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4.8 Summary 251

(4.7.16)

is easy to compute, since (t) may be determined from the measured joint
variables using (4.7.6).
The linear position error is simply given by
(4.7.17)

An orientation error e o (t) suitable for feedback purposes is more difficult to
obtain, but may be defined as follows.
Denote the rotation transformation portions of (4.7.14) and (4.7.3),
respectively, as R(t), R d (t). The orientation error can be expressed in terms of
a rotation of φ(t) rads about an Euler axis of k(t) that takes R(t) into R d (t)
(Appendix A). In fact, one may define the 3-vector

(4.7.18)

where e o(t) may be assumed small. With this definition, it can be shown that
e o(t) is found from T(t) and T d(t) using

(4.7.19)

The overall Cartesian error is now given by (4.7.4). Unfortunately, with this
definition of e o, it happens that e y is not the derivative of e y; however, the
control law (4.7.10) still yields suitable results. Alternative definitions of e y(t)
and (t) are given in [Wu and Paul 1982]; they are closely tied to the cross-
y
product matrix O, in Appendix A and require the selection of a sampling
period T.

4.8 Summary

In this chapter we showed how to generate smooth trajectories defining robot
end-effector motion that passes through a set of specified points. Then we
covered the important class of computed-torque controllers, which subsumes
many types of robot control algorithms. Both classical and modern control
algorithms are described by this class, so that computed torque provides a
bridge between older and more modern algorithms for motion control.
As special types of computed-torque algorithms, we mentioned PD control,
PID control, PD-plus-gravity, classical joint control, and digital control. Most
robot control algorithms are implemented digitally, and computedtorque
provides a rigorous framework for analyzing the effects of digitization and
the size of the sampling period. This is approached by considering digital

Copyright © 2004 by Marcel Dekker, Inc.