Leo spacecraft relative motion dynamics modeling and solving

Relative motion dynamics is the basis of spacecraft formation flying, simple and reasonable relative motion model is very important to navigation, guidance and control. CW equations were modeled under some assumptions, i.e. a circular reference orbit, linearized differential gravity, and without any perturbations. The applicability of CW equations are limited because the assumptions are always violated in practice, thus the needs of spacecraft formation flying mission, especially long-term formation flying, can not be met. Under the background of spacecraft formation flying, this paper presents an approximate relative motion dynamical model that considers the eccentricity of the reference orbit, the second order nonlinear differential gravity and the J2 perturbation. Numerical simulations are carried out to verify the accuracy. And comparisons are also made between this model and a class of models which were derived by directly adding the perturbations onto the right hand side of Lawden equations or CW equations. Results show that the model presented in this paper is more accurate. Furthermore, the perturbation theory is used to obtain the analytical solution to the model which ignores the effect of J2. Simulation results show that when the eccentricity is 0 to 0.05, the error of the analytical solution is less than 10% of the relative motion scale during five resolutions of the target spacecraft.