Optimal Cooperative Power-Limited Rendezvous Between Coplanar Circular Orbits

Minimum-fuel rendezvous of two power-limited spacecraft is investigated. Both vehicles are active and provide thrust to complete the rendezvous. Total propellant consumption is minimized. A direct-minimization method, direct collocation with nonlinear programming, is used to obtain cooperative rendezvous solutions in an inverse- square gravitational field. Unconstrained and constrained circular terminal orbits are considered. The optimal solutions depend upon the power-to-mass ratios of the spacecraft, the initial orbits, and the specified transfer time. Optimal cooperative rendezvous solutions are compared with optimal active-passive solutions and with previously reported linearized solutions. ISTORICALLY, spacecraft rendezvous has involved one thrusting (active) vehicle and one coasting (passive) vehicle. A cooperative power-limited (PL) rendezvous is an orbital maneuver in which two PL spacecraft are active. With both vehicles thrust- ing, a reduction in total propellant consumption is obtained when compared to the traditional active-passive rendezvous. The practical application of cooperative rendezvous occurs if both vehicles have comparable size and propulsive capability. It would make no sense to consider cooperative rendezvous between a small spacecraft and a space station. Early studies of cooperative control considered general linear or nonlinear systems with various performance indices.1"4 Recently, cooperative rendezvous methodology has been applied to spacecraft maneuvering. Prussing and Conway5 determined the optimal termi- nal maneuver for a cooperative impulsive rendezvous. Mirfakhraie6 and Mirfakhraie and Conway7 developed a method for determin- ing fuel-optimal trajectories for the fixed-time impulsive cooper- ative rendezvous. Coverstone-Carroll and Prussing8 obtained an- alytical solutions for fixed-time PL cooperative rendezvous using the Hill-Clohessy-Wiltshire linearized gravity field to approximate the inverse-square gravity field. In this article, fixed-time cooper- ative and active-passive (noncooperative) rendezvous solutions are obtained for two PL spacecraft in an inverse-square gravity field. Transfer times, maximum exhaust powers, and the initial orbits of the spacecraft are varied to determine their influence on the optimal cooperative solution. Optimal solutions subject to circular terminal- orbit constraints are also examined. Necessary Conditions for Optimal Solution