Homotopy Method with Initial Costate Evaluation for Low-Thrust Fuel-Optimal Homoclinic Halo Orbit Transfer

The problem of transferring a spacecraft between Halo orbits around Sun-Earth libration points using low-thrust propulsion is considered in this paper. The complete dynamics is established using Pontryagin's maximum principle (PMP) within the framework of the Sun-Earth circular restricted three-body problem (CRTBP), incorporating both state and costate variables. To solve the nonsmooth shooting equations with discrete bang-bang control, a homotopy continuation method is introduced, continuing the smooth energy-optimal controls to the discrete fuel-optimal control. Additionally, a particle swarm optimization (PSO) algorithm is employed to estimate initial costate values for energy-optimal problems, addressing the challenge of nonphysical initial costate estimation. Finally, the effectiveness of the homotopy algorithm, in conjunction with PSO costate initial evaluation, is demonstrated through numerical verifications of homoclinic Halo transfers, where multirevolution transfer and parameter analysis are subsequently considered. The proposed homotopy algorithm, supported by parallel computation, generates the global distribution of feasible transfer windows for different initial and end positions of Halo orbits, distributing to the selection of suitable one concerning fuel consumption, time, and their combination.