Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem

Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the double-homotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to an Earth-Moon $L_{1}$ halo orbit is presented to substantiate the effectiveness of the technique.