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150 Robot Dynamics

Note that , , and f d depend on q and , so that strictly speaking, the
Cartesian dynamics are not completely given in terms of However.
, and given y(t) we could use the inverse kinematics to determine
q(t), so that , , f d can be computed as functions of y and using computer
subroutines.

Structure and Properties of the Cartesian Dynamics
It is important to realize that all the properties of the joint-space dynamics
listed in Table 3.3.1 carry over to the Cartesian dynamics as long as J is
nonsingular [Slotine and Li 1987]. Note particularly that is symmetric
and positive definite. For a revolute arm the Jacobian has units of length and
is bounded. In that case, is bounded above and below.
Defining

(3.5.14)

it follows that

(3.5.15)
with

(3.5.16)

where V m was defined in Section 3.3.
It is easy to show that
(3.5.17)

is skew-symmetric. Indeed, use the identity

(3.5.18)

to see that

Copyright © 2004 by Marcel Dekker, Inc.