Optimal Low Thrust Trajectories to the Moon

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low thrust trajectories are demanding because realistic forces, due to oblateness, and third-body perturbations often dominate the thrust. Furthermore, because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large, because the trajectories are long, and very nonlinear because of the perturbing forces. This paper describes the application of the transcription method to compute an optimal low thrust transfer from an Earth orbit to a specified lunar mission orbit. It is representative of the SMART-1 or "Small Missions for Advanced Research in Technology" of the ESA scientific program J. Schoenmaekers, J. Pulido, and R. Jehn, Tech. report S1-ESC-RP-5001, European Space Agency, 1998. The spacecraft is deployed from an Ariane-5 into an elliptic Earth centered park orbit. The goal is to insert the spacecraft into a lunar orbit that is polar and elliptic and has its pericenter above the south pole. The spacecraft utilizes a solar electric propulsion system. The goal is to compute the optimal steering during the orbit transfer, which takes over 200 days, so that the fuel consumption is minimized. The vehicle dynamics are defined using a modified set of equinoctial coordinates, and the trajectory modeling is described using these dynamics. A solution is presented that requires the solution of a sparse optimization problem with 211031 variables and 146285 constraints. The trajectory we present requires two very long thrust arcs, and, consequently, the overall mission duration is much shorter than multiburn trajectories. Issues related to the numerical conditioning and problem formulation are discussed.