Ball & Beam Control System
By Stephen Salvatore
Ernest Esposito
Joseph Miller
Instructors:
Dr. Raul Ordonez
Dr. Ravi P. Ramachandran
Dr. Stephanie Farrell
Table of Contents
Section
I.
II.
III.
IV.
V.
VI.
Page
3
4
5
6-9
10
11
Title
Introduction
Problem Description
Requirements
Ball & Beam Control Theory
System Design
Team Organization
Page 2
Section I – Introduction
The ultimate goal of this experiment is to develop a control system and create the supporting
software for the CE106 ball and beam apparatus. In this experiment, a ball is free to roll along
the track of a beam that is pivoted at the center. Transducers on the ball and beam continuously
detect the ball position and the beam angle of the track. The control problem involves moving
the ball from one position on the track to another by controlling the track angle.
The control system is to be implemented using a computer interfaced to the ball and beam
system via a data acquisition card. Computer inputs are obtained from the voltage signals
proportional to the ball position and beam angle of the ball and beam. The computers output
voltage controls the motor that changes the beam angle. The block diagram below illustrates the
flow of data for the ball and beam experiment.
Inputs
Motor Signal
D to A
Computer
& Control
Software
Outputs
Beam Angle
Ball Position
A to D
Keithley DAQ
Card
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Ball & Beam
Experiment
Section II – Problem Description
The ball and beam apparatus is a simple system where a beam is tilted back and forth to
control the position of a ball at a desired position or reference, as shown in the figure below. For
instance, for a fixed beam angle the ball will accelerate along the beam at a rate that is
proportional to the beam angle. A system such as the ball and beam is an unstable system
because for a constant beam angle the ball does not stabilize at any fixed position. At constant
beam angle the ball moves at a constant acceleration until it reaches the end of the beam.
The beam angle transducer on the ball and beam apparatus consists of a servo potentiometer
mounted at the rear of the beam shaft. The signal is internally conditioned and brought out to the
Beam Angle socket mounted on the front panel. The range is between  10 V. A voltage signal
in the range 0 to 10 V applied to the Motor Drive Signal Input varies the speed of the motor. A
profiled cam attached to the motor shaft links the main beam via a secondary beam. Finally, two
parallel wires are stretched along the top of the beam. The wires support a steel ball and one of
the wires is connected to a voltage source. When the ball rests between the wires it allows a
fraction of the source voltage to be measured at the other wire. This voltage fraction provides a
voltage that is proportional to the ball position along the length of the wires. With conditioning,
an output signal is then generated which is proportional to the ball position on the beam. This
particular signal is available at the appropriate socket mounted on the front panel in the range of
 10 V.
Ball Position
Ball
Motor
Beam
Angle
Beam
The major control objective is to position the ball anywhere along the beam. This is a
difficult problem because the only actuator available is the beam itself and its rotation.
Moreover, this is a nonlinear, and unstable system.
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Section III - Requirements
1. The C++ software must have a controller module that is easy to edit so students can design
and test new controllers for each experiment.
2. All other software implementation (besides controller module) must be low maintenance.
3. Data must be acquired in at least 10 ms intervals because the ball and beam system's
dynamics have a time constant in the order of 1 ms.
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Section IV - Ball & Beam Control Theory
This section will present an overview of the control strategy to be used, describe the ball and
beam model, and discuss controller design.
Control Strategy: Cascade Control: The ball and beam system consists of a motor driven beam
with a ball free to roll on the beam. Manipulating the beam angle controls the ball position.
Since the beam angle is measured using a potentiometer and the motor drive input controls the
rate of change (velocity) of the beam angle, it is sensible to place the beam angle control loop
around the motor and drive amplifier. In other words, it is necessary to first construct a motor
angle controller that can position the motor shaft and beam at any angle required by the ball
position controller.
If the beam angle control loop is designed to respond rapidly and accurately to the reference
beam angle, then the control loop of the ball position can be designed be considering the actual
beam angle to be the control input. Now the ball position controller forms an outer control loop.
The inner loop is called the slave loop and the outer loop is called the master loop because the
master loop determines the reference input for the slave loop.
Ball and Beam Model: The ball and beam system can be modeled using the following block
diagram:
rp
yp
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where:
rp is the reference or desired value for the ball position on the beam.
yp is the actual ball position on the beam at a given instant in time.
ex is the calculated ball position error signal (ex = rp - yp).
 is the actual angle of the beam at a given instant in time.
ref is the reference angle generated by the ball position controller.
e is the calculated beam angle error signal (e = ref - ).
And u is the voltage signal that controls the unstable ball and beam system.
Motor Model: The drive motor tilts the beam through the action of a profiled cam attached to the
motor shaft. A retaining spring is attached to the beam to ensure that the cam follower is kept in
contact with the cam. We can model this motor with the following equation:
   1   2  Au
Where A, 1, and 2 are all constants that can be determined by conducting experiments with the
ball and beam apparatus. These experiments include collecting input versus output data for the
ball and beam motor. This data is used to construct a bode plot and derive the appropriate
constants for the ball and beam motor model. Also, u represents the control variable or function
for the control system.
Ball Model: The main co-ordinates for the ball and beam system are the ball position on the
beam and the beam angle . We can now write the Lagrangian (L) for the system:
L=U*-T
where U* is the kinetic energy and T is the potential energy in the system.
The potential energy is zero, since there are no springs in this system. Hence, the Lagrangian is
given by the kenetic energy which consists of the translational and rotational kinetic energy of
the ball as it moves on the beam. Now we can write the following formula:
L  U* 
1 2 1 2
mv  I
2
2
where m is the mass of the ball and I is the ball's moment of inertia.
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Now the translational velocity of the ball (v) and the angular velocity of the ball as it rolls ()
must be re-written in terms of x and . First, consider the angular velocity. The angular distance
rolled by the ball with respect to the vertical is given by the following formula:
    
x

r
Hence,

d x 
 
dt
r
where r is the rolling radius of the ball.
Also, the velocity (v) of the ball can be obtained by making a velocity diagram of the ball and
beam. This formula is given by:
 
2
v 2  x 2  x
Now, by substituting the previous two equations in the Lagrangian equation yields:


m 2
I  x

L
x  ( x) 2     
2
2 r

2
At this point we can write the expression for the moment of inertia of the ball:
I
2
mR2
5
where R is the ball radius.
The model of the system dynamics is obtained from the Lagrange equation:
d  l  L
 mg sin 
 
dt  x  x
and substituting L in this equation gives us the ball model for the ball and beam apparatus:
 2R2 
2 R 2 
1  2  x  2   x()2  g sin 
5r 
5r

where both R and r are characteristics of the ball.
State-space model: For this unstable ball and beam system it is useful simulate controllers using
numerical techniques. The state-space model is a method of producing mathematical
descriptions of control systems that do not rely on the Laplace transform, which we cannot write
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since this is a nonlinear system. This model defines the model using a system of first order
differential equations. In this particular case, we will be solving the nonlinear system of
differential equations using the Runge-Kutta method for simulation control design purposes.
To begin the state-space model we assign new variable names to the beam angle, beam
velocity, ball position, and ball velocity:
x1  
x  
2
x3  x
x4  x
Now we can begin to fill in the right side of each equation with the appropriate value in the ball
and beam system:
x1  x 2
x 2   1 x1   2 x 2  Au
x 3  x 4
g sin( x1 )  x3 ( x 2 ) 2  B( 1 x1   2 x 2  u )
x 4 
C
2
B
where
2R
2r 2
and C  1  B.
where R and r refer to different dimension of the ball.
Finally, we can write the expression for the control variable u:
u  C1 (C2 (e x )  x1 ),
where C2 is the outer loop (master) or ball position controller and C1 is the inner loop (slave) or
motor angle controller.
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Section V - System Design
A general system block diagram of the computer control is given below:
D to A
Control
Experiment
CPU
A to D
The computer will retrieve actual readings from the control experiment via the analog to digital
converter on the data acquisition card. The C++ software will process these readings and a
control voltage will be generated and sent to the digital to analog converter. This generated
voltage will in turn be sent to the ball and beam apparatus.
Software: All of the software developed for this project will be coded in C++.
Initialize D/A and A/D
converters
Set Reference Ball
Position rp(k)
Measure Ball Position
yp(k)
Calculate Position
Error
ex(k) = rp(k) - yp(k)
Generate Reference
Angle
ref(k) = C2(ex(k))
Measure Beam Angle
(k)
Calculate Angle Error
e(k) = ref(k) - (k)
Generate Control
Signal
u(k) = C1(e(k))
Page 10
Write control
voltage u(k) to the
motor
Algorithms: In the initial C++ software design a PID controller algorithm will be used. The
following flow chart outlines the algorithm of how the closed loop control would be
implemented on a computer.
Data Acquisition: In order to retrieve information from the various experiments, a Keithley
KPCL-3102 card will be utilized. Also, in order to plug into the various channels on the
computer interface a STP-68 screw terminal panel will be used connected by a CAB-305 cable.
Page 11
Section VI - Team Organization
Engine Speed Control Apparatus
Ernest Esposito - Electrical and Computer Engineering - Junior
Ball and Beam Control
Stephen Salvatore - Electrical and Computer Engineering - Senior
Motor Speed Control (Feedback)
Joseph Miller - Electrical and Computer Engineering - Junior
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