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188 Computed-Torque Control

It is important to realize that computed-torque depends on the inversion of
the robot dynamics, and indeed is sometimes called inverse dynamics control
In fact, (4.4.9) shows that (t) is computed by substituting d –u for in (4.4.2);
that is, by solving the robot inverse dynamics problem. The caveats associated
with system inversion, including the problems resulting when the system has
non-minimum-phase zeros, all apply here. (Note that in the linear case, the
system zeros are the poles of the inverse. Such nonminimum-phase notions
generalize to nonlinear systems.) Fortunately for us, the rigid arm dynamics
are minimum phase.
There are several ways to compute (4.4.9) for implementation purposes.
Formal matrix multiplication at each sample time should be avoided. In some
cases the expression may be worked out analytically. A good way to compute
the torque (t) is to use the efficient Newton-Euler inverse dynamics formulation
[Craig 1985] with d –u in place of (t).
The outer-loop signal u(t) can be chosen using many approaches, including
robust and adaptive control techniques. In the remainder of this chapter we
explore some choices for u(t) and some variations on computed-torque control.

PD Outer-Loop Design
One way to select the auxiliary control signal u(t) is as the proportional-plus-
derivative (PD) feedback,

(4.4.13)

Then the overall robot arm input becomes

(4.4.14)

This controller is shown in Figure 4.4.6 with K i=0. The closed-loop error
dynamics are
(4.4.15)
or in state-space form,

(4.4.16)

The closed-loop characteristic polynomial is

(4.4.17)

Choice of PD Gains. It is usual to take the n×n gain matrices diagonal so
that

Copyright © 2004 by Marcel Dekker, Inc.