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3.6 Actuator Dynamics 155

(3.6.11)

It is important to note that T is a matrix of motor electric time constants. In
the preceding subsection, these time constants were assume negligibly small
in comparison to the motor mechanical time constant.
To determine the overall dynamics of the arm plus do motor actuators,
eliminate τ between (3.6.1) and (3.6.10) to obtain an expression for I. Then,
differentiate to expose explicitly . Substitute these expressions into (3.6.9)
(see the Problems) to obtain dynamics of the form

(3.6.12)

The coefficient matrix D is given by

D(q)=TM’(q), (3.6.13)

so that it is negligible when L i are small.

Dynamics with Joint Flexibility
We have assumed that the coupling between between the actuators and the
robot links is provided through rigid gear trains with gear ratios of r i . In
actual practice, the coupling suffers from backlash and gear train flexibility
or elasticity. Here we include the flexibility of the joints in the arm dynamic
model, assuming for simplicity that r i =1.
This is not difficult to do. Indeed, suppose that the coupling flexibility is
modeled as a stiff spring. Then the torque mentioned in Equations (3.6.1),
(3.6.2) is nothing but
(3.6.14)
with B s =diag{b si }, K s =diag{k si }, and b si and k si the damping and spring constants
of the ith gear train. Thus the dynamical equations become

(3.6.15)

(3.6.16)

Copyright © 2004 by Marcel Dekker, Inc.