Finite-time stability of an underactuated tethered satellite system

The tethered satellite system has been extensively studied due to its widespread applications for space missions. However, the tether's oscillation draws back the performance of the system, and furthermore this oscillation is governed by an uncontrollable state. In this paper, an adaptive hierarchical second-order sliding mode control scheme is proposed to suppress the oscillation of space tether, and disturbance as well. Based on the dynamics equations, system energy of the tethered satellite system is further studied. A reference sliding variable is addressed to strengthen the coupling between the controllable and uncontrollable state. The proposed second-order sliding control scheme can suppress the oscillation and disturbance effectively, and meanwhile can weaken the control chattering. The closed-loop stability with the addressed references sliding variable is strictly verified via Lyapunov function. Finally, the proposed controller is verified by numerical simulations of both deployment and retrieval. With the comparison of other control scheme shows that the sub-satellite can be deployed and retrieved quickly and stably under the proposed scheme. Furthermore, the simulations in presence of disturbance prove the robustness of the proposed controller.