Periodic orbits of nonlinear relative dynamics along an eccentric orbit

This paper is concerned with formation acquisition and reconfiguration problems with an eccentric reference orbit. As a first step, the characterization problem is considered for all initial conditions that constitute periodic solutions of the nonlinear equations of relative motion. Under the condition that the inertial orbits of the leader and a follower are coplanar, initial conditions of all periodic relative orbits are generated in terms of parameters of their orbits and their initial positions. Then the inertial orbit of the follower is rotated successively around the two axes of its perifocal reference system, and the initial conditions of all noncoplanar relative orbits are derived. Based on these periodic relative orbits, formation acquisition and reconfiguration problems by a feedback controller are formulated. The main performance index of a feedback controller is the total velocity change during the operation. Using the property of null controllability with vanishing energy of the Tschauner-Hempel equations, suboptimal controllers are designed via the differential Riccati equation of the linear regulator theory of periodic systems.