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138 Robot Dynamics

which is bounded above for all θ 2 by

and below by

Since the arm is revolute and cos θ 2 is bounded above and below, M 2 and M 1
are constants. It is important to note that if the arm is revolute/prismatic (RP),
so that the joint variables are (θ 1 , a 2 ), the bounds are functions of q.

b. Bounds on the Coriolis and Gravity Terms
The bound v b on the Coriolis/centripetal vector is found using

whence v b=m 2a 1a 2.
Similarly, for the gravity bound,

Notice that if the arm is RP, then v b and g b are functions of q.

c. Coriolis/Centripetal Structural Matrices
We now list the various structural matrices for V(q, )discussed in this section.
Their computation is left as an exercise (see the Problems).

Copyright © 2004 by Marcel Dekker, Inc.