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

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.