Designing trajectories in the Earth--Moon system: a Levenberg--Marquardt approach

Abstract

Trajectory design in cislunar space under a high-fidelity ephemeris model (HFEM) is pursued through a nonlinear optimization perspective anchored on the transition of solutions from lower fidelity models, namely the Circular Restricted Three-Body Problem (CR3BP). The optimization problem is posed in the likeness of a multiple-shooting approach, aiming for segment-to-segment continuity while tracking proximity to the original CR3BP structures. The analysis of various formulations leads to the selection of an unconstrained least-squares problem for further investigation. The nonlinear optimization problem is convexified and the use of the Levenberg--Marquardt algorithm, as an alternative to the minimum-norm update equation found in most literature, is investigated for its control over the update step and inherent robustness. Additional techniques, such as adaptive weighting, are employed to further consolidate the behavior of the proposed algorithm in challenging scenarios. Numerical trials evaluate the adequacy of the methodology presented and compare it to the minimum-norm baseline over various application cases, including the generation of quasi-periodic trajectories and orbital transfers between them. The proposed technique is found to be a suitable alternative to the minimum-norm scheme, generally retaining better proximity to the original CR3BP trajectories and providing benefits in numerical robustness and stability. Moreover, the ease of including proximity objectives in a relaxed manner is shown to facilitate control over the shape of the final converged solution.