Series expansion-based state transition matrix for relative motion on eccentric orbits

This paper considers the formation dynamics with the target satellite in an elliptic orbit. A new state transition matrix for the linearized equations of relative motion is presented by series expansion and some mathematical transformations. The state transition matrix is applicable to any eccentricity elliptic reference orbit. Besides, the state transition matrix is just related to the trigonometric function of the true anomaly of the target satellite, so it is easy to calculate. With the state transition matrix, the contribution of three-order nonlinearity in the differential gravitational acceleration on the relative motion is estimated by a perturbation approach. Numerical simulations are included to evaluate the proposed methods.