Optimal round trip lunar missions based on the patched-conic approximation

Abstract

A study of optimal bi-impulsive trajectories of round trip lunar missions is presented in this paper. The optimization criterion is the total velocity increment. The dynamical model utilized to describe the motion of the space vehicle is a full lunar patched-conic approximation, which embraces the lunar patched-conic of the outgoing trip and the lunar patched-conic of the return mission. Each one of these parts is considered separately to solve an optimization problem of two degrees of freedom. The parameters to be optimized are two: the phase angle of the point at which the space vehicle reaches the edge of the Moon’s sphere of influence and the initial velocity at departure. The Sequential Gradient Restoration Algorithm is employed to achieve the optimal solutions. Analytical and numerical derivatives of expressions describing the lunar patched-conic approximations are utilized to ensure the results. The results based on the patched-conic approximation show a good agreement with the ones provided by literature, and the solution trajectories proved to be consistent with the image trajectories theorem.