This paper presents a novel six degree-of-freedom guidance and control algorithm for free-flyer robots equipped with a robotic manipulator operating on the International Space Station. The proposed algorithm combines a sliding mode controller for rotational motion and a model predictive controller for translational motion. A novel prediction-twisting sliding mode controller (PT-SMC) is introduced to stabilize configuration changes induced by the manipulator and correlated disturbances and to achieve a trade-off among control law flexibility, robustness, and accuracy. The proposed PT-SMC is shown to have improved stability compared to a first-order sliding mode controller. Model predictive control is investigated as a position controller to achieve formation flight between multiple free flyers in a constrained environment, and a leader–follower approach is used to move a triangular formation of robots. A control allocation problem is solved to address the subdivision of attitude and position controls between twelve thrusters. Simulation results of multiple robots approaching a target object demonstrate the improved performance of the proposed control approaches.