The precise analytic solution with the reference orbit eccentricity and perturbation effects is needed for the relative motion of formation flying satellites. Since Hill's equations have considerable errors and are insufficient for the long term prediction of the relative motion, the new approach, called geometric method, is proposed to obtain the state transition matrix as a precise solution under the effects due to the reference orbit eccentricity and the gravitational perturbations. Based on the transformation and the state transition matrix for the relative orbital elements, the geometric method gives a precise solution in closed form to 1 st order in J 2 for the non-circular reference orbit with mean orbital elements under the existence of the gravitational perturbation J 2. Finally, using the transformation matrix from the mean elements to the osculating elements in powers of the eccentricity, the state transition matrix of the relative motion with the osculating elements is derived without solving the differential equations. The results in this paper are based on the J 2 effects, but the approach could be extended to include other perturbing forces.

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An impulsive feedback control is developed to establish speci c relative orbits for spacecraft formation ying. The relative orbit tracking errors are expressed in terms of mean orbit elements. The feedback control, based on Gauss's variational equations of motion, allows speci c orbit elements or orbit element sets to be controlled with minimal impact on the remaining osculating orbit elements. This is advantageous when J 2 -invariant orbits are to be controlled, where only the argument of perigee and mean anomaly will drift apart at equal and opposite rates. The advantage of this impulsive feedback control, compared to optimal control solutions, is that it can operate with little computational effort and in a near-optimal manner, while requiring only a marginal penalty in fuel cost. When applied to the spacecraft formation ying problem, this control could also be used to perform general orbit corrections. Formulas are developed providing accurate estimates of the sensitivities of the mean semimajor axis and mean eccentricity with respect to the osculating inclination angle. With these sensitivities, the tracking error in semimajor axis, eccentricity, and inclination angle can be canceled within one orbit.

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