Hybrid QPSO and SQP Algorithm with Homotopy Method for Optimal Control of Rapid Cooperative Rendezvous

Abstract

The fuel-optimal rapid cooperative rendezvous problem between two spacecraft under finite thrust, based on the indirect method, was investigated and converted into a two-point boundary value problem (TPBVP) in this study by using Pontryagin’s maximum principle. First, normalization processing of initial costate variables with unknown scopes was carried out to restrict them on a unit hypersphere. The quantum particle swarm optimization (QPSO) algorithm was used to preliminarily search for the initial costate variables of the high-dimensional energy-optimal problem, and then the results obtained were further corrected by the sequence quadratic programming (SQP) algorithm. The preceding combinatorial optimization algorithm with normalization technique considerably increases the probability of finding the approximate initial values of the globally optimal solution. Based on these modified initial costates, the smooth energy-optimal results were transitioned to the desirable nonsmooth fuel-optimal results by the homotopy method. Through the combination of the preceding effective techniques, the following difficulties were successfully overcome: (1) The optimal control was a strongly nonlinear problem under continuous high thrust; (2) in a cooperative rendezvous, the terminal rendezvous orbit was unknown and the parameter variables were doubled, leading to high-dimensional control equations; and (3) the narrow convergence domain of the shooting function made the shooting process extremely sensitive to the initial guess of the costates. The simulation results demonstrate not only the feasibility of the indirect method in solving fuel-optimal cooperative rendezvous, but also the superiority over another orbit transfer optimization method, the hybrid method.