Linearized dynamics model for relative motion under a J₂- perturbed elliptical reference orbit

A method for directly establishing a linearized dynamics model of relative motion for a satellite formation flying on an arbitrary elliptical reference orbit is presented. The proposed linearized dynamics model of relative motion is intuitive and favorable to be utilized for designing formation control system, implementing guidance algorithm, and analyzing optimization problems etc. An analytical solution for the radius of a reference orbit affected by the J ₂ geopotential disturbance is used to approximate the actual reference orbit rather than directly use the unperturbed standard orbit. The accuracy of the analytical solution directly affects the accuracy of the linear relative motion model of a satellite formation. Thus, emphasis is placed on deducing an accurate analytical solution for a perturbed reference orbit radius, which is analyzed by adding perturbed motion in the radial direction to the corresponding unperturbed reference orbit radius. Furthermore, by considering time-varying angular velocity of an elliptical reference orbit, the analytical solution of a perturbed orbit radius is obtained in the true anomaly domain. In order to better match the actual force conditions of a satellite formation, a perturbed true anomaly and perturbed argument of perigee are adopted and substituted into the gradient of the J₂ disturbance force. Simulation results indicate the analytical solution for the radius of a perturbed elliptical reference orbit is accurate, and the proposed linear dynamics model of relative motion can track the high-fidelity simulated relative motion more accurately than previously proposed dynamics models, even for an elliptical reference orbit with high eccentricity.