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4.6 Optimal Outer-Loop Design 243

limits must be mapped using the feedback linearization input transformation
to determine the limits on the integrator outputs see [Bobrow et al. 1983].
Thus the saturation limits needed by the antiwindup compensator are
functions of the joint position q, the desired acceleration and so on. This
issue is explored in the problems.

4.6 Optimal Outer-Loop Design

In Section 4.4 we discussed computed-torque control, showing how to select
the inner control loop using exact techniques involving the inverse manipulator
dynamics, as well as by a variety of approximate means. We also discussed
several schemes for designing the outer linear feedback (tracking) loop. The
results of our discussions are summarized in Table 4.4.1. In this section we
intend to present a modern control optimal technique for selecting the outer
feedback loop. Modern optimal design yields improved robustness in the
presence of disturbances and unmodeled dynamics.
Several papers have dealt with “optimal” or “suboptimal” control of robot
manipulators [Vukobratovic and Stokic 1983], [Lee et al. 1983], [Luo and
Saridis 1985], [Johansson 1990]. Although they are not all based on a
computed-torque-like approach, we would like here to present the flavor of
this work by using optimal techniques to design the computed-torque outer
feedback loop.

Linear Quadratic Optimal Control
First, it is necessary to review modern linear-quadratic (LQ) design. Suppose
that we are given the linear time-invariant system in state-space form

(4.6.1)

with It is desired to compute the state-feedback
gain in

(4.6.2)

so that the closed-loop system

(4.6.3)

is asymptotically stable. Moreover, we do not want to use too much control
energy to stabilize the system, since in many modern systems (e.g., automobile,
spacecraft), fuel or energy is limited.

Copyright © 2004 by Marcel Dekker, Inc.