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EE 530
PROBLEM SET 1
DUE: 16 OCT 2006
Reading assignment: Ch 1 and Ch 2
The goal of the Frst problem set is to:
1. Review basic concepts and mathematical tools from EE 428 and EE 527.
2. Become familiar with the dynamic model of the rotary inverted pendulum system that will be used in future
problem sets to demonstrate adaptive control techniques.
Problem 1:
(25 points)
±igure 1 shows a photograph of the rotary inverted pendulum system, while ±igures 2 and 3 deFne several parameters
and variables in the system model.
±igure 1: The rotary inverted pendulum system.
±igure 2: Top view of the rotary inverted pendulum sys
tem.
±igure 3: Side view with pendulum in motion.
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View Full Document Application of Lagrangian mechanics results in the nonlinear ordinary diﬀerential equation model
(
J
eq
+
mr
2
)
¨
θ

mLr
cos(
α
)¨
α
+
mLr
sin(
α
)˙
α
2
+
B
eq
˙
θ
=
T
m
(1)
4
3
mL
2
¨
α

mLr
cos(
α
)
¨
θ

mgL
sin(
α
)=0
.
Figure 2 de±nes
θ
as the angular displacement of the rotating arm while Figure 3 de±nes
α
as the angular displacement
of the pendulum from its upright position.The variable
T
m
represents the output torque on the load from a DC motor
located in the SRV02 unit shown in Figure 2 and coupled to the rotating arm through a gear reduction system. Table
1 de±nes the constant parameters appearing in equation (1).
Symbol
Description
L
length to pendulum’s center of mass
m
mass of pendulum arm
r
rotating arm length
g
acceleration due to gravity
J
eq
equivalent moment of inertia of the motor system seen at the output of the gear reduction system
B
eq
equivalent viscous damping coeﬃcient seen at the output of the gear reduction system
Table 1: Parameters for the rotary inverted pendulum system.
The input to the inverted pendulum system is the armature voltage
V
m
applied across the DC motor within
the SRV02 system. The dynamic model of the DC motor simpli±es by neglecting the armature inductance as the
electrical time constant of the motor is more than two orders of magnitude smaller than the mechanical time constant.
The resulting relationship between the output torque and armature voltage is
T
m
=
η
m
η
g
K
t
K
g
±
V
m

K
g
K
m
˙
θ
²
R
m
,
(2)
where Table 2 de±nes the relevant constant parameters in the DC motor model.
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This note was uploaded on 07/23/2008 for the course EE 530 taught by Professor Schiano during the Fall '06 term at Pennsylvania State University, University Park.
 Fall '06
 SCHIANO

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