Path tracking and stability of a rolling controlled wheel on a horizontal plane by using the nonholonomic constraints

This paper studies the stabilization, tracking of a predefined trajectory and how to reach a desired set point for a wheel which is rolling on a horizontal plane without slipping. For this purpose, the wheel is controlled by small torque generated by internal servomechanisms whose dynamics can be neglected. An efficient procedure to determine the kinetic energy of the wheel is developed by introducing a set of reference systems, which in combination with the Lagrange equations with multipliers allow deriving the mathematical model of the rolling wheel. In this model, the Euler angles, the coordinates of the plane-wheel contact point and a control law of proportional+integral+derivative (PID) type provide an efficient computational procedure to track arbitrary trajectories. It is shown that the nonholonomic constraints are fulfilled with admissible reaction forces, even when the desired trajectory has cusp points. A circumference and a family of astroids are used as trajectories to verify the motion conditions derived from the energy conservation and dynamical equilibrium of the wheel along such trajectories. The results of the analytical calculations are corroborated through numerical simulations.