Feedback stabilization of the rolling sphere: An intractable nonholonomic system

A spherical rolling robot has several advantages over wheeled robots, such as enhanced mobility, orientational stability, compact and closed design, and capability of operations in hazardous environments. However, advances in the design and application of spherical mobile robots have been hindered due to complexity of their control problems. Of particular interest is the problem of feedback stabilization of a rolling sphere to an equilibrium configuration. The rolling sphere belongs to the class of nonholonomic systems which has been a popular area of research in the control systems community over the last decade. Although nonholonomic systems are usually controllable, they are not stabilizable to an equilibrium point using smooth static state feedback. This problem has been circumvented by development of techniques such as time-varying stabilization, discontinuous time-invariant stabilization, and hybrid stabilization. Nonetheless, the stabilization of a rolling sphere has remained an unsolved problem since its kinematic model cannot be reduced to the chained form; this renders all established nonholonomic motion planning and control algorithms inapplicable.; In this dissertation we present a feedback control law for stabilization of a rolling sphere to an equilibrium configuration. This control law, which to the best of our knowledge, is the first solution to the problem, stabilizes the sphere about an equilibrium point defined by the two Cartesian coordinates and three orientation coordinates of the sphere. In our formulation, the control inputs are two mutually perpendicular angular speeds in the moving reference frame of the sphere. These control actions individually cause the sphere to move in straight line and circular arc segments. Using an alternating sequence of these rudimentary maneuvers we achieve stabilization of the equilibrium configuration. We first develop an algorithm for partial reconfiguration of the sphere where evolution of one of the orientation coordinates is ignored. This algorithm, which we denote by the Sweep-Tuck algorithm, allows multiple solution trajectories of the sphere. We utilize this flexibility in achieving complete reconfiguration. In our discussion we first show the convergence of the configuration variables to the equilibrium under the proposed feedback law. Subsequently, we prove that the control algorithm stabilizes the equilibrium configuration of the sphere. Simulation results are presented to demonstrate the efficacy of the control strategy.