Page 203 -

186 Computed-Torque Control

(4.4.3)

To demonstrate the influence of the input (t)on the tracking error, differe-
ntiate twice to obtain

Solving now for in (4.4.2) and substituting into the last equation yields

(4.4.4)

Defining the control input function

(4.4.5)

and the disturbance function

(4.4.6)

we may define a state by

(4.4.7)

and write the tracking error dynamics as

(4.4.8)

This is a linear error system in Brunovsky canonical form consisting of n
2
pairs of double integrators 1/s , one per joint. It is driven by the control input
u(t) and the disturbance w(t). Note that this derivation is a special case of the
general feedback linearization procedure in Section 3.4.
The feedback linearizing transformation (4.4.5) may be inverted to yield

(4.4.9)

We call this the computed-torque control law. The importance of these
manipulations is as follows. There has been no state-space transformation in
going from (4.4.1) to (4.4.8). Therefore, if we select a control u(t) that stabilizes
(4.4.8) so that e(t) goes to zero, then the nonlinear control input given by
(t)(4.4.9) will cause trajectory following in the robot arm (4.4.1). In fact,
substituting (4.4.9) into (4.4.2) yields

Copyright © 2004 by Marcel Dekker, Inc.