Indirect Optimization of Three-Dimensional Multiple-Impulse Moon-to-Earth Transfers

This paper illustrates an indirect method to optimize multiple-impulse trajectories from circular lunar orbit to Earth. Optimization is performed in the circular restricted three-body problem, and the necessary optimality conditions are found through optimal control theory. In order to overcome the difficulty of initial adjoints estimation, a homotopic approach, which is based on an auxiliary optimization problem with known solution, is developed; this approach proves to be robust and efficient. Examples are presented for a range of lunar orbit orientations to assess the impact on velocity impulse requirements. Optimization results for trajectories with different number of impulses are also compared. The developed procedure can support fast and accurate evaluation of the transfer costs for Moon-to-Earth trajectories both in nominal conditions and for contingency plans.