MODELING AND CONTROL OF DUAL MOTOR

BALL & BEAM SYSTEM

HAO PENG YU

UNIVERSITI TEKNOLOGI MALAYSIA

MODELING AND CONTROL OF DUAL MATOR

BALL & BEAM SYSTEM

HAO PENG YU

A project report submitted in partial fulfilment of the requirements

for the award of the degree of Master of Engineering

(Electrical-Mechatronics and Automatic Control)

Faculty of Electrical Engineering

Unverisit Teknologi Malaysia

MAY 2011

iii

DEDICATION

To my beloved mother, father and family

谨此献给我所爱的父母，家人

iv

ACKNOWLEDGEMENT

First of all, I would like to express my sincere appreciation to my supervisor,

Assoc. Prof. Dr. Hj Mohd Fua’ad bin Haji Rahmat for his encouragement and

guidance as well as suggestions from the beginning to the end of this project.

I am grateful to all of the lecturers in CIED department FKE UTM, who have

opened my mind seeing the wonderful world of Control System and Instrumentation.

Also I would like to say thank you to all of my friends and colleagues that had help

me to complete this project.

Furthermore, I am also grateful to Universiti Teknologi Malaysia for

providing me the facilities to carry out this project.

Last but not least, I would like to express my heartiest appreciation to my

parents and my family for their support through my academic years.

v

ABSTRACT

Ball and beam balancer system is very popular and widely used in control

system engineering due to its simple modeling and easy understanding. The

conventional ball and beam balancer system is with a single motor (SMBB) and

recently a new variant of ball and beam balancer system with dual motors (DMBB)

has come up. The purpose of DMBB is similar to SMBB, which is to control the

position of the ball on the beam. Compared with SMBB, DMBB use two motors to

work together in order to achieve a satisfied control of the ball position. Before the

controller design, the mathematical model of the system needs to be derived. PD

controller and lead compensator are designed to perform the simulation with

MATLAB/Simulink. Two DC motors integrated with reduction gearbox is used as

the actuator in the prototype design and two incremental encoders work to form the

motor feedbacks. Interfaced with the computer through DAQ card, the designed

prototype is verified by LabVIEW. This prototype can be used to test certain control

strategies that SMBB cannot.

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ABSTRAK

Sistem bola dan pelantar seimbang sangat popular dan sering digunakan

dalam kejuruteraan sistem kawalan kerana rekabentuknya yang mudah.

Konvensional sistem bola dan pelantar seimbang adalah motor tunggal (SMBB) dan

baru-baru ini, sistem bola dan pelantar seimbang dengan dua motor (DMBB) telah

dibentuk. Tujuan DMBB lebih kurang sama dengan SMBB, iaitu untuk mengawal

kedudukan bola di atas pelantar. Berbanding dengan SMBB, DMBB mengguna dua

motor untuk mengimbang kedudukan bola dan mencapai kawalan yang memuaskan.

Sebelum pengawal direka, model matematik sistem harus diperoleh terlebih dahulu.

Kawalan PD dan kompensator direka untuk menjalankan simulasi dengan

MATLAB/Simulink. Dua DC motor integrasi dengan gearbox sebagai penggerak

dalam rekabentuk prototaip dan dua pengekod tambahan dijalankan untuk

membentuk tindakbalas motor. Rekabentuk prototaip disahkan melalui LabVIEW

dan hubung kait prototaip dengan komputer adalah melalui kad DAQ. Prototaip ini

boleh digunakan untuk menguji strategi kawalan tertentu yang SMBB tidak boleh

laksanakan.

vii

TABLE OF CONTENTS

1 INTRODUCTION 1

1.1 Background

1.2 Statement of the Problem

1.3 Objectives of the Study

1.4 Scope of the Study

1.5 Research Methodology

1.6 Thesis Outline

1

3

4

4

5

6

CHAPTER TITLE PAGE

DECLARATION

DEDICATION

ACKNOWLEDGEMENT

ABSTRACT

ABSTRAK

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS

LIST OF SYMBOLS

ii

iii

iv

v

vi

vii

x

xi

xiv

xvi

viii

2 LITERATURE REVIEW 8

2.1 Introduction

2.2 Previous Designs of Ball and Beam Balancer System

2.3 Synchronization

2.4 Cooperative Control

8

8

13

14

3 MODELING & SIMULATION 15

3.1 Introduction

3.2 Theoretical Analysis of DMBB Model

3.2.1 Assumptions

3.2.2 Model of DC Motor

3.2.3 Model of the Ball Moving on the Beam

3.2.4 Model of the Transmission Part

3.2.5 DMBB Model for Simulation

3.3 Simulation

3.3.1 Model Parameters

3.3.2 Controller Design of DMBB Model

3.3.2.1 Controller for DC Motors

3.3.2.2 Controller for Ball Position

3.3.3 Simulation by MATLAB / Simulink

15

15

16

18

22

24

26

27

27

29

30

34

38

4 PROTOTYPE DESIGN 41

4.1 Introduction

4.2 Components to Construct the Prototype

4.2.1 Actuator (DC Motor)

4.2.2 Motor Driver

41

42

42

44

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4.2.3 Sensors to Form the Feedback

4.2.3.1 Position Sensor

4.2.3.2 Rotary Displacement Sensor

4.2.4 Mechanical Parts

50

50

51

55

5 EXPERIMENT 57

5.1 Introduction

5.2 PCI-1711 DAQ Board

5.2.1 Introduction of PCI-1711 DAQ

5.2.2 Installation of the DAQ Board

5.2.3 Connections of the Prototype with PC through

5.3 LabVIEW Application

5.3.1 Introduction of LabVIEW

5.3.2 Program by LabVIEW

57

57

57

59

60

62

62

63

6 CONCLUSIONS AND RECOMMENDATIONS 67

6.1 Conclusion

6.2 Recommendation

67

68

REFERENCES 69

PCI-1711

x

LIST OF TABLES

TABLE NO TITLE PAGE

3.1 Parameters of the Ball & Beam Model 28

3.3 The Effect of Increasing Each of the PID Controller

Parameters

30

4.1 Parameters of Selected DC Motors 44

4.2 Logic Truth Table for LMD18200 47

4.3 Selected ICs to Make the Driver Circuit 49

5.1 Signals of the Prototype 60

xi

LIST OF FIGURES

FIGURE NO TITLE PAGE

1.1 Designed DMBB Prototype 2

1.2 Flow Chart of Research Methodology 6

2.1 ‘Ball on balancing beam’ built by Berkeley Robotics

Laboratory (Arroyo 2005)

9

2.2 ‘Ball and Beam Balancer’ built by The University of

Lakehead (Ambalavanar , Moinuddin & Malyshev

2006)

10

2.3 Ball and Beam module presented by ‘Quanser’ (Quanser

2006)

11

2.4 Ball and Beam System; a) Single Motor System, b) Dual

Motor System

11

2.5 Configuration of Dual Motor System 12

2.6 Dual-motor Ball and Beam System (Manavaalan

Gunasekaran and Ramprasad Potluri 2008)

12

3.1 Sketch Diagram of DMBB Initial State 17

3.2 Sketch Diagram of DMBB Dynamic 17

3.3 Sketch Diagram of DC Motor 18

3.4 Block Diagram of DC Motor Model 21

3.5 Sketch Diagram of Ball on Beam 22

3.6 Sketch Diagram of Force Analyze for Ball on Beam 22

3.7 Diagram of System Geometric Relationship of DMBB 25

xii

3.8 Geometric Relationship of Ball & Beam part and Motor

part

25

3.9 Derived DMBB Model 27

3.10 Block Diagram of the DMBB 29

3.11 Open-Loop Step Response of Motor Angle Output 32

3.12 Performance of PD Controller for DC Motor with Step

Input

33

3.13 Performance of PD Controller for DC Motor with Sine

Input

33

3.14 Block Diagram of Uncompensated Inner Loop System 35

3.15 Step Response of Uncompensated Inner Loop System 36

3.16 Bode Plot of Uncompensated Inner Loop System 36

3.17 Bode Plot of Open Loop System with Designed

Compensator

37

3.18 Simulation Block Diagram for the System 38

3.19 Simulation Result for the Ball Position 39

3.20 Simulation Result with Sine Wave Input 40

3.21 Simulation Result with Square Wave Input 40

4.1 Dimension of the Selected DC-Motor (from datasheet) 43

4.2 Illustration of Planetary-Gear System 43

4.3 Schematic of H Bridge 45

4.4 Operation of H-bridge; a) Left to Right Current Flow, b)

Right to Left Current Flow

45

4.5 Functional Block Diagram of LMD18200 46

4.6 Schematic of PWM Waveform; a) PWM with Duty

Cycle of 75%, b) PWM with Duty Cycle of 50%, c)

PWM with Duty Cycle of 75%

47

4.7 Circuit Diagram of Motor Driver 48

4.8 PCB of Motor Driver 49

xiii

4.9 Sketch Diagram of Position Sensor 51

4.10 Encoder Circuit Diagram 52

4.11 Quadrature Encoding of the Incremental Optical

Encoder; a) Forward (Clockwise) Encoding, b) Reverse

(Counter clockwise) Encoding

53

4.12 Encoder Feedback Signals Monitored by Oscilloscope 54

4.13 Mechanical Parts Details of DMBB Prototype 56

4.14 Overview of the Designed Prototype 56

5.1 PCI-1711 DAS Set 58

5.2 Installation Flow Chart of DAQ Card 59

5.3 Connections of the Prototype 61

5.4 LabVIEW 8.6 Program 62

5.5 Elements for LabVIEW from Advantech 64

5.6 Block Diagram of Main Program 65

5.7 Block Diagram of Sub Program for Motor1 65

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LIST OF ABBREVIATIONS

SMBB - Single Motor Ball & Beam Balancer System

DMBB - Dual Motor Ball & Beam Balancer System

DC - Direct Current

PC - Personal Computer

SISO - Single Input, Single Output

MIMO - Multiple Input, Multiple Output

P - Proportional

PD - Proportional-Derivative

PID - Proportional-Integral-Derivative

LQR - Linear Quadratic Regulator

MATLAB - Matrix Laboratory

LabVIEW - Laboratory Virtual Instrumentation Engineering Workbench

KVL - Kirchhoff’s Voltage Law

EMF - Electromotive Force

PM - Phase Margin

GM - Gain Margin

DIY - Do It Yourself

IC - Integrated Circuit

CMOS - Complementary Metal Oxide Semiconductor

DMOS - Double Diffused MOSFET

PWM - Pulse Width Modulation

Ch - Channel

xv

Ref - Reference

CCP - Capture/Compare/PWM

ECCP - Enhanced CCP

A/D - Analog to Digital

DAQ - Data Acquisition

AO - Analog Output

AI - Analog Input

DO - Digital Output

DI - Digital Input

DAS - Data Acquisition System

PCI - Peripheral Component Interconnect

SCSI - Small Computer System Interface

CD - Compact Disc

DLL - Dynamic Link Library

VI - Virtual Instrument

MOT - Motor

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LIST OF SYMBOLS

- Input voltage to DC motor

- Armature resistance of DC motor

- Armature inductance of DC motor

- Armature voltage of DC motor

- Armature current of DC motor

- Torque constant of DC motor

- Back EMF constant of DC motor

- Torque at the motor shaft

- Torque from load

- Gear ratio of reduction gearbox

- Effective moment of inertia

- Viscous friction coefficient

- Motor shaft angle speed output

- Motor shaft angle acceleration

- Gain constant of DC motor model

- Time constant of DC motor model

- Force on ball

m - Mass of the ball

r - Distance of the ball on the beam

g - Acceleration of gravity

- Beam’s tilt angle from the horizontal position

- Friction force

xvii

- Rotational angle of the ball

R - Radius of the ball

- Rotational radius of the ball

- Torque on the ball

- Inertia of the ball

- Angular acceleration of the ball

- Rotary angle of motor1

- Rotary angle of motor2

- Rotary angle of motor1 gear shaft

- Rotary angle of motor2 gear shaft

- Sum of the rotary angle of motor1 and motor2

L - length of the beam

l - Length of the arm

- Gain constant of ball beam model refer to the tilt angle of

beam

- Transfer function of DC motor integrated with gearbox

- Transfer function of the ball position refer to gear shaft

output

- New gain constant of DC motor after reduction.

- New Gain constant of ball beam model refer to the gearbox

shaft angle

- DC motor speed transfer function

u - Output of PID controller

e - Error signal

- Proportional gain

- Integral gain

- Derivative gain

- Transfer function of PID controller

- Integral time constant

- Derivative time constant

- Transfer function of motor controller

xviii

- Open loop transfer function for DC motor

- Transfer function of lead compensator

K - Gain constant of lead compensator

z - Zero of lead compensator

p - Pole of lead compensator

α - The specific value of p and z

- Necessary additional phase is required to give the desired

phase margin

- The inner loop transfer function

- Desired phase margin

- Makeup value of the phase margin

OS% - Overshoot

- Settling time

- Peak value

- Final value

- Output voltage from position sensor

- Voltage applied to position sensor

- Frequency of the PWM feedback signal from encoder

n - Rotary speed of the code disk

- Resolution of encoder

1

CHAPTER 1

1 INTRODUCTION

INTRODUCTION

1.1 Background

The ball and beam balancer system is a typical control system. It is also

called ‘balancing a ball on a beam’. This system can usually be found in most

university control labs. With the associated control design, it is an excellent platform

for testing and evaluating different control techniques (Rosales, R. A. 2004).

It is related to real control problems such as horizontally stabilizing an

airplane during landing or in turbulent airflow. The fundamental principles within

this control system can also be found in many industrial applications, such as precise

position control in production line (Wei Wang 2006).

The ball and beam balancer system has been built previously by many

organizations. Normally the system is with a single motor, also can be named as

single motor ball & beam system (SMBB). This application has been studied for

years and methods of control techniques have been discussed in many literatures.

There are two different basic models of SMBB dependent on the position of the

motor used to regular the beam position. For the first kind, the motor was amounted

2

in the middle of the beam; for the second kind, the motor was amounted in one of the

end of the beam.

Recent years, a new variant of this system has come up: system with dual

motors, dual motors ball & beam system (DMBB). This topic first appeared in

Aalborg University Esbjerg (2005). They have built the first DMBB prototype.

DMBB is a ball-beam system regulated by dual motors. In contrast to the SMBB,

each end of the beam in DMBB is driven by a motor. The two motors cooperatively

work to regular the beam angular which can control the ball position.

Figure 1.1 Designed DMBB Prototype

As shown in Figure 1.1, each end of the beam hangs from a level beam. The

level beam hangs from an arm. The arm is fixed with the shaft of the planetary gear.

Each planetary gear is amounted on the shaft of each of the two DC motors.

3

Such a setup can serve as a learning stage, for example, for the more difficult

dual-manipulator cooperative control task (Hisashi Osumi ans Tamio Arai 1994), and

for the more general multi-agent cooperative control (Daniel Cruz, James Mcclintock

2007).

1.2 Statement of the Problem

Same as SMBB, the aim of DMBB is to control the position of the ball to a

desired reference point. The control signal can be derived by feeding back the

position information of the ball. The angle signal from compensator goes to the two

DC motor controllers then generate voltage signal to go to the DC motors. At last the

torques generated from the two motors cooperatively drive the beam to rotate to the

desired angle. Thus, the ball can be located at the desired position. In contrast to the

SMBB, the difference is the two motors need to work cooperatively to achieve the

aim.

It is important to point out that the open loop of the system is unstable and

nonlinear. The problem of ‘instability’ can be overcome by closing the open loop

with a feedback controller. The nonlinear property is not significant when the beam

only deflects a small angle from the horizontal position. In this case, it is possible to

liberalize the system. However, the non-linearity becomes significant when the angle

of the beam from the horizontal is larger than 30 degrees, or smaller than -30 degrees.

In this case, a simple linear approximation is not accurate.

4

1.3 Objectives of the Project

The major job of the project is to model DMBB and design controller for the

DMBB model. Then build the prototype and verify it. Specifically, the aims are listed

in the following:

To derive the mathematical model of DMBB and build a simulation

model for this system.

To design controller for the DMBB model and see the simulation results.

To design the prototype of the DMBB.

1.4 Scope of project

Two DC motors of the DMBB system will control the ball position. It’s a

cooperative problem between both DC motors. In this project, the cooperative

control is without information exchange between the two motors and is accomplished

through predefined workload sharing.

Some assumptions are made when deriving the mathematic model of the

DMBB. And linear model is used. PD controller and lead compensator are used to

apply control to the DMBB model in simulation.

The DMBB prototype is built in this project and interfaced with PC by

PCI-1711 from Advantech.

5

1.5 Research Methodology

1. Literature review to understand the concept of a ball and beam balancer system is

conducted.

2. Search out previous and current projects of a ball and beam balancer system and

get ideas for designing the DMBB.

3. Derive mathematical model for DMBB and define mathematical model of the

controllers.

4. Perform simulation using MATLAB/SIMULINK.

5. Design the prototype and test it with labVIEW program.

6. Future work to test different kinds of control strategies.

All the methodology above can be summarized as shown Figure 1.2.

6

Figure 1.2 Flow Chart of Research Methodology

1.6 Thesis Outline

This project consists of six chapters including this chapter. The scope of each

chapter is explained as stated below:

Chapter 1

This chapter gives an introduction to the project report, objectives, problem

statement, scope of work and methodology adopted.

Literature

Review

Derive Mathematic

Model of DMBB

Prototype

Design

LabVIEW

Program

Design

Design

Requirement

Controller

Design

Perform

Simulation

Verified the

Prototype by

Experiment

Testing Different Control

Strategies (Future Work)

Future Work

7

Chapter 2

This chapter discusses the previous design of ball and beam system and also

the concept of synchronization and cooperative control.

Chapter 3

This chapter gives the analysis and modeling of the DMBB. With the

designed controller, simulation is performed.

Chapter 4

This chapter presents the design of DMBB prototype including the

mechanical parts and the electrical parts. The key components to construct the

prototype are given here.

Chapter 5

This chapter the connections between PC and prototype are given. And also

the designed DMBB prototype is verified by developing a program with LabVIEW.

Chapter 6

This last chapter presents the overall discussion and conclusion of this project.

Some recommendations and suggestions have been included for future works.

69

REFERENCES

Ambalavanar, S., Moinuddin, H. M. & Malyshev, A. (2006), Ball and Beam

Balancer, BA Thesis, Lakehead University.

Arroyo, S. (2005), Ball on Balancing Beam,

http://bleex.me.berkeley.edu/ME102/proj_archive/F03/Proj9/contactinfo.htm

Aalborg University Esbjerg (2005), Dual-Motor Ball–Beam System,

http://www.aaue.dk/~akbar/projects/Ball%20and%20Beam%20System.htm

Advantech, PCI-1711/1731 User’s Manual, www.advantech.com

Advantech, Device Driver User Manual for PCI-1711/L, www.advantech.com

Advantech LabVIEW Driver User’ s Guide, www.advantech.com

Advantech Device Driver Example Manual, attached in CD

Blekhman, P. S. Landa, and M. G. Rosenblum, Synchronization and chaotization in

interacting dynamical systems, ASME Appl Mech. Rev., vol. 48, pp. 733-752, 1995.

Daniel Cruz, James Mcclintock, Brent Perteet, Omar A.A Orqueda, Yuan Cao, and

Rafael Fierro., Decentralized cooperative control, IEEE Control Systems Magazine,

27(3), pp. 58-78, 2007.

70

Hisashi Osumi ans Tamio Arai, Cooperative control between two position-controlled

manipulators, In proceedings of the IEEE International Conference on Robotics and

Automation, volume 2, pp. 1509-1514, 1994.

Jimenez S and Wen Yu, Stable synchronization control for two ball and beam

systems, In International Conference on Electrical and Electronics Engineering, pp.

290-293, 2007.

Li-Qun Chen, A general formalism for synchronization in finite dimensional

dynamical systems, Chaos, Solitons and Fractals 19, pp. 1239–1242, 2004.

M. Gunasekaran and R. Potluri. Cooperative control of a dual-motor ball and beam

system. In India Conference, 2008. INDICON 2008. Annual IEEE, pp. 525–530,

volume 2, 2008.

M. Tomizuha, J. S. Hu, and T. C. Chiu, Synchronization of two motion control axes

under adaptive feedforward control, ASME J. Dynam. Syst., Meas. Control, vol. 114,

no. 6, pp. 196-203, 1992.

Microchip (2004), PICmicro® DC Motor Control Tips ‘n Tricks,

ww1.microchip.com/downloads/en/devicedoc/41233a.pdf

Quanser (2006), The ball and beam module,

http://www.quanser.com/english/downloads/products/Rotary/Quanser_SRV02_BallB

eam_PIS.pdf

Rosales, R. A. (2004), A ball-on-beam project kit, BA Thesis, Massachusetts Institute

of Technology.

71

Wei Wang (2007), Control of a Ball and Beam System, MS Thesis, The University of

Adelaide.

Wei Ren abd Beard R.W. and Atkins E.M., Information consensus in multivehicle

cooperative control, IEEE Control Systems Magazine, 27(2), pp. 71-82, 2007.