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264 Robust Control of Robotic Manipulators

of applicability of the simple PID controllers of Chapter 4, as a function of
the inherent lack of knowledge of the robot’s dynamics.
The controllers designed in this chapter may be analyzed using input-
output stability tools or state-space tools. In the input-output approach, the
stability of the controlled robot is shown using the small-gain theorem or
the passivity theorem. In the state-space approach, most designs are shown
to be stable using Lyapunov-based arguments. See Chapter 2 for an overview
of both approaches.
Consider the robot dynamics given in Chapter 3:

(5.1.1)

and assume that a desired trajectory in joint space is specified by the time
function . We will suppress the time dependence if no
ambiguity results. Let q d, d, d, and d be bounded functions of time. In a
fashion similar to Chapter 4, we assume the trajectory error e to have two
components:

(5.1.2)

The controllers of this chapter assume that measurements of q and are
available. As described in Section 3.5, variables other than q and may be
measured. The Cartesian computed-torque controllers of Section 4.7 provide
a setting where Cartesian trajectory is to be followed directly. We limit our
discussion to the case of joint measurements with the understanding that a
desired trajectory in another coordinate system may be followed by first
obtaining the corresponding joint trajectory then applying the methods of
this chapter.
We may, however, assume that the measurements p and p of q and p are
corrupted by a bounded noise, that is,

(5.1.3)

where ||w i ||≤c i .
In Section 5.2 we discuss the computed-torque-like controllers of Section
4.4 and study their robustness properties. The section is divided into
controllers whose robustness is deduced using Lyapunov stability and others
whose robustness relies on input-output stability. Nonlinear controllers which
are not necessarily derived from the computed-torque controllers are
presented in Section 5.3. These include controllers that exploit the passivity
of the robot dynamics and others which are variable-structure methods and
saturation controllers without particular emphasis on the special properties
of the robot. Finally, in Section 5.4 we review approaches that attempt to

Copyright © 2004 by Marcel Dekker, Inc.