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3.5 Cartesian and Other Dynamics 151

which is skew symmetric since is.
The friction terms in the Cartesian dynamics are

(3.5.19)

and they satisfy bounds like those in Table 3.3.1. Notice that in Cartesian
-1
coordinates the friction effects are not decoupled (e.g., J F vJ is not diagonal).
-T
The Cartesian gravity vector
(3.5.20)

is bounded.
The property of linearity in the parameters holds and is expressed as

(3.5.21)

where the known Cartesian function of robot functions is

(3.5.22)

and ϕ is the vector of arm parameters.

EXAMPLE 3.5–1: Cartesian Dynamics for Three-Link Cylindrical Arm
Let us show how to convert the joint space dynamics found in Example
3.2.3 to Cartesian dynamics. From Example A.3–1, the arm Jacobian is

(1)

whence its inverse is

(2)

From Example 3.2.3 the arm inertia matrix is

(3)

Copyright © 2004 by Marcel Dekker, Inc.