Optimal elliptic orbital rendezvous with continuous radial thrust on the chaser

Low-thrust technique is an important approach for spacecraft to accomplish orbital transfer and rendezvous. In most previous literatures for the optimal rendezvous problem, the thrust vector is aligned with the target's Local-Vertical Local-Horizontal (LVLH) frame. Moreover, current researches of radial thrust mainly focus on the orbital characteristics and the spacecraft escape condition, and there are few studies on the orbital rendezvous problem with radial thrust. This paper studies the optimal control problem for the time-fixed and time-free elliptic orbital rendezvous using continuous radial thrust on the chaser. Firstly, by using the relative direction cosine matrix, the relative motion equations for continuous radial thrust on the chaser can be obtained in the target's LVLH frame. As the direction cosine matrix is related to the system state variables, the resulting relative motion equations are nonlinear, and the classical Riccati method for linear equations is not suitable for this optimal control problem. Secondly, when the relative position is small, a dimension of the system is hardly controllable by analyzing the controllability of the system. Thirdly, by simplifying the uncontrollable state, the nonlinear optimal rendezvous problem is transformed into a two-dimensional optimal control problem with integral and terminal constraints. Finally, for the time-fixed and time-free orbital rendezvous, by using the maximal principle, the necessary conditions are derived for the minimum fuel control problem. The simulation results show that this proposed method in this paper has a high accuracy for the linear model, and the terminal error is small when the rendezvous angle is less than 3 rad for the nonlinear model; moreover, the terminal error becomes larger as the rendezvous angle increases.