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3.3 Structure and Properties of the Robot Equation 129

with the matrix coefficient given by

(3.3.16)

To find an equivalent expression for V m1 , note that

(3.3.17)

which may be written as

(3.3.18)

or as

(Note that ∂M/∂q i is symmetric.) Therefore,

(3.3.20)

whence using appropriate definitions we may write
(3.3.21)

It is also possible to write (see the Problems)
(3.3.22)

Since V v( ) is linear in it follows that V(q, ) is quadratic in q. In fact, it
can be shown (see the Problems) that

Copyright © 2004 by Marcel Dekker, Inc.