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4.4 Computed-Torque Control 205

and the disturbance is

(4.4.47)

This reduces to the error system (4.4.10) if exact computed-torque control
is used so that ∆=0, δ=0. Otherwise, the error system is driven by the
desired acceleration and the nonlinear term mismatch δ(t). Thus the tracking
error will never go exactly to zero. Moreover, the auxiliary control u(t) is
multiplied by (I -∆), which can make for a very difficult control problem.
Using outer-loop PD feedback so that u(t)=-K v e-K p e yields the error system

(4.4.48)

The behavior of such systems is not obvious, even if K v and K p are selected for
good stability of the left-hand side. There are two sorts of problems: first, the
disturbance term d(t), and second the function ∆(K v e+K p e) of the error and its
derivative.

PD-Plus-Gravity Controller
A useful controller in the computed-torque family is the PD-plus-gravity
controller that results when M=I, N=G(q)-q d , with G(q) the gravity term of
the manipulator dynamics. Then, selecting PD feedback for u(t) yields

(4.4.49)

This control law was treated in [Arimoto and Miyazaki 1984], [Schilling
1990]. It is much simpler to implement than the exact computedtorque
controller.
When the arm is at rest, the only nonzero terms in the dynamics (4.4.1)
are the gravity G(q), the disturbance d, and possibly the control torque .
The PD-gravity controller c, includes G(q), so that we should expect good
performance for set-point tracking, that is, when a constant q d is given so
that q d=0. The next result formalizes this. It relies on a Lyapunov proof
(Chapter 1) of the sort that will be of consistent usefulness throughout the
book, drawing especially on the skew-symmetry property in Table 3.3.1.
Thus it is very important to understand the steps in this proof.

Copyright © 2004 by Marcel Dekker, Inc.