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Lagrange-Euler Dynamics 131

(3.3.28)

for any matrices A and B. Note: It is not true that
Now we may use these various identities to show that

(3.3.29)

Thus
(3.3.30)

with
(3.3.31)

Note that, in general, V m1 ≠V m2 .
In terms of M and U, the arm dynamics may be written as
(3.3.32)

At this point we may prove an identity that is extremely useful in constructing
advanced control schemes. We call it the skew-symmetric property; it shows
that the derivative of M(q) and the Coriolis vector are related in a very particular
way. In fact,

(3.3.33)

since a matrix minus its transpose is always skew symmetric. This important
identity holds also if V m1 is used in place of V m2 .
It is important to note that the first equality in (3.3.33) holds because
multiplies . That is, , so that it is not necessarily true
that (M-2V m2 ) itself is skew symmetric. However, it is possible to define a
matrix such that

(3.3.34)

and

Copyright © 2004 by Marcel Dekker, Inc.