Relative dynamics and control of spacecraft formations in eccentric orbits

Formation flying is a key technology for both deep-space and orbital applications that involve multiple spacecraft. Many future space applications will benefit from using formation flying technologies to perform distributed observations (e.g., synthetic apertures for Earth mapping interferometry) and to provide improved coverage for communication and surveillance. Previous research has focused on designing passive apertures for these formation flying missions assuming a circular reference orbit. Those design approaches are extended and a complete initialization procedure for a large fleet of vehicles with an eccentric reference orbit is presented. The main result is derived from the homogenous solutions of the linearized relative equations of motion for the spacecraft. These solutions are used to find the necessary conditions on the initial states that produce T-periodic solutions that have the vehicles returning to the initial relative states at the end of each orbit, that is, v(t₀) = v(t₀ + T). This periodicity condition and the resulting initialization procedure are originally given (in compact form) at the reference orbit perigee, but this is also generalized to enable initialization at any point around the reference orbit. In particular, an algorithm is given that minimizes the fuel cost associated with initializing the vehicle states (primarily the in-track and radial relative velocities) to values that are consistent with periodic relative motion. These algorithms extend and generalize previously published solutions for passive aperture forming with circular orbits. The periodicity condition and the homogenous solutions can also be used to estimate relative motion errors and the approximate fuel cost associated with neglecting the eccentricity in the reference orbit. The nonlinear simulations presented clearly show that ignoring the reference orbit eccentricity generates an error that is comparable to the disturbances caused by differential gravity accelerations.