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148 Robot Dynamics

+
where J is the Moore-Penrose inverse [Rao and Mitra 1971] of the Jacobian
-1
J(q). If J(q) is square (i.e., p=n) and nonsingular, then J (q)=J (q) and we may
+
write
(3.4.37)

As we shall see in Chapter 4, feedback linearization provides a powerful
controls design technique. In fact, if we select u(t) so that (3.4.34) is stable
(e.g., a possibility is the PD feedback ), then the control
input torque τ(t) defined by (3.4.36) makes the robot arm move in such a way
that y(t) goes to zero.
In the special case y(t)=q(t), then J=I and (3.4.34) reduces to the linear
position/velocity form (3.4.22).

3.5 Cartesian and Other Dynamics

In Section 3.2 we derived the robot dynamics in terms of the time behavior of
q(t). According to Table 3.3.1,

(3.5.1)

or

(3.5.2)

where the nonlinear terms are
(3.5.3)
We call this the dynamics of the arm formulated in joint space, or simply the
joint-space dynamics.

Cartesian Arm Dynamics
It is often useful to have a description of the dynamical development of variables
other than the joint variable q(t). Consequently, define
(3.5.4)

with h(q) a generally nonlinear transformation. Although y(t) could be any
variable of interest, let us think of it here as the Cartesian or task space
position of the end effector (i.e., position and orientation of the end effector in
base coordinates).

Copyright © 2004 by Marcel Dekker, Inc.