<p>This paper proposes a model predictive control (MPC) algorithm for a small satellite to accomplish on-orbit inspection missions. The relative dynamics of satellite is modelled first. Then, multiple constraints are taken into account for the on-orbit inspection missions, including input saturation, obstacle avoidance, velocity limit, and task specifications. To precisely formulate the tasks, the signal temporal logic (STL) framework is employed, where an auxiliary function is required to be designed based on the robust semantics of STL formulas. Considering the impact of input saturation, the proposed algorithm designs the auxiliary function in the form of cube power function, and incorporate it into the optimization problem in MPC. After that, the terminal ingredients are designed, whose parameters can be efficiently calculated based on linear matrix inequality techniques. Finally, numerical simulation is applied to validate the effectiveness of the proposed control strategy.</p>