A linear time-varying model predictive control (LTV-MPC) method based on $SE(3)$ is proposed, to enhance the control precision and reduce energy consumption of the spacecraft. First, left-invariant principle of Lie group, Lie algebra $SO(3)$ , and other differential geometry theories are applied to extend LTV-MPC to $SE(3)$ . Then, considering the obstacle avoidance problem of the spacecraft in space, a suitable optimization function is selected to ensure smooth tracking of the desired working trajectory. By controlling the incremental output with the fastest convergence rate, the controller enables effective trajectory tracking. Finally, the effectiveness and applicability of the controller are verified by simulation experiments. These simulations validate the controller’s ability to achieve constraint satisfaction, optimize transient processes, and enhance control accuracy. The results provide compelling evidence of the controller’s potential for real-world spacecraft control applications.