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190 Computed-Torque Control
There are some upper limits on the choice for n [Paul 1981]. Although the
links of most industrial robots are massive, they may have some flexibility.
Suppose that the frequency of the first flexible or resonant mode of link i is
(4.4.23)
with J the link inertia and k r the link stiffness. Then, to avoid exciting the
resonant mode, we should select n < r /2. Of course, the link inertia J changes
with the arm configuration, so that its maximum value might be used in
computing r .
Another upper bound on n is provided by considerations on actuator
saturation. If the PD gains are too large, the torque τ(t) may reach its upper
limits.
Some more feeling for the choice of the PD gains is provided from error-
boundedness considerations as follows. The transfer function of the closed-
loop error system in (4.4.15) is
(4.4.24)
or if K v and K p are diagonal,
(4.4.25)
(4.4.26)
-1
We assume that the disturbance and M are bounded (Table 3.3.1), so that
(4.4.27)
—
—
with m and d known for a given robot arm. Therefore,
(4.4.28)
(4.4.29)
Now selecting the L 2—norm, the operator gain ||H(s)|| 2 is the maximum
value of the Bode magnitude plot of H(s). For a critically damped system,
(4.4.30)
Therefore,
Copyright © 2004 by Marcel Dekker, Inc.