Optimal Mirror Trajectories Using Continuous Thrust

Abstract

The theorem of mirror trajectories, proven almost six decades ago by Miele, states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits) there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). The theorem at hand regards feasible trajectories and proved extremely useful for investigating the spacecraft natural dynamics in the circular restricted problem of three bodies, by identifying special solutions, such as symmetric periodic orbits and free return paths. This theorem has recently been extended to optimal mirror trajectories, thus substantiating Miele’s conjecture based on numerical evidence. Unlike the theorem of mirror paths, which refers to natural (unpowered) orbital motion, the theorem of optimal mirror trajectories establishes the existence, characteristics, and optimal control time history of the returning path, once the outgoing optimal trajectory has been determined. This theorem applies to (i) finite-thrust trajectories, for which a limiting value of the thrust acceleration exists, (ii) constant-thrust-acceleration paths, (iii) impulsive trajectories, and (iv) artificial periodic orbits (that use very low thrust propulsion or solar sails). This work illustrates the theorem of optimal mirror trajectories applied to two cases of practical interest: (a) continuous, low-thrust orbit transfer, and (b) continuous-thrust lunar descent (with soft touchdown) and ascent (with final orbit injection). In both cases, the theorem allows the immediate and straightforward identification of the optimal control law of the returning path, once the outgoing optimal trajectory has been determined.