Nonlinear Programming of Low-Thrust Multi-Rendezvous Trajectories Using Analytical Hessian

TL;DR

This paper introduces a fast nonlinear programming method with analytical gradients for low-thrust multi-asteroid rendezvous missions, improving computational efficiency and accuracy in trajectory optimization.

Contribution

It derives analytical first- and second-order gradients for low-thrust rendezvous $ riangle v$, enabling efficient and precise nonlinear programming for multi-rendezvous trajectory design.

Findings

Mean relative error of $ riangle v$ approximation below 0.8% for main-belt asteroid transfers

Validated effectiveness on 9-asteroid rendezvous sequence with fuel and time optimization

Achieved consistent improvement on GTOC12 top-ranking sequences

Abstract

This study presents a fast nonlinear programming algorithm for low-thrust multi-asteroid rendezvous missions. The core contribution is the derivation of analytical formulations for both first- and second-order gradients of low-thrust rendezvous $\Delta v$ through an iterative Lambert-based $\Delta v$ estimator and their application to derive the Hessian matrix or nonlinear programming of the multi-rendezvous trajectory optimization problem. Numerical simulations demonstrate the method's accuracy, with mean relative errors of $\Delta v$ approximation below 0.8\% for main-belt asteroid transfers, with the analytical gradients matching those computed via the central difference method. The nonlinear programming algorithm's effectiveness is validated through a 9-asteroid rendezvous sequence under both fuel-optimal and time-optimal configurations. Additional validation on three top-ranking sequences from the 12th Global Trajectory Optimization Competition (GTOC12) shows consistent improvement over the original solutions. The proposed approach is well-suited for integration into global trajectory optimization algorithms for multi-spacecraft multi-target missions, offering high computational efficiency while maintaining precise objective function evaluation capabilities.