Maintaining periodic trajectories with the first-order nonlinear Hill's equations

An approach to provide active nonlinear control for the first-order nonlinear Hill's equations describing relative motion of two satellites in orbit about a spherical Earth is presented. Both the linearized and first-order nonlinear Hill's equations are controlled to remain close to specific invariant manifolds defined through the various system Hamiltonians. It is then shown that trajectories similar to the periodic trajectories of the linearized system can be maintained by the controlled nonlinear equations near invariant manifolds defined by the linearized system of Hill's equations. Forcing the nonlinear system towards an invariant manifold of the linearized system, with an appropriate choice of initial conditions, provides a significant reduction in the along-track drift of the first-order nonlinear Hill's equations as compared to the linearized equations. There is also a small drift reduction in the radial coordinate direction. The cross-track position suffers only a slight increase in the maximum amplitude of its oscillation.