Comparison of Indirect and Direct Methods for Operationally Constrained Low-Thrust Trajectories

This paper presents novel indirect and direct optimization methods for generating time-fixed, minimum-fuel heliocentric trajectories. To obtain higher-fidelity solutions, a variable-specific-impulse variable-thrust (VIVT) solar electric propulsion (SEP) system subject to solar power and duty-cycle constraints is considered. We also consider both regularized and nonregularized formulations of the indirect methods. We use an L2-norm-based regularization for smoothly embedding discontinuous events (e.g., throttle-control and power-input switches) to form a multiparameter family of smooth Hamiltonian boundary-value problems (BVPs), which are solved through a numerical continuation method. The solution to the regularized BVP is then used to initialize and solve an event-detection-based, nonregularized BVP. For the direct method formulation, a forward-shooting scheme is proposed with the introduction of an important power constraint to avoid overestimation of the performance of the VIVT SEP system. Two minimum-fuel interplanetary trajectories are solved with over 300 duty-cycle-driven thrust switches. The results of the both indirect and direct methods are compared. Initialized by the regularized solution, the nonregularized solution results in a noticeable improvement in the final mass value over the regularized indirect method and compared to the solutions obtained from the direct method.