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134 Robot Dynamics

(3.3.49)

Properties of the Gravity, Friction, and Disturbance
Properties of the Gravity Term G(q). According to (3.2.43) and (3.2.36),

whence using (3.3.11) twice reveals

(3.3.50)

A bound on the gravity term may be derived for any given robot arm. Thus

||G(q)||≤g b(q), (3.3.51)

where ||·|| is any appropriate vector norm and g b is a scalar function that may
be determined for any given arm (see Example 3.3.1). For a revolute arm, g b
is a constant independent of the joint vector q, but for an arm with prismatic
links, g b may depend on q. See the examples in Section 3.2 and Example
3.3.1 to verify these claims.

Properties of the Friction Term . The friction in the arm Equation (3.3.1)
is of the form
(3.3.52)

with F v the coefficient matrix of viscous friction, and F d a dynamic friction
term. The friction coefficients are among the parameters most difficult to
determine for a given arm and, in fact, (3.3.52) represents only an approximate
mathematical model for their influence. For more discussion, see [Craig 1988,
Schilling 1990].
Since friction is a local effect, we may assume that F ( ) is uncoupled
among the joints, so that

Copyright © 2004 by Marcel Dekker, Inc.