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5.2 Feedback-Linearization Controllers 291

or

(3)

We can immediately see that the joint position vector q is filtered through
the PID controller K 2 . Therefore, the differentiation of q is required unless
the measurement of is available. The behavior of the nonlinear closed-loop
system is shown in Figure 5.2.9, when =1 and =0, a 1=2, b 1=b 2=10, w 1=8,
and w 2=12. It is seen that initially, the torque effort and the trajectory errors
are too large. To understand the behavior of this controller, consider the
controller τ in the limit (i.e., as time goes to infinity). The output of K 1qd has
settled down to its final value qd and therefore the controller (3) becomes
equivalent to a PID compensator [see Chapter 4, equation (4.4.35)]. It seems
that a different structure for k 1 and K 2 is warranted because in the meantime,
the two-DOF controller preforms rather poorly. This is a characteristic of
the example rather than an inherent flow in the twoDOF methodology. As a
matter of fact, this structure has shown better performance than the one-
DOF PID compensator in [Sugie et al. 1988] for a set-tracking case. The
reader is encouraged to work the problems at the end of the chapter related
to this design in order to compare the performance of one- and two-DOF
designs.

We have this presented a large sample of controllers that are more or less
computed-torque based. We have shown using different stability arguments
that the computed-torque structure is inherently robust and that by
increasing the gains on the outer-loop linear compensator, the position and
velocity errors tend to decrease in the norm. This class of compensators
constitutes by far the most common structure used by robotics
manufacturers and is the simplest to implement and study. There are more
compensators that would fit into this structure while appealing to some
classical control applications. The PD and PID compensators may be
replaced with the lead-lag compensators. These are especially appealing
when only position measurements are available. Such designs are discussed
in [Chen 1989] in the discrete-time case. There is also some work being done
in the nonlinear observer area which is directly relevant to this problem
[Canudas de Wit and Fixot 1991]. We refer the reader to the observability
discussion in Section 2.11. We also suggest some of the problems at the end
of this chapter, which discuss further modification of the feedback-
linearization designs.

Copyright © 2004 by Marcel Dekker, Inc.