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250 Computed-Torque Control

(4.7.10)

A disadvantage with Cartesian computed-torque control is the necessity to
compute the inverse Jacobian. To avoid inverting the Jacobian at each sample
period, we might propose the approximate Cartesian computedtorque controller

(4.7.11)

where and are approximations to MJ and N, respectively. The error
-1
system for this control law is not difficult to compute (cf. the joint space
approximation in Table 4.4.1 and see the Problems).
A PD outer feedback loop yields
(4.7.12)

A special case of this control law is obtained by setting =I,
= which yields the Cartesian PD-gravity controller

(4.7.13)

The robustness properties of computed-torque control make this a successful
control law for many applications.
Simulations like those presented in this section could be carried out for
Cartesian computed-torque control. The basic principles would be the same
as for joint space computed-torque control (see the Problems).

Cartesian Error Computation
The actual Cartesian position may be computed from the measured joint
variables using the arm kinematics in terms of the arm T matrix

(4.7.14)

and the desired Cartesian position may likewise be expressed as (4.7.3). Then
a Cartesian position error and velocity error suitable for computedtorque
control may be computed as follows [Luh et al. 1980], [Wu and Paul 1982].
Define

(4.7.15)

with v(t), v d(t) the actual and desired linear velocity, and ω(t), ω d(t) the actual
and desired angular velocities. Then

Copyright © 2004 by Marcel Dekker, Inc.