Page 313 -

298 Robust Control of Robotic Manipulators

ones discussed here) by [Slotine 1985] and [Chen et al. 1990] remedied the
problem.
The first application of this theory to robot control seems to be in [Young
1978], where the set-point regulation problem ( d =0)was solved using the
following controller:

(5.3.4)

where i=1,…, n for an n-link robot, and r i are the switching planes,

(5.3.5)

It is then shown, using the hierarchy of the sliding surfaces r 1, r 2,…, r n and
given bounds on the uncertainties in the manipulators model, that one can
-
find and in order to drive the error signal to the intersection of the
+
sliding surfaces, after which the error will “slide” to zero. This controller
eliminates the nonlinear coupling of the joints by forcing the system into the
sliding mode. Other VSS robot controllers have since been designed.
Unfortunately, for most of these schemes, the control effort as seen from
(5.3.4) is discontinuous along r i=0 and will therefore create chattering, which
may excite unmodelled high-frequency dynamics. In addition, these
controllers do not exploit the physics of the robot and are therefore less
effective than controllers that do.
To address this problem, the original VSS controllers were modified in
[Slotine 1985] as described in the next theorem. Let us first define a few
variables to simplify the statement of the theorem. Let

(5.3.6)

where

THEOREM 5.3–2: Consider the controller

Copyright © 2004 by Marcel Dekker, Inc.