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266 Robust Control of Robotic Manipulators

or

(5.2.5)

Not that the notion above indicates that u(t) is the output of a system C(s)
when an input e(t) is applied. Note also from (5.2.4) that the different
joints or the robot are decoupled so that at this level, n SISO separate
controllers may be designed to control the n joints of the robot.
Unfortunately, the control law (5.2.2) cannot usually be implemented due
to its complexity or to uncertainties present in M(q) and N(q, ) and to the
presence of d and w i. Instead, one applies in (5.2.6) below where and
are estimates of M and N,

(5.2.6)

This in turn will reintroduce some coupling in the linear model and leads to
(Fig. 5.5.1)

(5.2.7)

Note first that , , and therefore are zero if = M and = N. In general,
however, the vector is a nonlinear function of both e and u and cannot be
treated as an external disturbance. It represents an internal disturbance of
the globally linearized error dynamics caused by modelling uncertainties,
parameter variations, external disturbances, friction terms, and maybe even
noise measurements [Spong and Vidyasagar 1987]. Most commercial robots
are in fact controlled with the controller given in (5.2.6) with choices of =
I and = 0 See, for example, [Luh 1983] and Section 4.4. The choice of
is validated by the powerful motors used to drive the robot links, and the
gearing mechanisms used to torque the motor output to an acceptable level,
while showing its speed down. The choice of is validated by keeping the
different motors from driving their links too fast, thus limiting the Coriolis
and centripetal torques. Such commercial controllers are known as “non-
model-based controllers” and have been used since the early days of robotics.

Copyright © 2004 by Marcel Dekker, Inc.