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224 Computed-Torque Control

inner nonlinear loop can be sampled more slowly than the outer linear feedback
loop. In view of the robustness properties of computed-torque control, this
works quite well in practice.
Guaranteed Performance on Sampling

It is usual in robot controls to design the controller in continuous time, providing
rigorous proofs of stability and error boundedness. However, when the controller
is implemented, a “small” sample period is selected and the stability is left to
chance and verified by simulation studies. That this “wishful thinking”
approach may not be so unreasonable is suggested by the next theorem. Define
.

THEOREM 4.5–1: Suppose that the PD digital control law (4.5.3) is applied
to the robot arm. Then for every L>0 there exists a T M such that for all
sampling periods T less than T M , each error trajectory which at some time to
satisfies ||e(t 0 )|| L has ||e(t)|| L+r for all t t 0 and any r>0.

Proof:
Using the digital control law yields the error system (Table 4.4.1)

(1)

where and

(2)

Defining this may be written in state-space form as

(3)

Applying now the outer-loop digital control

(4)

yields the closed-loop system

(5)

The feedback K is selected so that (A-BK) is stable; hence

Copyright © 2004 by Marcel Dekker, Inc.