Fundamental frequencies of satellite relative motion and control of formations

Expressions for the fundamental natural frequencies associated with J(2)-perturbed relative motion of satellites in near-circular orbits are derived. Special values of the orbit inclination, dependent on initial conditions, are obtained, for which the in-plane and out-of-plane fundamental frequencies remain equal to each other over an extended period of time, resulting in nonprecessing relative orbits. This result validates and generalizes a similar finding, based on numerical investigations by other researchers. The analysis is extended to the developments of accurate prediction and control models for fuel-optimal formation maintenance and intersatellite fuel balancing. Numerical simulation results are presented to demonstrate the accuracy of the developed models and the effects of frequency matching on control requirements.