Page 220 -

4.4 Computed-Torque Control 203

The carets denote design choices for the weighting and offset matrices. One
choice is =M(q), =N(q,q). The calculated control input into the robot arm
is c(t).
In some cases M(q) is not known exactly (e.g., unknown payload mass), or
N(q, ) is not known exactly (e.g., unknown friction terms). Then and
could be the best estimate we have for these terms. On the other hand, we
might simply wish to avoid computing M(q) and N(q,q) at each sample time,
or the sample period might be too short to allow this with the available
hardware. From such considerations, we call (4.4.43) an “approximate
computed-torque” controller.
In Table 4.4.1 are given some useful computed-torque-like controllers. As it
turns out, computed torque is quite a good scheme since it has some important
robustness properties. In fact, even if ≠M and ≠N the performance of
controllers based on (4.4.43) can be quite good if the outer-loop gains are selected
large enough. We study robustness formally in Chapter 4.
In the remainder of this chapter we consider various special choices of
and that give some special sorts of controllers. We shall present some
theorems and simulation examples that illustrate the robustness properties of
computed-torque control.

Error Dynamics with Approximate Control Law. Let us now derive the error
dynamics if the approximate computed-torque controller (4.4.43) is applied to
the robot arm (4.4.2). Substituting c(t) into the arm equation for (t) yields

Adding Mq d-Mq d to the left-hand side and Mu-Mu to the right gives

or

ë=u-∆u+d, (4.4.44)

where the inertia and nonlinear-term model mismatch terms are

(4.4.45)

(4.4.46)

Copyright © 2004 by Marcel Dekker, Inc.