Solution space exploration of low-thrust minimum-time trajectory optimization by combining two homotopies

Solving the low-thrust trajectory optimization problem with indirect methods has two main challenges including initial guessing and global optimum searching. To overcome these difficulties, two homotopies, including the thrust-homotopy and the longitude-homotopy, are combined to investigate the solution space of the low-thrust minimum-time trajectory optimization problem. The thrust-homotopy can be interpreted as a continuation on the maximum thrust magnitude, and the longitude-homotopy is proposed via a continuation on the final cumulative true longitude. Afterwards, the structure of the solution space is revealed by simulating a transfer scenario from the geostationary transfer orbit to the geostationary orbit. A bounded and smooth solution surface, where each point represents a solution with specific thrust and final cumulative true longitude, is formed by intersecting longitude-homotopy paths and thrust-homotopy paths. Based on the solution surface, the thrust-longitude-combined homotopic approach and a new hybrid homotopy are proposed to obtain local solutions by tracking homotopy paths from a solution of an easy problem, avoiding initial guessing. Finally, the longitude-homotopy is applied to search the global minimum-time solution from an obtained local solution.