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208 Computed-Torque Control

Assuming identical PD gains for each link, the PD-gravity control law for
the two-link arm is

(1)

This is easily simulated by commenting out several lines of code and making
a few other modifications in subroutine CTL(x) of Figure 4.4.2.
For critical damping, the PD gains are selected as

(2)

The results of the PD-gravity controller simulation for several values of w n
are shown in Figure 4.4.9. As ω n , and hence the PD gains, increases, the
tracking performance improves. However, no matter how large the PD gains,
the tracking error never goes exactly to zero, but is bounded about zero by a
ball whose radius decreases as the gains become larger. The performance for
ω n =50, corresponding to k p =2500, k v =100, would be quite suitable for many
applications.
It quite important to note that the dc value of the errors is equal to zero.
One can consider the gravity terms as the “dc portion” of the computed-
torque control law. If they are included in computing the torques, there will
be no error offset.
The associated control input torques are shown in Figure 4.4.10. It is
extremely interesting to note that the torques are smaller for the higher
gains. This is contrary to popular belief, which assumes that the control
torques always increase with increasing PD gains. It is due to the fact that
larger gains give “tighter” performance, and hence smaller tracking errors.
In view of the fact that the errors in Figure 4.4.9(a) have different magnitudes,
it would probably be more reasonable to take the PD gains larger for the
inner link than the outer link.

Classical Joint Control
A simple controller that often gives good results in practice is obtained by
selecting in (4.4.43)

(4.4.50)

Copyright © 2004 by Marcel Dekker, Inc.