ABSTRACT
This research is aimed at the modelling and structural analysis of the ball-on-sphere
system using bond graph technique. To achieve the bond graph model of the ball-onsphere
system, the various subsystems, storage elements, junction structures,
transformer elements with appropriate causality assignments and energy exchange that
make up the ball-on-sphere system were identified and modelled. The developed bond
graph model of the ball-on-sphere system overcame the computational complexities
involved using Euler-Lagrange modelling technique which is prone to modelling errors.
20-Sim software was used to validate the developed bond graph model. In the
developed bond graph model considering the effect of friction, the time of angular
position response of the ball ( ) achieved was 0.5253s while in the system model
without frictional effect, time of 0.5408s was achieved for the angular position response
of the ball ( ). This shows 2.9% improvement of the angular position response of the
ball considering frictional effect in the developed bond graph model. It was established
from the structural analysis that the developed model of the ball-on-sphere system was
controllable and observable. It was also determined from the structural analysis that the
system was invertible and input-output decouplable.
TABLE OF CONTENTS
TITLE PAGE I
DECLARATION II
CERTIFICATION III
DEDICATION IV
ACKNOWLEDGEMENT V
ABSTRACT VII
LIST OF APPENDICES XI
LIST OF FIGURES XII
LIST OF TABLES XIVV
LIST OF ABBREVIATIONS XV
CHAPTER ONE: INTRODUCTION
1.1 BACKGROUND 1
1.2 MOTIVATION 2
1.3 STATEMENT OF PROBLEM 3
1.4 AIM AND OBJECTIVES 4
1.5 DISSERTATION ORGANIZATION 4
CHAPTER TWO :LITERATURE REVIEW
2.1 INTRODUCTION 6
2.2 REVIEW OF FUNDAMENTAL CONCEPTS 6
2.2.1 Ball-on-Sphere System 6
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2.2.2 Mathematical Model of Ball-on-Sphere System 6
2.2.3 Bond Graph Modelling Technique 9
2.2.4 Causality Concept 12
2.2.5 20-Sim Software 17
2.2.6 Structural Analysis Concept 19
2.3 REVIEW OF SIMILAR WORKS 25
CHAPTER THREE :METHODS AND MATERIALS
3.1 INTRODUCTION 32
3.2 METHODOLOGY 32
3.3 BALL-ON-SPHERE MODELLING 33
3.3.1 Bond Graph Modelling of Ball-on-Sphere System 33
3.3.2 Ball-on-Sphere System Causalities Assignment 34
3.3.3 Ball-on-Sphere System Equations Derivation 35
3.3.4 Ball-on-Sphere Bond Graph Subcomponents 44
3.3.5 Simulink Model of the Ball-on-Sphere System 46
3.4 20-SIM VALIDATION OF BOND-ON-SPHERE SYSTEM MODEL 49
3.5 STRUCTURAL ANALYSIS OF THE BALL-ON-SPHERE SYSTEM 50
3.5.1 Ball-on-Sphere System Structural Controllability Analysis 50
3.5.2 Ball-on-Sphere System Structural Observability Analysis 51
3.5.3 Ball-on-Sphere System Inverse Model Analysis 53
3.5.4 Ball-on-Sphere System Input-Output Decoupling Analysis 55
3.6 STATE SPACE GENERATION OF THE BALL-ON-SPHERE MODEL 55
3.7 MATRIX APPROACH OF INVERTIBILITY AND DECOUPLING OF BALL-ON-SPHERE SYSTEM 58
CHAPTER FOUR: RESULTS AND DISCUSSION
x
4.1 INTRODUCTION 61
4.2 BALL-ON-SPHERE SYSTEM MODEL 61
4.2.1 Bond Graph Model of Ball-on-Sphere System 61
4.2.2 Bond Graph Causal Model of Ball-on-Sphere System 62
4.2.3 Mathematical Model of Ball-on-Sphere System 64
4.3 20-SIM VALIDATION OF BALL-ON-SPHERE SYSTEM MODEL 65
4.4 STRUCTURAL ANALYSIS OF THE BALL-ON-SPHERE SYSTEM 66
4.4.1 Ball-on-Sphere System Structural Controllability Analysis 66
4.4.2 Ball-on-Sphere System Structural Observability Analysis 68
4.4.3 Ball-on-Sphere System Bond Graph Inverse Model 70
4.4.4 Ball-on-Sphere System Input-Output Decoupling Analysis 71
4.5 NUMERICAL APPROACH TO THE BALL-ON-SPHERE SYSTEM ANALYSES 73
4.6 RESULTS OF BALL-ON-SPHERE MODEL ANALYSIS 74
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION
5.1 SUMMARY 76
5.2 CONCLUSION 76
5.3 LIMITATION 77
5.4 SIGNIFICANT CONTRIBUTIONS 77
5.5 RECOMMENDATIONS FOR FURTHER WORK 78
REFERENCES 79
xi
LIST OF APPENDICES
APPENDIX A 83
BALL-ON-SPHERE SYSTEM CONTROLLABILITY AND OBSERVABILITY ANALYSIS 83
APPENDIX B 85
BALL-0N-SPHERE MODEL ANALYSIS 85
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CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND
The ball-on-sphere system consists of the following basic components; a sphere, two
motors, and two friction wheels (Moezi et al., 2014) . The control objective of the
system is to balance the ball on top of the sphere by controlling the rolling of the sphere
along each of the two horizontal axes through friction wheels driven by motors. The
control of the system is a challenging task because of its non-linear, unstable and under
actuated nature (Ho et al., 2009).
The ball-on-sphere is an important class of balancing systems with applications from
robotics to transportation and aero-space in the following areas (Graf & Röfer, 2010):
1) missile guidance
2) modelling of a postural standing of human or humanoid robot
3) self-transport machine
4) modelling and simulation of the unstable system of a human or robotic upper limb
5) modeling and stabilization of space-ships and rockets
The ball-on-sphere system is as shown in Figure 1.1
SPHERE BALL
FRICTION
WHEEL
DC MOTOR
Figure 1.1: Ball on a Sphere System (Moezi et al., 2014)
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The dynamics of the ball-on-sphere system are nonlinear and complex and its parameters are interdependent in various directions; they have been considered to be two independent ball and wheel systems around the equilibrium point (Ho et al., 2009). Parameter identification is among the most difficult steps in model design phase, which is the most cause of model errors (Dauphin-Tanguy et al., 1999). The dynamics of ball-on-sphere system have been modeled using Euler-Lagrange modeling techniques. However, the existing method is limited with respect to not having detailed knowledge of the system’s parameters (Moezi et al., 2014). The dynamic equations of the ball-on-sphere system are non-linear and coupled, hence, it is normally difficult to obtain significant dynamics of the system using the various conventional numerical analytical tools.
Bond graph modeling technique captures most physical variables and dynamics of multi-domain systems. The bond graph model of the multi-domain systems help to overcome the mathematical complexities of conventional modelling techniques which are prone to errors (Borutzky, 2011).
The advantages credited to bond graph technique were explored in modelling the ball-on-sphere system in this work and in addition, were also used for the structural analyses of the system.
Several computer-based simulation tools are available for designing and simulating bond graphs. 20-Sim is a graphical modeling and simulation program which is suitable for generating and processing of dynamic systems (Alabakhshizadeh et al., 2011). The 20-Sim is used to simulate and analyze bond graph models.
1.2 MOTIVATION
The modelling of multi-domains systems for control analysis and diagnosis is a major step in the design and simulation of multi-domains system (Paynter, 1970). Multi-
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domain systems problem are interdisciplinary engineering systems problems, involving engineering systems in the domain of electrical, mechanical, chemical and amongst others. Modelling techniques such as variational and network graphic techniques do not have well defined uniform notations suitable for modelling and analyzing all type of physical domains systems (Yu & van Paassen, 2004). Bond graph technique provides uniform notation for physical systems and is based on energy and information flow in systems (Borutzky et al., 2006). Furthermore, the technique has the capacity to carry out structural analysis in order to deduce information on a variety of structural properties of the system being model (Sueur & Dauphin-Tanguy, 1991). The Euler-Lagrange technique used in modelling the ball-on-sphere system is tedious and prone to modelling errors due to its computational complexities and does not have the capacity to analyze dynamic behavior of the system. Hence, the ball-on-sphere system requires a modelling technique that captures the various physical components of the system and provides a simple and efficient means for structural analysis and generation of dynamic equations of the system. These motivated the use of bond graph technique in this work. The bond graph technique is used to model and structurally analyze the ball-on-sphere system in order to study the dynamics of the system. This is because the method is based on the energy characteristics of each constituent physical component that contribute to the entire system developed (Karnopp et al., 2012).
1.3 STATEMENT OF PROBLEM
Several attempts have been made to model the dynamics of the ball-on-sphere system. Researchers have used modelling technique such as Euler-Lagrange approach to model the system. However, the technique is prone to modeling error due to its computational complexities. The challenge had been that the dynamics of the system is nonlinear and complex and its parameters are interdependent on one another and are multi-directional
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(Ho et al., 2009). The existing model of ball-on-sphere system is limited for not having detailed knowledge of the system’s parameters (Moezi et al., 2014). Furthermore, the Euler-Lagrange technique does not have the capacity to carry out structural analysis of the information properties such as structural controllability and observability, inverse model and input-output decoupling of the ball-on-sphere model. Bond graph technique which is a suitable unifying concept that captures most physical variables and dynamics of the system was used to model the ball-on-sphere system. The bond graph technique was also used to carry out structural analysis of the ball-on-sphere system model properties in order to evaluate its dynamic behaviour.
1.4 AIM AND OBJECTIVES
The aim of this research is the modelling and structural analysis of the ball-on-sphere system using bond graph technique in order to study the dynamic behaviour of the system.
The objectives of the research are therefore as follows:
1) Development of bond graph model of the ball-on-sphere system.
2) Validation of the developed model in 1) using 20-Sim.
3) Implementation of the structural analysis of ball-on-sphere system using the bond graph technique in order to determine the structural controllability and observability, model inversion and input-output decoupling of the system.
1.5 DISSERTATION ORGANIZATION
The general introduction has been presented in Chapter One. The rest of the chapters are structured as follows: In Chapter Two, the literature review which comprises the review of similar works and the review of fundamental concepts pertinent to the research were presented. The fundamental concepts include ball-on-sphere system and its mathematical modelling, bond graph technique and structural analysis amongst
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others. Chapter Three explains the bond graph technique method and procedures used in modelling and carrying out structural analysis of the ball-on-sphere system. The developed ball-on-sphere system bond graph model, the derived mathematical models describing the developed model and results of the structural analysis of the developed model were presented in Chapter Four. Finally, conclusion and recommendations of further work makes up the Chapter Five. The list of cited references and MATLAB codes in the appendix are provided at the end of this dissertation.
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