TABLE OF CONTENTS Afamily of vector fields 
on a manifold 
is sometimes called a dynamical polysystem. The system is symmetric if for every 
is also in 
.
The family of dynamical polysystems we will study are control affine nonlinear control systems, written
| (12.6) |  |
The vector field g 0 is called the drift vector field, defining the natural unforced motion of the system, and the g i, i = 1 m, are linearly independent control vector fields. The control vector u belongs to the control set 
, and u( t) is piecewise continuous. If g 0 = 0, the system is called drift-free or driftless. Kinematic systems (such as the unicycle) may be drift-free, but second-order systems (such as the PBWT) are not.
The control system for the unicycle is written 
, where u 1 is the driving speed and u 2 is the steering control.
The control system for the PBWT is written 
, where u 1 is the thrust force at thruster 1 and u 2 is the force at thruster 2.
We will consider two classes of control sets:
: This class of control sets includes any control set 
containing the origin of 
in the interior of its convex hull. In other words, the control set positively spans 
any point in 
can be...
