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A stable hierarchical sliding-mode control method for a class of second-order underactuated systems is presented. The ideas behind the method are as follows. First, the underactuated system is divided into two subsystems. For each part a first-level sliding surface is defined. For these two first-level sliding surfaces, a second-level sliding surface is defined. The sliding-mode control law is derived using Lyapunov law. The control law can drive the subsystems toward their sliding surfaces and attain their desired values, and implement antidisturbance control. The asymptotic stability of all the sliding surfaces is proved theoretically, and simulation results show the controller's validity and its adaptive abilities for all kinds of extraneous disturbances.
