It is a great challenge to maintain the tether system in gravitational stable formation and avoid tumbling failure if it is perturbed by a large orbital propulsion force. This paper studies the dynamic stability and control of a tethered space-tug system for space debris removal during and after the propulsion stage of the Hohmann transfer. In the present work, it is assumed that the magnitude of the propulsion force is constant whereas the propulsion direction is controllable. The perturbation of the orbital propulsion force to the tether libration and the tug attitude motion is eliminated through a two-stage control strategy. In the first stage, a control minimization problem is established with the propulsion time selected as the cost function and the operational constraints considered, such as bounded motion both for the tether libration and the tug attitude motion, positive tension and the required velocity increment of the Hohmann transfer. The optimization problem is solved to find a reference control profile for the control inputs including the reel-in/out acceleration of tether, the control toque and the direction of the propulsion force acting on the tug during the propulsion stage that ensures a final gravitational stable of the tethered space-tug system. A sliding mode controller with an adaptation laws for the upper bounds of the norm of the uncertainty is proposed as a closed-loop control for tracking the resulted reference trajectory. Then in the second stage, the residual vibration of the tethered system after propulsion stage is removed by a tether length rate regulation based on the sliding controller. Finally, the effectiveness of the propose control strategy is carefully examined by numerical simulation.
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