Very low-thrust trajectory optimization using a direct SQP method

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, aerodynamic drag, and third-body perturbions 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 the solution of very low-thrust orbit transfers. 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 for a representative transfer using a spacecraft with a thrust acceleration of approximately 1.25×10−7 km/s2. This transfer requires over 578 revolutions, and leads to a sparse optimization problem with 416 123 variables and 249 674 constraints. Issues related to the numerical conditioning and problem formulation are discussed.