-range rendezvous scenarios, utilizing a linear relative motion model. The study derives fundamental properties of the reachable set under maximum thrust-magnitude constraint, demonstrating its convexity and leveraging this property to simplify the optimization problem. Analysis of the reachable set envelope proves that optimal thrust magnitude consistently operates at maximum value and the analytical expression for optimal thrust direction is obtained using Pontryagin's maximum principle. Based on the developed geometric reachable set theory, two optimal rendezvous methods are proposed for minimum-time trajectories and determining minimum thrust-magnitude requirements. One theoretical contribution emerges in the geometric interpretation of the costate vector in the indirect method as the inward normal vector of the reachable set's supporting hyperplane. Numerical examples are presented to demonstrate that the proposed method for the reachable set is accurate and the developed approaches for optimal orbital rendezvous are effective.