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214 Computed-Torque Control

(4.4.55)

with the closed-loop characteristic polynomial

(4.4.56)

The PD gains can be selected to obtain a suitable natural frequency and
damping ratio, as we have seen.
At steady state, the only nonzero contribution to the disturbance d(t) is the
neglected gravity G(q). Table 3.3.1 shows that the gravity vector is bounded
by a known value g b for a given robot arm. Therefore, at steady state,

Using the final value theorem, the steady-state tracking error for joint i
using PD control is bounded by
(4.4.57)
Therefore, for set-point tracking where a final value for q d is specified and
PD control with a large k p might be suitable.
As a matter of fact, the next results show that PD control is often very
suitable even for following a desired trajectory, not only for set-point control.
It is proven in [Dawson 1990].

THEOREM 4.4–2: If the PD control law (4.4.54) is applied to each joint and
e(0)=0, (0)=0, the position and velocity tracking errors are bounded within a
ball whose radius decreases approximately (for large k v)
as .

This result gives credence to those who maintain that PD control is often
good enough for practical applications.
Of course, the point is that k v cannot be increased without limit without
hitting the actuator torque limits. Other schemes to be discussed in the book
allow good trajectory following without such large torques.
In [Paul 1981] are discussed several methods for modifying the PD control
law to obtain better performance. These include gravity compensation [which
yields exactly controller (4.4.49)], and acceleration feedforward, which
amounts to using

(4.4.58)

Copyright © 2004 by Marcel Dekker, Inc.