Page 206 -

4.4 Computed-Torque Control 189

(4.4.18)

Then
(4.4.19)

and the error system is asymptotically stable as long as the K vi and K pi are
all positive. Therefore, as long as the disturbance w(t) is bounded, so is the
error e(t). In connection with this, examine (4.4.6) and recall from Table
-1
3.3.1 that M is upper bounded. Thus boundedness of w(t) is equivalent to
boundedness of d (t).
It is important to note that although selecting the PD gain matrices diagonal
results in decoupled control at the outer-loop level, it does not result in a
decoupled joint-control strategy. This is because multiplication by M(q) and
.
addition of the nonlinear feedforward terms N (q, q) in the inner loop scrambles
the signal u(t) among all the joints. Thus, information on all joint positions
.
q(t) and velocities q(t) is generally needed to compute the control (t) for any
one given joint.
The standard form for the second-order characteristic polynomial is
(4.4.20)

with the damping ratio and n the natural frequency. Therefore, desired
performance in each component of the error e(t) may be achieved by selecting
the PD gains as

(4.4.21)

with , n the desired damping ratio and natural frequency for joint error i. It
may be useful to select the desired responses at the end of the arm faster than
near the base, where the masses that must be moved are heavier.
It is undesirable for the robot to exhibit overshoot, since this could cause
impact if, for instance, a desired trajectory terminates at the surface of a
workpiece. Therefore, the PD gains are usually selected for critical damping
=1. In this case

(4.4.22)

Selection of the Natural Frequency. The natural frequency n governs the
speed of response in each error component. It should be large for fast responses
and is selected depending on the performance objectives. Thus the desired
trajectories should be taken into account in selecting n. We discuss now
some additional factors in this choice.

Copyright © 2004 by Marcel Dekker, Inc.