Comparison of the Legendre–Gauss pseudospectral and Hermite–Legendre–Gauss–Lobatto methods for low-thrust spacecraft trajectory optimization

Low-thrust spacecraft propulsion systems enable fuel-efficient trajectories through space but the resulting trajectory optimization problems can be challenging. Such problems often use direct collocation methods for transcribing the optimal control problem into a nonlinear programming problem. In this work, the Hermite–Legendre–Gauss–Lobatto (HLGL) and the Legendre–Gauss pseudospectral (PS) direct collocation methods have been compared for a minimum-time low-thrust Earth-to-Mars transfer problem. It is demonstrated that trajectories with different time domains can be effectively compared using points that correspond to the same normalized times. Various metrics that describe the computational cost as well as the errors with respect to a reference trajectory have been used. For the HLGL method, the number of subintervals ( m ) and the order of the interpolating polynomial ( n ), can be varied such that the number of nodes ( N ) stays constant. The distribution of the number of such ( m , n ) pairs is analysed using the concept of divisor functions from number theory and an algorithm for determining all the possible combinations for any N value is developed. In addition, a method is presented by which the Edelbaum trajectory is used to set bounds, scale the problem, and generate a suitable initial guess. The primary contribution of this paper involves a detailed comparison of the performance of all the HLGL pairs with the PS method for nodes in the range N = 6 to N = 40 for the minimum-time low-thrust Earth-to-Mars transfer.