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4.4 Computed-Torque Control 199

Selecting diagonal control’ gains

(4.4.40)

gives

(4.4.41)

By using the Routh test it can be found that for closed-loop stability we require
that

(4.4.42)

that is, the integral gain should not be too large.

Actuator Saturation and Integrator Windup. It is important to be aware of an
effect in implementing PID control on any actual robot manipulator that can
cause serious problems if not accounted for. Any real robot arm will have
limits on the voltages and torques of its actuators. These limits may or may
not cause a problem with PD control, but are virtually guaranteed to cause
problems with integral control due to a phenomenon known as integrator
windup [Lewis 1992].
Consider the simple case where =k i ε with ε(t) the integrator output. The
torque input (t) is limited by its maximum and minimum values max and min .
If k i ε(t) hits max , there may or not may not be a problem. The problem arises
if the integrator input remains positive, for then the integrator continues to
integrate upwards and k i ε(t) may increase well beyond max . Then, when the
integrator input becomes negative, it may take considerable time for k i ε(t) to
decrease below max . In the meantime is held at max , giving an incorrect
control input to the plant.
Integrator windup is easy to correct using antiwindup protection in a digital
controller. This is discussed in Section 4.5. The effects of uncorrected windup
are demonstrated in Example 4.4.4.
The next example shows the usefulness of an integral term when there are
unknown disturbances present.

EXAMPLE 4.4–2: Simulation of PID Computed-Torque Control
In Example 4.4.1 we simulated the PD computed-torque controller for a
two-link planar arm. In this example we add a constant unknown

Copyright © 2004 by Marcel Dekker, Inc.