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206 Computed-Torque Control

THEOREM 4.4–1: Suppose that PD-gravity control is used in the arm
dynamics (4.4.1) and d =0, q d =0. Then the steady-state tracking error. e=q d -
q is zero.
Proof:
1. Closed-Loop System
Ignoring friction, the robot dynamics are given by
(1)

When q d=0, the proposed control law (4.4.49) yields the closed-loop dynamics
(2)

2. Lyapunov Function
Select now the Lyapunov function
(3)

and differentiate to obtain
(4)

Substituting the closed-loop dynamics (2) yields

(5)
Now, the skew symmetry of the first term gives

(6)

The state is , so that V is positive definite but only negative
semidefinite. Therefore, we have demonstrated stability in the sense of
Lyapunov, that is, that the error and joint velocity are both bounded.

3. Asymptotic Stability by LaSalle’s Extension
The asymptotic stability of the system may be demonstrated using Barbalat’s
lemma and a variant of LaSalle’s extension [Slotine and Li 1991] (Chapter
2). Thus it is necessary to demonstrate that the only invariant set contained in
the set V=0 is the origin.
Since V is lower bounded by zero and nonpositive, it follows that V
approaches a finite limit, which can be written

Copyright © 2004 by Marcel Dekker, Inc.