Stabilizing motion relative to an unstable orbit: Applications to spacecraft formation flight

A control law is derived and analyzed that stabilizes a class of unstable periodic orbits in the Hill restricted three-body problem. The control law is derived by stabilizing the short-time dynamics of motion about a trajectory by the use of a feedback law specified by the instantaneous eigenvalue and eigenvector structure of the trajectory. This law naturally generalizes to a continuous control law along an orbit. By applying the control to an unstable periodic orbit, we can explicitly compute the stability of the control over long periods of time by computing the monodromy matrix of the periodic orbit with its neighborhood modified via the control law. For the case of an unstable halo periodic orbit in the Hill restricted three-body problem, we find that the entire periodic orbit can be stabilized. The resulting stable periodic orbits have three distinct oscillation modes in their center manifold. We discuss how this control can be applied to formation flight about a halo orbit. Some practical implementation issues of the control are also considered. We show that the control acceleration can be provided by a low-thrust engine and that the total fuel cost of the control can be quite reasonable.