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286 Robust Control of Robotic Manipulators

stability result of [Spong and Vidyasagar 1987]. It was shown in [Becker
and Grimm 1988], however, that the 2 stability of the error cannot be
guaranteed unless the problem is reformulated and more assumptions are
made. It effect, the error will be bounded, but it may or may not have a finite
energy. In particular, noisy measurements are no longer tolerated for 2
stability to hold. We next present an extension of the ∞ stability result that
applies to dynamical compensators similar to the one described in Theorem
5.2.3 but without the requirement that 2 =0.

THEOREM 5.2–4: The error system of (5.2.30) is bounded if
∞

=0

and

(1)

Proof:
An extension of the small-gain theorem. See [Becker and Grimm 1988]
for details.

A study of (1) reveals that the following desired characteristics will help
satisfy the inequality:

1. A large µ 1 due to a large M(q).
2. A small 1 and a 2 close to 1, which will result from a large-gain
compensator C.

3. Small i ’s, which will result from a good knowledge of N.
4. A small c due to a small d .

Note that Craig’s conditions in Theorem 5.2.2 are recovered if 1 =max 11 ,
12 and 2 k= 11 k p + 12 k v .
On the other hand, assuming that d =0 and 2 =0, the 2 stability of e was
shown in [Becker and Grimm 1988] if

(5.2.36)

where as given in Lemma 2.5.2. This
controller is summarized in Table 5.2.4.

Copyright © 2004 by Marcel Dekker, Inc.