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This work establishes a general-purpose method for computing the exact boundary of the reachable set in a specified direction, with additional constraints imposed on the terminal velocities. Reachable sets with rest boundary conditions, that is, zero-velocity constraints imposed at initial and final times, are of interest for formation flying and spacecraft rendezvous and docking applications. The rest-to-rest reachability problem is extended to compute reachable neighboring circular orbits and a modified problem formulation is developed to exclusively compute the maximum and minimum reachable circular orbits. Reachable sets are computed by applying classical optimal control techniques to a maximum position optimization problem. A solution method for the optimization problem is outlined, which makes use of a Newton’s method solving schema. Additional constraints are added to the problem to compute maxima along specified directions, allowing for the characterization of the reachable set in as many directions as desired. Example reachable sets are computed to illustrate the use of this method on various systems, including an application to relative motion.
Natherson et al. (Wed,) studied this question.