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Stabilization of Nonholonomic Systems 203

(v) Then the smooth control law

u 1 (x 1 , x 2 )
u = u(x 1 , x 2 ) =
u 2 (x 2 )

locally exponentially stabilizes the system (5.12).

The hypotheses of Theorems 5.3, 5.4, and 5.5 seem very restrictive.
However, it is possible to transform several smooth nonholonomic systems
in such a way that the aforementioned hypotheses are automatically fulfilled.

5.3.2 The σ Process

In this section we discuss the use of nonsmooth coordinates changes to trans-
form continuous systems into discontinuous ones. We consider a choice of
coordinates system in which, to a small displacement near a fixed point, there
corresponds a great change in coordinates. The polar coordinates system pos-
sesses such a property; however the cartesian to polar transformation requires
transcendental functions; therefore, when not needed, we avoid using the polar
coordinates, usinganotherprocedure: theso-calledσ process(seeReference45,
where the σ process is used to resolve singularities of vector fields).
Mainly, the σ process consists of a nonsmooth rational transformation, but,
with abuse of notation, we denote with the term σ process every nonsmooth
coordinates transformation possessing the property of increasing the resolution
around a given point.

Example 5.2 Consider the two dimensional system with one control

˙ x = g 1 (x, y)u, ˙ y = g 2 (x, y)u

and perform the coordinates transformation

z x
= (x, y) =
w y/x
The resulting system is

g 2 (z, zw) − wg 1 (z, zw)
˙ z = g 1 (z, zw)u, ˙ w = u (5.16)
z

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 203 — #17