Orbital Rendezvous and Flyaround Based on Null Controllability with Vanishing Energy

Hill's equations describe the relative motion of a chaser with respect to the target spacecraft in a circular orbit. They possess periodic solutions that form the relative orbits of the chaser. In this paper it is shown that Hill's equations with three independent control accelerations are null controllable with vanishing energy. Based on this property, the relative orbit transfer problem is formulated as a linear quadratic regulator problem and a feedback control with arbitrary small L2 norm is obtained via the Riccati equation. The design method is then extended to Tschauner - Hempel equations that describe the relative motion of the chaser along an eccentric orbit. It is shown that the controlled Tschauner - Hempel equations are also null controllable with vanishing energy and a feedback control with arbitrary small L2 norm is designed using the periodic solution of the Riccati differential equation. Numerical simulations of Hill's equations as well as Tschauner - Hempel equations are given and feedback controls with good performance are obtained.