Warm-Start Multihomotopic Optimization for Low-Thrust Many-Revolution Trajectories

Traditionally, fuel-optimal problems for low-thrust spacecraft transfers are connected with energy-optimal problems to increase the convergence possibility of indirect methods. However, for low-thrust many-revolution missions, it is still hard to resolve the energy-optimal problems in a fast and reliable manner due to the strong nonlinearity. To address this issue, a warm-start multihomotopic optimization approach is proposed in this article, wherein the Sundman transformation and multihomotopic techniques are used to connect the energy-optimal problems to linear ones with analytical solutions. This article focuses on the following three contributions. First, the energy-optimal control problems are transformed into true longitude-depended energy-optimal problems based on the homotopy to dynamical model and Sundman transformation, and the equivalence relations of the solutions before and after transformation are obtained analytically. Second, the problems are further connected with time-free energy-optimal problems based on a homotopy to the time constraint. Third, the time-free problems are linearized near a nominal trajectory, and the corresponding analytical solutions are obtained. Starting with the analytical solutions, the algorithm can gradually iterate back to the solutions of the original energy- and fuel-optimal problems. Since the whole transformation process is lossless (optimality is preserved under the transformation process), the developed warm-start multihomotopic algorithm enjoys the advantages on reliable convergence and high computational efficiency. Numerical simulations of Earth-orbit transfer missions from GTO to GEO are conducted by comparing with traditional methods, and the results are given to substantiate the effectiveness of the proposed warm-start multihomotopic method.