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294 Robust Control of Robotic Manipulators

the other hand, if one can show the passivity of the system, which maps to
a new vector r which is a filtered version of e, a controller that closes the
loop between -r and will guarantee the asymptotic stability of both e and .
This indirect use of the passivity property was illustrated in [Ortega and
Spong 1988] and will be discussed first.
Let the controller be given by (5.3.1), where F(s) is a strictly proper, stable,
rational function, and K r is a positive-definite matrix,

(5.3.1)

Substituting (5.3.1) into (5.1.1) and assuming no friction [i.e., F( )=0], we
obtain

(5.3.2)

Then it may be shown that both e and are asymptotically stable. In fact,
choose the following Lyapunov function:

Then

Substituting for M(q) from (5.3.2), we obtain

Therefore, r is asymptotically stable, which can be used to show that both e
and e are asymptotically stable [Slotine 1988]. This approach was used in
the adaptive control literature to design passive controllers [Ortega and Spong
1988], but its modification in the design of robust controllers when M, V m
and G are not exactly known is not immediately obvious. Such modifications
will be given in the variable-structure designs, but first, we present a simple
controller to illustrate the robustness of passive compensators.

Copyright © 2004 by Marcel Dekker, Inc.