Hierarchical Sliding Mode Control Design for Stabilization of Underactuated Wheeled Acrobot System

This paper presents the hierarchical sliding mode control (HSMC)--based stabilization problem of the wheeled Acrobot (WAcrobot) system, which combines an actuated wheel that rolls in the horizontal plane with an Acrobot consisting of two inverted pendulum links. These links are driven by an actuator at the second joint and freely rotate in a vertical plane. The stabilization problem of the WAcrobot, previously addressed using switch-based controllers in many studies, is solved in this work by designing a single-stage controller without requiring switching steps or time. The design approach comprises two parts. First, the system is reduced to a cascaded nonlinear model in finite time using the inherent dynamic coupling relationship. Second, a two-loop control scheme is employed: the outer loop ensures that the actuated state variables track the desired response, while the inner loop forces them to exhibit an asymptotically stable nature. After the desired design of both loops, the asymptotic stability of the overall dynamics is achieved. Finally, the theoretical analysis is validated using MATLAB. Simulation results demonstrate that the proposed controller effectively stabilizes the WAcrobot system at the equilibrium points.