Lyapunov-based Low-energy Low-thrust Transfers to the Moon

Abstract

This paper investigates the numerical computation of low-fuel low-thrust Earth-Moon transfers in a full ephemeris model incorporating the gravitational influence of the Sun, the Moon and all planets of the solar system plus the solar radiation pressure perturbation based on a sample spacecraft area and mass. First, an initial velocity increment to be provided by the launcher is computed. This initial impulse puts the spacecraft on the counterpart in the full ephemeris model of a stable invariant manifold defined in the Sun-Earth Circular Restricted Three-Body Problem. Then, after a coast arc, a closed-loop thrust law is applied to bring the spacecraft to the target lunar orbit. This control law is based on Lyapunov control theory. More precisely, a control-Lyapunov function is defined as the weighted quadratic distance between the first five equinoctial orbital elements of the spacecraft in a Moon-centered reference frame and those defining the target lunar orbit. The control is computed in such a way so as to make the time derivative of the control-Lyapunov function as negative as possible. Numerical results are provided first for a transfer with constant maximum thrust. Then, it is shown that unlike in the case of an open loop control, concentrating the thrust in the vicinity of the perilune and the apolune increases the transfer duration but without reducing the fuel consumption. This is largely due to the uncontrolled effect of the perturbations acting on the spacecraft during the coast arcs. Finally, the robustness of the guidance law against unexpected engine shutdown events is demonstrated.