The Dynamics of Formation Flight About a Stable Trajectory

Abstract

This paper investigates the dynamics of a formation of spacecraft traveling in the center manifold about a stable periodic orbit. We consider a halo orbit in the Hill problem that has been stabilized by a closed-loop control. The monodromy matrix of the resulting orbit has all of its eigenvalues of the unit circle with nonzero imaginary parts. Furthermore, by adjusting the feedback control gain it is possible to induce large winding numbers, leading to high frequency oscillations about the orbit with periods much shorter than the period of the orbit. The description of motion in this system over relatively short time spans is investigated using the theory of linear dynamical systems. To describe the motion we approximate the time-varying linear dynamics with a “local” time-invariant system. The “orbital elements” of these linear solutions are defined and used to describe the possible motions for a formation of spacecraft. The error introduced by this approximation is evaluated and shown to be relatively small. Applications of the theory to spacecraft interferometry are noted.