Satellite relative motion using differential equinoctial elements

For precise control, to minimize the fuel consumption, and to maximize the lifetime of satellite formations a precise analytic solution is needed for the relative motion of satellites. Based on the relationship between the relative states and the differential orbital elements, the state transition matrix for the linearized relative motion that includes the effects due to the reference orbit eccentricity and the gravitational perturbations is derived. This method is called the Geometric Method. To avoid any singularities at zero eccentricity and zero inclination, equinoctial variables are used to derive the relative motion state transition matrices for both mean and osculating elements. This approach can be extended easily to include other perturbing forces.