New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion

Abstract

This paper addresses energy-optimal guidance and control of satellite relative motion for formation flying and rendezvous using impulsive maneuvers. To meet the requirements of future multisatellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are accounted for by splitting the optimal reconfiguration into guidance (long-term) and control (short-term) layers. Both problems are cast in relative orbit element space, which allows the simple inclusion of secular and long-periodic perturbations through a state-transition matrix and the translation of the energy-optimal optimization into a minimum-length path-planning problem. From the choice of state variables, both guidance and control problems can be solved in closed form, leading to optimal, predictable maneuvering schemes for simple onboard implementation. Besides generalizing previous work, this paper finds five new in-plane and out-of-plane closed-form solutions to the relative motion control problem in unperturbed eccentric and perturbed near-circular orbits. A delta-v lower bound is formulated, which provides insight into the optimality of the control solutions. Finally, the functionality and performance of the new maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and systematic comparison with primer vector optimal control.