Nonlinear station-keeping control of unstable orbits near collinear libration points

A new nonlinear optimal tracking control method is proposed and analyzed which stabilizes a class of unstable periodic orbits near collinear libration points. Since such libration point orbits are, in general, unstable, spacecraft moving on these orbits must use some form of trajectory control to remain close to their nominal paths. In this study, the control law based on optimal control theory gives a closed-form suboptimal feedback solution to solve this nonlinear control problem. For the case of an unstable orbit which has in space the shape of a curved eight in the Hill restricted three-body problem, we find that the actual trajectory tracks a nominal orbit given in the form of the Fourier series with good accuracy. It is shown 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.