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3.6 Actuator Dynamics 153

The actuator coefficient matrices are all constants given by

(3.6.3)

where the ith motor has inertia J Mi , rotor damping constant B Mi , back emf
constant K bi , torque constant K Mi , and armature resistance R ai .
The gear ratio of the coupling from the ith motor to the ith arm link is r i ,
which we define so that

q i =r i q Mi or q=Rq M . (3.6.4)

If the ith joint is revolute, then r i is a dimensionless constant less than 1. If q i
is prismatic, then r i has units of m/rad.
The actuator friction vector is given by

F M =vec{F Mi }

with F Mi the friction of the ith rotor.
Note that capital “M” denotes motor constants and variables, while V m is
the arm Coriolis/centripetal vector defined in terms of Christoffel symbols.
Using (3.6.4) to eliminate q M in (3.6.2), and then substituting for τ from
(3.6.1) results in the dynamics in terms of joint variables

(3.6.5)

or, by appropriate definition of symbols,

(3.6.6)

Properties of the Complete Arm-Plus-Actuator Dynamics. The complete
dynamics (3.6.6) has the same form as the robot dynamics (3.6.1). It is very
easy to verify that the complete arm-plus-actuator dynamics enjoys the same
properties as the arm dynamics that are listed in Table 3.3.1 (see the Problems).
In particular, V’ is one-half the difference between and a skew-symmetric
matrix, all the boundedness assumptions hold, and linearity in the parameters
holds. Thus, in future work where we design controllers, we may assume that
the actuators have been included in the arm equation in Table 3.3.1

Copyright © 2004 by Marcel Dekker, Inc.