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128 Robot Dynamics

Kronecker Product Analysis of V(q, ) Let us first examine the term V(q,
) from the point of view of the Kronecker product [Brewer 1978], defined for
two matrices as
(3.3.9)

where A has elements a ij and [a ij B] means the np×mq block matrix composed
of the p×q blocks a ij B. Thus, for we, have

For matrices A(q), B(q), with , define the matrix derivative as

(3.3.10)

Then we may prove the product rule

(3.3.11)

with I n the n×n identity.
Now we may examine the Coriolis/centripetal vector V(q, )[Koditschek
1984, Gu and Loh 1988]. Using (3.3.11) twice on (3.2.42), we may obtain

(3.3.12)

or
(3.3.13)
where

(3.3.14)

(3.3.15)

Copyright © 2004 by Marcel Dekker, Inc.