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248 Computed-Torque Control

4.7 Cartesian Control

We have seen how to make a robot manipulator track a desired joint space
trajectory q d(t). However, in any practical application the desired trajectories
of a robot arm are given in the workspace or Cartesian coordinates. An important
series of papers dealt with resolved motion manipulator control [Whitney 1969];
[Luh et al. 1980], [Wu and Paul 1982]. There the joint motions were resolved
into the Cartesian coordinates, where the control objectives are specified. The
result is that an operator can use a joystick to specify Cartesian motion (e.g.,
for a prosthetic device), with the arm following the specified motion. Older
teleoperator devices used joysticks that directly controlled the motion of the
actuators, resulting in long training times and very awkward manipulability.
There are several approaches to Cartesian robot control. For instance, one
might:

1. Use the Cartesian dynamics in Section 3.5 for controls design (see the
Problems).
2. Convert the desired Cartesian trajectory y d(t) to a joint-space trajectory
q d(t) using the inverse kinematics. Then use the joint-space computed-
torque control schemes in Table 4.4.1.
3. Use Cartesian computed-torque control.

Let us discuss the last of these.

Cartesian Computed-Torque Control
This approach begins with the joint space dynamics
(4.7.1)

In Section 3.4 we discussed a general feedback-linearization approach for
linearizing the arm dynamics with respect to a general output. In this section
the output we are interested in is the Cartesian error

(4.7.2)

with y d(t) the desired Cartesian trajectory and y(t) the end-effector Cartesian
position.
The problems associated with specifying the Cartesian position of the end
effector are covered in Appendix A. There we see that y(t) is not necessarily a
6-vector, but could in fact be the 4×4 arm T matrix. Then y d(t) is a 4×4 matrix
given by

Copyright © 2004 by Marcel Dekker, Inc.