Shape-Based Approach for Low-Thrust Earth-Moon Trajectories Initial Design

A robust finite Fourier series (R-FFS) approach is developed for fast generation of Earth-moon trajectories using continuous low thrust. Each component of the position vector is approximated using a finite Fourier series as a function of time; these approximations are then used to design a trajectory that satisfies the equations of motion and the constraints, at discrete points, as well as the problem boundary conditions. The R-FFS method leverages the three-body problem characteristics to achieve all the required plane change without the use of propulsion. The trajectory is divided into phases: escape, intermediate, and capture. The escape phase is further divided into segments. The phase of the trajectory near the L1 Lagrange point is designed first and is always a thrust-free phase. This thrust-free phase is optimized to achieve the required plane change, enabling planar trajectories in the other phases. The initial guess needed by the solver, in the escape and capture phases, is generated using an analytic approximation developed in this paper. The numerical results show that the R-FFS can generate three-dimensional transfers to high lunar orbits, low lunar orbits, and halo orbits, while meeting constraints on the maximum thrust level of the engine.