Due to the under-actuated feature, the reference signals using the combination of the system outputs, whose number is larger than that of reference signal, are designed so that the number of control inputs and sliding surfaces is the same, and that the uncontrollable mode is indirectly controlled. Under the uncertain environment, the sliding-mode under-actuated control (SMUC) with the satisfaction of suitable condition is designed to asymptotically track the reference signal. Otherwise, a bounded tracking result is obtained for the mild condition. In this situation, an on-line fuzzy modeling for the uncertainty is employed to design a fuzzy model-based sliding-mode under-actuated control (FSMUC) to improve the system performance; e.g., the bounded tracking result of SMUC becomes an asymptotical tracking. The proposed hybrid fuzzy sliding-mode under-actuated control (HFSMUC) combining SMUC and FSMUC with a transition can be applied to a class of generalized, under-actuated and uncertain nonlinear systems, e.g., the trajectory tracking control of a differential mobile robot (DMR). Finally, the simulations of the HFSMUC system are presented to confirm the efficiency and effectiveness of the proposed control.
