Direct-to-indirect mapping for optimal low-thrust trajectories

Abstract

Optimal, many-revolution spacecraft trajectories are challenging to solve. A connection is made for a class of models between optimal direct and indirect solutions. For transfers that minimize thrust-acceleration-squared, primer vector theory maps direct, many-impulsive-maneuver trajectories to the indirect, continuous-thrust-acceleration equivalent. The mapping algorithm is independent of how the direct solution is obtained and requires only a solver for a boundary value problem and its partial derivatives. A Lambert solver is used for the two-body problem in this work. The mapping is simple because the impulsive maneuvers and co-states share the same linear space around an optimal trajectory. For numerical results, the direct coast-impulse solutions are demonstrated to converge to the indirect continuous solutions as the number of impulses and segments increases. The two-body design space is explored with a set of three many-revolution, many-segment examples changing semimajor axis, eccentricity, and inclination. The first two examples involve a small change to either semimajor axis or eccentricity, and the third example is a transfer to geosynchronous orbit. Using a single processor, the optimization runtime is seconds to minutes for revolution counts of 10 to 100, and on the order of one hour for examples with up to 500 revolutions. Any of these thrust-acceleration-squared solutions are good candidates to start a homotopy to a higher-fidelity minimization problem with practical constraints.