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156 Robot Dynamics

The structure of these equations is very different from the rigid joint arm
described in Table 3.3.1. We discuss the control of robot manipulators with
joint flexibility in Chapter 6 (see [Spong 1987]). The next example shows the
problems that can occur in controlling flexible joint robots.

EXAMPLE 3.6–1: DC Motor with Flexible Coupling Shaft
To focus on the effects of joint flexibility, let us examine a single armature-
controlled do motor coupled to a load through a shaft that has significant
flexibility. The electrical and mechanical subsystems are shown in Figure
3.6.1.
The motor electrical equation is

(1)

with i(t), u(t) the armature current and voltage, respectively. The back emf
is .

The interaction force exerted by the flexible shaft is given by
f= , where the shaft damping and spring constants
are denoted by b and k. Thus the mechanical equations of motion may be
written down as

(2)

(3)

with subscripts m and L referring, respectively, to motor parameters and
load parameters. The load inertia J L is assumed constant. The definitions
of the remaining symbols may be inferred from the foregoing text.
To place these equations into state-space form, define the state as

(4)

with the motor and load angular velocities. Then

Copyright © 2004 by Marcel Dekker, Inc.