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.**