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5.2 Feedback-Linearization Controllers 273

(3)

Note that this is basically a computed-torque-like PD controller. A simulation
of the robot’s trajectory is shown in Figure 5.2.2. We also start our simulation
at =0. The effect of increasing the gains is shown in Figure 5.2.3,
which corresponds to the controller

(4)

Note that at least initially, more torque is required for the higher-gains case
(compare Figs. 5.2.2c and 5.2.3c) but that the errors magnitude is greatly
reduced by expanding more effort.

There are other proofs of the uniform boundedness of these static controllers.
In particular, the results in [Dawson et al. 1990] provide an explicit expression
for the bound on e in terms of the controller gains. In the interest of brevity
and to present different designs, we choose to limit our development to one
controller in this section.
As discussed in Section 4.4, a residual stead-state error may be present
even when using an exact computed-torque controller if disturbances are
present. A common cure for this problem (and one that will eliminate constant
disturbances) is to introduce integral feedback as done in Section 4.4. Such a
controller may again be used within a robust controller framework and will
lead to similar improvements if the integrator windup problem is avoided
(see Section 4.4).
In the next section we show the stability of static controllers similar to the
ones designed here and use input-output stability methods to design more
general dynamic compensators.

Input-Output Designs

In this section we group designs that show the stability of the trajectory
error using input-output methods. In particular, we present controllers that
show and stability of the error. We divide this section into a subsection
∞
2
that deals with static controllers such as the ones described previously,

Copyright © 2004 by Marcel Dekker, Inc.