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202 Computed-Torque Control

disturbance to the arm dynamics and compare PD to PID computed
torque.
Thus let the arm dynamics be
(1)

with d a constant disturbance with 1 N-m in each component. This could
model unknown dynamics such as friction, and so on. The value of 1 N-m
represents quite a large bias.
Adding 1 to the computation of the nonlinear terms N1 and N2 in
subroutine arm(x, xp) in Figure 4.4.2 and using the PD computed-torque
controller with k p=100, k v=20 yields the error plot in Figure 4.4.7(a).
There is a unacceptable residual bias in the tracking error due to the
unmodeled constant disturbance, which is not accounted for in the
computed-torque law. The largest error is 0.033 rad, somewhat less
than 2 deg.
Adding now an integral-error weighting term with k i=500 yields the
results in Figure 4.4.7(b), which show a zero steady-state error and are
quite good.
The associated control torques are shown in Figure 4.4.8, which shows
that the torque magnitudes are not appreciably increased by using the
integral term.
To simulate the PID control law, it is necessary to add two additional
states to x in subroutine arm(x, xp). It is convenient to call them x(5) and
x(6), so that the lines added to the subroutine are

xp(5)=e(1).
xp(6)=e(2). (2)

Thus it is now necessary to compute e(1) and e(2) in subroutine arm(x, xp)
for integration purposes.

Class of Computed-Torque-Like Controllers
An entire class of computed-torque-like controllers can be obtained by
modifying the computed-torque control law to read

(4.4.43)

Copyright © 2004 by Marcel Dekker, Inc.