Bounding Homotopy Method for Minimum-Time Low-Thrust Transfer in the Circular Restricted Three-Body Problem

Solving minimum-time low-thrust orbital transfer problems in the three-body problem by indirect methods is an extremely difficult task, which is mainly due to the small convergence domain of the optimal solution and the highly nonlinear nature associated with the three-body problem. Homotopy methods, the principle of which is to embed a given problem into a family of problems parameterized by a homotopic parameter, have been utilized to address this difficulty. However, it is not guaranteed that the optimal solution of the original problem can be obtained by most of the existing homotopy methods. In this paper, a new bounding homotopy method is proposed, by which the continuous homotopy path can be constructed and the optimal solution of the original problem is guaranteed to be found. In the parameter bounding homotopy method, an initialized problem with much higher thrust is constructed and a state-of-the-art parameter bounding homotopy approach is utilized to connect separated homotopy branches outside the predefined domain of the homotopic parameter. Furthermore, multiple optimal solutions of the original problem can be obtained if the homotopic approach continues after the first solution, among which the best solution can be figured out. Finally, numerical solutions of minimum-time low-thrust orbital transfers from GEO to Moon orbit and from GTO to halo orbit in the circular restricted three-body problem are provided to demonstrate the effectiveness of the homotopy method.