The Impact of Second-Order Analytical Derivatives on Regularized Direct Multiple Shooting Method for Impulsive Spacecraft Trajectory Optimization

Abstract

While the first-order derivatives of an objective function and constraints have often been analytically provided in the gradient-based optimization algorithms, the joint use of the second-order derivatives can further improve the computational efficiency and robustness. This paper implements the first- and second-order analytical derivatives in the direct multiple shooting-based, regularized method of minimizing the fuel expenditure for impulsive space trajectories. The high-order dynamical information expresses the Hessian matrix of the Lagrange function in the nonlinear programming problem. The result is an efficient tool for robustly computing fuel-optimal, multi-impulse trajectories in the regularized framework of removing singularities associated with null thrust impulses from the derivatives of the objective function. The computational performance is compared for the cases of optimizing impulsive transfers between cislunar periodic orbits within the regularized framework implementing the analytical derivatives in different ways over various initial guess trajectories and computational conditions. The results indicate the superiority of adopting both the first- and second-order analytical derivatives in terms of the efficiency and robustness.