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276 Robust Control of Robotic Manipulators

the error signals was shown using a static controller. The norms used in
(5.2.8)–(5.2.10) are then ∞ norms. The development of this controller starts
with assumptions (5.2.8), (5.2.9), and a modification of (5.2.10) to
(5.2.20)

This assumption is justified by the fact that N is composed of gravity and
velocity-dependent terms which may be bounded independent from the
position error e [see (5.1.1)]. We shall also assume that =0. Let us
then choose the state-feedback controller (5.2.12) repeated here for
convenience:
(5.2.21)

The corresponding input-output differential equation
(5.2.22)

A block diagram description of this equation is given in Figure 5.2.4. Consider
now the transfer function from η (taken as an independent input) to e:

(5.2.23)

or

(5.2.24)
It can be seen that K v and K p are both diagonal, with , a critically
damped response it achieved at every joint [see (4.4.22), (4.4.30), and
(3.3.32)]. The infinity operator gains of P 11(s) and P 12(s) are (see Lemma
2.5.2 and Example 2.5.8)

(5.2.25)

where

(5.2.26)

Consider then the following inequalities:

Copyright © 2004 by Marcel Dekker, Inc.