Geometric–PBC Approach for Control of Circular Ball and Beam System∗

In this paper, a nonlinear control law is proposed for the curved ball and beam system with the ball at the upright position. The stabilization of such a system is challenging, as compared to conventional ball and (straight) beam system, due to the presence of two unstable equilibrium points and the gyroscopic forces appearing in its dynamics. Further, the system belongs to the interesting class of underactuated mechanical systems. We propose Geometric Passivity Based Control (PBC) methodology for synthesizing stabilizing control law for the system under consideration. The power flow between the controller and the unactuated dynamics is established by manipulating the mechanical connection. An additional dissipation term is included in the control law to deal with the gyroscopic forces. The main advantage of the proposed controller design strategy is that it does not involve solving any Partial Differential Equations (PDEs) or nonlinear transformations. The simulation results are presented to validate the control law. The system parameters used in simulation adheres to the physical model constraints.