Constrained Lyapunov Stabilization Based on Gauss Variational Equations: From Spacecraft Orbital Transfers to Rendezvous

Lyapunov feedback laws can be constructed for performing orbital transfer maneuvers based on Gauss Variations Equations (GVEs), while satisfying the specified state and control constraints. These state and control constraints are enforced using barrier functions and saturation, respectively, while the reference governor, employed as a convergence governor, is utilized to avoid getting stuck at spurious equilibria that may be created by barrier functions. In this article, these Lyapunov feedback laws are extended to rendezvous maneuvers where not only five orbital elements are matched to the prescribed target values but also the true anomaly matches its (time-dependent) target value. The modification involves altering the commanded semi-major axis with an outer-loop feedback law, also designed using Lyapunov techniques. We illustrate the resulting safe closed-loop rendezvous maneuvers in simulations for the conventional thrust-based propulsion and for Lorenz force-based propulsion. In the latter case, only the current through the tether is controlled subject to the current limits to accomplish prescribed orbital transfer and rendezvous maneuvers.