Stabilization and positioning control of a rolling disk by using the Bogdanov–Takens bifurcation

On the basis of Bogdanov–Takens bifurcation, we obtain two nonlinear control laws for stabilization and positioning of a rolling disk on a horizontal plane at a desired set point. The first control law is applied to reach a set of predetermined points from previously calculated initial conditions, which unavoidably implies a small position error. To overcome this problem, a second procedure based on a nonlinear inverse control law is examined by using adequate leaning, tilting and rolling moments. The system satisfies nonholonomic constraints, it is underactuated, and the analysis of the Bogdanov–Takens bifurcation provides the admissible parameter values to reach the desired set point. Our analysis allows stabilization and positioning of the rolling disk through small control torques (neglecting the internal servomechanism dynamics) with adequate reaction forces satisfying the nonholonomic conditions. The results of the analytical calculations are verified by full numerical simulations.