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

TL;DR

This paper tackles the problem of designing spacecraft trajectories in the Earth–Moon system by transitioning from low-fidelity CR3BP solutions to high-fidelity HFEM dynamics. It proposes a Levenberg-Marquardt–based nonlinear least-squares framework with an update damping term, fixed-time multiple-shooting, and adaptive weighting to ensure monotone convergence and robustness even when the initial guess is poor. The approach supports proximity constraints to control the final trajectory shape and is shown to outperform a baseline minimum-norm update in challenging cases, including quasi-periodic transitions and transfers between distant orbits. The results demonstrate practical benefits for cislunar mission design, offering a lightweight, robust method capable of incorporating proximity objectives and shaping trajectories under HFEM without excessive computational burden.

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 approach is found to outperform the baseline in applications where the initial guess is poor and the ease of including proximity constraints provides benefits in control over the shape of the converged solution.

Figures (9)

• Figure 1: Multiple shooting example of a CR3BP initial guess under the HFEM.
• Figure 2: LM implementation considering adaptive weighting.
• Figure 3: Construction of the initial guess from a CR3BP solution.
• Figure 4: CR3BP L2 Halo orbit considered for transition into the HFEM. Dimensions to scale, including Earth and Moon sizes.
• Figure 5: Application of the algorithms to transition a CR3BP L2 Halo orbit into a 50 revolution quasi-periodic counterpart in the HFEM considering four patch points.