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3.5 Cartesian and Other Dynamics 149

The derivation of the Cartesian dynamics from the joint-space dynamics is
akin to the feedback linearization in Section 3.4. Differentiating (3.5.4) twice
yields
(3.5.5)

(3.5.6)

where the Jacobian is

(3.5.7)

3
The Cartesian velocity vector is , with v R the linear
3
velocity and ω R the angular velocity. Let us assume that the number of
links is n=6, so that J is square. Assuming also that we are away from workspace
singularities so that |J|≠0, according to (3.5.6), we may write
(3.5.8)
which is the “inverse acceleration” transformation. Substituting this into (3.5.2)
yields

T
Recalling now the force transformation τ=J F, with F the Cartesian force vector
(see Appendix A) we have
(3.5.9)

This may be written as

(3.5.10)
where we have defined the Cartesian inertia matrix, nonlinear terms, and
disturbance by
(3.5.11)

(3.5.12)

(3.5.13)

Equation (3.5.9)–(3.5.10) gives the Cartesian or workspace dynamics of the
robot manipulator.

Copyright © 2004 by Marcel Dekker, Inc.