Exponential Stabilization of the Rolling Sphere:
An Intractable Nonholonomic System
Department of Mechanical Engineering
Michigan State University, East Lansing, MI 48824
The geometry of rotations is central to the analysis of diverse problems in mechanics involving rolling motion and the rolling sphere captures the complexity of the control problems of many of these systems. The rolling sphere is a classical example of a nonholonomic mechanical system for which standard nonlinear control methods cannot be easily applied for stabilization to an equilibrium state. This follows from a well-known result, which establishes that there exists no smooth static state feedback which renders the equilibrium state of a closed loop nonholonomic system asymptotically stable. To circumvent this problem, researchers have developed control strategies that can be classified under smooth time-varying stabilization, piecewise-smooth time-invariant stabilization, and hybrid stabilization. These regimented control strategies are however applicable to a class of nonholonomic systems that can be transformed into a special form known as the “chained form”. The systems that cannot be converted to chained-form have intrinsic difficulties associated with their design of stabilization strategies and are referred to as “defective” or “intractable”. The rolling sphere is one such system and requires stabilization strategies to be custom designed.
We will present a piecewise-smooth controller for exponential stabilization of the sphere about an arbitrary desired configuration, defined by its two Cartesian coordinates and three orientation coordinates. Our work is based on a kinematic model that exploits the singularity of Euler angle representation of the equilibrium and definition of control inputs in a rotating reference frame that results in two fundamental motion primitives, namely, straight line motion and motion along a circular arc. We will use these motion primitives to first develop the “Sweep-Tuck” algorithm for partial reconfiguration of the sphere. In partial reconfiguration, the Cartesian coordinates will be converged to the origin but all the orientation coordinates will not be reconfigured. We will subsequently extend the algorithm for complete reconfiguration. The stability of the equilibrium under the Sweep-Tuck algorithm and exponential convergence will be established separately. Simulation results of partial and complete reconfiguration as well as stable behavior will be presented. If time permits, we will discuss implementation issues and hardware details of a spherical mobile robot platform to which we intend to apply the control algorithm.
Friday, October 11, 2002
3:30 – 4:30 p.m.