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282 Robust Control of Robotic Manipulators

4. 2 close to 1, which may also be obtained with a large-gain compensator
C.

Note that in the limit, and as the gain of C(s) becomes infinitely large, 1
goes to zero. This will then transform condition (1) to

(5.2.34)

It is also seen from (5.2.33)–(5.2.34) that increasing the gain k of C(s) will
decrease 1 , therefore decreasing ||e|| ∞ . A particular compensator may now
be obtained by choosing the parameter Q(s) to satisfy other design criteria,
such as suppressing the effects of . One can, for example, recover Graig’s
compensator, by choosing C(s)=-K so that the control effort is given by

u=Ke. (5.2.35)

Then note that conditions (5.2.28) and (1) are identical if 2 =0 and
2 = 11 k p + 12 k v . Also note from (2) that a smaller d results in a smaller tracking
error. In fact, if e=0 and 0=0, the asymptotic stability of the error may be
shown. Finally, note that the presence of bounded disturbance will make the
bound on the error e larger but will not affect the stability condition (1).
This controller is summarized in Table 5.2.3.
The factorization approach gives the family of all one-degree-of-freedom
stabilizing compensators C(s). The design methodology is illustrated for the
two-link robot in the next example.

EXAMPLE 5.2–3: Dynamic Controller (Input-Output Design)
Let G v(s) of Example 4.2.1 be factored as

(1)

where N(s), D(s), N(s), and D(s) are matrices of stable rational functions.
We can then find

(2)

Copyright © 2004 by Marcel Dekker, Inc.