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208 Autonomous Mobile Robots

but, in general, the matrix g 21 (ξ 1 , ξ 2 ) is not defined at ξ 1 = 0, that is, fulfills
condition (5.15).

The presented discontinuous stabilization approach has been exploited in
the control of underactuated spacecraft in Reference 36 and has been given
an interesting geometric interpretation in Reference 48 and related references.
Finally, in Reference 49 and related works, it has been shown that the proposed
approach can be interpreted in terms of a state-dependent time-scaling.

5.4 CHAINED SYSTEMS AND POWER SYSTEMS
From this section onward, we focus on two special classes of nonholonomic
systems: chained systems and power systems. They occupy a special place in the
theory of nonholonomic control. Many nonholonomic mechanical systems can
be represented by, or are feedback equivalent to, kinematic models in chained
form or in power form. Chained systems have been introduced in Reference 7,
where sufficient conditions for (local) feedback equivalence to chained forms
have also been given. Power systems have been introduced in Reference 50.
Therein, it has also been shown that chained systems and power systems are
globally feedback equivalent. Chained systems 15 are described by equations of
the form

˙ x 1 = u 1
˙ x 2 = u 2
˙ x 3 = x 2 u 1
(5.22)
.
.
.
˙ x n = x n−1 u 1 .

Power systems are described by equations of the form

˙ x 1 = u 1

˙ x 2 = u 2
˙ x 3 = x 1 u 2
(5.23)
.
.
.
1 n−2
˙ x n = x u 2 .
(n − 2)! 1
15 In the terminology of Reference 7, Equations (5.22) describe a one-chain single generator system.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 208 — #22