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3.3 Structure and Properties of the Robot Equation 133

(3.3.42)

(3.3.43)

(3.3.44)

where all sums are over the number of joints n. Now, the Lagrange equation
shows that the arm dynamics are expressed componentwise as

(3.3.45)
with n the number of joints.
By interchanging the order of summation and taking advantage of symmetry,

(3.3.46)

Therefore, we may define

(3.3.47)

and write the arm dynamics as

(3.3.48)

The cyclic symmetry of the v ijk is what allows us to derive the important
properties of the Coriolis/centripetal vector V(q, )which corresponds to the
second term in this equation. The quantities v ijk are known as Christoffel
symbols (of the first kind) [Borisenko and Tarapov 1968].
The matrix defined in (3.3.36) has components v kj given by

Copyright © 2004 by Marcel Dekker, Inc.