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130 Robot Dynamics

(3.3.23)

for appropriate definition of V i (q) [Craig 1988]. Indeed, the V i (q) are symmetric
n×n matrices.
Since V(q, ) is quadratic in , it can be bounded above by a quadratic
function of . That is,
(3.3.24)

with v b (q) a known scalar function and ||·|| any appropriate norm. For a
revolute arm, v b is a constant independent of q. See Examples 3.2.2 and
3.3.1, where the quadratic terms in are multiplied by sin θ 2 , whose magnitude
is bounded by 1. On the other hand, for an arm with prismatic joints v b (q)
may be function of q; see Examples 3.2.1 and 3.2.3, where V(q, ) has a term
in r multiplying the quadratic terms .
To assist in determining v b (q) for a given robot arm, note that so that
so that

where (q) is defined in (3.3.23). Therefore, for a revolute arm

(3.3.25)

We may note that

(3.3.26)

This is an n -vector consisting of all possible products of the components of .
2
This and (3.3.19) allow us to demonstrate that

(3.3.27)

In this proof, we also need the identity

Copyright © 2004 by Marcel Dekker, Inc.