Spacecraft Relative Motion Dynamics and Control Using Fundamental Modal Solution Constants

Abstract

This paper explores expressing the relative state in the close-proximity satellite relative motion problem in terms of fundamental modal solution constants. The nominal uncontrolled relative state can be expressed in terms of a weighted sum of fundamental and geometrically insightful modal motions. These fundamental motions are obtained using the Lyapunov–Floquet theory. In the case that the dynamics are perturbed by the action of a controller or by unmodeled dynamics, the weights on each fundamental solution are allowed to vary as in a variation-of-parameters approach, and in this manner function as state variables. This methodology reveals interesting insights about satellite relative motion and enables elegant control approaches. This approach can be applied in any dynamical environment as long as the chief orbit is periodic, and this is demonstrated with results for relative motion analysis and control in the eccentric Keplerian problem and in the circular restricted three-body problem. Some commentary on the extension of the methodology beyond the periodic chief orbit case is also provided. This is a promising and widely applicable new approach to the close-proximity satellite relative motion problem.