Lyapunov-Based Guidance for Orbit Transfers and Rendezvous in Levi-Civita Coordinates

Abstract

This paper considers planar orbit transfers and rendezvous problems around a central body using Lyapunov stability theory. The model used is the Levi-Civita transformation of the planar two-body problem. One of the advantages of working in these transformed coordinates is that the solution to the unperturbed equations of motion is that of a simple harmonic oscillator, so the analytical solution is known at all times during coast phases. We design a closed-loop guidance scheme for orbit transfers from any initial elliptical orbit to any final elliptical orbit using a spacecraft with thrust–coast capabilities. A similar procedure is performed to design a control law for rendezvous with any desired target spacecraft. The proposed Lyapunov functions give rise to asymptotically stabilizing control laws. The algorithms designed are robust to initial and final conditions, computationally fast, and no restrictions are imposed on the magnitude of the thrust. The guidance scheme is also effective in a dynamic model where unmodeled perturbations are present.