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3.3 Structure and Properties of the Robot Equation 127

kinetic energy is

(3.3.5)

Some expressions for M are given in the next subsection.
Another vital property of M(q) is that it is bounded above and below. That
is,

µ 1I≤M(q)≤µ 2I (3.3.6)

with µ 1, and µ 2 scalars that may be computed for any given arm (see Example
3.3.1). When we say that µ 1I≤M(q), for instance, we mean that (M(q)-µ 1I) is
positive semidefinite. That is,

x (M-µ 1I)x≥0
T
n
for all xε R .
Likewise, the inverse of the inertia matrix is bounded, since

(3.3.7)

If the arm is revolute, the bounds µ 1 and µ 2 are constants, since q appears only
in M(q) through sin and cos terms, whose magnitudes are bounded by 1 (see
Examples 3.2.2 and 3.3.1). On the other hand, if the arm has prismatic joints,
then µ 1 and µ 2 may be scalar functions of q. See Example 3.2.1, where M(q) is
2
bounded above by µ 2=mr (if r>1).
The boundedness property of the inertia matrix may also be expressed as

m 1≤||M(q)||≤m 2, (3.3.8)

where any induced matrix norm can be used to define the positive scalars m 1
and m 2.

Properties of the Coriolis/Centripetal Term
A glance at (3.2.42) reveals a problem that, if not understood, can make the
study of robot dynamics confusing. Simplification of this V(q, ) term would
require taking the derivative of a matrix [i.e., M(q)] with respect to the n-
vector q. However, such derivatives are not matrices, but tensors of order
three-that is, they must be represented by three indices, not two. There are
several ways to get around this problem, involving several definitions of
some new quantities.

Copyright © 2004 by Marcel Dekker, Inc.