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114 Robot Dynamics

shall show how to determine q(t) given the control inputs n(t) and f(t) by
computer simulation in Chapter 4.
Given our discussion on forces and inertias it is easy to identify the terms
in the dynamical equations. The first terms in each equation are acceleration
terms involving masses and inertias. The second term in (9) is a Coriolis
term, while the second term in (10) is a centripetal term. The third terms are
gravity terms.
c. Manipulator Dynamics

By using vectors, the arm equations may be written in a convenient form.
Indeed, note that

(11)

We symbolize this vector equation as

(12)

Note that, indeed, the inertia matrix M(q) is a function of q (i.e., of θ and r),
the Coriolis/centripetal vector V(q, ) is a function of q and , and the gravity
vector G(q) is a function of q.

EXAMPLE 3.2–2: Dynamics of a Two-Link Planar Elbow Arm
In Example A.2–2 are given the kinematics for a two-link planar RR
arm. To determine its dynamics, examine Figure 3.2.4, where we have
assumed that the link masses are concentrated at the ends of the links.
The joint variable is
q=[θ 1 θ 2] T (1)

and the generalized force vector is
(2)

Copyright © 2004 by Marcel Dekker, Inc.