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278 Robust Control of Robotic Manipulators

2. A good knowledge of N, which will translate into small i ’s.

3. A large µ 1 or a large inertia matrix M(q).

4. A trajectory with a small d .
5. Robots whose inertia matrix M(q) does not vary greatly throughout its
workspace (i.e. µ 1 ≈µ 2 )), so that a is small. Note that a small a is needed
to guarantee that at least < 1 in (5.2.28). This will translate into
the severe requirement that the matrix M should be close to the inertia
matrix M(q) in all configurations of the robot.

The controller is summarized in Table 5.2.2.
These observations are similar to those made after inequality (6) and are
illustrated in the next example.

EXAMPLE 5.2–2: Static Controller (Input-Output Design)

Consider the nonlinear controller (5.2.6), where

(1)

Therefore,

(2)

Condition (1) is then satisfied if k v >720. This of course is a large bound that
can be improved by choosing a better . A simulation of the closed-loop
behavior for k p=225 and k v=30 is shown in Figure 5.2.5. The errors
magnitudes are much smaller than those achieved with the PD controllers of
Example 5.2.1 with a comparable control effort. This improvement came
with the expense of knowing the inertia matrix M(q) as seen in (1).

Dynamic Controllers

The controllers discussed so far are static controllers in that they do not
have a mechanism of storing previous state information. In Chapter 4 and in
this chapter, these controllers could operate only on the current position and
velocity errors. In this section we present three approaches to show the
robustness of dynamic controllers based on the feedback-linearization

Copyright © 2004 by Marcel Dekker, Inc.