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4.4 Computed-Torque Control 207

(7)

We now invoke Barbalat’s lemma to show that V goes to zero. For this, it is
necessary to demonstrate the uniform continuity of V. We see that
(8)

The demonstrated stability shows that a and q are bounded, whence (2) and
-1
the boundedness of M (q) (see Table 3.3.1) reveal that q is bounded. Therefore,
V is bounded, whence V is uniformly continuous. This guarantees by Barbalat’s
lemma that V goes to zero.
It is now clear that goes to zero. It remains to show that the tracking
error e(t) vanishes. Note that when q=0, (2) reveals that
(9)

Therefore, a nonzero e(t) results in nonzero and hence in q≠0, a contradiction.
Therefore, the only invariant set contained in {x(t)|V=0} is x(t)=0. This finally
demonstrates that both e(t) and vanish and concludes the proof.
Some notes on this proof are warranted. First, note that the Lyapunov
function chosen is a natural one, as it contains the kinetic energy and the
“artificial potential energy” associated with the virtual spring in the control
law [Slotine and Li 1991]. K p appears in V while K v appears in V.
The proof of Lyapunov stability is quick and easy. The effort comes about
in showing asymptotic stability. For this, it is required to return to the system
dynamics and study it more carefully, discussing issues such as boundedness
of signals, invariant sets, and so on. The fundamental property of skew
symmetry is used in computing V. The boundedness of M (q) is needed in
-1
Barbalat’s lemma.
The next example illustrates the performance of the PD-gravity controller
for trajectory tracking. In general, if q ≠0, then the PD-gravity controller
d
guarantees bounded tracking errors. The error bound decreases, that is, the
tracking performance improves, as the PD gains become larger. This will be
made rigorous in Chapter 4.

EXAMPLE 4.4–3: Simulation of PD-Gravity Controller
In Example 4.4.1 we simulated the exact computed-torque control law on
a two link planar manipulator. Here, let us simulate the PD-gravity
controller. We shall take the same arm parameters and desired trajectory.

Copyright © 2004 by Marcel Dekker, Inc.