Multiphase Homotopic Approaches for Best Solution to Low-Thrust Geocentric Trajectories

Low-thrust trajectories between geocentric orbits usually involve long durations and many orbital revolutions, leading to trajectory optimization problems with multiple local optima. This work proposes multiphase homotopic approaches to find the best solution to the low-thrust geocentric time-optimal and fuel-optimal problems with transfer and rendezvous boundary conditions. For the time-optimal problem (TOP), based on the thrust magnitude homotopy, a highly efficient three-phase homotopic approach is introduced to find the best solution to the transfer TOP by modifying the terminal longitude search and introducing a medium thrust. For the fuel-optimal problem (FOP), a similar three-phase approach for finding the best solution to the transfer FOP is proposed by designing two types of hybrid homotopy. In addition, the boundary condition homotopy is developed to obtain the best solutions to the rendezvous TOP and FOP, with the best solutions to the transfer problems as the initial guess. Owing to the medium thrust and multiple well-designed homotopies, the solving process for each scenario can be initialized by random guesses, and the best solution can be obtained quickly. Time-optimal and fuel-optimal examples from a geostationary transfer orbit to a geostationary orbit are investigated to substantiate the efficiency of the proposed approaches.