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4.7 Cartesian Control 249

(4.7.3)

containing the desired orientation (n d(t), o d(t), a d(t)) and position p d(t) of the
end effector with respect to base coordinates. On the other hand, y(t) could be
specified (nonuniquely) using Euler angles as a 6-vector, or using quaternions
as a 7-vector, or using the encoded tool configuration vector which gives y(t)
6
R .
Although there are problems with specifying y(t) as a 6-vector, the
Cartesian error is easily specified (see the next subsection) as the 6-vector

(4.7.4)

with e p(t) the position error and e o(t) the orientation error. Thus equation
(4.7.2) is generally valid only as a loose notational convenience.
Let us assume that
(4.7.5)

with h(q) the transformation from q(t) to y(t), which is a modification of the
kinematics transformation, depending on the form decided on for y(t). Then
the associated Jacobian is J=¶h/¶q and
(4.7.6)

Now, the approach of Section 3.4, or a small modification of the derivation
in Section 4.4, shows that the computed-torque control relative to e y (t) is
given by

(4.7.7)

which results in the error system
(4.7.8)

with the disturbance

(4.7.9)

We call (4.7.7) the Cartesian computed-torque control law.
The outer-loop control u(t) may be selected using any of the
techniques already mentioned for joint-space computed-torque control (see
Table 4.4.1). For PD control, for instance, the complete control law is

Copyright © 2004 by Marcel Dekker, Inc.