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3.3 Structure and Properties of the Robot Equation 135

(3.3.53)

with f i(·) known scalar functions that may be determined for any given arm.
We have defined the vec{·} function for future use.
The viscous friction may often be assumed to have the form
(3.3.54)

with v i known constant coefficients. Then F v =diag{v i }, a diagonal matrix with
entries v i . The dynamic friction may often be assumed to have the form

(3.3.55)
with k i known constant coefficients and the signum function defined for a
scalar x by

(3.3.56)

Then
(3.3.57)
with K d =diag{k i } the coefficient matrix of dynamic friction and the signum
function defined for a vector x by
(3.3.58)

A bound on the friction terms may be assumed of the form
(3.3.59)

with v and k known for a specific arm and ||·|| a suitable norm.
Another friction term that may be included in F( )is the static friction,
which has components of the form
(3.3.60)

where k si is the coefficient of static friction for joint i and is a small positive
parameter. We shall generally ignore this term.

Copyright © 2004 by Marcel Dekker, Inc.