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5.3 Nonlinear Controllers 295

THEOREM 5.3–1: Consider the control law (1)

(1)

where (s) is an SPR transfer function, to be chosen by the designer, and the
external input u 2 is bounded in the norm. Then is asymptotically stable,
2
and e is Lyapunov stable.
Proof:
Using the control law above, one gets from Figure 5.3.1,

(2)

By an appropriate choice of (s) and u 2, one can apply the passivity theorem
and deduce that and r are bounded in the norm, and since (s) is SPR
-1
2
(being inverse of an SPR function), one deduces that is asymptotically
stable because

(3)

This will imply that the position error e is bounded but not its asymptotic
stability in the case of time-varying trajectories . In the set-point
tracking case, however (i.e., =0), and with gravity precompensation, the
d
asymptotic stability of e may be deduced using LaSalle’s theorem. The
robustness of the closed-loop system is guaranteed as long as (s) is SPR
and that u 2 is 2 bounded, regardless of the exact values of the robot’s
parameters.

The controller is summarized in Table 5.3.1.

Table 5.3.1: Passive Controller

Copyright © 2004 by Marcel Dekker, Inc.