Hovering control scheme to elliptical orbit via frozen parameter

In this paper, we analyze the hovering control operation of a target in an elliptical orbit based on frozen parameter. The relative dynamics of spacecraft in eccentric orbits can be described as linear time-varying systems with periodic parameters. Those periodic parameters are the orbital radius, angular velocity and angular acceleration. The time-varying characteristics of these periodic parameters were analyzed via a performance index which is obtained analytically with small fixed time interval. Then, the frozen time interval was obtained with a fixed performance index using a numerical method. A series of constant coefficient linear systems are then obtained to represent the periodic linear system, namely the relative motion dynamics. In the frozen time interval, the hovering control scheme was designed based on a Linear Quadratic Regulator (LQR). Numerical simulations showed that the hovering control performance is good, except the vicinity near the apogee and perigee, both of which exhibit a relative large performance index. Decreasing the performance index from 0.01 to 0.001 reduces the position errors from 5 m to 0.5 m during steady state. It is shown that this approach can save propellant and eliminate chatter during the convergence compared to Control Lyapunov Functions (CLF) approach.