Dynamics and Control of Satellite Relative Motion in Elliptic Orbits using Lyapunov-Floquet Theory

Abstract

Rendezvous and proximity operations involving satellites operating near each other have been performed since the Gemini missions. A better understanding of the dynamics of satellites in close proximity could be helpful for both mission planners and operators as well as for algorithm development. With this increased understanding, future satellite missions may involve an impromptu fly-around of another satellite and a greater focus on satellite autonomy.

In satellite proximity operations, it is desirable to express the relative motion of a deputy satellite with respect to the chief satellite. For satellites in elliptic orbits, this relative motion can be described by the time-varying Linearized Equations of Relative Motion. Making an assumption that the chief satellite is in a circular orbit results in the linear time-invariant Hill-Clohessy-Wiltshire equations. These equations allow the relative-motion solution to be intuitively visualized. Due to the abundance of algorithms in the literature based on the Hill-Clohessy-Wiltshire equations, the focus of this dissertation is to explore the relationship between relative motion in elliptic orbits and the Hill-Clohessy-Wiltshire equations.

The major contributions of this work are twofold. First, three time-varying coordinate transformations are derived which relate the Hill-Clohessy-Wiltshire equations to the Linearized Equations of Relative Motion. These transformations show that the Hill-Clohessy-Wiltshire equations are able to exactly capture the time-varying dynamics. Second, the literature does not contain infinite-horizon continuous-thrust control for satellites in elliptic orbits. A time-varying gain matrix was constructed using the time-varying coordinate transformations, optimal-control theory, and a control method from the literature. This control is able to drive the deputy's position toward rendezvous with an elliptic chief.