Application of Evolutionary Algorithms to Engineering Design

Application of Evolutionary Algorithms to Engineering Design – Kevin Hayward

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Application of Evolutionary Algorithms to

Engineering Design

Kevin Hayward, BE BSc

This thesis is presented for the degree of Doctor of

Philosophy of the University of Western Australia.

School of Mechanical Engineering

Submitted in 2007


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Dedication

This thesis is dedicated to Kevin Clark, my grandfather. He has

shown me that hard-work and principles can go a long way. I am

honoured to have been given his name.


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Declaration

This thesis does not contain work that I have published, nor work

under consideration for publication. This thesis is completely the

result of my own work, and was substantially conducted during the

period of candidature, unless otherwise stated in the thesis.

Signature………………………………………..


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Abstract

The efficiency of the mechanical design process can be improved by the use of

evolutionary algorithms. Evolutionary algorithms provide a convenient and robust

method to search for appropriate design solutions. Difficult non-linear problems are often

encountered during the mechanical engineering design process. Solutions to these

problems often involve computationally-intensive simulations. Evolutionary algorithms

tuned to work with a small number of solution iterations can be used to automate the

search for optimal solutions to these problems. An evolutionary algorithm was designed

to give reliable results after a few thousand iterations; additionally the scalability and the

ease of application to varied problems were considered. Convergence velocity of the

algorithm was improved considerably by altering the mutation-based parameters in the

algorithm. Much of this performance gain can be attributed to making the magnitude of

the mutation and the minimum mutation rates self-adaptive. Three motorsport based

design problems were simulated and the evolutionary algorithm was applied to search for

appropriate solutions. The first two, a racing-line generator and a suspension kinematics

simulation, were investigated to highlight properties of the evolutionary algorithm:

reliability; solution representation; determining variable/performance relationships; and

multiple objectives were discussed. The last of these problems was the lap-time

simulation of a Formula SAE vehicle. This problem was solved with 32 variables,

including a number of major conceptual differences. The solution to this optimisation was

found to be significantly better than the 2004 UWA Motorsport vehicle, which finished

2nd in the 2005 US competition. A simulated comparison showed the optimised vehicle

would score 62 more points (out of 675) in the dynamic events of the Formula SAE

competition. Notably the optimised vehicle had a different conceptual design to the actual

UWA vehicle. These results can be used to improve the design of future Formula SAE

vehicles. The evolutionary algorithm developed here can be used as an automated search

procedure for problems where performance solutions are computationally intensive.


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Table of Contents

Abstract .............................................................................................................. 4

Table of Contents ............................................................................................... 5

Acknowledgements ............................................................................................. 8

Thesis ................................................................................................................. 9

1 Introduction .................................................................................................. 9

1.1 Evolutionary Algorithms ...................................................................... 10

1.2 Design ................................................................................................ 11

1.3 Evolutionary Algorithms in Motorsport Design .................................... 12

1.4 Outline of Following Chapters ............................................................ 15

1.5 Statement of Original Contribution ..................................................... 17

2 Design ........................................................................................................ 18

2.1 Introduction ......................................................................................... 18

2.2 Decision Making ................................................................................. 19

2.3 Types of Design ................................................................................. 21

2.4 Product Development Models ............................................................ 22

2.4.1 Product Development Tools and Techniques ................................... 27

2.5 Racing Car Design ............................................................................. 28

2.6 Race Car Design Procedures ............................................................. 29

2.6.1 Determination of Design Constraints ................................................ 31

2.6.2 Determination of Design Specifications ............................................ 31

2.6.3 Conceptual Design ........................................................................... 32

2.6.4 Preliminary Design ............................................................................ 32

2.6.5 Detailed Design ................................................................................ 33

2.7 Practical Design Considerations ......................................................... 34

2.8 Evolution of Race Cars ....................................................................... 35

3 Evolutionary Algorithms ............................................................................. 37

3.1 Introduction to Simulated Evolution .................................................... 37

3.2 Optimisation Problems ....................................................................... 38

3.3 Evolutionary Algorithm Process .......................................................... 43


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3.3.1 Population Seeding............................................................................ 45

3.3.2 Population Evaluation ........................................................................ 45

3.3.3 Parent Selection ................................................................................ 45

3.3.4 Population Renewal ........................................................................... 46

3.3.5 End Criteria and Algorithm Completion .............................................. 47

3.4 Solution Representation ..................................................................... 47

3.5 Self-Adaptation ................................................................................... 49

3.6 Fast Evolutionary Algorithms .............................................................. 50

3.7 Multi-Objective Algorithms .................................................................. 52

3.8 Application of Evolutionary Algorithms ............................................... 53

4 Development of an Evolutionary Algorithm ................................................ 54

4.1 Considerations for Algorithm Development ........................................ 55

4.2 Test Problem Set ............................................................................... 56

4.3 Method ............................................................................................... 58

4.3.1 Algorithm Parameter Tuning .............................................................. 59

4.3.2 A Note on Starting Mutation Rates .................................................... 61

4.4 Population Size .................................................................................. 63

4.5 Number of Parents ............................................................................. 70

4.6 Introducing and Studying Selective Pressure ..................................... 71

4.7 Investigating Mutation Strength and Distribution ................................ 73

4.8 Investigating Minimum Mutation Rate ................................................ 81

4.9 Investigating Cauchy Distribution ....................................................... 84

4.10 Controlling Mutation Parameters ........................................................ 94

4.11 Conclusion ....................................................................................... 106

4.12 Developed Evolutionary Algorithm ................................................... 107

5 Practical Application of Evolutionary Algorithms ...................................... 109

5.1 Ideal Path Generator ........................................................................ 110

5.1.1 Path Definition ................................................................................. 111

5.1.2 Vehicle Model .................................................................................. 113

5.1.3 The Problem .................................................................................... 114

5.2 Suspension Kinematics .................................................................... 131

5.2.1 Model Details ................................................................................... 131


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5.2.2 Different Representations ............................................................... 133

5.2.3 Determining Parameter Relationships ............................................ 138

5.2.4 Multiple Objectives .......................................................................... 144

5.2.5 Problems with System Level Designing .......................................... 150

6 Evolving Racing Cars ............................................................................... 151

6.1 The Formula SAE Competition ......................................................... 152

6.2 Experiment Setup ............................................................................. 154

6.3 Non-Conceptual Optimisation Results .............................................. 156

6.3.1 Discussion ...................................................................................... 161

6.4 Conceptual Optimisation Results ..................................................... 167

6.4.1 Discussion ...................................................................................... 172

6.5 Comparison to Existing Vehicles ...................................................... 175

6.6 Parameter Sensitivity ....................................................................... 177

6.7 Conclusion ........................................................................................ 180

7 Conclusion ............................................................................................... 182

8 Recommendations for Future Work ......................................................... 185

9 Bibliography ............................................................................................. 187

10 Appendix A: Lap Time Simulation ............................................................ 197

10.1 Simulation Requirements ................................................................. 197

10.2 Lap Time Simulation ......................................................................... 198

10.2.1 Program Structure ....................................................................... 200

10.2.2 Engine Model .............................................................................. 203

10.2.3 Drive-train .................................................................................... 204

10.2.4 Brakes ......................................................................................... 207

10.2.5 Aerodynamics ............................................................................. 208

10.2.6 Wheel Loading ............................................................................ 209

10.2.7 Suspension Geometry ................................................................. 210

10.2.8 Tyre Modelling ............................................................................. 212

10.2.9 Tyre Model .................................................................................. 215

11 Appendix B: Chapter 7 Non-Conceptual Results ..................................... 226

12 Appendix C: Chapter 7 Conceptual Results ............................................. 232


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Acknowledgements

More than five years has passed since beginning this thesis. During that time I found just

about every source of distraction I could. I built racing cars, worked overseas, and

married the love of my life. If it hadn’t been for the constant support, encouragement and

reprimands from a number of people in my life I would not have been able to complete

the work. I would particularly like to thank the following people for their support:

• Angus Tavner, my supervisor, for his incredible patience and valuable

feedback.

• Peter Hayward, my father, for giving me a love of the applied sciences. I

walk in his footsteps.

• Christine Hayward, my mother, I am blessed to have been raised by her.

• Alma Clark, my Nan, the strong base of our family tree.

• Peter and Kaye Pearson, for fostering my interest in knowledge.

• Lachlan Tomlin, my colleague and friend, for his wise council.

• Jodi-Lee Hayward, my wife, for sharing my life with me.

My greatest thanks are reserved for God, who has given me the chance to do this work,

and has provided such wonderful people around me. I also wish to state that I see no irony

in using the theory of evolution while proclaiming God as our creator. If I can use

evolutionary techniques to help design racing cars, I see no issue with the creator of the

universe being able to use evolution for much more difficult design problems. Although I

suspect his algorithms may be a little bit more advanced than those shown in this thesis.

Then I saw all that God has done. No one can comprehend what goes on under the sun.

Despite all his efforts to search it out, man cannot discover its meaning. Even if a wise

man claims he knows, he cannot really comprehend it. (Bible: NIV translation,

Ecclesiastes 8:17)


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Thesis

The author proposes that evolutionary algorithms are a convenient and robust method to

automate the search for appropriate design solutions to increase the efficiency of the

mechanical design process.

1 Introduction

This thesis demonstrates that it is possible to develop evolutionary algorithms that are

simple, fast and robust in order to apply them effectively within a general design

methodology. Initially a short study was conducted into what constitutes a general design

methodology, and to identify where optimisation routines might be applied. This was

followed by research into the history and current state of evolutionary algorithms. An

evolutionary algorithm was designed and tuned for use on complex problems, given

limited available computation time. The algorithm was applied to a number of complex

problems to gauge its performance and to make observations about its application. Finally

the evolutionary algorithm was successfully applied to a difficult design problem that the

author had previously attempted to solve using traditional techniques.

The particular emphasis of this work has been application of evolutionary algorithms to

the design of mechanical systems in the motorsport industry. This industry provides a

suitable challenge, because high-performance complex systems must be developed within

short time frames, in a constantly changing environment. The lessons learnt about the

design and application of appropriate evolutionary algorithms for motorsport design

problems can, of course, be applied to other fields of industry.

The evolutionary algorithm developed for this work was applied to three different

motorsport design problems. Each was chosen for both its application to vehicle design,

and what it could indicate about the evolutionary algorithm. The first problem was to find

the ideal path for a given track layout. The second problem was to design appropriate

kinematics for a racing car suspension system. The final problem was the conceptual


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design of a racing car. It should be noted that each of these problems occurs in a different

stage during overall design of a racing vehicle. Ideal paths are likely to be needed when

the car is in use, and a racing team is attempting to find the best way around a particular

track. The kinematics design of suspension occurs during the detailed design of vehicle

sub-components. Finally, the conceptual design of a racing car is amongst the first steps

of vehicle design.

As the following document will describe, an effective algorithm was developed that had

performance at least equivalent to other similar evolutionary algorithms. It was used

effectively for each of the 3 problems, which spanned different stages of the design

process. All of this is evidence that evolutionary algorithms are a convenient and robust

method to automate the search for appropriate design solutions to increase the efficiency

of the mechanical design process.

1.1 Evolutionary Algorithms

Evolutionary algorithms use simulated biological evolution models to solve optimisation

problems. They are proposed as a way to find solutions close to global maxima/minima

for complex problems in a much shorter time than would be required by evaluating all

possible solutions. A number of potential solutions to a given problem are created. Each

solution is evaluated against a known performance (or fitness) function. A new

population is created, based on the best solutions of the previous population that have

been slightly modified. The processes of evaluation and population renewal are repeated.

Following this concept of ‘survival of the fittest’, better solutions to the problem are

constantly being created.

It is only in the last few decades that computational power has advanced to the point

where evolutionary algorithm techniques are practical, and the field has grown rapidly in

response. However, computational speed is still a significant issue and likely to remain so

in the foreseeable future. The quality of the result used is highly dependent on the number

of iterations that can be performed and the efficiency of the algorithm in creating superior

solution candidates.


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1.2 Design

Engineering design is a decision making process to devise systems, components, or

processes to meet desired needs. (Ertas & Jones 1993) A number of different models have

been devised in order to facilitate decision making throughout the product development

from the initial market research phase to the sale of the final items. These processes are

discussed in Chapter 2 with some discussion of the application of these processes to

racing car design. The use of evolutionary algorithms allows some automated iteration of

the decision making process to allow for a considerable number of appropriate solutions

to be analysed within the time available.

A product can be considered static or dynamic. A dynamic product is one in which

conceptual changes are often required, while allowing the possibility of a marked increase

in product performance. Conversely, a static product requires only incremental changes,

with lower potential performance increases. (Hollins et al. 1990) Evolutionary algorithms

can be applied to both types of products successfully as is shown in Chapter 6.

Limitations in design time often force products that should be considered dynamic to be

treated as static products. Increased efficiency in the design process can help alleviate this

situation. The author contends that the use of evolutionary algorithms in these situations

is one such way of increasing the efficiency of the design process.

A product development model should be scalable, extensible, adaptable, and able to be

applied incrementally. (Sum, 1992) This creates a general purpose model that is

applicable to a large number of problems. A similar approach should be taken with

evolutionary algorithms, so that they can be used at different points within the design

process framework.


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1.3 Evolutionary Algorithms in Motorsport Design

Motorsport design offers an ideal situation to apply evolutionary algorithms. Design

problems involving racing cars and racing car components are often complex and time

consuming. There is also a fundamental limitation in the time available. New vehicle

designs are produced constantly and governing rules are updated regularly. The restriction

of time forces the simplification of complex design problems and limits the number of

conceptual design alternatives that can be analysed; this leads to racing vehicles often

becoming static products. This is seen in many classes of racing, where vehicles appear to

converge to a common design. Hotten (1998) states that ‘racing is driven by a relentless

search for fractional improvements in lap times’.

“But the ethos of speed extends beyond the public theatre of competition.

Formula 1 in the current era is a race of development led by the outfit that can

progress faster than the competition. Anyone who stands still, even for a moment,

will be found slipping helplessly down the grid order. The race is on to develop

faster wind tunnel programs and faster computational processes that can iterate

down to the theoretically ideal machine more quickly than the rivals.”

(Armstrong-Wilson 2005)

This comment shows that the design of an effective racing car is a problem of

optimisation, controlled by the speed at which that optimisation process operates.

Designing and tuning a racing car is a difficult problem. In an example of a setup sheet

for a racing car Glimmerveen (2004) presents over 50 parameters that need to be

recorded. This set is just a subset of the parameters required for the design of a vehicle.

However if there were just two possibilities for each of these 50 parameters there would

be approximately 1.1 x 1015 different solutions. To put that in perspective, if one solution

was tested every second, it would take approximately 36 million years to test all possible

solutions. Clearly an exhaustive search of all possible solutions is not practical, even in a

simulation context.


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Modern racing is incredibly competitive. Sponsors desire immediate results, and testing is

becoming more costly, with increased limitations being placed on testing by motorsport

regulatory bodies. Rouelle (2007) uses these points to illustrate that there is an increasing

demand for:

• Understanding of race car dynamics

• Testing efficiency

• Race driver self-coaching tools

• Quick, realistic, easily understandable decision making tools

• Vehicle dynamics and lap time simulation software

It is the final two points that are of particular interest here. Armstrong-Wilson (2002) also

indicates the need for fast decision making:

“In motorsport, however, a decision that has not been taken can be

enormously damaging as it prevents progress and quickly leaves you

trailing behind. Time is the enemy and consequently a very different

culture has evolved within successful teams where everyone understands

that decisions need to be taken quickly. This can apply to everything from

finalising the qualifying set-up to choosing which engine to buy for next

season. It is accepted that some wrong choices will be made, but not to let

those hurt the objective any more than they need to.”

This highlights the importance placed on quick decision making throughout the whole

design process. A host of simulation software has been developed in order to increase the

efficiency of the decision making process. Wagstaff (2005) reviews a number of

currently-available commercial lap-time simulation tools and comments on their use as a

quick tool to assess vehicle parameter changes without costly testing. However an

important point is made:


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“… despite the capabilities of simulation software, it is important to emphasise that

simulation is not a magic solution. It cannot design a racecar, or automatically

define optimum set-up, and in no way replaces the designer or engineer.”

This suggests that while current tools are able to reduce testing time they do not radically

improve the efficiency of the decision making process. The designer is still responsible

for guiding the software through the iterative process. Further comment is made on one of

the packages:

“With Pi-Sim, basic set-up parameters such as gear ratios or wing settings can

be established in advance of visiting the track. Performing a ‘parameter sweep’

may even automatically optimise some settings. This is a process that uses the

power of batch run simulation to generate predictive data for a pre-defined

combination of settings in order to select the optimal combination. An example is

the selection of optimal front/rear wing positions in order to optimise the

downforce/drag relationship, thus minimise lap time. The engineer can also

experiment with the effect of various wing settings on the car’s balance to

compare the benefits of running different aerodynamic configurations for

qualifying and race.”

This approach shows a great dependency on the engineer to make decisions. Given the

sheer number of possible vehicle parameter combinations previously mentioned, it is

necessary for the designer to decide which variables are most important in order to

effectively use a parameter sweep technique; with only a small number of variables able

to be analysed at a time. Using this approach the speed of the process becomes limited by

the ability of the designer to make and enter the required decisions about which variables

to study, and in what order. Hence, evolutionary algorithms appear well-suited to the

design of racing cars. Race car design is driven almost completely by the single objective

of faster lap times. Current implementations of evolutionary algorithms favour single

objective functions as discussed in Chapter 3. Once appropriate performance simulations

have been developed, evolutionary algorithms can be used as a practical automated

searching method; decreasing the dependency on the engineer to guide the search for

superior solutions.


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1.4 Outline of Following Chapters

Chapter 2 – Design

This chapter involves a literature review of both general and motorsport-specific design

methods. A general design methodology based on Pugh (1991) was chosen as a basis for

continued work, primarily based on its simplicity and coverage of all areas of design from

ideation to product use and sales.

Chapter 3 – Evolutionary Algorithms

This chapter involves a literature review of evolutionary algorithms. A focus was made

on developed algorithms with significant history of use, or algorithms that deviate only

slightly from a well developed base. This chapter also includes a discussion on the

procedures and parameters of evolutionary algorithms and how these relate to their

application.

Chapter 4 – Development of an Evolutionary Algorithm

This chapter details the design and tuning of an evolutionary algorithm for use in a design

procedure. The algorithm was designed with the aid of a suite of test functions with

significant numbers of variables, constrained by allowable computation time. This work

highlighted the importance of mutation parameters in an evolutionary algorithm, and in

the process of tuning produced parameter values separately from generally applied values.

Significant improvement in the algorithm was found by both controlling minimum rates

of mutation, a parameter often ignored as a source of performance improvement.

Additional improvements were also found by allowing some of the evolutionary

parameters to change throughout an evolutionary run. These advantages were attributed to

the fact that allowable computation time had been reduced, and tend to be overlooked in

the development of many evolutionary algorithms.


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Chapter 5 – Practical Applications

This chapter covers the application of the algorithm developed in Chapter 4 to two

example problems. The first problem was finding an ideal path (i.e. fastest) for a given

racetrack. The problem was chosen as it is a common problem in motorsport, it has a

large number of variables, the solution is not immediately obvious, and it gives a good

visual demonstration of the performance of evolutionary algorithms. The second problem

was the kinematics design of a double A-arm suspension system. This problem was

chosen because it was work that was being carried out at the time during the design of a

racing car, and showed the potential of the information gained during use of the

evolutionary algorithm to aid in determining performance relationships. In addition, the

kinematics design problem is typically a multi-objective design problem; this showed the

immediate limitations of using a simple evolutionary algorithm for this type of problem.

Chapter 6 – Evolving Racing Cars

This chapter used the Lap Time Simulation described in Appendix A (Chapter 10) with

the evolutionary algorithm developed in chapter 4. In this application, the environment

was varied as well as allowing different types of parameters to be optimised. Analysis of

the results clearly showed advantages to using evolutionary algorithms during the

conceptual design stage. Excellent results were found for the optimised vehicles. Further

effort was devoted to analysing how to use information gained during the running of an

evolutionary algorithm and how the important relationships between design parameters

and overall performance could be visualised.

Chapter 7 and 8 – Recommendations and Conclusions

Chapter 7 covers a few future recommendations for continuing the work. The main

recommendations are to test further modifications to the evolutionary algorithm

developed in this thesis, as well as investigate improved data-treatment techniques to

gather more information from the results of the optimisation. Chapter 8 summarises the

thesis and the major points discussed in the work.


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1.5 Statement of Original Contribution

The novel aspects of this work focus on the development and practical application of an

evolutionary algorithm for use with realistic engineering problems. A detailed study and

tuning of evolutionary algorithms was conducted for small numbers of evaluations.

Identifying mutation parameters as having the primary effect on performance led to the

unique adoption of self-adaptive mutation parameters for both the strength of mutation

and the minimal allowable mutation rate.

Additionally the thesis provides a unique treatment of the overall design of race-cars

using evolutionary algorithms. This involved assessing both how useful the application of

these algorithms is, as well as using the results of the optimisation process to determine

global relationships.

It is worth noting that all of the simulations and code used within the thesis were created

by the author.


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2 Design

In order to define the requirements of a suitable evolutionary algorithm it was necessary

to investigate formal design processes, including both the design process and product

development models. Particular attention was paid to the design process as applied to

racing cars, given that is was the focus of the problems to which the algorithm was

applied (Chapter 5 and 6).

2.1 Introduction

Engineering design is the process of devising a system, component or process

to meet desired needs. It is a decision making process (often iterative), in

which the basic sciences, mathematics, and engineering sciences are applied

to convert resources optimally to meet a stated objective.

(Ertas & Jones, 1993 pg. 2: Quote from Accreditation Board for Engineering

and Technology1)

Design is a decision making process to meet a stated objective optimally. Starkey (1992)

comments on the position of decision making in the design process:

The main task of the engineering designer is decision making. At every stage

and at every level in the design process, the designer has to make a single

choice from a number of alternative courses of action presented. Every

decision made will significantly influence the way in which the design will

develop from that point on. A ‘good’ decision will ensure satisfactory

technical and economic progress: a ‘bad’ one will almost certainly hinder

further progress.

1 Accreditation Board for Engineering and Technology, In. Annual Report for the year ending September

30, 1988, New York, 1988.


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The success of a given design process is undoubtedly measured by the quality of the final

result, and the speed with which it was delivered. It is in this context that evolutionary

algorithms could prove to be a useful tool to the engineering designer.

The importance of time to market has recently been shown to be

responsible for over 30% of the total profit to be made from a product

during its life-cycle. Booker (Booker et. al. 2001)

There are a variety of approaches to engineering design. It is not the author’s intention to

provide an exhaustive review of the engineering design field. The following sections will

cover the general properties of design and product development models, concluding with

the example of design processes adapted to race car design. A more comprehensive

treatment of the development of engineering design theories can be found in Chapter 3 of

Design Science (Hubka and Eder, 1996)

2.2 Decision Making

Given that the design process is focused on decision making, it is valuable to note the

types of decision, and the general process of solving them. Starkey (1992) provides three

different types of decision within the design process:

• Fundamental Decisions – Few in number, but are of great importance to

whether a design is ‘good’ or ‘bad’. Fundamental decisions are made during

the start of a design process, and are often unchangeable without substantial

redesign.

• Intermediate Decisions – These are extensions of, and supplement,

fundamental decisions. Many intermediate decisions may spring from one

fundamental or other intermediate decision. These decisions may be changed,

but often not without difficulty and expense.

• Minor Decisions – Occur in vast numbers, and are most often concerned with

design details. These can usually be changed with minimal difficulty and

expense.


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Starkey also notes that the fundamental decisions are made early in the design process

where knowledge of the problem is limited, and the amount of data available is at its

minimum. Hence great caution must be used during this stage.

Figure 2-1 - Decision Making Process (Pahl and Beit z 1996)

Figure 2-1 shows the general process for finding solutions. In effect, this is the process

that leads to making a decision. The phases of creation and evaluation are most relevant

to the application of evolutionary algorithms. The creation phase requires the

development of solutions within defined constraints, while the following stage evaluates

the performance of the created solution. Evolutionary algorithms can be used to automate

these steps, allowing a large number of solutions to be analysed within a defined problem.

This process is covered in more detail in the next chapter.

The evaluation of a given solution can be a difficult problem in itself. Sen and Yang

(1998) examine multiple-criteria decisions within engineering design. Criteria may be

either subjective or objective, and the choice and prioritisation of appropriate criteria is an

important task for the designer. Sen and Yang (1998) also mention that complexity in


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problems is most often dealt with by some form of decomposition, where difficult

problems are split into a number of simpler set of problems. This thesis will focus on

solving single criteria problems that result from the decomposition of larger design goals.

Section 5.2.5 discusses some of the potential problems associated with this approach.

2.3 Types of Design

Pahl and Beitz (1996) make mention of three broad classifications of design, namely:

• Original Design – Development of a new system to perform a new task or one

that has been solved by other means previously.

• Adaptive Design – Using known and established solution principles to meet

new requirements.

• Variant Design – Sizes and arrangements of aspects of a given system are

altered, however the solution principle and its application remain the same.

Birmingham et al. (1997) make the comment that the majority of design falls within the

scope of variant design. This is primarily because of the low risk associated with it.

Products can be considered static or dynamic (Hollins et al. 1990). Designing for static

products only requires incremental changes, while design for dynamic products often

requires new concepts to be considered. Hollins also notes that the performance of a

product increases more rapidly during dynamic phases. Available design time was the

first factor listed that determines whether a product is static or dynamic. Limited design

time leads to static products, while adequate design time allows for dynamic products.


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2.4 Product Development Models

There is some contention regarding the beginning and the end of the design process.

Some contend that design starts with market research and ends with manufacture (Pugh

1991). Others contend that design begins at the requirements definition stage and ends

with solution documentation (Pahl and Beitz, 1996). To avoid confusion the term

‘Product Development’ will be used. Since product initiation occurs somewhere in the

market research stage and ends with delivery of the final item, this will lead to the use of

a model similar to Pugh’s.

Design methods or ‘philosophies’ have been extensively researched and

documented, but they are far from a product panacea. The primary

purpose of these methods is to formalize the design process and

externalise design thinking. (Booker et. al. 2001)

No one product development will suit all applications perfectly, however a general set of

guidelines for a successful model does exist. Sum (1992) presents the characteristics that

a product development process should exhibit:

• Scalability – The size of organisations change constantly.

• Potential to be introduced incrementally – Any new product development

model is likely to be introduced to smaller sections of an organisation and then

spread to other areas.

• Extensibility – New features of product development, such as new tools and

techniques, are likely to emerge. These need to be incorporated into the existing

product development model.

• Adaptability – Situations vary in different organisations. Uniform product

development processes do not capitalize on the strengths of, or address the

possible weaknesses of different enterprises.

Figure 2-2 shows a product development process offered by Pugh (1982). This is very

similar to the process offered by Pahl & Beitz (1996) Figure 2-3. Clearly there is an early


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stage of specification followed by conceptual design, leading to more detailed design.

Pugh places more importance on design as a part of the process from market research to

product delivery. Pahl and Beitz focus on the specification to the detailed design delivery.

It is clear that the model proposed by Pahl and Beitz is an expanded subset of the steps

outlined by Pugh.

Application of Evolutionary Algorithms to Engineering Design

Figure 2-2 – Design Activity Model


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Design Activity Model (Pugh, 1982)

Kevin Hayward


Figure


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Figure 2-3 - Design Process ( Pahl & Beitz, 1996


Pahl & Beitz, 1996 )


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An increased work effort near the beginning of a product development process has many

benefits that are outlined below: (Booker et. al. 2001)

• It is easy to influence the customer

• Design changes are easy

• The cost of change is low

• Management involvement is more cost effective

It should be noted that the design process is conveniently split into various stages from

design initiation to completion. However Haque and Pawer (1998) mention that

following these functional stages sequentially results in long development times and

problems with product quality. Traditional sequential product development models are

being replaced by more efficient team-based concurrent engineering models. Pugh’s

model addresses this by stating that each of the functional stages affects every other stage

(Pugh, 1991). Booker summarises the advantages of this approach:

• Reduced time to market

• Reduced engineering costs due to the reduction in reworking of designs

• Better responsiveness to market needs

• Reduced manufacturing costs.

He also states a few of the disadvantages:

• Increased overheads – the teams require their own administration support

• Costs of co-location – people being relocated away from their functions to be

with the team

• Cultural resistance

• Inappropriate application – it is not a panacea for development problems, as

poor conceptual designs will not be improved by using concurrent methods.


27

The change to concurrent methods has been made possible by changing the role of design

engineers. The approach requires multi-functional design teams and the use of new and

existing design tools (Russell and Taylor 1995).

2.4.1 Product Development Tools and Techniques

Many tools and techniques have been developed in order to facilitate the tasks involved

during product development. Huang (1996) highlights the main engineering activities that

should be aided:

• Gather and present facts about products and processes

• Clarify and analyse relationships between products and processes

• Measure performance

• Highlight strengths and weaknesses and compare alternatives

• Diagnose why an area is strong or weak

• Provide redesign advice on how a design can be improved

• Predict what-if effects

• Carry out improvements

• Allow iteration to take place

Some typical techniques include Failure Mode Effects and Analysis (FMEA), Quality

Function Development (QFD), Design for Assembly / Design for Manufacture (DFA /

DFM) and Design of Experiment (DOE), and each can affect different stages of product

development.


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2.5 Racing Car Design

The first organised motorsport event was a reliability trial between Paris and Rouen in

1894. In the United Kingdom alone this has grown to a 4.6 billion pound (A$10.5b)

industry involving over 4000 companies (MIA 2007). Top-level racing teams employ

hundreds of engineers with typical design times of a product around 6 months. Designing

and building a professional race car is a serious and rapid engineering endeavour.

Vehicles are designed with systematic design procedures using a variety of advanced

tools and techniques. This section will outline the fundamental design objectives of

racing cars, properties of systematic design procedures and typical design tools, and the

application of these techniques to racing car design.

Race Car Vehicle Dynamics (Milliken and Milliken, 1995) begins with the following

line:

The overall technical objective in racing is the achievement of a vehicle

configuration, acceptable within the practical interpretation of the rules,

which can traverse a given course in a minimum time (or at the highest

average speed) when operated manually by driver utilising techniques

within his/her capabilities.

In practice this goal is likely to be idealistic, as it does not include any restrictions placed

by resource constraints. Furthermore, the overall performance of a vehicle is likely to be

judged after a full season of racing and will be affected by reliability, maintainability and

adjustability, as well as vehicle speed. A good example of these trade-offs is found in

Roberts et al. (2006), which details the design of a modern customer race car.

Racing is considered a high technology field and computers are used extensively in racing

car design, even at amateur levels of the sport. The most common programs generally fall

into one of three categories: simulations; design tools; and data acquisition. Recently

more advanced computer programs have been used to aid decision-making based on the

principles of artificial intelligence. For example in Formula 1, expert systems are used to


29

determine appropriate pit-stop strategies (Purnell 1998). Suggestions have been made to

enhance this type of program to include much more system information than is currently

used. Such an approach attempts to model a vehicle as a whole and not as a set of

subsystems. This allows designers and race engineers to limit the separation of vehicle

systems required to facilitate design and tuning work.

2.6 Race Car Design Procedures

Race car design has been carried out by both amateur and professional engineers.

Anecdotal evidence suggests that the less engineering-orientated designer will design

vehicles based on accumulated experience. This approach places severe limitations on the

ability to advance quickly in the field. It has been generally accepted in the more

professional forms of motorsport that a much more scientific and systematic approach to

design is required.

Early attempts to improve racing vehicle design still drew heavily on accumulated

experience. Michael Costin, a development Engineer at Lotus Cars Ltd., co-authored a

text with David Phipps (Costin and Phipps, 1967) that detailed racing and sports car

chassis design in the early 1960’s. The final chapter entitled “Designing a Motor Car”

gives an indication of how top-level designers in the motorsport industry approached the

problem in the past:

However experienced the designer, there is a basic sequence in which he

goes about his task. Before any detail design work is begun it is necessary

to establish the over-all specification of the car.

This shows great similarity to the design process outlined by Pugh (1991) in the

suggestion of using a product design specification as a basis to work from. Costin and

Phipps also alert the designer to the fact that detail design work may cause initial design

ideas to change. Essentially, the mechanical design of the vehicle is broken down into a

series of sequential processes. Such an approach shows significant experience with the

process, and enables the design of a complex vehicle to be performed by a small group of


30

people. This is indicative of the way racing car design was achieved during the early

years of motorsport. Given the increase in complexity of modern racing cars, top race

teams can no longer field successful cars designed by similarly sized teams.

The process outlined by Costin and Phipps can be directly compared to a more modern

approach offered by Milliken & Milliken (1995). The authors offer a more sobering view

of racing car design in their chapter entitled “Race Car Design” than presented in the

opening chapter.

It is important to recognise the relationship between available resources

and expectations …When starting a design project; it is well to assess

one’s resources to avoid frustration and compromise at a later time.

The chapter opens with the statement that “design is not sequential but rather one of

multiple stages of revision and refinement”. Milliken and Milliken detail a design

procedure that shows a lot of similarities to Pugh (1991). The process can be broken

down into the following major steps, which as mentioned before, may involve multiple

revisions.

• Determination of Design Constraints

• Determination of Design Specifications

• Conceptual Design

• Preliminary Design

• Detailed Design

These stages are discussed in more detail in the following sections.


31

2.6.1 Determination of Design Constraints

Constraint determination is one of the results of the market analysis stage given by Pugh

(1991). In the case of a racing vehicle, the primary constraints are those offered by the

competition rules, other constraints include those due to available resources and

processes. Milliken & Milliken (1996) state that success in competition requires working

at the absolute limits of the constraints. Given the requirement to determine constraints

accurately, it is clear that extensive market research and analysis is necessary.

2.6.2 Determination of Design Specifications

Pugh (1991) explains the application of a Product Design Specification as discussed

previously. Milliken & Milliken (1996) suggest a similar process for race car design.

They suggest that a specification is an outline of the detailed objectives of the vehicle.

These include, but are not limited to, performance goals, handling objectives, structure

type, ergonomics, safety concerns, tyres, and which features are to be adjustable.

Milliken & Milliken (1996) state that this is an area “where parameter studies (using

whole vehicle computer modelling) can be very helpful” . The focus of the text is

primarily on the technical attributes of the vehicle. It appears likely that the application

of a more detailed specification system, such as that outlined by Pugh (1991), could

provide benefits. By dealing with other aspects such as manufacturing processes and

maintenance, there may be opportunities to improve time efficiency and overall cost that

in the long run may improve the performance of the vehicle by freeing up resources.

It should be noted here that while Milliken & Milliken’s process is similar to Pugh’s

when defining constraints and specification, they do not distinguish between the two

different steps. This, perhaps, shows more similarity to the first step of the design

procedure outlined by Pahl and Beitz (1996). This step, named ‘Clarification of the

Task’, presupposes that market research and analysis has been done. Pugh appears to

offer a more complete, and hence potentially more useful, design process.


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2.6.3 Conceptual Design

Milliken & Milliken (1995) mention the need for conceptual design considerations for a

particular vehicle. However they do not explain the processes involved in conceptual

design. Given the nature of the text, which is a study of race car dynamics, such a

discussion is possibly unwarranted. It is also possible that little discussion is given

because many race cars are based on permutations of previous models. This is observable

in the field, where substantial conceptual changes in high levels of motorsport are

infrequent. This could indicate that the limits of the constraints of the rules have already

been met closely, or that conceptual changes occur primarily on an individual component

basis. Furthermore as described in Section 1.3 there is a constant pressure to make fast

decisions in motorsport.

However it would appear unwise to disregard this part of the design process. As outlined

in Section 2.4, the cost of early change is low, while at the same time offering maximum

potential to improve final product performance. Clearly in some race series such as Solar

Car Races, Land Speed Records, or the Student-based Formula SAE there is a great need

for conceptual design. Even in higher level motorsport (e.g. Formula 1, LeMans, or

Nascar) every time the environment changes there is likely to be a need for some

conceptual work. This environment change could be in the form of new and/or modified

tracks, rule changes, and new technologies. All these changes have occurred on a regular

basis in professional motorsport.

2.6.4 Preliminary Design

The goal of preliminary design, as outlined by Milliken and Milliken (1996), is to fix the

general arrangement in the car such that packaging constraints are met. This step

involves estimation of component dimensions and weights. Milliken & Milliken suggest

that physical mock-ups are appropriate at this stage. This stage is very similar to the

embodiment design phase outlined by Pahl and Beitz (1996). Embodiment design is

described as a complex process in which many actions have to be performed

simultaneously, and that alteration and additions in one area can have repercussions for


33

existing design in others. It is clear that this stage, whatever one chooses to call it, must

be highly iterative.

2.6.5 Detailed Design

Following preliminary design, more detailed design of all the vehicle systems is required

before manufacture. However Milliken & Milliken (1996) make an important note:

Decisions must be made on the amount of detailing versus shop floor

design. Even for top level teams, the resources and time available do not

permit the level of detailed design used for production cars or aircraft.

The other side of this argument is presented by Costin and Phipps (1967):

Whatever the car and whatever the use to which it is to be put, it is

important that a complete design study is made before any work is begun,

to avoid the risk of major changes being necessary in the final stages of

construction. And even more important that all detail work is completed

before the car is considered “finished” – in the case of a one-off – or

“passed for production”. Impatience to get the car on the road has been

the downfall of many a good design.

New racing car designs are required to be implemented within short spaces of time.

Given that racing seasons are usually run annually there is often a need for designs to be

put into manufacture within less than one year. Hence available time is one of the critical

vehicle constraints.


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2.7 Practical Design Considerations

In order to facilitate the design of race cars the task is usually broken up into major

subsystems. (Wright 2003, Milliken & Milliken 1996, Costin & Phipps 1967) Examples

of these areas are given below:

• Engine

• Transmission

• Aerodynamics

• Brakes, Suspension, and Steering

• Tyres

• Systems: Hydraulics, Electrics, and Electronics

While it is clear that interaction between engineers working on these areas is required, a

certain amount of autonomy is necessary due to time and resource constraints. This

decomposition of a complex problem into simpler sub problems is considered a common

approach to difficult design problems (Sen and Yang, 1998).

Furthermore, to ensure adequate levels of vehicle performance, a variety of empirical

relationships are used where modelling and simulation are inaccurate, too complex, too

time consuming, or too expensive. These rules are developed both theoretically and

empirically. Ultimately, there are a number of criteria that determine a successful race

car. Some of these are quantitative, such as price and overall speed, while others are

qualitative such as handling and responsiveness. This differentiation between criteria, are

discussed by Sen and Yang (1998). Methods are continually developed that allow some of

the subjective data to be treated objectively.

The strict time constraints also need to be considered for both the manufacture and testing

of components as well as the design process. Often a design with increased performance

must be abandoned due to the time required to manufacture or test. A good example of

this is seen in Formula SAE. Most teams use dampers designed for mountain bikes, due

to a lack of available units specifically suited to FSAE, and while the design of a suitable


35

unit is not particularly difficult the large requirement for testing and precise

manufacturing precludes it as an option for most teams.

2.8 Evolution of Race Cars

There is an evolutionary pattern apparent in race car design. When a successful design

change is made to one car, there is generally only a small amount of time before the

majority of the field adopts the change. This can lead to a rapid change in the design of

the cars. One such example was the first Formula 1 cars to use sliding skirt ground

effects. In 1978 Lotus introduced the concept and won the championship, by 1979 (the

next generation of cars) other teams had adopted and refined the concept and Lotus did

not win a single race for the season. (Lawrence, 2002) Alternately unsuccessful changes

are usually short lived. This mimics the idea of survival of the fittest and vehicles tend to

converge on concepts within a given environment. This approach seems more natural than

the design of vehicles based on first principles, which could be considered impractical

given the vast number of design variables.

It is worth noting a few other characteristics of this design approach. First, there are

generally large initial gains, but as time progresses the performance increases become

smaller and smaller. Secondly a lot of the gains are found outside of accepted norms, such

as the introduction of slicks and wings. Thirdly the rate of performance increases is also

linked to the design environment. This includes, but is not limited to, the resources

available and the competition rules. The trends of evolutionary design can be seen in the

following example of Formula 1 fastest qualifying times at Monza from 1950.


Figure 2- 4

A few notes should be made regarding this graph. In 1972 chicanes were added to the

track, which caused an increase in qualifying time. In 1976 the F1 regulations were

altered, causing the increase in qualifying time. In 1988 turbo

causing the increase in qualifying time. From 1994 onwards the rules have been unstable,

making it difficult to perceive the improvement of the vehicles. However it is clear tha

the performance of the vehicles is highly environment dependant. Furthermore,

improvements in vehicle speed decrease in magnitude as time progresses.

2 Note the peaks in the fifties and sixties are due to a wet track during qualifying.


36

4 - Monza Pole Time (1965-2003)



using the increase in qualifying time. In 1988 turbo-charged cars were banned,


making it difficult to perceive the improvement of the vehicles. However it is clear tha


improvements in vehicle speed decrease in magnitude as time progresses.2

Note the peaks in the fifties and sixties are due to a wet track during qualifying.

Kevin Hayward



charged cars were banned,


making it difficult to perceive the improvement of the vehicles. However it is clear that


Note the peaks in the fifties and sixties are due to a wet track during qualifying.


37

3 Evolutionary Algorithms

Important practical problem classes where evolutionary algorithms yield

solutions of high quality include engineering design applications involving

continuous parameters … (Bäck et. al. 1997)

This chapter covers a review of standard evolutionary algorithms, including a discussion

on the procedures and parameters of evolutionary algorithms and how these relate to their

application.

3.1 Introduction to Simulated Evolution

Evolutionary algorithms use models of biological evolution to solve optimisation

problems and have been in existence since the 1950s. Darwin was the first to formalise

the theories of evolution in the 19th century (Darwin 1859), and the mechanisms of

evolutionary algorithms can clearly be traced back to this original work. These

mechanisms include reproduction, mutation, recombination, natural selection and survival

of the fittest. Lack of computational power and shortcomings of the methods limited the

application of the field until the 1970’s. Recent advances in computational performance

has lead to rapid growth in the development and application of evolutionary algorithms,

as they are adapted to a large number of different problems in different scientific fields.

Bäck et al. state that the most significant advantage of using evolutionary search lies in

the gain of flexibility and adaptability to the task at hand, in combination with robust

performance … and global search characteristics. (1997)

Bäck and Schwefel (1993, 1996), Bäck et al. (1997) and Fogel (1994) provide much of

the introductory material for evolutionary algorithms referenced in this chapter. These

papers outline the generalised algorithm and the mechanisms used. The author chose to

focus on real-valued representations of evolutionary algorithms, namely evolution

strategies, and evolutionary programming. Solution representations and the reasons for

choosing this focus are discussed in Section 3.4.


38

3.2 Optimisation Problems

There are two fundamental objectives for any optimisation algorithm; finding the best

possible solution, and doing so in the minimum time possible. Finding the optimum

involves searching the solution space for the best candidate. The solution space, or search

space, is defined as the set containing all the possible solutions for a given problem.

Evolutionary algorithms are one of many classes of algorithms that have been proposed to

search large search-spaces for difficult problems.

It is important to differentiate between small and large problems, as well as between

difficult and simple problems. A problem can be considered small where there are few

possible solutions. In these cases, every possible solution can be tested and the optimum

is easily found. The case of every candidate being tested is referred to as a brute force

search. The number of possible solutions is almost always related to the number of

variables, or dimensions. Generally, more variables involve more possible solutions. For

example if for each variable of a given problem there were two possible values then the

number of solutions would increase as a power of 2 for the number of variables, as shown

in Figure 3-1. This simple relationship indicates only problems of low-dimensionality

allow for all the possible candidates to be considered.


Figure

For the purposes of this thesis a problem will be considered simple if the shape of the

solution space naturally directs an algorithm to the optimum.

1-dimensional problems are presented from the test fu

first (Test Function 1)

performance (Figure 3

uni-modal function. In this case simple gradient descent algorithms can be used to find

the optimum effectively.


39

Figure 3-1 - Number of Solutions v s. Number of Variables


solution space naturally directs an algorithm to the optimum. To help illustrate this

dimensional problems are presented from the test functions given in

first (Test Function 1) is a simple quadratic relationship between the variable and

3-2). This type of function has only one minimum, and is

modal function. In this case simple gradient descent algorithms can be used to find

the optimum effectively.


s. Number of Variables


To help illustrate this, two

nctions given in Section 4.2. The

is a simple quadratic relationship between the variable and

on has only one minimum, and is termed as

modal function. In this case simple gradient descent algorithms can be used to find


Figure 3-2 - 1 Dimensional Version of Test Function 1

The second (Test Function 9) has a cosine function superimposed on a quadratic

relationship (Figure 3-3). This function shows a single global minimum, but many local

minima. This is termed a multi-

gradient descent method.


40

2)( xxf =

Equation 3-1

1 Dimensional Version of Test Function 1

(Test Function 9) has a cosine function superimposed on a quadratic

). This function shows a single global minimum, but many local

-modal function and cannot be solved with a simple

Kevin Hayward

(Test Function 9) has a cosine function superimposed on a quadratic

). This function shows a single global minimum, but many local

and cannot be solved with a simple


Figure

The problem with simple gradient descent methods can be described with the aid of

Figure 3-4. The potential solution is shown at point p. There is a local minimum to the

left of the candidate, and a g

method would result in the solution at point A.

heuristic algorithm. T

process on the pretext

solutions. This approach gives the potential for the evolutionary algorithm to ‘hill

in order to reach a global minimum, in this case at point B.

also able to solve problems more suited to a gradient descent method, albeit with less

efficiency. In the case where the shape of the solution space is unknown

shape should be assumed as being possible, hence a heuristic method, such as an

evolutionary algorithm, should be used.


41

10)2cos(10)( 2 +−= xxxf π

Equation 3-2

Figure 3-3 - 1 Dimension al Version of Test Function 9


. The potential solution is shown at point p. There is a local minimum to the

left of the candidate, and a global minimum to the right. A simple gradient descent

method would result in the solution at point A. Evolutionary algorithms

heuristic algorithm. Temporarily inferior solutions are permitted to exist

process on the pretext that further optimisation from these new points may lead to better

This approach gives the potential for the evolutionary algorithm to ‘hill

in order to reach a global minimum, in this case at point B. Evolutionary algorithms are

e to solve problems more suited to a gradient descent method, albeit with less

efficiency. In the case where the shape of the solution space is unknown


ary algorithm, should be used.


al Version of Test Function 9


. The potential solution is shown at point p. There is a local minimum to the

lobal minimum to the right. A simple gradient descent

Evolutionary algorithms are a type of

to exist during the search

that further optimisation from these new points may lead to better

This approach gives the potential for the evolutionary algorithm to ‘hill-climb’

Evolutionary algorithms are

e to solve problems more suited to a gradient descent method, albeit with less

efficiency. In the case where the shape of the solution space is unknown, a multi-modal



Figure 3

Consideration must also be given to the time taken to calculate the performance function

for one candidate. In some cases the performanc

using elementary mathematical functions

provide a number of these examples. In other cases computation of the performance of a

candidate can take quite a long time to calculate, engineering problems such as

structural analysis are one such example.

algorithm.

EvaluationTotal TnT = (

Equation 3-3 shows the relationship governing the total time required to perform an

optimisation; TEvaluation is the time taken for each function evaluation,

overhead time for each evaluation,

algorithm, while n is the number of evaluations

functions, the time taken by overheads

discussed in this thesis the time taken to evaluate each candidate is considered


42

3-4 - Local and Global Minima


In some cases the performance of a candidate can be quickly calculated

functions. The test functions outlined in S


take quite a long time to calculate, engineering problems such as

analysis are one such example. This has a direct effect on the running time of an

eadFixedOverhheadScaledOverEvaluation TT ++ )

Equation 3-3

shows the relationship governing the total time required to perform an

is the time taken for each function evaluation, TScaledOverhead

overhead time for each evaluation, TFixedOverhead is a fixed overhead required to run the

n is the number of evaluations. For simple functions, such as

functions, the time taken by overheads can be significant. However for the situations

ime taken to evaluate each candidate is considered

Kevin Hayward


e of a candidate can be quickly calculated

Section 4.2


take quite a long time to calculate, engineering problems such as complex

This has a direct effect on the running time of an

ead

shows the relationship governing the total time required to perform an

ScaledOverhead is the

is a fixed overhead required to run the

such as the test

significant. However for the situations

ime taken to evaluate each candidate is considered


43

significantly larger than the overheads required, hence the total time can be approximated

as Equation 3-4.

EvaluationTotal nTT =

Equation 3-4

When reference is made of application of evolutionary algorithms to a large difficult

problem the following assumptions are made:

• There are too many possible solutions to permit a brute force search.

• The shape of the solution space is unknown and not assumed to be conducive to

the use of simple gradient descent methods.

• The time taken for each function evaluation is significantly large.

3.3 Evolutionary Algorithm Process

The most common Evolutionary computation techniques are genetic algorithms,

evolutionary programming, evolution strategies, and genetic programming.

(Michalewicz, 1996) The procedures differ in data representation, methods of varying

potential solutions and methods for selecting solutions for further development. (Fogel,

1994) The basic structure of the algorithm, which follows a neo-Darwinian model, is

shown in Figure 3-5. (Bäck and Schwefel 1996, Michalewicz 1996, Bäck et. al. 1997)

Each step is discussed in the following sub-sections (3.3.1 to 3.3.5).


Figure 3-5 -

It should be noted that the selection and

different methods. However it is clear from this flow diagram that the evolution is the

result of the selection, creation of new populations, and evaluation processes

many times. (Bäck et. al. 1997)

Michalewicz (1996) states that any evolutionary approach

contain the following components:

• A genetic representation of solutions to the problem.

• A way to create an initial population of solutions.

• An evaluation function, to rate solutions according to their ‘fitness’.

• ‘Genetic’ operators that alter the genetic composition of children during

reproduction.

• Values for the parameters (population size, probabilities of applying genetic

operators, etc.).


44

- Evolutionary Algorithm Process

It should be noted that the selection and population renewal step can be reversed for

thods. However it is clear from this flow diagram that the evolution is the

result of the selection, creation of new populations, and evaluation processes

Michalewicz (1996) states that any evolutionary approach to problem solving must

contain the following components:

A genetic representation of solutions to the problem.

A way to create an initial population of solutions.

to rate solutions according to their ‘fitness’.

that alter the genetic composition of children during

Values for the parameters (population size, probabilities of applying genetic

Kevin Hayward

step can be reversed for

thods. However it is clear from this flow diagram that the evolution is the

result of the selection, creation of new populations, and evaluation processes occurring

to problem solving must

that alter the genetic composition of children during

Values for the parameters (population size, probabilities of applying genetic


45

The main differences between the different types of evolutionary algorithms lie in the

representation of individuals, the design of the variation operators, and the

selection/reproduction mechanism. (Bäck et. al. 1997)

3.3.1 Population Seeding

Evolutionary computation techniques maintain a population of individuals, each of which

represents a potential solution to the problem. (Michalewicz 1996) Creation of the initial

population usually involves creating random solutions within a given search space. If

self-adaptive algorithms are used, search parameters are also randomly seeded. This is

discussed in more detail in Section 3.5. A population may also be seeded with the results

of another algorithm or from pre-existing data.

Population size can have a marked effect on algorithm performance. Evolutionary

algorithms can be created where the population is maintained at one; however the use of a

population size greater than one is one of the key features of evolutionary computation.

Population-based evolutionary algorithms can drastically decrease the required

computation time and decrease the failure rate of an algorithm (He and Yao 2002).

3.3.2 Population Evaluation

Each individual is evaluated against a predefined function and assigned a fitness (or

performance) value that is the result of the function. For minimization problems, a lower

error function value indicates a superior solution, and vice-versa for maximization

problems. This function is the only link between the problem and the algorithm and must

be as close to a perfect representation as possible. (Michalewicz 1996)

3.3.3 Parent Selection

A set of individuals is chosen from the population to provide parents for the next

generation. Selection is based solely on the fitness values of the individuals. There are a

number of different selection operators that fall into one of two categories; deterministic

or probabilistic selection.


46

Deterministic methods only select the best individuals of the current population to form

the parent set. If n parents are required from a population of λ members, the population

would be sorted based on performance and the top n members would form the parent set.

This is known as an elitist selection criterion.

Probabilistic selection allows a chance for inferior individuals to become part of the

parent set. Some of these methods include:

• Proportional selection – Probability of selection is proportional to individual

fitness

• Ranking Methods – Probability of selection is proportional to the rank of an

individual in the population

• Tournament selection – A number of random individuals compete for selection

and the fittest is selected. (Michalewicz 1996)

Selection should be designed such that both sufficient optimisation progress and genetic

diversity to escape local minima are maintained. Standard selection methods only use the

fitness of individuals. Hutter and Legg (2006) present a selection scheme where the

sparseness of populated regions is taken into account.

3.3.4 Population Renewal

The parent set created from the selection process is used to produce a new population set.

In this process new individuals are created. The two main methods of altering the parent

population are through recombination and mutation. These two methods can be used

together or in isolation. Recombination involves mixing the properties of two or more

parents together to create a child. Mutation involves varying properties of an individual

randomly. In the creation of new population members unfeasible solutions may be

produced. This must be taken into account by the algorithm by allowing these solutions to

exist temporarily in the population or by repairing them. Where recombination exists

there is deemed to be sexual reproduction, where it is absent and only mutation is used as

an operator it is deemed to be asexual reproduction.


47

The new population can be created as a population solely of children solutions created by

the parents. This technique is denoted as (λ, n) where λ is the size of the population and n

is the number of parents. In this situation the best solution from one generation is not

passed to the next. Conversely the new population may be created as the combination of

the children solutions and the original parent set. This technique is denoted as (λ + n).

This allows the best of a generation to compete for selection in the following generation.

Combining this latter technique with a deterministic selection process guarantees that the

fittest individual from any following generation is at least as good as the fittest individual

in the current generation.

3.3.5 End Criteria and Algorithm Completion

The end criterion is met when a predetermined condition is reached. This usually occurs

when a sufficiently low error value is arrived at, or the available computational resources

have been exhausted. At the completion of the algorithm the best solution from the final

generation is taken as the final solution. Alternatively, if the best solution found

throughout the run, was recorded and it is better than the best individual in the final

generation, it may be used as the final solution.

3.4 Solution Representation

There are two main methods of genetic representation. Genetic algorithms represent

individual solutions as single binary strings. In order to apply an error function these

binary strings need to be ‘decoded’. Binary strings are chosen on the assumption that they

resemble strands of DNA, which are modified during natural evolution. Evolutionary

algorithms and evolutionary programming techniques tend to represent individuals as a

vector of real numbers. This representation does not follow the natural example as

closely, but avoids the need for encoding and decoding functions within the algorithm.

There has been much research that covers the advantages and disadvantages of both

representations. Michalewicz (1996) made the following comment:


48

It is common knowledge that for numerical optimization problems one should

use an evolutionary strategy of genetic algorithm with floating point

representation, whereas some versions of genetic algorithm would be the best

to handle combinatorial optimization problems.

Parameters which represent the variables subject to optimisation are termed the phenotype

space. The operators in the algorithm itself are termed the genotype space. (Bäck et. al.

1997) The real numbers of the evolutionary algorithms are an attempt to match the

genotype space as closely as possible to the phenotype space. This avoids the possibility

of coding functions introducing additional mathematical difficulties which can hinder the

search process. However by using a standard algorithm, such as the genetic approaches,

and custom decoding algorithms, it becomes possible to draw upon empirical and

theoretical results in order to reuse appropriate parameters, which in turn can reduce the

need for algorithm testing. (Bäck et. al. 1997)

There are no cases in nature where a genotype and a phenotype have a one-to-one

relationship. In reality there exists Plieotropy where one gene can affect a number of

phenotypic traits, and Polygeny where one phenotypic trait is defined by many genes. In

nature the phenotype will be a complex non-linear function of the genetic structure.

(Fogel 1994) Some of this complexity can be avoided with appropriate parameters used to

represent an individual. A practical example of this is discussed in Section 5.2.2 where

two representations for the same problem were tested, with one noticeably improving the

performance of the evolutionary algorithm.

When self-adaptation is used (see Section 3.5) an individual is usually represented by a

pair of real-value vectors. The first vector represents the location of the individual in the

search space, while the second vector represents the control parameters that allow self-

adaptation. A real-valued representation allows for precise adjustment of the control

parameters of the evolutionary algorithm. (Liang and Leung 2002)


49

3.5 Self-Adaptation

Self-adaptation is used in all the main paradigms of evolutionary computation. (Toussaint

and Igel, 2002) These algorithms represent individuals as two-part entities. The first part

is a vector of values that define the individual itself. The second part is a vector of search

parameters. As the algorithm is run both vectors are optimized. The end result is an

algorithm that searches for both the optimum solution and the optimum search parameters

for a given time in the search procedure.

During the evolutionary process the ideal tuning parameters change as the potential

solutions get closer to the optimum solution. It has been found that the optimal mutation

strength is defined as Equation 3-5. (Fogel 1994)

Equation 3-5

Where is the Euclidean distance between the current and the optimum solutions.

As the Euclidean distance3 between the current solution and the optimum is reduced, the

mutation strength also needs to be reduced. For a given problem where the location of the

optimum is unknown the Euler distance cannot be calculated, this will be the case for any

non-test problem. A heuristic schedule can be used to reduce this step size. Self-

adaptation was introduced by Schwefel (1981) as a way to reduce the step size over time.

This allows the mutation parameters to adapt to arbitrary circumstances.

3 The Euclidean Distance is shortest distance between two points in n-dimensional space. Mathematically

defined as: ( ) ( )2211 nn uvuv −++− K , where v and u are two points in n-dimensional space.

n

xx −=

*224.1σ

xx −*


50

Self-adaptive algorithms may be isotropic or non-isotropic. Isotropic algorithms alter all

individual parameters by the same standard deviation, while non-isotropic algorithms

allow for individual parameters to be altered at different rates. For isotropic algorithms

the following occurs:

)1,0('.'

))1,0(.exp(.' 0

Nxx

N

ii στσσ

+==

Equation 3-6

For non-isotropic algorithms this changes to:

)1,0('.'

))1,0(.)1,0('.exp(.'

Nxx

NN

iii

iii

σττσσ

+=+=

Equation 3-7

Toussaint and Igel (2002) make the point that neutrality is a necessity for efficient self-

adaptation. A neutral variation is defined as one where the genotype is altered without

affecting the phenotype. This occurs quite regularly in nature as the genotype-phenotype

mapping is quite redundant. Toussaint and Igel contend that this neutrality allows for

optimisation of the search parameters (self-adaptation parameters) without having to

sacrifice fitness.

3.6 Fast Evolutionary Algorithms

A number of other algorithms have been developed in an attempt to increase speed over

classical evolutionary algorithms (e.g. Liang and Leung 2002, Hyeon-Joong Cho et al.

1998, Yao & Yong Liu 1997). An increase in process speed can be defined as finding

equivalent solutions with less problem evaluations, or superior solutions for the same

number of evaluations. The work begun by Yao & Liu (1997) was worth studying as

simple modifications to the classical evolutionary strategies yielded significant

performance advantages.


51

Yao & Liu suggested the use of Cauchy distributions to improve the convergence speed

of self-adaptive evolution strategy algorithms. This was followed with the Cauchy

distributions applied to evolutionary programming for the same reasons. (Yao et al. 1999)

Compared to a Gaussian distribution, a Cauchy distribution allows the probability of

larger mutation intervals. They have shown that this can improve an algorithm’s

resistance to becoming trapped in local minima, hence improving the global optimisation

process. The trade-off for this is the degradation of the algorithm’s fine-tuning ability. To

overcome this they have proposed that the Cauchy distribution could be mixed with a

Gaussian distribution to utilise the best properties of both distributions. The probability

distribution function for both the Cauchy and Gaussian distributions are shown in

Figure 3-6.

Figure 3-6 – Cauchy and Gaussian Probability Densit y Functions

Probability Density Functions

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-5 -4 -3 -2 -1 0 1 2 3 4 5

Cauchy Gaussian


52

The claim is made that the long tail increases the probability of the Cauchy-modified

algorithms being able to ‘jump’ out of a local minimum. Empirical studies involving 23

test functions appear to support this modification having a positive effect. (Yao et al.

1997)

This was a promising avenue of development for this thesis, as it showed a considerable

increase of performance with only simple changes to an established algorithm. Liu and

Yao (2002) discuss different approaches to further improve performance, including

changing the parameters of the Cauchy distribution, as well as introducing a mix of

Cauchy and Gaussian distributions into the same algorithm.

3.7 Multi-Objective Algorithms

Often, problems are created where the solution needs to meet a number of criteria, or

objectives. The algorithms discussed in this chapter, and in this thesis deal with single

objective problems. Section 5.2.5 shows a situation where using single objective

algorithms has practical limitations.

Multi-objective optimisation is notably more complex than the single objective

optimisation of the ‘classic’ evolutionary optimisation and is not within the scope of this

work. There are a number of papers that cover techniques that can be used for problems

with multiple objectives. (Valenzuela 2002, Socha & Kisiel-Dorohinicki 2002, Costa and

Oliveira 2002,


53

3.8 Application of Evolutionary Algorithms

Evolutionary algorithms have been applied to a number of different problems. Of

particular interest to this thesis the algorithms have been applied to a number of

motorsport problems. (Mühlmeier and Müller 2002, Mitchell et al. 2004, Castellani &

Franceschini 2003) They are applicable to so many problems because they do not make

assumptions about the environment they are operating in. Jin (2005) presented a survey

on techniques used to improve evolutionary optimization in uncertain environments. Four

classes of uncertainties are mentioned:

• Noise: The fitness evaluation is subject to noise.

• Robustness: The design variables are subject to perturbations after the optimal

solution has been determined.

• Fitness Approximation: Fitness functions are often approximated, where they are

very expensive to evaluate, or an analytical fitness function is not available.

• Time-Varying Fitness Functions: The fitness function is dependent on time.

Jin notes that when applying evolutionary optimisation to design problems the use of

approximate fitness functions is unavoidable; the search for robust solutions is also

necessary.


54

4 Development of an Evolutionary Algorithm

Development and tuning of an evolutionary model is an important part of the process of

applying evolutionary techniques to a particular problem. Bäck et al. (1997) state:

In our opinion, evolutionary algorithms should not be considered as off-

the-peg, ready-to-use algorithms but rather as a general concept which

can be tailored to most of the real-world applications that are often

beyond solution by means of traditional methods. Once a successful EC4 -

framework has been developed it can be incrementally adapted to the

problem under consideration, to changes of the project requirements, to

modifications of the model, and to the change of hardware resources.

An evolutionary algorithm was developed by the author to deal with difficult engineering

design problems. The specification of the algorithm is listed below:

• A close-to-ideal solution is to be found within the calculation time.

• The location and the value for the ideal solution to the problem are

unknown.

• Calculation time is easily predicted, such that it can be accurately scheduled,

with an upper bound set at one day.

• Algorithm is scalable (both in number of dimensions and evaluations).

• Algorithm is robust for a variety of problem types.

• Algorithm must deal with a mix of discrete and continuous variables.

• Solution space boundaries are well-known by the user and may not be

exceeded.

• Some tolerance is expected for the product of the design process.

4 Evolutionary Computation


55

• Algorithm must be readily programmed using currently available

programming tools, and must run on an average desktop computer.

• Algorithm is simple to apply to a variety of problems.

A few assumptions about the algorithm and its use are briefly given below:

• Calculation time for solving each potential solution is considerable.

• Calculation times of the evolutionary algorithm operations are insignificant when

compared with solving each potential solution.

• End user will not alter the algorithm.

The following chapter details the design and tuning of an algorithm that meets the

requirements and assumptions given above. Tuning typical algorithm parameters, the

application of Gaussian and Cauchy distributions, and methods to control minimum

mutation rates were all investigated. This algorithm is based on existing algorithms and

is known to give performance increases for certain applications.

4.1 Considerations for Algorithm Development

Recent studies into evolutionary algorithms have been conducted using substantial

computing power. For example, when investigating different random number

distributions, Yao & Liu (1997) compares algorithms to solve a series of 30 variable

problems (as well as problems with fewer variables) with between 150,000 and 2,000,000

function evaluations. If using a simulation with a function evaluation time of

approximately 30 seconds this would result in a computation time of between 52 and 695

days. This sort of time requirement would limit the usefulness of the algorithms when

used as part of an engineering design process.

To develop an algorithm suitable for design optimisation, a few key assumptions were

made. The most important of these assumptions was that the computation time for

evaluation of the fitness function was not insignificant. Complex simulations were used

as fitness functions for design studies. One example is a lap time simulation which has an


56

approximate running time of 30 seconds per evaluation. This affects the design of the

algorithm, because the number of fitness evaluations must be limited to complete the

optimisation process in reasonable time. For example, setting the total computation time

at 24 hours would allow approximately 3000 evaluations. The limited number of fitness

evaluations indicates that it is more likely that the end criteria of the algorithm will be the

exhaustion of available computation time rather than finding the optimal point within the

solution space.

Another assumption for the algorithm was that the boundaries of the solution space are

well known and may not be exceeded. This would allow simple solution repair methods

to be used. This assumption is based on the theory that the evolutionary techniques would

be used by professional engineers already capable of writing and/or using complex

engineering simulations.

An assumption was also made that a solution within a small Euclidean distance from the

optimum would be treated as an optimum point. This was based on the idea that the given

solution would have to be implemented in the final engineered product; therefore some

degree of tolerance would be appropriate.

Hence the purpose of the algorithm was to find acceptable solutions to complex problems

(approximately 30 variables) within 3000 function evaluations. Ideally the algorithm

should be suitable for a large variety of problems, as well as performing credibly given

different numbers of variables and/or function evaluations. This would avoid having to

tailor the evolutionary approach to different parts of the design process.

4.2 Test Problem Set

It is common for algorithms to be tuned using test functions that are computationally

quick, but allow different search space properties to be analysed. Test problems were

selected as a subset of the 23 presented by Yao et al. (1999). These are single and multi-

modal problems with a known solution location. Only the 13 functions that exhibited an

easily scalable number of variables were chosen.


57

Uni-modal Functions:

Multi-modal Functions:

∑=

=n

iixxf

1

21 )(

∏∑==

+=n

ii

n

ii xxxf

112 )(

∑ ∑= =

=

n

i

i

jjxxf

1

2

13 )(

{ }nixxf i ≤≤= 1,max)(4

( )[ ]∑−

=+ −+−=

1

1

22215 )1(100)(

n

iiii xxxxf

∑=

+=n

iixxf

1

26 )5.0()(

[ )∑=

+=n

ii randomixxf

1

47 1,0)(

( )( )∑=

−=n

iii xxxf

18 sin)(

( )[ ]∑=

+−=n

iii xxxf

1

29 102cos10)( π

exn

xn

xfn

ii

n

ii ++

−

−−= ∑∑

==

202cos1

exp1

2.0exp20)(11

210 π

1cos4000

1)(

11

211 +

−= ∏∑==

n

i

in

ii

i

xxxf

( ) ( ) ( )[ ] ( ) ( )∑∑=

−

=+ +

−++−+=

n

ii

n

inii xuyyyy

nxf

1

1

1

21

221

212 4,100,10,1sin1011sin10)( πππ

( )14

11 ++= ii xy


58

This test problem set, containing both uni-modal and multi-modal problems, is used in

order to represent problems of varying levels of complexity. Each of the test functions

can be scaled to any number of variables. Each test function was treated as equally

important when determining the performance of the algorithm.

4.3 Method

For each of the test problems/functions, final solution vectors produced by the algorithm

were compared with their Euclidean distance from the optimal solution vector. In order

to allow even comparison between differing types of problems, and reduced complexity

in algorithm coding, solution vectors were normalised. Each of the solution vectors was

scaled to provide equivalent results to the original equation. An example of the process

for function 1 is given below:

where,

Equation 4-1

A simple repair function was used according to the following function:

( ).

,

,

,)(

,0

,)(

,,,

ax

axa

ax

axk

axk

mkaxu

i

i

i

mi

mi

i

−<≤≤−

>

−−

−=

( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )∑∑=

−

=+ +

+−++−+=

n

ii

n

innii xuxxxxxxf

1

1

1

21

221

213 4,100,5,2sin113sin113sin1.0 πππ

∑=

=n

iixxf

1

21 )(

( )5.0ˆ200ˆ +×= sx

[ ]nx 100,100ˆ −∈

[ ]ns 1,0ˆ∈


59

Equation 4-2

The algorithm was based around a standard Evolutionary Strategy Algorithm (Bäck, 95).

The following section shows how appropriate algorithm parameters were found.

4.3.1 Algorithm Parameter Tuning

A typical non-isotropic self-adaptive evolutionary algorithm has the following parameters

that need to be determined:

λλλλ The number of parents

µ The number of children (population size)

ττττ Mutation operator allows for individual changes of “the mean step sizes”

ττττ’ Mutation operator allows for an overall change in mutability

e A minimum mutation size

With tournament selection an additional selection pressure parameter is needed:

q The selective pressure

If a Cauchy distribution is used in place of the Gaussian then an additional parameter that

defines the shape of the distribution curve is also needed:

t Cauchy scale parameter

=>=<=

Otherwisess

ss

ss

,

1,1

0,0


60

The parameters were tuned sequentially based on how important each parameter was

perceived to be. This also allows some investigation into the behaviour of each parameter.

The following tests were conducted:

• Varying the size of the population

• Varying the number of parents

• Investigation of selective pressure for tournament selection

• Investigation of mutation rates

• Investigation into minimum mutation rate

• Investigation using Cauchy distributions

• Investigation into controlling mutation parameters during the optimisation process

Each of the 13 test functions, given in Section 4.2, was used to determine the behaviour

of the algorithm for a given value of each parameter. Performance was based on a ranking

system developed for this thesis as follows:

• Lower, upper, and step values were determined for each of the parameters.

• Starting at the lowest value, the algorithm was run 20 times for each of the test

functions. The performance measure was taken as an average of the best values

found during each of the 20 runs.

• The step value was added to the parameter value and another run was performed.

This was repeated until the highest value for the parameter had been tested.

• The output of this process was a vector of performance values associated with a

vector of the parameter value that was tested. The values of the parameter were

then sorted and scaled linearly according to how well they performed. This would

scale from 0 for the worst to 100 for the best.

• The ranks for each of the functions were averaged to give one curve of

performance vs. the parameter.


61

A significant benefit of this process is that the distance of the peak performance value

from 100 will give an idea of how well the algorithms agree. A peak performance value

of 100 will mean that all the same parameter value gave the best results with all of the

functions. Anything lower shows a discrepancy. One disadvantage of this process is that

some information about the closeness of a solution to the ideal is lost. However, ranking

prevents order of magnitude differences in performance distorting the information gained.

For an initial test of the algorithm, the selection process was set as deterministic. This

results in the fittest µ individuals being selected at each step. The minimum mutation rate

was set to an arbitrarily low value of 10-4.

4.3.2 A Note on Starting Mutation Rates

Bäck and Schwefel set initial mutation sizes to 3.0. Given the decision to normalise

variables this would be too high, this is discussed using the equation given in section 3.5:

Equation 4-3

Where is the Euclidean distance between the current and the optimum solutions.

The optimum standard deviation may be determined for the starting position. In the one-

dimensional case we can assume that the optimum is randomly placed, as in the current

solution:

Figure 4-1 - Euclidean Distance

n

xx −=

*224.1σ

xx −*


62

The average Euclidean distance between x and an unknown optimum can be shown to be

1/3. This can be extended into n dimensions by showing the starting Euclidean distance to

the optimum has a mean of:

Equation 4-4

The resultant standard deviation, according to Equation 4-3, will then be:

Equation 4-5

However at this stage it should be noted that this is the mean case for an unknown

optimum location. For the case of the test functions, the minimums are known to exist in

the centre of the solution space. In this case the average starting Euclidean distance is:

Equation 4-6

With a lower resultant standard deviation, according to Equation 4-3, of:

Equation 4-7

For tuning the algorithm, it was assumed that the position of the minimum was unknown

and that the standard deviations are initially set according to Equation 4-5. For the case of

30 variables this would result in a standard deviation of approximately 0.13, much

smaller than the 3.0 proposed by Bäck and Schwefel. While self-adaptive mechanisms

would undoubtedly reduce the initial mutation rate quite quickly it was assumed that

3

n

n3

224.1=σ

2

n

n2

224.1=σ


63

increased initial performance of the algorithm could be achieved by selecting a more

appropriate value as discussed.

4.4 Population Size

Population size and the ratio of parents to population size were deemed the two most

important parameters in this situation. Given a limited number of evaluations, the

population size will determine the number of times there is a new generation as described

by the following equation.

Equation 4-8

The first test involved investigating the performance of a given population size with the

number of evaluations fixed at 3000. Population size was varied from 10 to 200 in

intervals of 5. The algorithm was run with each population size 20 times, and the results

were averaged for each test function. The results for population size for the 13 test

functions were ranked on performance and averaged. The relative performance of the

population size was determined by ranking the results of each population size and then

averaging for the 13 test functions. Figure 4-2 shows that performance of the algorithm

generally decreased as the population size was increased. However the worst performance

was seen when the population was very low.

µsEvaluation

sGeneration

nn =


Figure 4-2 : Performance vs. Population Size

A population of between 30 and 70 appears to offer the best average performance.

However it was also worth investigating how the ideal population size varies with the

number of allowed evaluations. For th

retained. Population was varied from 20 to 100 in intervals of 10. Each population size

was run 20 times and the results averaged for each test function. The number of

evaluations was varied from 500 to 100

ranking system are shown in Figure


64

33

: Performance vs. Population Size

population of between 30 and 70 appears to offer the best average performance.

s also worth investigating how the ideal population size varies with the

r of allowed evaluations. For this test all of the parameters used above

. Population was varied from 20 to 100 in intervals of 10. Each population size


s was varied from 500 to 100,000. The results processed through a

Figure 4-3.

Kevin Hayward

population of between 30 and 70 appears to offer the best average performance.

s also worth investigating how the ideal population size varies with the

used above were

. Population was varied from 20 to 100 in intervals of 10. Each population size


processed through a similar


65

Figure 4-3 - Performance vs. Population Size

Apart from the outlying result of 5000 evaluations, there did not appear to be any

dependence on the ideal population size when the number of evaluations was changed.

The optimum population size appeared to be around 30. However in the interests of

providing a robust algorithm a population size of above 30 would appear to be a safer

approach. It is unclear why the results for 5000 evaluations were significantly different.

Figure 4-4 shows a graph of the best population size for a given number of evaluations.

The outlier produced for 5000 evaluations is clearly visible.


Figure 4-4 - Best Population Size vs. Evaluations

It would appear that there is some

number of evaluations. However the relationship does not appear to be monotonic and

applicable to algorithm scaling. Another possibility is that for low numbers of evaluations

the evolutionary process is not efficient enough to give useful

optimum population size. Looking at the performance for each of the functions against

the number of evaluations supports this theory.


66

Best Population Size vs. Evaluations

would appear that there is some relationship between the best population size



the evolutionary process is not efficient enough to give useful enough results to


the number of evaluations supports this theory.

Kevin Hayward

the best population size and the



ts to determine



67

Figure 4-5 - Performance vs. Evaluations

As can be seen in Figure 4-5, there is a marked improvement in the results of the

algorithm for most of the functions once the number of evaluations is increased beyond

10,000. Of note is the fact that the more complex multi-modal problems, functions 7-13,

are still some distance from the global minimum for the given evaluation time. (Function

8 is excluded from the log-based graph as it returns negative values). However a similar

trend is seen for function 8 as shown in Figure 4-6.


Figure 4-6 - Performance vs. Evaluations (Function 8)

Given that the purpose of developing this algorithm

limited numbers of evaluations, the population siz

Another test was conducted in order to determine the dependence of population size on

the number of variables in the test function

evaluations was set to 3000. Population size was varied from 2

10. The number of variables was varied from 5 to 30 in increments of 5. The results are

shown in Figure 4-7.


68

Performance vs. Evaluations (Function 8)

Given that the purpose of developing this algorithm was to get best performance from

the population size was set at 50.


in the test function, to a maximum of 30 variables. The number of

evaluations was set to 3000. Population size was varied from 20 to 100 in increments of

umber of variables was varied from 5 to 30 in increments of 5. The results are

Kevin Hayward

was to get best performance from a


, to a maximum of 30 variables. The number of

0 to 100 in increments of

umber of variables was varied from 5 to 30 in increments of 5. The results are


Figure

Figure


69

Figure 4-7 - Performance vs. Number of Varia bles

Figure 4-8 - Ideal Population vs. Number of Variables


bles

Ideal Population vs. Number of Variables


70

There does not appear to be any noticeable correlation between the number of variables

and the population size. The above figures suggest that the population size should be set

to between 40 and 60 for robustness, further confirming previous findings.

In conclusion, the tests indicated that the size of the ideal population does not vary

considerably with changing numbers of variables or available evaluations. A population

size of 50 was chosen for the algorithm.

4.5 Number of Parents

The ratio of parents to children will determine how many times the mutation operators are

used on unique individuals. Back and Schwefel (1993) suggest a ratio of:

Equation 4-9

A test was conducted to investigate the relationship between the number of parents and

children, and whether the ratio given in Equation 4-9 was suitable. The parameters for the

test were set as above with the population size set to 50. The number of parents was

varied from 1 to 15 in steps of 1. The results are shown in Figure 4-9.

7

1=λµ


Figure

This data shows the ideal number of parents being 6, with no real lack of performance in

the neighbouring region. The previously mentioned ratio of 1/7 offered by Schwefel is

close to this range as it would result in 7 pare

the performance difference between 6 and 7 parents appears minimal it was decided to

maintain Schwefel’s suggested ratio and continue tests with 7 parents.

4.6 Introducing and Studying Selective Pressure

The selection process

population renewal. Investigating selective pressure gives insight into how restrictive

selection should be. A high value for this pressure leads to deterministic selection where

only the best candidates are used to create the next generation. Low selection pressure

allows for less “fit” individuals to remain in the population for the renewal process.


71

Figure 4-9 - Performance vs. Number of Parents



close to this range as it would result in 7 parents for a population size of 50. Given that


tain Schwefel’s suggested ratio and continue tests with 7 parents.

Introducing and Studying Selective Pressure

on process determines which population members are



the best candidates are used to create the next generation. Low selection pressure



erformance vs. Number of Parents



nts for a population size of 50. Given that


tain Schwefel’s suggested ratio and continue tests with 7 parents.

Introducing and Studying Selective Pressure

s which population members are retained to initiate



the best candidates are used to create the next generation. Low selection pressure



In the previous sections a deterministic selection process was used. A

to determine whether a tournament selection process would improve the evolutionary

process, and the ideal selective pressure

size at 7. All other parameters were kept the same as previous tests.

graph appeared quite noisy. Filtering

shown in Figure 4-10.

Figure 4-10 - Performance vs. Selective Pressure

There appears to be reduced performance for lower selection pressures. The performance

remains roughly constant once selection pressure reaches between 60 and 70. This

levelling off is most likely due to the fact that once the selection pressure reaches a

certain value the algorithm will act

In order to analyse the effects of selection pressure for higher numbers of iterations

number of evaluations was increased by a factor of 10 to 30

Figure 4-11.


72

In the previous sections a deterministic selection process was used. A test was cond


ideal selective pressure. The population size was set to 50 with parent

size at 7. All other parameters were kept the same as previous tests. The per

graph appeared quite noisy. Filtering with a 7 point moving average yielded the graph

Performance vs. Selective Pressure

performance for lower selection pressures. The performance

once selection pressure reaches between 60 and 70. This

ling off is most likely due to the fact that once the selection pressure reaches a

the algorithm will act as though it were deterministic.

In order to analyse the effects of selection pressure for higher numbers of iterations

number of evaluations was increased by a factor of 10 to 30,000. This is shown in

Kevin Hayward

test was conducted


. The population size was set to 50 with parent

The performance

with a 7 point moving average yielded the graph

performance for lower selection pressures. The performance

once selection pressure reaches between 60 and 70. This

ling off is most likely due to the fact that once the selection pressure reaches a

In order to analyse the effects of selection pressure for higher numbers of iterations, the

This is shown in


Figure 4-11

The results gained by increasing the number of evaluations indicated that there was less

of a performance loss for runn

appear to be any advantage in implementing anything but a high selective pressure

number of evaluations given

implement a tournament se

increased the complexity of the algorithm. For the following tests a deterministic

selection process was maintained.

4.7 Investigating Mutation Strength and Distribution

Mutation is the only operator

algorithm. The two mutation

on a variable level, and

made the following suggestions for


73

- Performance vs. Selection Pressure (30,000 Evaluati ons)


of a performance loss for running lower selection pressures. However there still did not

appear to be any advantage in implementing anything but a high selective pressure

number of evaluations given. Following these results the decision was made not to

implement a tournament selection process. While appearing to offer no advantages


selection process was maintained.

Investigating Mutation Strength and Distribution

utation is the only operator that alters individuals in the population

mutation parameters tested in this investigation

on a variable level, and τ’, which operates on the individual. Bäck and Schwefel

uggestions for these parameters:


Performance vs. Selection Pressure (30,000 Evaluati ons)


ing lower selection pressures. However there still did not

appear to be any advantage in implementing anything but a high selective pressure for the

. Following these results the decision was made not to

lection process. While appearing to offer no advantages, it


Investigating Mutation Strength and Distribution

that alters individuals in the population for a standard

parameters tested in this investigation are τ, which operates

ck and Schwefel (1993)


74

Equation 4-10

Equation 4-11

This is based on the following equations:

Equation 4-12

Equation 4-13

Whether the suggestion of 1 as the scaling factor is ideal in these circumstances was

unknown. It was considered likely that higher than normal mutation rates would be

needed to increase convergence speed for the limited number of evaluations. This

investigation was carried out in two parts. The first was allocating a number for mutation

strength, ‘a’. The second was investigating whether mutation should be more focused on

variable or individual mutation. To study this, the idea of introducing a mutation

distribution value, ‘b’, was conceived. When equal to 0, only individual variables would

be mutated, when equal to 1 the individual would alter variables globally. These

parameters changed Equation 4-10 and Equation 4-11 to the following:

n2

1=τ

n2

1'=τ

n2

1∝τ

n2

1'∝τ


The results of varying the mutation strength, ‘a’,

Figure

A mutation strength of approximately 4.3 showed the best p

13 test functions.

The two mutation operators also define how much influence mutation

variables or all the variables.

The effect of altering the balance of the two parameters was investigated


75

Equation 4-14

Equation 4-15

of varying the mutation strength, ‘a’, are shown in Figure

Figure 4-12 - Performance vs. Mutation Strength

of approximately 4.3 showed the best performance averaged over the

The two mutation operators also define how much influence mutation

variables or all the variables. τ is an individual variable operator,

The effect of altering the balance of the two parameters was investigated

n

ba

2

)1(2 −=τ

n

ab

2

2'=τ


Figure 4-12:

Performance vs. Mutation Strength

erformance averaged over the

The two mutation operators also define how much influence mutation has on individual

, while τ' acts globally.

The effect of altering the balance of the two parameters was investigated. In order to do


this, the mutation multiplier was set to 4.3 according to the results found previously

results of varying the mutation distribution, ‘b’, are

Figure 4-13 - Performance vs. Mutation Distribution

These results indicate that the average performance is increase

applied slightly more globally. Given these results a mutation distribution of 0.55 was

chosen. This yields multipliers for

For scaling purposes it is also worth looking at the mutation operators with respect to the

number of variables as well as the number of allowed evaluations. Both

already functions of the number of variab

variables was set at 10, 20 and 30 and the results were compared. The results for the

mutation strength are shown in Figure


76

the mutation multiplier was set to 4.3 according to the results found previously

arying the mutation distribution, ‘b’, are shown in Figure 4-13.

Performance vs. Mutation Distribution

These results indicate that the average performance is increased when the mutation is


chosen. This yields multipliers for τ and τ' of 3.9 and 4.7 respectively.


number of variables as well as the number of allowed evaluations. Both τ

already functions of the number of variables. In order to verify this, the number of


Figure 4-14.

Kevin Hayward

the mutation multiplier was set to 4.3 according to the results found previously. The

d when the mutation is



and τ' are

the number of



77

Figure 4-14 - Performance vs. Mutation Strength

It appears from this example that mutation strength does not need to be altered as a

function of the number of variables. The results for the mutation distribution are shown in

Figure 4-15.


78


Again there is good correlation between the graphs for different numbers of variables.

These two tests support the validity of the traditional functions that link the mutation

variables τ and τ' to the number of variables in a problem.

Another test was conducted to determine whether there was any relationship between the

two mutation parameters and the number of evaluations allowed. The number of

evaluations was varied between 300 and 30,000. Figure 4-16 shows the effect of

evaluations on the optimum mutation multiplier.


Figure

This graph indicates that the magnit

of the variables but also

numbers of evaluations it is feasible that the mutation multiplier may approach the value

of one that is given in

distribution, and the results are shown


79

Figure 4-16 - Optimum Mutation Multiplier vs. Evaluations

This graph indicates that the magnitude of the mutation parameters is not only a function

of the variables but also of the number of evaluations available. Given vastly increased


of one that is given in Bäck (1995). The test was also repeated for th

he results are shown in Figure 4-17.


Optimum Mutation Multiplier vs. Evaluations

ude of the mutation parameters is not only a function

the number of evaluations available. Given vastly increased


. The test was also repeated for the optimum mutation


Figure 4-17 - Optimum Mutation Distribution vs. Evaluatio

The optimum mutation distribution does not appear to be a function of the number of

evaluations. There is a slight difference for low numbers of evaluations tending towards

increased global change of variables. However if we look at the performance for

the runs (Figure 4-18) we can see that even at low numbers of evaluations

performance is acceptable at a mutation distribution of 0.55.


80

Optimum Mutation Distribution vs. Evaluatio ns



increased global change of variables. However if we look at the performance for

) we can see that even at low numbers of evaluations,

performance is acceptable at a mutation distribution of 0.55.

Kevin Hayward



increased global change of variables. However if we look at the performance for each of

, algorithm


81


4.8 Investigating Minimum Mutation Rate

The final parameter to be investigated of the minimum parameter set is e, which

represents the minimum allowable mutation rate. The minimum mutation rate is usually

set to an arbitrarily small number to prevent any mutation value from reaching zero, while

at the same time playing little to no part in the actual performance of the algorithm. Given

a small number of generations, the author believed that this parameter could be used to

prevent premature convergence to a local minimum. Keeping a higher minimum mutation

rate would force the algorithm to mutate the population more. The range of values that

would be ideal for different numbers of evaluations was investigated. For the following

tests the other parameters were set according to the results given in the previous tests. The

minimum mutation was varied by a base of 10. The results are shown in Figure 4-19.


Figure 4-19 - Performance vs. Minimum Mutation

From this graph it appears that the performance of the algorithm is dependent on the value

set for the minimum mutation, with an optimum

interesting finding because the minimum mu

mutation from reaching zero and making any further evaluations useless. However there

is little mention in the literature

Understandably if e is set too high it h

hence e is usually set very low

performance may also suffer if e is set too low. This warranted further investigation. As

for the mutation parameters discu

the number of evaluations was studied.

The number of variables was varied between 5 and 30. The results are

Figure 4-20.


82

Performance vs. Minimum Mutation


set for the minimum mutation, with an optimum e of approximately 10-2.5. This is an

the minimum mutation is seen as a method of preventing the


in the literature of the effects of this parameter on performance.

Understandably if e is set too high it hinders the process by not allowing any fine tuning,

low. However, these results indicated that algorithm


discussed above, the effects of the number of variables and

the number of evaluations was studied.

The number of variables was varied between 5 and 30. The results are

Kevin Hayward


. This is an

tation is seen as a method of preventing the


of the effects of this parameter on performance.

inders the process by not allowing any fine tuning,

results indicated that algorithm


the effects of the number of variables and

The number of variables was varied between 5 and 30. The results are shown in


83

Figure 4-20 - Performance vs. Minimum Mutation

As can be seen there does not appear to be any significant change in the optimal minimal

mutation due to the number of variables. This is probably due to the fact that the

parameter e acts on mutation parameters for individual variables.

A further test was conducted to investigate the effect of the number of evaluations on the

minimum mutation. The number of evaluations was varied between 300 and 60,000. The

graph of the optimum value for e is shown in Figure 4-21.


Figure 4-21 - Optimum Minimum Mutation vs. Evaluations

A clear relationship between e and the number of evaluations is apparent. The size of

decreases as the number of evaluations increases. As the number of evaluations increases

the more effective the algorithm is at finding solutions closer to the optimum.

results approach the optimum solution,

to fine-tune the result. It appears that this relationship holds for the minimum mutation as

well, indicating that its value does have a bearing on the mutation properties of the

algorithm.

Thus it appears that the optimum values

function controlled by the number of evaluations available.

4.9 Investigating Cauchy Distribution

Yao et al. (1997, 1999) has proposed the use of replacing a Gaussian with a Cauchy

distribution for the mutation operators. This is intended to increas

of the algorithm because Cauchy

This is shown in Figure 4-22.


84

Optimum Minimum Mutation vs. Evaluations

A clear relationship between e and the number of evaluations is apparent. The size of

of evaluations increases. As the number of evaluations increases

the more effective the algorithm is at finding solutions closer to the optimum.

results approach the optimum solution, the lower the required mutation has to be

he result. It appears that this relationship holds for the minimum mutation as


Thus it appears that the optimum values of all three of the mutation parameters

the number of evaluations available.

Investigating Cauchy Distribution

has proposed the use of replacing a Gaussian with a Cauchy

istribution for the mutation operators. This is intended to increase the convergence speed

of the algorithm because Cauchy distributions will allow higher values for mutations.

Kevin Hayward

A clear relationship between e and the number of evaluations is apparent. The size of e

of evaluations increases. As the number of evaluations increases

the more effective the algorithm is at finding solutions closer to the optimum. As the

mutation has to be in order

he result. It appears that this relationship holds for the minimum mutation as


of all three of the mutation parameters may be a

has proposed the use of replacing a Gaussian with a Cauchy

e the convergence speed

istributions will allow higher values for mutations.


85

Figure 4-22 – Cauchy and Gaussian Probability Densi ty Functions

There is some evidence that this simple change can yield quite valuable results (Yao et

al., 1997 & 1999). The Cauchy distribution introduces an additional parameter t. t

behaves as a scaling factor as shown in Figure 4-23.


86

Figure 4-23- Cauchy Distribution Scaling

A test was conducted to determine an appropriate Cauchy distribution scaling value; other

parameters were defined as found in the preceding sections. The results are shown in

Figure 4-24.


There appears to be a slight advantage with t set to around 1.5.

two distributions they were compared using a student t

was a true statistical difference. A larger value for T indicates a larger statistical

difference in the mean


87

Figure 4-24 - Performance vs. t

There appears to be a slight advantage with t set to around 1.5. In order to compare the

ons they were compared using a student t-test to determine whether there


difference in the mean.


In order to compare the

to determine whether there



88

Table 4-1 shows the results of the Cauchy distribution against the Gaussian for the

individual performance of the functions at this value for t.

Table 4-1 - Gaussian vs. Cauchy Distribution

Function Gaussian Cauchy

T µ σ µ σ

1 1.93E+04 5.63E+03 1.73E+04 7.23E+03 -1.52

2 5.92E+01 1.19E+01 5.17E+01 1.24E+01 -3.09

3 4.83E+04 1.43E+04 4.68E+04 1.10E+04 -0.62

4 7.40E+01 8.77E+00 7.83E+01 1.11E+01 2.13

5 3.28E+07 1.55E+07 2.94E+07 2.32E+07 -0.84

6 1.82E+04 6.30E+03 1.57E+04 4.19E+03 -2.27

7 1.37E+01 6.89E+00 1.12E+01 5.59E+00 -1.99

8 -7.55E+03 5.21E+02 -7.55E+03 5.39E+02 0.07

9 1.84E+02 2.61E+01 1.86E+02 2.65E+01 0.30

10 1.80E+01 9.63E-01 1.80E+01 1.22E+00 0.10

11 1.59E+02 4.61E+01 1.20E+02 2.20E+01 -5.45

12 4.42E+07 4.38E+07 3.85E+07 3.13E+07 -0.75

13 1.07E+08 9.93E+07 9.92E+07 7.00E+07 -0.43

This is summarized in a bar chart in

Figure 4-25.


These results indicate t

distribution. However for most of the functions

quite clear, the improvement

Following this, a test was conducted to determi

parameter t altered with the number of evaluations.


89

Figure 4-25 - Gaussian vs. Cauchy Distribution

These results indicate that the Cauchy distribution generally outperforms the Gaussian

distribution. However for most of the functions, although the statistical difference is

the improvement in mean is quite moderate.

a test was conducted to determine whether the

parameter t altered with the number of evaluations.


Gaussian vs. Cauchy Distribution

hat the Cauchy distribution generally outperforms the Gaussian

the statistical difference is

ne whether the optimum value of


Figure 4 -

It can be seen that there is a general drop in the

number of evaluations increases. The shape of the graph is similar to the shape of the

ideal mutation rates shown above

value of t is probably dropping as a way to control the effective

A study was also conducted to show how the ideal value for t alter

variables. The graph below shows the results.


90

-26 - Optimum t vs. Evaluations

It can be seen that there is a general drop in the optimum value of parameter t as

. The shape of the graph is similar to the shape of the

ideal mutation rates shown above (Figure 4-16, Figure 4-21). In this respect

ropping as a way to control the effective rate of mutation.

to show how the ideal value for t altered with the number of

variables. The graph below shows the results. (Figure 4-27)

Kevin Hayward

parameter t as the

. The shape of the graph is similar to the shape of the

. In this respect, the ideal

of mutation.

with the number of


91

Figure 4-27 - Performance vs. t

As with many of the previous tests, the ideal value of t did not appear to be a function of

the number of variables for the range tested.

The values of the parameters for both the Cauchy and Gaussian distributions were held

constant and tested for 300 and 30,000 evaluations to give an indication of robustness.

The results for 300 evaluations are shown in Table 4-2.


Table 4-2 - Gaussian vs. Cauch

Function Gaussian

µ

1 5.46E+04

2 2.46E+08

3 8.78E+04

4 8.37E+01

5 1.99E+08

6 5.48E+04

7 7.37E+01

8 -3.93E+03

9 3.79E+02

10 2.03E+01

11 4.83E+02

12 3.78E+08

13 7.05E+08

Figure 4-28 - Gaussian vs. Cauchy for 300 Evaluations

In this case it is the Gaussian distribution which appears to b

distribution. The results from 30,000 evaluations are shown


92

Gaussian vs. Cauch y Distribution (300 Evaluations)

Gaussian Cauchy T

σ µ σ

9.61E+03 5.26E+04 8.69E+03 -1.05

7.31E+08 3.73E+08 7.21E+08 0.87

2.44E+04 7.11E+04 2.32E+04 -3.50

7.52E+00 8.54E+01 8.90E+00 0.98

4.15E+07 1.75E+08 4.16E+07 -2.83

6.91E+03 5.83E+04 8.08E+03 2.38

2.09E+01 7.58E+01 2.06E+01 0.51

4.71E+02 -3.86E+03 4.56E+02 0.72

2.38E+01 3.81E+02 2.45E+01 0.35

2.68E-01 2.03E+01 2.58E-01 0.49

6.36E+01 5.00E+02 5.14E+01 1.49

1.10E+08 3.85E+08 1.13E+08 0.33

2.67E+08 7.91E+08 3.34E+08 1.43

Gaussian vs. Cauchy for 300 Evaluations

In this case it is the Gaussian distribution which appears to be outperforming the Cauchy

000 evaluations are shown in Table 4-3.

Kevin Hayward

e outperforming the Cauchy


Table 4-3

Function

1

2

3

4

5

6

7

8

9

10

11

12

13

Figure 4- 29

Again it appears that the Gaussian distribution

may indicate that the Gaussian distribution gives more robust results. Another plausible


93

3 - Gaussian vs. Cauchy Distribution (30,000 Evaluations)

Gaussi an Cauchy

µ σ µ σ

1.07E+00 1.86E+00 1.81E+01 3.73E+01

2.78E-01 2.04E-01 7.17E-01 3.41E-01

8.67E+03 3.07E+03 8.41E+03 3.68E+03

5.91E+01 3.83E+01 5.19E+01 3.67E+01

8.98E+02 9.86E+02 5.79E+04 2.50E+05

2.37E+00 3.87E+00 4.33E+01 1.16E+02

5.11E-01 1.85E-01 6.91E-01 5.75E-01

-1.07E+04 4.13E+02 -1.07E+04 2.70E+02

3.40E+01 7.21E+00 2.96E+01 5.80E+00

2.39E+00 4.32E+00 4.11E+00 6.92E+00

5.64E-01 4.07E-01 8.45E-01 3.64E-01

6.43E-01 1.05E+00 1.03E+00 3.12E+00

6.57E+02 2.93E+03 1.38E+00 1.70E+00

29 - Gaussian vs. Cauchy Distribution for 30,000

Again it appears that the Gaussian distribution outperforms the Cauchy distribution. This



Evaluations)

T

3.73E+01 3.21

01 7.81

3.68E+03 -0.37

3.67E+01 -0.97

2.50E+05 1.62

1.16E+02 2.50

01 2.12

2.70E+02 0.25

5.80E+00 -3.39

6.92E+00 1.50

01 3.63

3.12E+00 0.84

1.70E+00 -1.58

30,000 Evaluations

outperforms the Cauchy distribution. This



94

explanation for these results is that the other parameters for the algorithm have been tuned

to suit the Gaussian distribution. The T values for each test are summarised in the chart

below (Figure 4-30).

Figure 4-30 - Gaussian vs. Cauchy Distribution Summ ary

It is clear that merely substituting the Cauchy distribution in place of the Gaussian, in this

situation, is not an immediate way to increase performance. Previous tests in this thesis

would indicate that while the Cauchy distribution may be used to alter mutation

parameters favourably, similar results can be achieved through simple algorithm tuning.

4.10 Controlling Mutation Parameters

As stated in Section 4.1, the algorithm was designed to run for a set number of

evaluations. The tests conducted above indicated that the performance of the algorithms

increased if the size of the mutation was decreased for an increased number of

evaluations. For a fixed population size this is the same as stating that performance

increased as the mutation was decreased for an increased number of generations. Given

that the end point was known, the author proposed that altering mutation parameters

could be achieved, similar to simulated annealing (Kirkpatrick et. al. 1983). The two

mutation parameters to be tuned were the mutation multiplier, and the minimum mutation

T for 13 Functions(-ve T Shows Cauchy is Better)

-8

-6

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10 11 12 13

Function

300 Evaluations 3,000 Evaluations 30,000 Evaluations


rate. These control the magnitude of the mutation and were found to be significantly

dependent on the number of generations. Initial tests were made into linearly altering

these values over time.

investigated.

The particular situation imposed by complex engineering problems

termination criteria of the algorithm would be the exhaustion of available computation

time. Because the number

mutation parameters as functions of the number of generations with a known start and

ending value. A simple linear function was implemented to test the validity of this theory.

The starting and ending values for both th

mutation were set using the empirical data found in the above tests.

are shown below (

Figure 4-31 & Figure


95


the number of generations. Initial tests were made into linearly altering

these values over time. Setting these values as self-adaptive

The particular situation imposed by complex engineering problems


number of evaluations can be fixed, it is possible to



The starting and ending values for both the mutation multiplier and the minimum

mutation were set using the empirical data found in the above tests.

Figure 4-32).

Figure 4-31 - Minimum Mutation vs. Generation



the number of generations. Initial tests were made into linearly altering

adaptive parameters was also

The particular situation imposed by complex engineering problems is that the likely


it is possible to control these



e mutation multiplier and the minimum

mutation were set using the empirical data found in the above tests. These linear graphs

Minimum Mutation vs. Generation


Figure 4-32 -

The results compared to the tuned algorithm using the Gaussian distribution

Table 4-4.

Table 4-4 - Original vs. Linear Mutation Rate Alterations

Function Gaussian

µ

1 1.93E+04

2 5.92E+01

3 4.83E+04

4 7.40E+01

5 3.28E+07

6 1.82E+04

7 1.37E+01

8 -7.55E+03

9 1.84E+02

10 1.80E+01

11 1.59E+02

12 4.42E+07

13 1.07E+08


96

- Mutation Strength vs. Generation

The results compared to the tuned algorithm using the Gaussian distribution are shown

Original vs. Linear Mutation Rate Alterations

Gaussian Linear T

σ µ σ

5.63E+03 1.46E+04 4.41E+03 -4.59

1.19E+01 4.58E+01 1.02E+01 -6.07

1.43E+04 3.85E+04 1.09E+04 -3.88

8.77E+00 6.57E+01 7.24E+00 -5.18

1.55E+07 1.75E+04 5.09E+03 -14.91

6.30E+03 1.54E+04 5.35E+03 -2.34

6.89E+00 9.92E+00 5.82E+00 -2.93

5.21E+02 -7.62E+03 5.11E+02 -0.60

2.61E+01 1.70E+02 2.18E+01 -2.90

9.63E-01 1.80E+01 1.06E+00 0.16

4.61E+01 1.38E+02 4.75E+01 -2.28

4.38E+07 2.00E+04 5.53E+03 -7.14

9.93E+07 1.80E+04 4.56E+03 -7.59

Kevin Hayward

are shown in


Figure

The proposed algorithm has notable improvements over the more con

algorithm. In order to try and eliminate the need for initial and final values for the

mutation multiplier and the minimum mutation rate they were tested as self

parameters. The minimum and maximum values for both the minimum mutat

mutation strength were altered to allow a greater range of values. Minimum mutation rate

was limited to between

and 5. The results are


97

Figure 4-33 - Linear Mutation Control vs. No Control

The proposed algorithm has notable improvements over the more con


mutation multiplier and the minimum mutation rate they were tested as self

The minimum and maximum values for both the minimum mutat


was limited to between 10-4.5 and 10-1. The mutation strength was limited to between 3

results are shown in Table 4-5.


Linear Mutation Control vs. No Control

The proposed algorithm has notable improvements over the more conventional tuned


mutation multiplier and the minimum mutation rate they were tested as self-adaptive

The minimum and maximum values for both the minimum mutation and the


. The mutation strength was limited to between 3


Table 4-5 – Original vs. Self Adaptive Mutation Rate Modificati on

Function Gaussian

µ

1 1.93E+04

2 5.92E+01

3 4.83E+04

4 7.40E+01

5 3.28E+07

6 1.82E+04

7 1.37E+01

8 -7.55E+03

9 1.84E+02

10 1.80E+01

11 1.59E+02

12 4.42E+07

13 1.07E+08

Figure 4-34 - Self

The improvement between the original algorithm and the self

is also quite significant. For some of the functions the improvement was many orders of


98

Original vs. Self Adaptive Mutation Rate Modificati on

Gaussian Self -Adaptive T

σ µ σ

5.63E+03 5.89E+03 3.24E+03 -14.57

1.19E+01 3.70E+01 1.03E+01 -10.01

1.43E+04 2.47E+04 6.11E+03 -10.73

8.77E+00 5.83E+01 6.96E+00 -9.92

1.55E+07 1.11E+04 4.61E+03 -14.91

6.30E+03 5.55E+03 2.30E+03 -13.29

6.89E+00 3.14E+00 2.27E+00 -10.25

5.21E+02 -7.93E+03 6.42E+02 -3.21

2.61E+01 1.71E+02 3.26E+01 -2.27

9.63E-01 1.45E+01 2.11E+00 -10.71

4.61E+01 5.17E+01 2.42E+01 -14.59

4.38E+07 1.38E+04 3.34E+03 -7.14

9.93E+07 1.15E+04 4.66E+03 -7.59

Self -Adaptive Control vs. No Control

The improvement between the original algorithm and the self-adaptive mutation sc


Kevin Hayward

adaptive mutation scheme



magnitude. The table below

that of the linear alteration of mutation parameters:

Table 4-

Function

1

2

3

4

5

6

7

8

9

10

11

12

13

Figure 4-35 -


99

The table below (Table 4-6) shows the self-adaptive approach compared to

that of the linear alteration of mutation parameters:

-6 - Linear vs. Self- Adaptive Mutation Rate Modi

Linear Self - Adaptive

µ σ µ σ

1.46E+04 4.41E+03 5.89E+03 3.24E+03

4.58E+01 1.02E+01 3.70E+01 1.03E+01

3.85E+04 1.09E+04 2.47E+04 6.11E+03

6.57E+01 7.24E+00 5.83E+01 6.96E+00

1.75E+04 5.09E+03 1.11E+04 4.61E+03

1.54E+04 5.35E+03 5.55E+03 2.30E+03

9.92E+00 5.82E+00 3.14E+00 2.27E+00

-7.62E+03 5.11E+02 -7.93E+03 6.42E+02

1.70E+02 2.18E+01 1.71E+02 3.26E+01

1.80E+01 1.06E+00 1.45E+01 2.11E+00

1.38E+02 4.75E+01 5.17E+01 2.42E+01

2.00E+04 5.53E+03 1.38E+04 3.34E+03

1.80E+04 4.56E+03 1.15E+04 4.66E+03

- Self Adaptive Mutation Control vs. Linear Mutation Con


adaptive approach compared to

Adaptive Mutation Rate Modi fication

T

3.24E+03 -11.30

1.03E+01 -4.30

6.11E+03 -7.78

6.96E+00 -5.19

4.61E+03 -6.65

2.30E+03 -11.98

2.27E+00 -7.67

6.42E+02 -2.71

3.26E+01 0.10

2.11E+00 -10.62

2.42E+01 -11.44

3.34E+03 -6.78

4.66E+03 -7.10

Self Adaptive Mutation Control vs. Linear Mutation Control


100

The self-adaptive algorithm appeared to be the better-performing of the two mutation

controlling schemes. This is a beneficial situation, as the self-adaptive scheme does not

depend as strongly on the initial and final values required of the linear scheme. However

boundary conditions for the mutation parameters must still be set and these were set

similar to the values used for the initial and final values for the linear scheme. Hence, the

self-adaptive scheme is still not isolated from the testing used to improve algorithm

tuning.

The self-adaptive algorithm was tested against the conventional algorithm for both 300

and 30,000 evaluations. The parameters for the conventional algorithm were set to those

found earlier in testing. The results for 300 evaluations are shown in Table 4-7.


Table 4-7 - Original vs. Self

Function

1

2

3

4

5

6

7

8

9

10

11

12

13

Figure 4-36 - Self Adaptive Control vs. Empirical Best for No Con trol (300 Evaluations)


101

Original vs. Self -Adaptive Mutation Modification (300 Evaluations)

Gaussian Self -Adaptive

µ σ µ σ

5.13E+04 4.84E+03 4.71E+04 5.09E+03

1.91E+08 7.51E+08 6.25E+04 8.72E+03

7.82E+04 2.62E+04 7.15E+04 9.07E+03

8.26E+01 4.05E+00 8.02E+01 5.56E+00

1.71E+08 4.04E+07 4.89E+04 7.36E+03

5.30E+04 7.69E+03 4.60E+04 4.11E+03

6.53E+01 1.73E+01 7.05E+01 1.76E+01

-3.86E+03 5.00E+02 -3.83E+03 5.17E+02

3.71E+02 2.34E+01 3.77E+02 1.57E+01

2.02E+01 1.64E-01 2.03E+01 2.11E-01

4.74E+02 5.86E+01 4.76E+02 5.66E+01

3.30E+08 1.13E+08 5.15E+04 8.63E+03

6.70E+08 1.87E+08 5.09E+04 5.65E+03

Self Adaptive Control vs. Empirical Best for No Con trol (300 Evaluations)


Adaptive Mutation Modification (300 Evaluations)

T

5.09E+03 -4.22

8.72E+03 -1.80

9.07E+03 -1.70

5.56E+00 -2.39

7.36E+03 -29.86

4.11E+03 -5.69

1.76E+01 1.49

5.17E+02 0.32

1.57E+01 1.38

01 4.21

5.66E+01 0.19

8.63E+03 -20.61

5.65E+03 -25.34

Self Adaptive Control vs. Empirical Best for No Con trol (300 Evaluations)


102

Again the performance advantage of the self-adaptive parameters is clear for most of the

functions. It is also worth noting that the performance of the self-adaptive function for

functions 5, 12, and 13 is better with 300 evaluations than the conventional algorithm is

with 3000 evaluations.

The results for increasing the number of evaluations to 30,000 are shown in Table 4-8.

Table 4-8 - Original vs. Self-Adaptive Mutation Mod ification (30,000 Evaluations)

Function Gaussian Self -Adaptive

T µ σ µ σ

1 3.83E-02 3.35E-02 1.62E+00 6.47E+00 1.72

2 1.99E-02 2.78E-02 1.98E-02 5.79E-02 -0.01

3 8.41E+03 2.43E+03 3.25E+03 3.11E+03 -9.24

4 2.45E+01 6.25E+00 1.25E+01 6.05E+00 -9.74

5 5.83E+02 8.72E+02 1.92E+02 2.04E+02 -3.09

6 6.81E-02 1.25E-01 1.56E-01 2.67E-01 2.11

7 3.27E-01 8.67E-02 2.35E-01 1.63E-01 -3.55

8 -1.03E+04 5.95E+02 -9.85E+03 7.63E+02 2.98

9 6.02E+01 1.76E+01 8.76E+01 1.92E+01 7.44

10 7.55E+00 5.94E+00 2.07E+00 2.87E+00 -5.88

11 9.81E-02 9.67E-02 1.83E-01 2.54E-01 2.21

12 3.15E-01 3.02E-01 4.43E-01 1.05E+00 0.83

13 5.74E-01 7.14E-01 4.04E-01 1.15E+00 -0.89


Figure 4-37 - Self Adaptive Control vs. Empirical Best for No Con trol (

Again the self-adaptiv

functions. However the advantages are not as noticeable. It is worth noting that the

apparent poor performance of the

high outlier amongst the 50 runs.

shown in Table 4-9.


103

Self Adaptive Control vs. Empirical Best for No Con trol (

adaptive algorithm appears to outperform the Gaussian on some of the


apparent poor performance of the self-adaptive algorithm for function 1 is due to a single

st the 50 runs. The minimum value found in any of the 50 runs


Self Adaptive Control vs. Empirical Best for No Con trol ( 30,000 Evaluations)

e algorithm appears to outperform the Gaussian on some of the


daptive algorithm for function 1 is due to a single

he minimum value found in any of the 50 runs is


104

Table 4-9 - Original vs. Self-Adaptive (Best Soluti on in 50 Runs)

Function Gaussian Adaptive

1 7.77E-04 2.12E-06

2 1.35E-03 4.67E-08

3 4.45E+03 2.11E+02

4 1.59E+01 2.07E+00

5 3.24E+01 8.92E-01

6 4.82E-03 1.60E-05

7 1.90E-01 7.53E-02

8 -1.12E+04 -1.13E+04

9 2.39E+01 5.28E+01

10 2.28E+00 6.99E-05

11 1.45E-03 3.51E-06

12 5.15E-04 4.27E-07

13 1.72E-02 4.82E-04

Apart from function 9, it was the self-adaptive algorithm that found the lowest value. This

indicates that the self-adaptive routine may offer improved performance but not as

consistently. It is also worth noting that the parameter values for the Gaussian algorithm

were changed according to the test results in the above sections for each of the evaluation

runs, according to the values found empirically in the above investigations. However the

self-adaptive algorithm parameters were set to the same value for each of the evaluation

tests. This indicates that the self-adaptive algorithm may be less sensitive to poor

implementations. Repeating the test for 30,000 evaluations but keeping the parameters for

the conventional algorithm set as for 3000 evaluations yielded the results shown in

Table 4-10.


Table 4-10 - Original vs. Self

Function

1

2

3

4

5

6

7

8

9

10

11

12

13

Figure 4-38


105

Original vs. Self -Adaptive Without Changing Algorithm Properties

Gaussian Linear

µ σ µ σ

1.07E+00 1.86E+00 1.62E+00 6.47E+00

2.78E-01 2.04E-01 1.98E-02 5.79E-02

8.67E+03 3.07E+03 3.25E+03 3.11E+03

5.91E+01 3.83E+01 1.25E+01 6.05E+00

8.98E+02 9.86E+02 1.92E+02 2.04E+02

2.37E+00 3.87E+00 1.56E-01 2.67E-01

5.11E-01 1.85E-01 2.35E-01 1.63E-01

-1.07E+04 4.13E+02 -9.85E+03 7.63E+02

3.40E+01 7.21E+00 8.76E+01 1.92E+01

2.39E+00 4.32E+00 2.07E+00 2.87E+00

5.64E-01 4.07E-01 1.83E-01 2.54E-01

6.43E-01 1.05E+00 4.43E-01 1.05E+00

6.57E+02 2.93E+03 4.04E-01 1.15E+00

- Self Adaptive Control vs. No Control ( 30,000


Adaptive Without Changing Algorithm Properties

T

.47E+00 0.57

02 -8.61

3.11E+03 -8.76

6.05E+00 -8.50

2.04E+02 -4.96

01 -4.05

01 -7.91

7.63E+02 6.72

1.92E+01 18.53

2.87E+00 -0.44

01 -5.61

1.05E+00 -0.95

1.15E+00 -1.58

30,000 Evaluations)


106

As can be seen with this case there are only two functions that the conventional algorithm

out-performs the self-adaptive one, namely 8 and 9. (Note that due to the low T value

associated with the comparison for functions 1, no statistical difference is apparent). This

is evidence of the robustness of allowing the mutation parameters to be self-adaptive.

From the tests conducted on controlling the mutation parameters it is clear that significant

performance gains can be made. It is beyond the scope of this work to exhaustively test

all the possibilities that this finding introduces into the algorithm manipulation.

4.11 Conclusion

The tuning process clearly had a beneficial effect on improving the average performance

of the algorithm. Key observations are:

• The ratio between population size and the number of generations is of significant

importance where the total number of evaluations is restricted.

• Probabilistic selection methods (e.g. Tournament Selection) do not appear to offer

advantages where the total number of evaluations is restricted.

• The size of mutation and the minimum mutation rate are critical to the

performance of the algorithms and are closely linked to the number of evaluations

available.

• Substituting Cauchy in place of Gaussian random number distributions offers no

discernable performance increase in this situation.

• Controlling mutation rates (including the minimum mutation rate) throughout a

run dramatically improves the algorithm’s convergence rate. This is best done

self-adaptively.

• The number of variables (for the range tested) has little effect on the values of the

tuned parameters.


107

The tuned and modified algorithm clearly gave better results than the traditional

algorithm for the conditions set on the test. However it is worth noting that the algorithm

has been designed with convergence being given priority over the quality of the final

solution. If the algorithm were to be used outside of the ranges of the tests there is no

precedent for performance5. This is in keeping with the “No free lunch theorem” (Wolpert

and Macready, 1995):

"[...] all algorithms that search for an extremum of a cost function perform

exactly the same, when averaged over all possible cost functions."

In essence the algorithm is tailored to be useful for problems involving small numbers of

evaluations with a non-trivial number of variables.

4.12 Developed Evolutionary Algorithm

The developed algorithm was based on a standard Evolutionary Strategy Algorithm

(Bäck, 95). The following properties were set for the algorithm:

• The population size was set to 50. This appeared to be an appropriate value up to

the tested 100, 000 evaluations, which was much higher than the number of

evaluations expected for a typical design problem.

• The number of parents was set to 7. This maintains the ratio suggested by

Schwefel (1993) as discussed in section 4.5.

• Deterministic selection was adopted for the reasons outlined in section 4.6.

The most notable differences with the algorithm developed by the author involve the

control of the mutation parameters. The equations were introduced to control τ and τ’

which are global and individual mutation parameters. The equations are as follows:

5 No interpolation was used between points; only tested values were used for the algorithm parameters.


108

Equation 4-16

Equation 4-17

The variable ‘a’ was termed the mutation strength parameter, and the variable ‘b’ was

termed the mutation distribution algorithm. The mutation variable, ‘b’, was set at 0.55,

while the mutation strength, ‘a’, was controlled through self-adaption (between the values

3 and 4). The minimum mutation rate ‘e’ was also controlled self-adaptively (between the

values 10-2 and 10-3). The use of self-adaptive control of mutation and mutation rates

higher than normally used differentiate the developed algorithm over traditional

approaches.

n

ba

2

)1(2 −=τ

n

ab

2

2'=τ


109

5 Practical Application of Evolutionary Algorithms

This chapter discusses the application of evolutionary algorithms to two example

problems. Both problems are ones that have well-developed methods to find solutions.

They are presented here as an aid to discuss the behaviour and methods of applying

evolutionary algorithms

There are a number of practical issues that arise when attempting to apply evolutionary

algorithms to real design problems. Two test problems were investigated in this chapter to

cover some of these issues. The first was the optimisation of a driving line around a given

track profile. This problem was chosen to study the quality of the solutions produced by

the optimisation algorithm and to discuss the trade-off between the quality and reliability

of the result. Continuous improvement is clearly shown throughout the optimisation

process; however there are also significant differences in the final result of repeated

optimisations.

The second problem chosen was the kinematics design of a double A-arm suspension

system. This was chosen to study the effects of different variable representations, issues

involved with problems that exhibit multiple design objectives, as well as the ability to

use data gained throughout an optimisation to determine relationships between individual

variables and performance. Different representations of exactly the same problem were

shown to have markedly different performance characteristics between optimisations. An

example is shown where problem-specific knowledge is applied to the solution

representation to improve the performance of the optimisation process. Multiple

objectives for the problem were studied and the weaknesses of applying a single objective

solver to a multiple objective problem were shown. Each tested solution during the

optimisation problem was recorded so that possible relationships between performance

and individual variables could be studied after the optimisation process was complete.

The results of simply graphing performance against individual variables proved

promising and useful relationships were found.


110

5.1 Ideal Path Generator

Evolutionary optimisation processes rely on random numbers. While the focus of the

algorithm tuning has been on robustness, there are no guarantees that the solutions found

in a particular run are close to the optimum. This creates the problem of how to use the

available number of evaluations. For example, if for a given problem it is estimated that

100 evaluations may be made in the available time before a solution is needed, the

question becomes whether it is better to conduct the optimisation process once with 100

evaluations, twice with 50 evaluations each and take the best, three times with 33

evaluations each and take the best and so on. This is markedly different from the tuning

process, in which the average result of the optimisation process conducted many times is

taken as being representative of the algorithm’s performance. When applied to real

problems it is likely that the best performance of any of the times the optimisation was

conducted would be used as the solution to the problem. To study this issue the algorithm

was applied to a program designed by the author to optimise the driving line around a

racetrack.

The path generator outlined in this section optimises the racing line of a constant-width

track defined by the program operator. The program is used to input virtual paths into a

lap-time simulator, as well as studying the driving lines for different vehicle

configurations and tracks.

Path generators can be especially useful when teams are required to run on new or

modified race tracks. For example in 2004 a new race in China was introduced to the

Formula 1 Calender. Each of the participating teams received track data from which they

could run simulations to predict vehicle performance. This allowed teams to better

optimise their vehicles for the race. In order for this to succeed it is necessary to estimate

a driving line. Mühlmeier & Müller (2002) show optimisation of a racing line through

small sections of racetracks. Similarly they use evolutionary optimisation techniques;

however the algorithm used is specifically modified to suit the ideal racing line problem.


111

5.1.1 Path Definition

A track is defined by an n x 3 matrix, where n is the number of track sections. Each

section is defined by a curvature, length and resolution value. The resolution value

determines how many points the algorithm will analyse within the section. Increasing the

resolution values of each section increases the accuracy of the vehicle model at the cost of

computation time. This process will create a matrix in the following form:

Equation 5-1

The number of sections is directly related to the number of control points by the

following relationship:

Equation 5-2

Increasing the number of control points for a given track increases fine control of the path

at the cost of increased computation time. Each control point consists of two variables,

position and gradient. The position is defined by a percentage value from the left hand

side of the track to the right hand side (i.e. r=0 for the left hand edge of the track and r=1

for the right hand edge). The gradient represents the rate of lateral change from one side

of the track to the other. Paths are defined by fitting cubic splines to successive control

points. Splines are used as they behave in a smoother manner than polynomials; cubic

splines have the lowest curvature in the spline family, resulting in a smooth racing line.

Each cubic spline is defined as follows:

Equation 5-3

Where r is the lateral position (relative to the edges of the track) and t is the percentage of

section completion (i.e. t = 0 at the start of the section t = 1 at the end of the section). The

coefficients of the cubic splines are calculated as follows:

nnn RLK

RLK

.........111

1int += nN sControlPo

dctbtatr +++= 23


112

Equation 5-4

Equation 5-5

Equation 5-6

Equation 5-7

Where:

r0 - lateral position at the beginning of the section E(0,1)

r1 - lateral position at the end of the section E(0,1)

g0 - rate of lateral movement at the beginning of the section

g1 - rate of lateral movement at the end of the section

The path along the splines, which are defined in a track-based co-ordinate system, is

converted to Cartesian co-ordinates through the following transformations:

Equation 5-8

From these co-ordinates a curvature and length are created at for each point of the racing

line according to the following equations that utilize both the previous point and the

following point:

- Previous point

- Current point

- Following point

)(2 0101 rrgga −−+=

)(32 0101 rrggb −+−−=

0gc =

0rd =

)( LeftRightLeft xxrxx −+=

),( 11 yx

),( 22 yx

),( 33 yx


113

Equation 5-9

Equation 5-10

Equation 5-11

Equation 5-12

Curvature at each point is defined as:

Equation 5-13

Length at each point is approximated as the linear distance:

Equation 5-14

5.1.2 Vehicle Model

A simple quasi-static vehicle model has been implemented in order to reduce computation

time. The model is based around a user defined velocity-dependent g-g diagram. An

example of such a diagram is shown in Figure 5-1. The given example is based on a

Formula SAE vehicle with a small amount of aerodynamic down-force.

213

213 )()( yyxxa −+−=

212

212 )()( yyxxb −+−=

223

223 )()( yyxxc −+−=

)2

(cos222

1

bc

acb −+= −θ

θsin2=K

cL =


Figure 5-1 –

As can be seen the available longitudinal acceleration decreases with velocity. This is due

to the limit of tractive force available from the power

rolling resistance and aerodynamic drag. Available braking and lateral accelerations

slightly increase with velocity due to increased grip from aerodynamic down

5.1.3 The Problem

A track section was designed that conformed to the 2005 Formula SAE rules

The section was sized to keep the number of variables close to 30

the algorithm was tuned (Chapter

marked:


114

– Velocity Dependent g-g Diagram


vailable from the power-train, compounded by


slightly increase with velocity due to increased grip from aerodynamic down-force.

esigned that conformed to the 2005 Formula SAE rules (SAE 2005)

The section was sized to keep the number of variables close to 30, because this was how

the algorithm was tuned (Chapter 4). The track is shown below with the se

Kevin Hayward


train, compounded by increased


force.

(SAE 2005).

, because this was how

. The track is shown below with the segments


115

Figure 5-2 - Track Profile

This track section has a road width of 4.5m. However in order to account for the width of

a vehicle, the road width was reduced to 3m in this example. There are 15 segments in the

track section with 161 calculation points. This yields a problem with 32 variables and an

unknown optimum solution. The track starting point is at (0,0). The following graphs

indicate how the optimisation process occurred through a single run of 2000 generations

(with a population size of 50).


116

Figure 5-3 - Paths after 1 generation

Here are the paths for the first generation as shown by the green lines. This shows the

seeding of the population. The paths clearly follow no similar path and are so random that

nearly the whole of the track is covered.


117

Figure 5-4 - Paths after 10 generations

This figure shows the same run after 10 generations (500 evaluations) have been

completed. Already distinct paths have started to form.


118

Figure 5-5 - Paths after 100 Generations

This figure shows the same run after 100 generations (5000 evaluations). The paths had

converged considerably and a cleaner racing line can be seen.


119

Figure 5-6 - Paths after 1000 generations

This figure shows the same run after 1000 generations. There was little difference of the

paths at this point. The racing line has been improved over the previous figure; it is

closer to the inside edge of the hairpin and there are fewer direction changes through the

slalom area.

Hairpin

Slalom Section


120

Figure 5-7 - Paths after 2000 Generations

After the allotted 2000 generations (100,000 evaluations) there is no visible difference

between the paths. The racing line appears only slightly changed from the line given after

1000 generations (Figure 5-6).


121

The best path (i.e. the minimum time) found during the run is shown in Figure 5-8:

Figure 5-8 - Best path found during run

While the line shown in Figure 5-8 appears to be a good one, it does appear that near the

beginning of the straight there could be a little improvement. However the vehicle starts

at a velocity of zero. Hence the curvature at the low velocities will not produce a high

lateral g-force requirement. This results in almost full longitudinal g-force still being

available. It is likely that any further improvement in the time of the path in this section of

the track would be minimal. The time taken to traverse this path was 9.42 seconds.


122

This path was found in the final generation of the optimisation process. This shows that

improvements were being made right until the end of the process. Figure 5-9 shows a

graph of the number of successful generations against the number of generations

completed. In this case a successful generation was considered one in which a new

optimum solution was found (i.e. the time taken was reduced). The figure shows that the

ratio of successful generations to the number of generations completed is relatively

constant throughout the process. A line of best fit is shown and the correlation value (R2)

is quite close to one. In this case roughly 60% of the generations produced a better

optimised solution.

Figure 5-9 - Cumulative Successful Generations

Figure 5-10 shows the number of successful generations for periods of 100 generations.

This helps to show the rate at which new optimum solutions were being produced. Near

the start of the algorithm new solutions were being produced by approximately 50% of

the generations. This is most likely due to a ‘start-up’ phase, as search parameters are

being evolved to suit the problem. This increased to over 60% of the generations for most

of the optimisation process, with a reduction to fewer than 40% towards the very end of

Cumulative Successful Generations

y = 0.6353x

R2 = 0.9974

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Generation

Suc

cess

ful G

ener

atio

ns


the process. As the solution gets closer to the optimum, it becomes more unlikely that the

random mutations will produce improvements to the path

successful generation

Figure

The time taken to traverse the optimised line

travelling along the middle of the track

below:

All three of these obvious paths perform well below that of the path produced usi

optimisation algorithm.


123


random mutations will produce improvements to the path, explain

rate.

Figure 5-10 - Successful Generations per 100 Generations

e time taken to traverse the optimised line can be compared to the

elling along the middle of the track, the inside edge, and the outside edge

Racing Line Time (s)

Optimised 9.42

Middle 11.86

Inside 11.72

Outside 12.08

All three of these obvious paths perform well below that of the path produced usi

optimisation algorithm.



xplaining the reduction in the

Successful Generations per 100 Generations

can be compared to the simple cases of

, the inside edge, and the outside edge as shown

All three of these obvious paths perform well below that of the path produced using the


124

With a problem of 32 variables it was not feasible to do a direct search within reasonable

time. Testing just two options for each of the variables would have required 232

evaluations (4.3 x 109) and is not likely to produce a solution acceptable as being close to

the global minimum. To compare to the algorithm, which used 105 evaluations, the same

number of random solutions was tested. The best path found is shown below:

Figure 5-11 - Best Random Path

The best random path found in 100,000 evaluations gave a time of 11.70 seconds. This

was only marginally better than the inside edge. A random approach was tested, not

because it was expected to produce a good result, but rather to show the poor performance


of non-directed trial and error. It is entirely possible in this case that a linear algorithm or

alternate optimisation algorithm would work well.

It is interesting to note the rate of improvement of the solution found by u

evolutionary algorithm. The following graph shows the relationship between the time the

vehicle takes to traverse the given path and the number of generations. It was created by

averaging the results from 5 different runs.

For this optimisation the algorithm had found a better solution than the random approach

(using 100,000 evaluations) within 10 generations (500 evaluations).

evolutionary algorithm

exceeds the performance of pure trial and error.

The relative improvement

percentage of the current generation to the best value found overall


125

directed trial and error. It is entirely possible in this case that a linear algorithm or

alternate optimisation algorithm would work well.

It is interesting to note the rate of improvement of the solution found by u



averaging the results from 5 different runs.

Figure 5-12 - Path Time vs. Generations


000 evaluations) within 10 generations (500 evaluations).

evolutionary algorithm depends on to some extent on random number values

exceeds the performance of pure trial and error.

The relative improvement over time can also be shown as a graph indicating the

percentage of the current generation to the best value found overall


directed trial and error. It is entirely possible in this case that a linear algorithm or

It is interesting to note the rate of improvement of the solution found by using the




000 evaluations) within 10 generations (500 evaluations). While the

to some extent on random number values, it far

can also be shown as a graph indicating the

percentage of the current generation to the best value found overall.


Figure 5-13 - Percentage of Improvement vs. Generations

From this we can see that the improvement is continuous, while the optimisation process

produces the most striking results early in the process. T

first generation to the last is 3.40 seconds. Zooming in to the later generations we can see

that the process is still occurring quite clearly.


126

Percentage of Improvement vs. Generations


produces the most striking results early in the process. The difference in time from the


that the process is still occurring quite clearly.

Kevin Hayward


he difference in time from the



Figure 5

Figure 5-14shows a more linear optimisation with much smaller improvements for the

final 25% of the optimisation process. The time range between the

and the end is 0.01 seconds


127

5-14 - Path Time vs. Ge nerations (Last 500 Generations)

shows a more linear optimisation with much smaller improvements for the

of the optimisation process. The time range between the

seconds


nerations (Last 500 Generations)

shows a more linear optimisation with much smaller improvements for the

of the optimisation process. The time range between the 75% completed point


Figure 5-15 - Path Time vs. Generations (Last 50 Generations)

Figure 5-15 shows the final 5% of the generations and it is clear that improvements are

still occurring. However the time range between

seconds. At this stage the improvements appear to be negligible. However it is worth

noting the end results from the 5 different runs.

Table 5-1

The difference from quickest to slowest of the different runs (0.0609 seconds) was larger

than the average improvement over the last 1000 generations (0.0394 seconds). If w

at the results after half the generations we get the following table:


128

Path Time vs. Generations (Last 50 Generations)

5% of the generations and it is clear that improvements are

ime range between 95% the end point is now only 0.0006


noting the end results from the 5 different runs.

1 - Best Time for Each of 5 Runs

Run Time

1 9.3752

2 9.4228

3 9.4002

4 9.3619

5 9.4079


than the average improvement over the last 1000 generations (0.0394 seconds). If w

at the results after half the generations we get the following table:

Kevin Hayward

5% of the generations and it is clear that improvements are

5% the end point is now only 0.0006



than the average improvement over the last 1000 generations (0.0394 seconds). If we look


Table 5-2

By this stage one of the runs has already reached a value superior to

of the other runs after

optimisation. It should be noted that the difference between the quickest and slo

at this stage is 0.1751 seconds. This is almost three times the difference between the runs

after 2000 generations. The difference in times between the best and worst of the runs is

shown below:

Figure 5-16 - Time Between Maximum and Minimum Runs vs. Generatio ns

Figure 5-16 shows that as the number of generations increases

runs decreases.


129

2 - Best Time for Each of 5 Runs (After 1000 Generation s)

Run Time

1 9.4190

2 9.5433

3 9.4166

4 9.3682

5 9.4183

this stage one of the runs has already reached a value superior to

of the other runs after the full 2000 generations. This shows the random nature of the

optimisation. It should be noted that the difference between the quickest and slo



Time Between Maximum and Minimum Runs vs. Generatio ns

shows that as the number of generations increases, the variability between the


Best Time for Each of 5 Runs (After 1000 Generation s)

this stage one of the runs has already reached a value superior to that achieved by any

the full 2000 generations. This shows the random nature of the

optimisation. It should be noted that the difference between the quickest and slowest runs



Time Between Maximum and Minimum Runs vs. Generatio ns

the variability between the


130

This shows that there is an inherent trade-off when applying evolutionary algorithms to

problems. One option is to put all the evaluations into a single run knowing that if the

number of evaluations is enough for the given problem, the variability between runs will

be insignificant. The other option is to split the available evaluations up into multiple runs

on the basis that the benefit of spreading the odds and having a single good run may

outweigh the detriment of limited computation time. The second approach should be used

with caution, because if the number of evaluations is split up too much, then the result

from any of those could be worse than any individual longer run. For the example above,

if we look at each of the five runs at the 20% time mark we get the following table:

Table 5-3 - Best Time for each of 5 Runs (After 400 Generations)

Run Time

1 9.5871

2 9.7300

3 9.5182

4 9.4648

5 9.4907

Each of the 5 runs is worse in performance to any of the single runs where 5 times the

computation was given.

Since the optimisation process runs without human intervention, it is possible that a single

run of a significant number of evaluations may be performed. Design can continue using

this data while an unattended computer can be used to perform further runs that may be

used as confirmation of the previous results, or may allow for some minor design changes

at a later stage.


131

5.2 Suspension Kinematics

The following section gives an example of applying the evolutionary algorithm to the

geometrical design of a vehicle’s suspension system. Representation of variables,

determination of objective functions, and the problems of dealing with designs at the

system level were considered. Techniques to identify key design variables were also

investigated.

Each of the wheels of a vehicle has six possible degrees of freedom. These are derived

from the three translational degrees of freedom and the three rotational degrees of

freedom. A suspension system is designed to constrain 5 of the degrees of freedom. The

6th degree of freedom is controlled with a spring/damper system. The path of the wheel

through this degree of freedom is determined by the geometry of the suspension system.

Kinematic programs are used for the study of these paths. Examples of commercially

available tools include ADAMS, Mitchell, and 3d suspension analyser. The author

managed a team during 2006 that produced OptimumK, a racing car kinematics program

that was released for sale in 2007. These packages model the suspension system in three

dimensions. For ease of analysis, links can be considered rigid. For a race car with well-

designed links and rose-joints, this is a reasonable assumption.

5.2.1 Model Details

A 3-dimensional kinematics program for a double A-arm suspension system was

developed. Double A-arms are the most common suspension arrangement for vehicles

specifically built as race-cars. In the case of Formula SAE vehicles, double A-arm

suspension systems are used on the front and rear of the car with very few exceptions. A

graphical representation of the model for a single wheel of a vehicle is shown in Figure

5-17.


Figure 5-17 - 3D Kinematics Model Showing a Single Corner

The model was extended so that a 3

of the whole vehicle could be analysed. The model has been used as a tool to aid in the

suspension kinematic design of three race cars for the University of Western Australia

Motorsport team. This model was used for the example

simulation model is shown alongside the final vehicle in


132

3D Kinematics Model Showing a Single Corner

that a 3-dimensional representation of the suspension system



This model was used for the examples in this chapter. The rendered

simulation model is shown alongside the final vehicle in Figure 5-18.

Kevin Hayward

on of the suspension system



s in this chapter. The rendered


Figure 5- 18

5.2.2 Different Repre

Bäck et al. (1997) make the following comment

solve problems:

Expert knowledge about the problem needs to be incorporated into the

representation and the operators in order to guide the search process and

increase convergence velocity

In order to demonstrate this, a simple problem was set up showing two different

representations of the variables involved.

One of the important issues of suspension design is the camber of the wheel throughout

its travel. A simple objective was defined

25mm of vertical wheel

difference between the camber gain and the desired camber gain.


133

18 - 2004 UWAM Racing Car (Vehicle & Kinematic Model)

Different Representations

make the following comment about using evolutionary algorithms to



increase convergence velocity.


representations of the variables involved.


A simple objective was defined so that camber would increase 1 degree for

of vertical wheel movement. The error function was defined as the absolute

difference between the camber gain and the desired camber gain.


2004 UWAM Racing Car (Vehicle & Kinematic Model)

about using evolutionary algorithms to





that camber would increase 1 degree for

movement. The error function was defined as the absolute


Figure 5-19 - Camber Gain with Suspension Travel

Figure 5-19 is a depiction of the camber gain due to suspension travel. Camber is defined

as the angle between the vertical and the wheel angle. Negative camber is when

the wheel is tilted towards the inside. Typically a double A

increase the negative camber with upwards movement of the wheel.

The representation for the A-arms

dimensional problem. Geometry of the spring and rocker system had no bearing on the

camber, so was ignored. These simplifications created a problem that had 4 points in 2

dimensional space to be determined. The 8 variables in this problem were represented in

two different ways. One method was defining each of the x and y co

points as a variable. This is shown graphically


134

Camber Gain with Suspension Travel

is a depiction of the camber gain due to suspension travel. Camber is defined

as the angle between the vertical and the wheel angle. Negative camber is when

the wheel is tilted towards the inside. Typically a double A-arm suspension system will

increase the negative camber with upwards movement of the wheel.

arms was set so that it could be considered a two

problem. Geometry of the spring and rocker system had no bearing on the

so was ignored. These simplifications created a problem that had 4 points in 2


ferent ways. One method was defining each of the x and y co-ordinates of the 4

points as a variable. This is shown graphically in Figure 5-20.

Kevin Hayward

is a depiction of the camber gain due to suspension travel. Camber is defined

as the angle between the vertical and the wheel angle. Negative camber is when the top of

arm suspension system will

was set so that it could be considered a two-

problem. Geometry of the spring and rocker system had no bearing on the

so was ignored. These simplifications created a problem that had 4 points in 2-


ordinates of the 4


135

Figure 5-20 - x-y Representation

The second method introduced different variables to define the points involved. The 8

variables used to determine the points consisted of:

• The swing arm length (SAL)

• The roll centre (RC)

• x co-ordinate for point 2

• y co-ordinate for point 2

• The upright height

• The king-pin inclination

• x co-ordinate for point 1

• The length of the upper arm as a percentage of the lower arm

This representation introduces accepted vehicle suspension parameters. This is shown

graphically in Figure 5-21.


136

Figure 5-21 - SAL Representation

The second representation (SAL) shows the application of problem specific knowledge.

While both representations fully define the mechanism, the second representation presents

variables as they would be treated by a designer. For example, the location of the

instantaneous centre gives a large indication of how a four-bar linkage will move, because

it is an approximation of the point about which the tyre will rotate. Including the swing-

arm length and the roll centre height as variables allows for the instantaneous centre to be

defined by two variables instead of the eight required in the x-y representation.

Both representations were used with the same error function defined above. The

evolutionary algorithm developed in Chapter 4 was applied to both, 50 times. The results

are given below.


137

Table 5-4 - Comparison between Representations

SAL Representati on

Mean 3.54E-08

Std. Dev. 1.08E-07

x-y Representation

Mean 1.10E-06

Std. Dev. 3.65E-06

t-value -2.06

The results clearly showed improvement when the problem was set up with variables

designed with problem-specific knowledge. The average error for the two different

representations plotted against the number of generations is shown below.

Figure 5-22 - Average Error vs. Generations (For 2 Different Parameter Representations)

This shows the improved convergence velocity for the SAL representation over the x-y

representation, hence the inclusion of problem specific knowledge in the representation of

the variables improves the performance of the evolutionary optimisation.

Average Error vs. Generations

-8

-7

-6

-5

-4

-3

-2

-1

00 10 20 30 40 50 60

Generation

Log(

Err

or)

SAL Representation xy Respresentation


5.2.3 Determining Parameter Relati

Recording the process of the evolutionary algorithm produces useful information

addition to a final problem solution. In the previous example we can look at the variation

of the different variables over time. The graph below

performance of all the points covered by a single run for the variable defining the swing

arm length. An optimum swing arm length of around 1500 was seen. Furthermore a clear

and expected relationship can be seen between the

Figure 5-23 - Performance vs. Swing Arm Length

The other variables did not appear to have a similar relationship. For example

variable that is common to both of the representations

Figure 5-24 shows the performance of each point plotted against this variable.


138

Determining Parameter Relationships

Recording the process of the evolutionary algorithm produces useful information


of the different variables over time. The graph below (Figure 5-23)


An optimum swing arm length of around 1500 was seen. Furthermore a clear

relationship can be seen between the camber gain and the swing arm length.

Performance vs. Swing Arm Length

not appear to have a similar relationship. For example

h of the representations is the x co-ordinate for point 1.

shows the performance of each point plotted against this variable.

Kevin Hayward

Recording the process of the evolutionary algorithm produces useful information, in


) shows the


An optimum swing arm length of around 1500 was seen. Furthermore a clear

camber gain and the swing arm length.

not appear to have a similar relationship. For example, one such

ordinate for point 1.


The log of the error was plotted to clearly show where the best performing solutions

existed. In this case the best solutions were found when the x

its maximum. However it appears from this that there is no strong relationship between

performance and this variable

only one optimal solution is computed. In order to support such relationships it is useful

to perform more than one run with the algorithm. The previous two graphs are r

below with five different runs shown on the same graph.


139

Figure 5-24 - Performance vs. x1

f the error was plotted to clearly show where the best performing solutions

In this case the best solutions were found when the x-coordinate of point 1 was at

However it appears from this that there is no strong relationship between

this variable. It should be noted that only running the algorithm


to perform more than one run with the algorithm. The previous two graphs are r

below with five different runs shown on the same graph.


f the error was plotted to clearly show where the best performing solutions

coordinate of point 1 was at

However it appears from this that there is no strong relationship between

running the algorithm once,


to perform more than one run with the algorithm. The previous two graphs are reproduced


Figure 5-25 - Performance vs. Swing Arm Length (5 Runs Overlaid)


140

Performance vs. Swing Arm Length (5 Runs Overlaid)

Kevin Hayward


Figure

By conducting multiple runs of the algorithm there is further confirmation that there is a

strong relationship between performance and SAL

was close to 1500, and the shape of the relationsh

is further evidence that no

co-ordinate of point 1.

variables produced similar results t

relationships between performance and any one of the variables when the problem was

solved in its x-y co-ordinate representation.

This is a very simple analysis of the variables and will only really

there is a distinct relationship between one of the variables and performance. However,

should such a relationship be found, it can become very useful in determining the

behaviour of the system for different conditions. For example

above it was hypothesized that there was a strong relationship between the swing arm


141

Figure 5-26 - Performanc e vs. x1 (5 Runs Overlaid)


strong relationship between performance and SAL. In each run the final value of the SAL

was close to 1500, and the shape of the relationship appears consistent. Furthermore

is further evidence that no unique relationship exists between the performance and the x

ordinate of point 1. Different optimum values were found for each of the 5 runs.

variables produced similar results to the x coordinate of point 1. There were n

between performance and any one of the variables when the problem was

ordinate representation.

This is a very simple analysis of the variables and will only really



behaviour of the system for different conditions. For example, f



e vs. x1 (5 Runs Overlaid)


. In each run the final value of the SAL

ip appears consistent. Furthermore there

relationship exists between the performance and the x

Different optimum values were found for each of the 5 runs. Other

There were no noticeable

between performance and any one of the variables when the problem was

This is a very simple analysis of the variables and will only really aid the designer when



from the results shown



length and the camber properties of the system. A further hypothesis was made that a

relationship could be formed linking the different swing arm len

amounts of camber gain. To test these, the evolutionary algorithm was used to find

solutions for a variety of camber gains ranging from 0.2 to 2 degrees for 25mm of vertical

suspension travel. The graph below shows the average SAL of 5

the camber gains.

Figure 5-27 - Swing Arm Length vs. Camber Gain

A clear relationship was seen linking the Swing Arm Length and the amount of camber

gain. The graph also showed that for low camber gains

arm length was not satisfactory to continue the relationship.

Given knowledge of the problem this relationship was not surprising. The swing arm

length determines how far the instantaneous centre


142


relationship could be formed linking the different swing arm lengths with different



suspension travel. The graph below shows the average SAL of 5 different runs for each of

Swing Arm Length vs. Camber Gain


hat for low camber gains, the available range for the swing

arm length was not satisfactory to continue the relationship.


length determines how far the instantaneous centre of the suspension system is away from

Kevin Hayward


gths with different



different runs for each of


the available range for the swing


of the suspension system is away from


the wheel. If we model the wheel as being attached to a beam connected to this point we

get the following geometric relationship for the camber gain (for 25mm travel) of the

wheel:

Plotted this against the previous graphs yielded the following:

Figure

The values determined through the theoretical model matched tho

algorithm quite well. Longer swing arm lengths seem to give slightly higher camber gains

than expected by the theoretical model. This is likely to be the result of the length

difference between the upper and lower arms.


143



Equation 5-15

Plotted this against the previous graphs yielded the following:

Figure 5-28 - Swing Arm Length vs. Camber Gain

The values determined through the theoretical model matched tho



difference between the upper and lower arms. This showed that the algorithm may be

)sin(

25

CamberGainSAL=




Swing Arm Length vs. Camber Gain

The values determined through the theoretical model matched those of the evolutionary



hat the algorithm may be


144

used as a tool to test theoretical models as well as pointing towards relationships that may

not be known by the user.

5.2.4 Multiple Objectives

Continuing to use the kinematic design of suspension in this example causes a number of

issues to be raised. The algorithm has been shown to be effective at determining solutions

for different camber gain values, and useful to test hypotheses concerning problem

variables. However, as with many engineering design problems, this example cannot

easily be formed to provide an appropriate single objective function. In this case the

camber gain for the suspension is not the only objective that will determine performance:

there are a number of properties of a suspension system that may determine the overall

performance of the vehicle. Some examples include:

• Roll centre movement as a result of suspension movement

• Toe change versus vertical travel

• Track change due to suspension movement

• Caster change due to suspension movement

• Ackermann effects in steering geometry

It is difficult to form these issues into a single objective suitable for the evolutionary

algorithms. There are a number of methods for dealing with multiple objectives, which

are considered beyond the scope of this work. (Section 3.7)

If strong relationships between some variables can be found (as shown in the previous

section) some simple methods of dealing with multiple objectives can be found. To

demonstrate this, the previous example was extended. In addition to desiring a camber

gain of 1 degree for 25mm of travel, another objective was added requiring a

minimization of roll centre movement for a given roll angle. In order to simulate what

would occur during a roll movement, two sides of the suspension were required. The


vehicle was assumed to be symmetrical and the roll was defined by one side having a

vertical movement of 25mm while the other had a vertical movement of

error function was defined as the absolute difference between the original roll

height and the roll centre height after the roll motion.

algorithm yielded the following result.

The initial roll centre height was

relationship to performance. These are shown below for 5 overlaid runs.

6 This is a simplification as roll will exhibit uneven vertical movements for the right and left wheels;

however it was considered a reasonable approximation for non

close to the ground.


145


vertical movement of 25mm while the other had a vertical movement of

error function was defined as the absolute difference between the original roll

height and the roll centre height after the roll motion. Applying the evolutionary

algorithm yielded the following result.

Figure 5-29 - Performance vs. Generations

The initial roll centre height was the only variable that appeared to have a significant


This is a simplification as roll will exhibit uneven vertical movements for the right and left wheels;

however it was considered a reasonable approximation for non-compliant suspensions with roll centres



vertical movement of 25mm while the other had a vertical movement of -25mm.6 The

error function was defined as the absolute difference between the original roll centre

Applying the evolutionary

the only variable that appeared to have a significant


This is a simplification as roll will exhibit uneven vertical movements for the right and left wheels;

compliant suspensions with roll centres


Figure 5- 30

The roll centre height is usually decided independent

on the lateral load transfer of the vehicle during cornering.

the roll centre to reduce movement is not a desirable outcome.

function, no consideration was made

these issues, both the swing arm length and the roll centre were set at known values and

the problem repeated with a reduced number of variables. The results are shown below

when the SAL was set to 1540mm (according to the relationship found in

and the roll centre to an arbitrary height of 25mm.


146

30 - Performance vs. Roll Centre

ecided independently of the geometry, as it has an effect

on the lateral load transfer of the vehicle during cornering. Hence defining the position of

the roll centre to reduce movement is not a desirable outcome. Using this simple objective

onsideration was made of the camber properties. In order to account for

both the swing arm length and the roll centre were set at known values and


AL was set to 1540mm (according to the relationship found in section

and the roll centre to an arbitrary height of 25mm.

Kevin Hayward

as it has an effect

Hence defining the position of

Using this simple objective

. In order to account for

both the swing arm length and the roll centre were set at known values and


section 5.2.3)


147

Figure 5-31 - Performance vs. Generations (Paramete rs Unrestricted vs. Restricted)

This showed that by restricting the changes in the number of variables that can be altered,

the amount of roll centre movement was increased. However it should be noted that the

camber gain was close to what was originally desired. For example the candidate for the

last run had a camber gain of 0.9981 degrees per 25mm vertical wheel travel. This was

within 0.2% of the desired camber gain value and would be insignificant in vehicle

construction. In addition to finding a valid solution, there was some indication that there

may be a relationship between the relative lengths of the upper and lower arms and the

overall performance. The graph below shows this.


Figure 5-32 - Perfo rmance vs. Ratio of Upper and Lower Arm Lengths

A final test involved fixing all the variables apart from this single variable to ascertain

whether it may be used as an independent variable to affect the roll centre movement. The

SAL and the RC were set as above. All other variables were set arbitrarily in the middle

of their available range. The following


148

rmance vs. Ratio of Upper and Lower Arm Lengths

all the variables apart from this single variable to ascertain


above. All other variables were set arbitrarily in the middle

of their available range. The following relationship was found.

Kevin Hayward

rmance vs. Ratio of Upper and Lower Arm Lengths

all the variables apart from this single variable to ascertain


above. All other variables were set arbitrarily in the middle


Figure 5- 33

This example shows a clear relationship. However

quite a lot more effort. Given these results, it is probable that while the other variables

have an effect on the roll centre movement

arm may be altered to find a reasonable solution that minimises roll centre movement.

Using this approach has revealed a significant amount of information

It also highlights the difficulties that additional objective functions introduce

Familiarity with the problem can help to determine some of these relationships with the

aid of the information gathered by the evolutionary process. While it is clear that

relationships were easy to determine by looking at the performance against indivi

variables, it takes some knowledge to be able to determine which relationships can be

acted upon. The example in this section is the relationship between the roll centre height

and the roll centre movement. The inexperienced designer m


149

33 - Performance vs. Ratio of Upper and Lower Arm Length s

e shows a clear relationship. However, finding this relationship involved


ffect on the roll centre movement, they can be fixed and the length of the upper


Using this approach has revealed a significant amount of information

It also highlights the difficulties that additional objective functions introduce



relationships were easy to determine by looking at the performance against indivi

it takes some knowledge to be able to determine which relationships can be


and the roll centre movement. The inexperienced designer might set the rol


Performance vs. Ratio of Upper and Lower Arm Length s

finding this relationship involved


and the length of the upper


Using this approach has revealed a significant amount of information about the problem.

It also highlights the difficulties that additional objective functions introduce.



relationships were easy to determine by looking at the performance against individual

it takes some knowledge to be able to determine which relationships can be


set the roll centre height


150

in a location to deliver minimum movement. While being correct for one particular

objective, it would cause problems with other objectives.

5.2.5 Problems with System Level Designing

The suspension kinematic problem is a good example of a real problem faced during

vehicle design. As mentioned in the previous section, there are a number of criteria that

define a successful suspension system. This immediately leads to multiple objectives. For

a racing car the case can be made that a more appropriate single objective problem exists

that involves the kinematic objectives outlined in this section.

Typically, goal of designing a racing car is to achieve the minimum lap time, for a given

number of laps (Section 2.5). When designing a suspension system, a number of

assumptions are made to determine what suspension properties will lead to a vehicle with

a minimum lap time. Often these assumptions are based on empirical evidence and

include some amount of subjectivity. In this chapter both a camber gain and a roll-centre

movement objective were analysed. In both cases an arbitrary ideal was decided. Ideally

the suspension problem could be included as part of an optimisation run that includes the

whole vehicle. That way the designer is freed from having to define multiple objectives

and their ideal values. This is also the case with many of the other design areas involved

with the design of a racing car (Section 2.7).


151

6 Evolving Racing Cars

In Chapter 5, evolutionary algorithms were applied to problems to show the potential

performance of the technique, how key variables in a problem can be identified, issues

with variable representation, and some of the limitations of using the algorithms. In this

section the evolutionary algorithms are applied to optimise parameters that can effectively

define a whole vehicle. This shows the application of the algorithms in the early stages of

the design process: concept evaluation; and preliminary design.

The lap-time simulation (10 Appendix A: Lap Time Simulation) was designed to be a

computationally quick approximation, so does not offer the accuracy required to make

detailed design decisions. However, the following section will show that application of

evolutionary algorithms yield valuable information as to the direction to take in vehicle

design, as well as giving an idea of how sensitive the performance of a vehicle would be

to various parameters.

This section applies an evolutionary algorithm to the problem of designing a Formula

SAE vehicle. The author has been involved in the formula SAE competition for 5 years

with the University of Western Australia (UWA) Motorsport team. During that time

UWA vehicles have won design awards 3 times in international competitions. This should

allow adequate comparison to be made between the results of the optimisation and a

world-class vehicle. The Formula SAE competition provides a good case study for the

following reasons:

• The rules are not very restrictive and allow for a range of conceptual differences

between cars;

• Vehicles must undergo a number of different vehicle tests that can be simulated

readily, and require different vehicle configurations for success;

• A commonly accepted “best” solution to the rules is unlikely to exist, because of

the youth of the competition and the complexity created by conflicting

requirements of the different tests, as well as it being a non-professional

competition.


152

6.1 The Formula SAE Competition

The Formula SAE (FSAE) rules start with the following statement of purpose:

The Formula SAE ® Series competitions challenge teams of university

undergraduate and graduate students to conceive, design, fabricate and compete

with small, formula style, autocross racing cars. To give teams the maximum design

flexibility and the freedom to express their creativity and imaginations there are

very few restrictions on the overall vehicle design. Teams typically spend eight to

twelve months designing, building, testing and preparing their vehicles before a

competition. [Formula SAE Rules 2007, Section1.1]

Thus the competition focuses on design flexibility, for non-professional teams, in short

time frames. During the competition the teams put their vehicles through a series of

dynamic tests. These are:

• Skidpan – Vehicles are timed over a constant radius circle in both the clockwise

and anti-clockwise direction.

• Acceleration – Vehicles are timed from a standing start along a 75m straight.

• Autocross – Vehicles are timed from a standing start over a short autocross track

which includes a range of features such as chicanes, straights, and curves as

defined in the rules.

• Endurance – Vehicles are timed over 22km (14 miles) of a track similar to an

autocross track. In addition, the fuel usage of each vehicle is measured to calculate

a fuel economy score.

For a vehicle to be successful in the competition it is necessary to perform credibly in all

the dynamic events. However, given the difference between events it is also clear that

different parameters of the vehicle will have different effects on the performance of the

vehicle. For example brakes are likely to be important for the autocross and endurance,

but are unlikely to be required at all in the acceleration and the skidpan events.


153

In order to investigate the design parameters of a vehicle, the evolutionary process was

applied to the parameters for each of these events individually. This showed that the ideal

parameters for each individual event are different. In addition, the parameters for the

vehicle were tuned for all the events according to the cumulative scores given in the rules

(SAE 2007):

Dynamic Events

Skid-Pad 50

Acceleration 75

Autocross 150

Fuel Economy 50

Endurance 350

Total Points 675

The points for each individual competition are evaluated by the following formulas:

Equation 6-1

Equation 6-2

Equation 6-3

Equation 6-4

5.21)/184.6(

1)/184.6(5.47

2min

2

+−−

=T

TSkidpan your

5.31)/8.5(

1)/8.5(5.71

min

+−−

=T

TonAccelerati your

5.71)/(

1)/(5.142

minmax

max +−−

=TT

TTAutocross your

1)/(

1)/(50

minmax

max

−−

=VV

VVEconomy your


154

Equation 6-5

6.2 Experiment Setup

The vehicle was assumed to be symmetric with the same tire model for all four tyres, and

the evolutionary algorithms were programmed to alter the following vehicle variables:

• Mass

• Centre of Gravity Height

• Weight Distribution (Front-to-Rear)

• Front-to-Rear Spring Stiffness Distribution (% Front)

• Front & Rear Spring Progression

• Front & Rear Anti-Roll Bar Stiffness

• Front & Rear Anti-Roll Bar Progression

• Front & Rear Roll-Centre Heights

• Differential Torque Bias Ratio

• Brake Balance (Front-to-Rear)

• Front Sprocket Teeth

• Rear Sprocket Teeth

• Front and Rear Initial Cambers

• Front and Rear Camber Coefficients

• Caster Angle (Front)

• Engine Power (Multiplier)

• Upper Gear Change RPM

• Lower Gear Change RPM

• Frontal Area

• Coefficient of Drag

• Coefficient of Lift (-ve)

• Centre of Pressure Height

• Centre of Pressure (Front-to-Rear)

501)/(

1)/(300

minmax

max +−−

=TT

TTEndurance your


155

This is a total of 29 parameters, and while it in no way represents all possible vehicle

parameters, it makes for a sizeable problem. It is customary for designers to aim for the

extreme limits of some of these variables. For example, designers generally aim to have a

vehicle that is as light as possible, with the lowest drag coefficient, and the highest engine

power. These design goals are clearly demonstrated in the results. However they are

included in the optimisation for two reasons: firstly an idea of the sensitivity of the

vehicle performance can be found; and secondly it is possible that there will be cases

where the designer is mistaken in these assumptions.

Following this study, another optimisation was run using similar parameters with a few

conceptual packages in which the designer has a clear choice between substantially

different options

• Option 1 – A different engine package is used that uses half of the fuel, produces

half of the power, but weighs 40kg less

• Option 2 – The designer is able to choose between using an aerodynamic package

or not. Choosing wings means the vehicle is able to produce 40kg downforce at

60kph, with a fixed weight penalty of 40kg. The winged vehicle also has an

increase in drag, and a raised centre of pressure height.

• Option 3 – A different set of tyres are used that have a higher peak grip, but are

more sensitive to camber angles.

These conceptual differences are very close to some of the variety in existing Formula

SAE vehicles. The numbers were based on anecdotal evidence. In the simulation, each of

these options was defined as either on or off.

The track for the autocross and endurance events was determined using onboard data

acquisition equipment on the University of Western Australia’s 2004 Formula SAE

vehicle. The data was collected during a test session prior to the 2005 United States

competition. The same data was used for both events. However in order to simulate the


higher speeds seen in the endurance event

curvature of the track for that event.

6.3 Non-Conceptual Optimisation Results

The following section summarises the results for optimisation

options (smaller engine, aerodynamic package, different tyres)

provided in Appendix B: Chapter 7

Figure

Figure 6-1 shows the best skidpan times for each of five optimisation runs. The average

of the five runs was 5.053s as shown by the dashed line, the best time was 5.041s, and the

range between best and worst runs was 0.028s (0.55%).


156

endurance event, a scaling factor was applied to decre

curvature of the track for that event.

Conceptual Optimisation Results

The following section summarises the results for optimisation without the conceptual

options (smaller engine, aerodynamic package, different tyres). Tabulated results are

Chapter 7 Non-Conceptual Results.

Figure 6-1 - Skidpan Times

shows the best skidpan times for each of five optimisation runs. The average

s was 5.053s as shown by the dashed line, the best time was 5.041s, and the


Kevin Hayward

a scaling factor was applied to decrease the

without the conceptual

Tabulated results are


s was 5.053s as shown by the dashed line, the best time was 5.041s, and the


Figure 6-2 shows the best acceleration times for each of five optimisation runs. The

average of the five runs was 4.339s as shown by the dashed line, the best time was

4.333s, and the range between best and worst runs was 0.014s (0.32%).

Figure 6-3 shows the best autocross times for each of five optimisation runs. The average

of the five runs was 21.388s as shown by the dashed line, the best time was 21.371s, and

the range between best and worst runs was 0.048s (0.22%).


157

Figure 6-2 - Acceleration Times

he best acceleration times for each of five optimisation runs. The



Figure 6-3 - Autocross Times

shows the best autocross times for each of five optimisation runs. The average


e between best and worst runs was 0.048s (0.22%).


he best acceleration times for each of five optimisation runs. The



shows the best autocross times for each of five optimisation runs. The average



Figure

Figure 6-4 shows the best endurance times for each of five optimisation runs. The ave

of the five runs was 18.556s as shown by the dashed line, the best time

the range between best and worst runs was 0.0

the time per lap.

In order to calculate the overall score

the minimum times for skidpan, acceleration, autocross, and endurance were taken as the

minimums found in the optimisations given above. This results in the following table:

EventSkidpanAcceAutocrossEndurance

Table

Fuel economy score was taken as the maximum 50 points.


158

Figure 6-4 - Endurance Times

shows the best endurance times for each of five optimisation runs. The ave

s as shown by the dashed line, the best time was 18.534

the range between best and worst runs was 0.054s (0.29%). The times shown represent

In order to calculate the overall score according to the equations presented in section



Event Time (s) Skidpan 5.041 Acceleration 4.333 Autocross 21.371 Endurance 18.534

Table 6-1 - Minimum Times

Fuel economy score was taken as the maximum 50 points.

Kevin Hayward

shows the best endurance times for each of five optimisation runs. The average

was 18.534s, and

The times shown represent

d in section 6.1




Figure 6-5 shows the best overall score for each of five optimisation runs. The average of

the five runs was 653.5 points as shown by the dashed line, the best score was 657.6

points, and the range between best and worst runs wa


159

Figure 6-5 – Overall Scores

shows the best overall score for each of five optimisation runs. The average of


points, and the range between best and worst runs was 7.7 point (1.18%).


shows the best overall score for each of five optimisation runs. The average of


s 7.7 point (1.18%).


160

Figure 6-6 – Overall Score vs. Generations

Figure 6-6 shows the improvement of the overall score against the number of generations.

The pattern shown is very similar to that seen in section 5.1. Both the minimum and

maximum scores for each of the runs are shown as upper and lower bound on the graph.

Near the beginning of the optimisation process there is a large variation of 45 points

(8.1%) between the minimum and maximum. By the end of the optimisation process the

difference is only 7.7 points (1.2%). This shows the reduction in the variation between the

different optimisation runs as the number of generations increase. It is expected that if

more computation time was applied, the variation between the runs could be further

reduced.

Score vs. Generation

540

560

580

600

620

640

660

1 6 11 16 21 26 31 36 41 46 51 56

Generation

Sco

re

Average

Minimum

Maximum


161

Run Skidpan Acceleration Autocross Endurance Fuel Total 1 44.9 62.8 147.6 344.6 50.0 649.9 2 44.4 67.2 149.5 346.6 50.0 657.6 3 45.2 63.8 147.5 343.4 50.0 649.9 4 44.9 65.1 148.1 347.8 50.0 656.0 5 43.7 70.6 146.7 342.9 50.0 653.9

Average 44.6 65.9 147.9 345.0 50.0 653.5 Range 1.5 7.9 2.8 5.0 0.0 7.7

Table 6-2 - Overall Score for Individual Runs

Table 6-2 shows the individual scores that made up the best candidate found in each of

the five runs. The best score for each event is highlighted. It is interesting to note that in

no individual run was the best score found for every event.

6.3.1 Discussion

The results show that each event requires different parameter values for an optimum

solution. Each of the events represents a real world racing challenge. The acceleration

event mimics a drag race, the skidpan is a real world test and could be compared to oval

track racing, while the autocross and endurance mimic track racing. However the results

from the overall optimisation will likely yield the best design directions for the vehicles.

The following table shows the simulated overall scores for the best of each optimisation.

Car Optimised for:

Skidpan (/ 50)

Acceleration (/75)

Autocross (/150)

Endurance (/350)

Fuel (/50)

Total (/675)

Skidpan 50 35 113 287 50 535 Acceleration 0 75 38 159 50 322 Autocross 44 58 150 349 50 651 Endurance 43 59 149 350 50 652 Overall Score 44 67 149 347 50 658

Table 6-3 - Overall Scores for Different Vehicles

Each of the optimised cars scored highest for its respective event. Cars optimised for the

Autocross and the Endurance event also perform very well overall. This was not

unexpected, as a car designed to compete well on a typical race track must have the

ability to both accelerate and corner well. However, a car tuned solely to corner well

does not need to brake or accelerate well, and a car tuned solely for acceleration does not

need to brake or corner well. An extreme example of this is a Drag Racing car, which

corners very poorly.


162

Some of the parameters required to tune for a particular event are fixed in the

manufacture of a car. An example in Formula SAE is the Centre of Gravity Fore-Aft

location. A table below shows the centre of gravity positions for the best vehicle in each

event.

Car Optimised for: Rear Weight % Skidpan 0.51 Acceleration 0.65 Autocross 0.60 Endurance 0.62 Overall 0.61

Table 6-4 - Ideal Rear Weight Bias

This shows that a car that corners well has close to equal weight front-to-rear, whereas a

car that accelerates well has a high rear weight bias, and a good track car lies somewhere

between these two.

Some other vehicle parameters are made to be adjustable. Results from the optimisation

of all the events can lead to an idea of how much adjustment is needed. For example we

can look at the initial rear camber.

Car Optimised for:

Rear Initial Camber

Skidpan -1.30 Acceleration 0.00 Autocross -0.76 Endurance -0.68 Overall -0.90

Table 6-5 - Ideal Rear Camber

In this case if the vehicle was constructed with rear camber tuneable between 0 and -1.3

degrees then the car can be set up optimally for each of the events. However care should

be taken not to treat variables independently. In this example, the rear camber angle of a

vehicle is also affected by the camber compensation, a parameter that is generally not

tuneable. In this example the ideal acceleration vehicle had no camber compensation,

while the other events required significant camber compensation. In this case, one


possible solution would be to increase the available tuning range

determined during later design stages.

As shown in Section 5.2.3

to overall performance

vehicle. For each of the optimisation runs minimum vehicle mass was ideal. This

relationship can clearly be seen by plotting all of the solutions as shown below.

It should be noted at this point that the mass was taken as the sum of the vehicle and

driver mass. A minimum vehicle mass of 220kg was taken, which represented the

approximate full fuel mass of the 2004 UWA FSAE vehicle

suit and helmet) was taken as 85kg.

7 Other FSAE cars are lighter than the 220kg of the 2004 UWA FSAE entry, including later entries from the

same team. However this number was used as minimising the mass was a primary design goal of the team


163

solution would be to increase the available tuning range

determined during later design stages.

5.2.3, relationships that directly link the value of certain parameters

all performance may be found. One example in this test was the overall mass of the



Figure 6-7 - Performance vs. Mass



uel mass of the 2004 UWA FSAE vehicle7. A driver mass (including

suit and helmet) was taken as 85kg.

Other FSAE cars are lighter than the 220kg of the 2004 UWA FSAE entry, including later entries from the



solution would be to increase the available tuning range; this would be

directly link the value of certain parameters

. One example in this test was the overall mass of the





. A driver mass (including

Other FSAE cars are lighter than the 220kg of the 2004 UWA FSAE entry, including later entries from the



164

This graph is from optimising the overall competition. In order to show it is a

minimisation problem the score found was subtracted from the maximum possible score.

This is represented on the Y axis with the title of Available Points. The line represents

the lower boundary of the set of points. This relationship is clearly one-way and would

indicate that further reductions in vehicle mass will likely yield higher performance.

Other parameters show relationships that are not as direct. The following example shows

the performance relationship for the Final Drive Ratio8.

and it represented the lowest mass achievable by the team at the time. The maximum mass is close to the

value of the 2001 UWA FSAE entry. As such; this mass range spans from the first to the last UWA FSAE

vehicle that the author was actively involved in designing. 8 The final drive ratio is calculated by dividing the number of rear sprocket teeth by the number of front

sprocket teeth.


These relationships not only show an ideal value but they give an idea of the sensitivity of

the result. For example

3. In addition, it shows that

it. Two dimensional plots can be made for each of the parameters against performance. It

is then very easy to visually assess whether a clear relationship can be found.

These graphs can also

environments. For this example we can look at the Torque Bias Ratio (TBR) of the

differential. For the autocross event

the Skidpan event the lowest TBR v

In a very simple analysis one could conclude that lower TBR values are better for

skidpan, and the higher better for autocross. For example

of these two values (~3.5)

design decision could easily be disproved by looking at the following graphs.


165

Figure 6-8 - Performance vs. Final Dr ive Ratio


the result. For example, the Final Drive Ratio in Figure 6-8 gives an ideal value

it shows that decreasing the drive ratio is more detrimental than

Two dimensional plots can be made for each of the parameters against performance. It

very easy to visually assess whether a clear relationship can be found.

These graphs can also help to determine how variables are linked to particular


differential. For the autocross event, a reasonably high TBR of around 6 was optimal. For

the Skidpan event the lowest TBR value of 1 (representing an open differential) was ideal.


skidpan, and the higher better for autocross. For example, designing a TBR in the middle

of these two values (~3.5) might end up working reasonably in both environments. Such a



ive Ratio


gives an ideal value of about

more detrimental than increasing

Two dimensional plots can be made for each of the parameters against performance. It

very easy to visually assess whether a clear relationship can be found.

how variables are linked to particular


a reasonably high TBR of around 6 was optimal. For

alue of 1 (representing an open differential) was ideal.


designing a TBR in the middle

might end up working reasonably in both environments. Such a



Figure 6-9 -

Figure 6- 10


166

- Torque Bias Ratio for Autocross

10 - Torque Bias Ratio for Skidpan

Kevin Hayward


167

What is clear in these graphs is that for the autocross event the time gets worse either side

of the ideal value of around 6. However for there to be any real improvement in skidpan

performance the TBR would have to be less than 2. Hence the original suggestion that

some value in-between 1 and 6 would be ideal seems highly unlikely. Further evidence

for this is that the ideal TBR value for the overall competition was found to be close to 6.

Care must be taken not to dismiss variables that do not show clear relationships to

performance. In the example of the lap time simulation (10 Appendix A: Lap Time

Simulation), an important calculation in each step of the simulation required the

computation of the roll moment distribution. This value determines how lateral load

transfer is distributed between the front and rear axles, in turn having a noticeable effect

on vehicle performance. In the optimisation run there are 9 different variables that affect

this calculation:

• Spring Stiffness Distribution (% Front)

• Front & Rear Spring Progression

• Front & Rear Anti-Roll Bar Stiffness

• Front & Rear Anti-Roll Bar Progression

• Front & Rear Roll-Centre Heights

Dependent variables like these are unlikely to show a clear relationship between

performance and their given value, but performance may still be very sensitive to changes

in any one of these variables because of their influence on roll moment distribution.

6.4 Conceptual Optimisation Results

The following section summarises the results for optimisation where the three conceptual

options (smaller engine, aerodynamic package, different tyres) were included. This was

achieved by adding a continuous variable with values between 0 and 1 for each of the

options. If the given value was less than 0.5 then the option was not used, if it was greater

than 0.5 it was. This creates a discrete on/off variable for the optimisation routine.

Tabulated results are provided in Appendix C: Chapter 7 Conceptual Results.


Figure

Figure 6-11 shows the best skidpan times for each of five

of the five runs was 5.066s as shown by the dashed line, the best time was 5.0

range between best and worst runs was 0.0

Figure

Figure 6-12 shows the best acceleration times for each of five optimisation runs. The

average of the five runs was 4.3

4.337s, and the range between best and worst runs was 0.01


168

Figure 6-11 - Skidpan Times


s as shown by the dashed line, the best time was 5.055


Figure 6-12 - Acceleration Times

shows the best acceleration times for each of five optimisation runs. The


s, and the range between best and worst runs was 0.014s (0.32%).

Kevin Hayward

optimisation runs. The average

55s, and the

shows the best acceleration times for each of five optimisation runs. The

s as shown by the dashed line, the best time was


Figure 6-13 shows the best autocross times for each of five optimisation runs. The

average of the five runs was 21.

21.79s, and the range between best and worst runs was 0.

Figure 6-14 shows the best endurance times for each o

average of the five runs was 18.

18.95s, and the range between best and worst runs was 0.


169

Figure 6-13 - Autocross Times

shows the best autocross times for each of five optimisation runs. The


s, and the range between best and worst runs was 0.33s (1.51

Figure 6-14 - Endurance Times

shows the best endurance times for each of five optimisation runs. The


s, and the range between best and worst runs was 0.1s (0.53%).


shows the best autocross times for each of five optimisation runs. The

hed line, the best time was

1.51%).

f five optimisation runs. The

s as shown by the dashed line, the best time was

%).


In order to calculate the overall score according to the equations present



EventSkidpanAccAutocrossEndurance

Table

Fuel economy was taken as 50 points where the smaller engine was used, and 25 points

where the larger engine was used.

Figure

Figure 6-15 shows the best overall score for each of five optimisation runs. The average

of the five runs was 613.7 points as shown by the dashed line,

points, and the range between best and worst runs was


170

In order to calculate the overall score according to the equations presented in section



Event Time (s) Skidpan 5.055 Acceleration 4.337 Autocross 21.790 Endurance 18.950

Table 6-6 - Minimum Times


Figure 6-15 – Overall Scores

shows the best overall score for each of five optimisation runs. The average

points as shown by the dashed line, the best score was 621.1

points, and the range between best and worst runs was 29.6 points (4.82%).

Kevin Hayward

ed in section 6.1




shows the best overall score for each of five optimisation runs. The average

core was 621.1


171

Figure 6-16 – Overall Score vs. Generations (Concep tual Optimisation)

Figure 6-16 shows the improvement of the overall score against the number of

generations. The pattern shown is very similar to that seen in section 5.1 and for the non-

conceptual optimisation. Both the minimum and maximum scores for each of the runs are

shown as the upper and lower bounds on the graph. Near the beginning of the

optimisation process there is a large variation of 37.5 points (7.2%) between the

minimum and maximum. By the end of the optimisation process the difference is still

quite high at 29.6 points (4.8%). It is expected that if more computation time was applied,

the variation between the runs could be further reduced.

Score vs. Generation

490

510

530

550

570

590

610

630

1 6 11 16 21 26 31 36 41 46 51 56

Generation

Sco

re

Average

Minimum

Maximum


172

Run Skidpan Acceleration Autocross Endurance Fuel Total 1 42.2 57.7 148.5 345.8 50.0 619.2 2 41.1 57.2 147.3 345.8 50.0 616.4 3 42.8 58.0 148.1 346.3 50.0 620.2 4 38.0 67.3 138.3 323.0 50.0 591.5 5 41.8 58.2 148.4 347.8 50.0 621.1

Average 41.2 59.7 146.1 341.7 50.0 613.7 Range 4.8 10.1 10.2 24.8 0.0 29.6

Table 6-7 - Overall Score for Individual Runs

Table 6-7 shows the individual scores that made up the best candidate found in each of

the five runs. The best score for each event is highlighted. As with the non-conceptual

optimisation, no individual run gave the best score found for every event.

6.4.1 Discussion

The first note of interest is the increased range between optimisation runs for each event.

This is most notably observed for the overall competition score. For the non-conceptual

optimisation the final range between the best and worst runs was 7.7 points. In the

conceptual optimisation the range was 29.6 points. This was caused by one of the five

runs in the conceptual analysis having a different conceptual choice. This shows the

sensitivity of the concepts which is discussed in section 6.6.

The following table analyses how a vehicle optimised for each event would perform

overall.

Car Optimised for:

Skidpan (/50)

Acceleration (/75)

Autocross (/150)

Endurance (/350)

Fuel (/50)

Total (/675)

Skidpan 50 0 87 253 50 441 Acceleration 18 75 66 204 25 388 Autocross 42 51 150 349 25 617 Endurance 42 53 149 350 25 619 Overall 42 58 148 348 25 621

Table 6-8 - Overall Scores for Different Vehicles ( Conceptual Optimisation)

The results end up similar to those seen in the previous section. Each optimised car

performs optimally in its respective event.


With the different conceptual options

performance have been altered. The best example of these is the mass of the vehicle. This

relationship is shown below.

Table 6

Two of the options involved changing the mass of the vehicle. This in turn has affected

the graph. The ideal mass is now no longer the minimum mass. This is an important

design discovery because


173

With the different conceptual options, some of the relationships betwe


relationship is shown below.

6-9 - Performance vs. Mass (Conceptual Optimisation)

options involved changing the mass of the vehicle. This in turn has affected


because performance may not always increase as mass is decreased


some of the relationships between variables and


Performance vs. Mass (Conceptual Optimisation)

options involved changing the mass of the vehicle. This in turn has affected


as mass is decreased.


174

At this stage a design decision for the different options is much more informed and is

summarised below:

Option 1: Different Engine

The lower mass of the engine was certainly advantageous for skidpan, however the lack

of power meant that each other event required the original heavier, and more powerful

engine to increase performance. This option would not be used because overall

performance would be reduced.

Option 2: Addition of an Aerodynamic Package

The higher mass due to the aerodynamic package meant that it would not feature on a car

designed for acceleration. However for all other events the higher mass was more than

compensated for by the increase in grip due to down-force. This option should be used.

Option 3: Different Tyre

The tyre with higher grip, but increased camber sensitivity appeared to be the better of the

two tyres for all of the events. This option would be used.

Each of these options was similar to design decisions made within the University of

Western Australia Formula SAE team. In these particular cases the different engine was

not chosen, which appeared to be the right choice. Wings were not added to the car,

which appeared to be the wrong choice. A tyre with lower camber sensitivity was chosen,

however this decision was made on the basis of tyre wear encountered during testing, a

variable not included in the simulation. It would appear from these results that the team

should investigate the addition of wings to future vehicles as an avenue to increase

performance.


175

6.5 Comparison to Existing Vehicles

There is value in comparing the results of the simulation to the specifications of an

existing vehicle. The vehicle used for comparison will be the 2004 University of Western

Australia (UWA) Formula SAE vehicle. The vehicle was quite successful, winning the

Design Award (for the best designed vehicle) and coming 2nd overall in the 2005 United

States Formula SAE Competition. (This was the largest competition of the year with 140

teams.) The vehicle was the fourth (and last) UWA vehicle in which the author was

involved in the design process. During that time the on-track performance of each

successive vehicle improved.

The following graph shows the comparison of the autocross simulation and the original

data.

Table 6-10 - Simulated vs. Actual Vehicle Velocity

Vehicle Velocity

0

5

10

15

20

25

Vel

ocity

(m

/s)

Actual (22.68s) Simulated (22.30s)


176

The correlation is reasonable, considering the simulation was designed with limited

accuracy to increase computational speed. It is also not surprising that the simulated time

is faster. The author, the driver of the vehicle, has had prior experience in the car and its

predecessors; however he is not a professional driver. This point is highlighted as the

simulation and optimisation process will give an idea of the capabilities of the vehicle, the

actual final performance is likely to be different. The simulation always runs at the limit

of the vehicle performance envelope, while the driver is not able to do that.

A similar graph can be used to compare the simulations of the existing vehicle and one

that has been optimised using the evolutionary algorithms.

Table 6-11 - Optimised vs. UWA Vehicle (Autocross)

The optimised vehicle shows a simulated improvement of 0.61s (3%). The following

table shows the difference in competition scores between the optimised car and the UWA

vehicle.

Vehicle Velocity

0

5

10

15

20

25

Vel

ocity

(m

/s)

Optimised (21.69s) UWA Vehicle (22.30s)


177

Car Skidpan

(/50) Acceleration

(/75) Autocross

(/150) Endurance

(/350) Fuel (/50)

Total (/675)

UWA Vehicle 33 60 134 306 25 559 Optimised 42 58 148 348 25 621

Table 6-12 - Optimised vs. UWA Vehicle (Overall Sco re)

The difference in scores of 62 points is very significant. Notably, the optimised vehicle

outperformed the UWA Vehicle in all but one event. This is significant for two reasons:

first it offers an idea of how to make changes in the design of the car to improve

performance; secondly it shows that the evolutionary optimisation process was able to

provide a solution that was superior to 4 years of design processes that did not use these

methods.

6.6 Parameter Sensitivity

Throughout this and previous sections, there has been some mention of determining the

sensitivity of the final result to various parameters. This is invaluable in the design

process as it provides knowledge to the designer about the required manufacturing

tolerances, as well as suggesting tuning ranges for particular parameters. Furthermore as

the design process continues, opportunities may arise to alter the boundaries of some of

the tuned parameters. For example if engine power was shown to be very important to the

final result, the design team might investigate forced induction systems.

In the previous sections there were examples of plotting parameter values for all of the

tested solutions against their relative performance. This has been shown to provide insight

into various relationships. Other methods include finding the range and/or standard

deviation for a particular variable over a number of evolutionary runs. For example we

can look at the Brake Bias Parameter for the non-conceptual optimisation of the vehicle.

The values, range and standard deviation are given below.


178

Run Brake

Balance 1 1.18 2 0.93 3 1.18 4 1.02 5 0.96

Average 1.05 Range 0.26

Std. Dev. 0.10

Table 6-13 - Brake Balance for 5 Runs

Clearly there is not much variation between each of the runs. This indicates that the ideal

brake balance is likely to be within the range provided. Secondly, the standard deviation

would indicate that approximately 95% of the brake balance solutions would fall between

±2 standard deviations of the mean. This would give 95% of the values between 0.85 and

1.25. This gives an idea of the tuneable range for this parameter. Such information could

be useful immediately for the starting parameters for the detailed design of an appropriate

pedal box and brake bias adjustment device. This is supported by the previous method of

plotting the performance against the parameter values as shown below (Figure 6-17).


This process can be repeated for any of the relevant variables.

varying each parameter of the final solution

performance of the vehicle. This is a procedure shown

parameters of the best solution of the conceptual optimisation were altered to test

sensitivity. The table below shows the

This gives an idea of wh

greatest margin. However, caution must be

parameters have been optimised to suit the original conceptual choice. In reality

performance decreases


179

Figure 6-17 - Ideal Brake Bias Range

This process can be repeated for any of the relevant variables. Another approach

each parameter of the final solution to determine its relative effect on the

performance of the vehicle. This is a procedure shown by Crolla et al. (2002).


. The table below shows the effects of deciding on different concepts

Configuration Score Difference Original 621 0 Different Engine 535 -86 Remove Wings 586 -35 Different Tyre 615 -6

Table 6-14 - Conceptual Sensitivity

This gives an idea of which of the conceptual options affects the performance by

margin. However, caution must be exercised because

parameters have been optimised to suit the original conceptual choice. In reality

performance decreases caused by changing concepts is likely to be smaller than indicated.


Another approach involves

ts relative effect on the

by Crolla et al. (2002). To test this,


different concepts.

ich of the conceptual options affects the performance by the

exercised because the rest of the vehicle

parameters have been optimised to suit the original conceptual choice. In reality, the

by changing concepts is likely to be smaller than indicated.


180

This method should mainly be used as a post optimisation check rather than a decision

making process. The process can be used on other parameters as well. A few examples

are given below.

Configuration Score Difference Original 621 0 Mass (+5%) 602 -19 Mass (-5%) 641 20 Engine Power (+5%) 629 8 Engine Power (-5%) 612 -9 Weight Dist. (+5% Rearwards) 616 -5 Weight Dist. (-5% Rearwards) 619 -2 Final Drive (+5%) 619 -2 Final Drive (-5%) 621 0 Caster (+5%) 621 0 Caster (-5%) 621 0

Table 6-15 - Selected Parameter Sensitivity

This analysis could be carried out with all parameters, and if percentages of change are

small enough, some linear optimisation of the final solution can be found. This method

can also be used as a way to rank the effect of individual parameters on the final solution.

For further detail on this method see Crolla et al. (2000).

6.7 Conclusion

This chapter has shown how evolutionary algorithms can be effectively used in the design

process for a difficult problem. Good results were found within short time frames. The

final result of the optimisation showed a simulated improvement of close to 10% over an

already existing and successful UWA vehicle. The process highlighted some conceptual

changes that could be made to future vehicles to improve performance. The optimisation

process was also time-effective, as a single run of the algorithm for the full competition

took approximately 1.5 hours of computation time on the author’s desktop computer, well

within the specifications for the algorithm outlined in Chapter 4. The algorithm was also

able to find good solutions for the different environments of the different competition

events. This would indicate that if conditions were to change, the process could be

repeated quickly with good results.


181

Using the evolutionary process also has the advantage of providing a large amount of data

from throughout the search process. If this data is recorded, simple data mining and

visualization techniques can be used to determine the relationship of parameters to the

overall performance of the problem.

There were a number of different types of parameters used in the same optimisation

routine without problems. There were Boolean variables such as the choice of concepts,

discrete variables such as sprocket sizes, and continuous variables such as spring

stiffness. There were parameters that appeared to have a direct link to performance, as

well as parameters that interacted with each other to affect performance. Despite this

variation the algorithm gave good robust results.

The evolutionary algorithm approach appears to be very effective during the conceptual

design phase. In this particular case, many of the parameters in the design process are set

very early on, but have a large effect on overall performance. A clear example would be

the choice of engines. The evolutionary algorithms could be used to make quick and

accurate decisions during the early stages of design.


182

7 Conclusion

Evolutionary algorithms are a convenient and robust method to automate the search for

appropriate design solutions to increase the efficiency of the mechanical design process.

A study of current design processes indicated that the evolutionary algorithm’s main use

would be in some automation of the decision making process. The following

requirements of an evolutionary algorithm used to increase design efficiency were

determined:

• Easy to apply to the relevant problem

• Quick

• Reliable

• Scalable

• Ability to adapt to a large number of problems

An algorithm was developed that met these requirements, and was successfully applied to

three motorsport design problems. Using the algorithms to test conceptual changes, which

would occur at the very start of the design process, was shown to provide significant

performance increases to the particular problem of racing car design. This would indicate

that the earlier the algorithms are used, the better the final result.

In order to apply the evolutionary algorithm all that was necessary was to replace one line

of code with a call to a predefined performance function. The designer must supply the

performance function, including a relevant representation of the variables. Representation

of the variables being optimised is very important. Poor variable representation reduces

the performance of the optimisation. The goal should be to try, where possible, to achieve

a direct mapping between performance and the variable. Representation of a design as a

number of relevant variables, and a method to assess its performance are both required in

the product development method, so is quite separate from the application of the

evolutionary algorithms. Once this has been created, the application of the evolutionary

algorithm is quite simple.


183

Time required to perform the optimisation process was reduced by tuning the algorithm to

give decent results for a relatively small number of iterations. Significant increases in the

convergence speed of the algorithm were found first by using tuning parameters of

different values than those traditionally prescribed. In addition, evolutionary parameters

involved with the control of mutation strength, and distribution were added and made

self-adaptive. These showed significant improvements over traditional algorithms for the

limited number of iterations available.

The algorithm was tuned to be reliable using a mix of uni-modal and multi-modal test

functions. In addition the trade-off between assigning computation to increase

performance or reliability was studied. The more computation time applied to a single

optimisation run, improves the performance, however computation should be split

between multiple runs to improve the reliability of the result.

Scalability of the algorithm was a natural extension of the traditional evolutionary

algorithms. Any number of variables can be used, as the most important tuning

parameters are defined as functions of the number of variables. Testing showed good

results for the algorithm with between 5 and 30 variables without changing any of the

tuning parameters. It is expected that further testing could be conducted and that good

results would be found for an even larger number of variables.

The same evolutionary code was used without modification for each of the three different

motorsport problems, as well as the suite of 13 test functions. This represented a variety

of conditions; large and small problems, uni and multi-modal problems, discrete and

continuous variables, dependent and independent variables, and a range of complexity of

the problems. Focusing on self-adaptive features in the algorithm as well as tuning for a

variety of different test functions, allowed the creation of an algorithm that works

credibly in a number of different situations. By normalising the input variables of the

evolutionary algorithm, a common form was introduced for any variable relevant to the

optimisation. This allowed treatment of any continuous or discrete variable of any

specified range. In each case exactly the same evolutionary code was used, with the


184

replacement of the relevant performance function. In each case the algorithm gave good

results. It is expected that many more problems could be approached with exactly the

same algorithm.

Of particular interest for the design process was the ability to analyse the data created by

the optimisation process. Relationships of the variables with respect to performance can

be analysed. Much of the data gained is in areas of the search space that offer high

performance potential. This allows the designer to set more suitable ranges of the

variables along with finding clear one-to-one relationships between particular variables

and the performance of the design. This is an area in which much work could still be

done.

Particular note should be made about problems with multiple design objectives, as they

can occur quite frequently in real-world situations. While multi-objectives are

cumbersome to deal with using the simple algorithms presented in this thesis, it is

possible to apply weightings to particular objectives to find a suitable result. The Formula

SAE competition provides one such example. The car must be designed to perform well

in five different events, but its final score is a weighted average of the five events. Also

where possible it may be better to frame the multiple objectives into a larger problem that

has only one objective. In Chapter 5 suspension geometry was discussed with a number

of objectives. In reality this is part of a larger problem of lap-time speed. The same

geometrical problem could be included in a lap time solution which has a single objective.

In conclusion, an evolutionary algorithm was developed with unique mutation control

methods; that was successfully applied to a number of design problems. In each case both

the efficiency of the process and the quality of the final solution were improved.


185

8 Recommendations for Future Work

There are many avenues where future work may be conducted. This work has focused on

the tuning and application of a particular type of evolutionary algorithm. Other

evolutionary algorithms (or other non-linear multi-modal solvers) could be tested to

provide a comparison between algorithms. However the “No free lunch theorem”

(Wolpert and Macready, 1995) would indicate that on average, algorithms will perform

equally over a complete set of problems. Also given the number of possible approaches to

solving these types of problems it is unlikely that a fair comparison of algorithms could

be achieved.

Of more interest would be the further development of modifications to the algorithm

already presented. During the development of the algorithm it was evident that using

tuning parameters with different values to the accepted norm allowed greater performance

for restricted computational time. In particular, self-adaptation of the mutation parameters

gave a notable increase in performance for the conditions tested. Further work could be

conducted to test these modifications for more problems, and under different amounts of

available computation.

Given that the design process for complex systems can continue over extended periods of

time it may be desirable to implement a continuous evolutionary algorithm. This could

allow the current best results to be displayed at any given time as a design reference point.

Furthermore modifications to the environment or simulation models can be incorporated

as new information about the problem is discovered.

Another area where time might be saved is in the introduction of models with varying

levels of complexity. Early in the evolutionary search process simpler models could be

adopted that provide a rough approximation of the problem, but are quick. As the process

continues the model can be made more accurate as fine-tuning of the solution is needed.


186

Robustness of the final solutions could be introduced as a part of the performance

evaluations. Non-robust solutions would incur a performance penalty. This could be a

way to introduce the idea of design tolerances into the optimisation. Achieving this will

require some local approximation of the search-space to avoid increasing the computation

time significantly.

During the design process it appears that much use could be made of the analysis of the

data gained throughout the optimisation process. In this thesis there was some coverage of

this by plotting parameters against performance, as well as comparing parameter values

between the results of the same optimisation performed a number of times. More

advanced data-mining techniques could be employed to better determine possible

relationships between parameters and performance. Of particular interest would be in

determining where performance may have a relationship with a number of dependent

variables.

The main recommendation of the author is to continue applying these techniques to new

problems. Through application of the algorithm the true strengths and weaknesses of this

approach will be revealed.


187

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10 Appendix A: Lap Time Simulation

This appendix provides discussion on the lap time simulation created by the author. There

are two primary goals in studying vehicle dynamics. The first is to improve the

performance envelope of a vehicle while the second is to improve the comfort of the

occupants (including driver) of the vehicle. These two goals are often in conflict with

each other and must be balanced for a particular application. In motorsport the overriding

concern is the performance envelope, whiling maintaining the minimum level of driver

comfort to allow effective control of the vehicle. As previously stated in section 2.5, the

primary goal of race car design is the minimisation of the time taken to traverse a given

circuit. Computation methods to determine lap times have been developed over the last 50

years. They have become a useful method of applying the theories of vehicle dynamics to

improve race car performance. Hacker et al. (2000) note that little work has been done,

however, on how to effectively use these simulations in conjunction with advanced

optimisation techniques in a systematic way.

10.1 Simulation Requirements

There are three fundamental requirements for the development of a lap time simulation to

be used effectively with evolutionary algorithms. These are listed below:

• Returns a single value for vehicle performance

• Computationally quick

• Represents the real-world problem with acceptable accuracy

Possibly the most critical requirement of the simulations are that they are computationally

quick. If the simulation is not quick to perform it is difficult to apply an evolutionary

approach as the optimisation technique is highly dependent on the iterative step. The

speed required is linked to the level of accuracy achievable in the simulations. Racing car

simulation is an inherently complex problem. Racing car tyres operate near the limits of

adhesion. This makes the dynamics of race cars extremely non-linear.(Gadola et al. 1996)


198

However it is important that the simulations exhibit a minimum level of accuracy such

that results do not contradict what would be expected in the physical situation. As

available computation power is increased it is expected that more complex models may be

used to improve the accuracy of the results.

In order to use a simulation program effectively, it is necessary to provide accurate inputs

and validate the results. Purnell (1998) states:

“Accurate simulation demands a great deal of quantitative detail about the

car and this in turn is influencing the direction of the test and measurement

equipment used. Mathematical modelling of the car from simple force

balance models to highly sophisticated lap time simulation or dynamic

response modelling requires accurate data and lots of it. The more realistic

the simulation demanded, the more data is required.”

The simulation model developed must represent the available data. For conceptual

studies simpler models can be used, with the understanding that the results are an

indication of performance, rather than an exact value for what the car can achieve.

The models used in the simulation should reflect this.

10.2 Lap Time Simulation

“… the simulation engineer will demand editing tools to play with the

parameter values to determine their importance to performance, tools to

find optimum values, presentation software and database managers to

handle flow in and out of the simulation software.” (Purnell, 1998)

A Lap time simulator is mainly used during two stages of vehicle development (Crolla &

Deakin, 2000); the design process of a new vehicle, and setup of an existing vehicle.

During the tuning, or setup, of the vehicle, predictions can be made regarding vehicle

adjustments such as changes to aerodynamic trim, springs, dampers, suspension geometry

etc. Using a lap time simulator in this case may reduce the number of iterations that need


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to be performed to maximise a race car’s performance for a given track. The second main

use for such a simulator is during the design stage of a vehicle in order to study the effects

of alterations to particular design parameters. It is apparent that assessing small changes

to a pre-existing vehicle would require increased accuracy in both vehicle and circuit

models over simulations used in the early design stages of a vehicle. Siegler et al. (2000)

states:

“... it has a use during the initial design phase after which parameters, such

as the centre of gravity position, cannot be changed. The vehicle can then be

produced so that the fundamental design parameters are close to the

optimum values.”

According to Milliken & Milliken (1995) it is probable that Mercedes Benz were the first

to analytically simulate the performance of a race car around a circuit as early as the late

1930’s. It was clear that by the 1950’s the technique had become more regularly used.

Reference is made to the Cornell Aeronautical Laboratory (CAL) using such analysis to

aid in designing a race circuit at Watkins Glen between 1954 and 1956. Early efforts at

lap time simulation were based on steady state equations calculated by hand. Modern

computing has allowed for increased model complexity and accuracy.

It is difficult to ascertain the current state of the field. Lap time simulators are usually

designed and implemented by individual race teams or companies. Given the competitive

nature of the motorsport industry this information rarely enters the public domain. Deakin

and Crolla (2000) note:

In the 40+ years that the codes have been in use, few papers have been

written which describe how a racing car can be made to go faster and

what design or set-up parameters should be changed to achieve this, such

is the secrecy that enshrouds motor racing.


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There are differing levels of model complexity that may be used in lap time simulation. It

is logical that the more complex models require increased computation time. Hence a

trade-off must be made between simulation accuracy and computation time. This

compromise varies for different applications. There are many procedures that can be used

to model a vehicle’s performance over a given circuit. Steady-state, quasi-static and

transient strategies were compared during cornering by Siegler et al. (2000). These three

strategies give an idea of the differing levels of complexity that are used in lap time

simulations. Allen (1994) states that “a model must have sufficient complexity for a given

application but should not be overly complicated”. The reasoning behind minimising

complexity where possible aids in reducing software coding, checking and validation

efforts, also aiding in reducing errors. Candelpergher et al. (2000) also states that one of

the goals of their lap-time simulator was to have a high computation speed to enable the

race engineer to estimate the effect of various setup settings in the shortest possible time.

10.2.1 Program Structure

“When attempting to capture the dynamics of an automobile in a computer-

based simulation, the amount of detail available for inclusion is almost

limitless. In the preliminary design stage, however, most of these details are

of little or no importance. These details, no matter how well researched, can

never completely overcome shortcomings inherent to the basic design.”

(Hacker et al. 2000)

The lap time simulator written for this thesis was designed for parameter studies during

the early stages of racing car design. The complexity of the simulation reflects the need

for fast computation rather than absolute accuracy. A few of the key simulation properties

below:

• Vehicle model is quasi-static

• No driver model is included

• Tracks are assumed to be perfectly flat with uniform grip properties


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These have important implications on the results gained from the simulation. Since the

model is quasi-static transient dynamics of the vehicles will be ignored and response to

inputs will be assumed to be instantaneous. This will not allow the program to be used to

study the effects of vehicle response or to enable selection of desirable damper

characteristics. The lack of a driver model assumes that the driver is able to operate the

vehicle at the limit of its performance 100% of the time. This is an unrealistic situation

and does not account for vehicle parameters that may result in a vehicle that is difficult to

control. Likewise assuming track as being perfectly flat with uniform grip is also an

unrealistic situation. However to correct these assumptions would require markedly

increased computation time for the simulation. Were this simulation to be used in a

design cycle it would be likely that once basic vehicle parameters have been decided

more accurate vehicle models could be used in the detail design stages.

Gadola et al. (1996) applies the same assumptions above. Sieglar et al. (2000) also notes

that present simulations (From the year 2000) find the cornering ability of the vehicles

using the quasi-static solution. This allows quick and efficient simulations without

requiring a complicated time-dependent solution. Furthermore, Sieglar et al. show close

correlation between the results of the quasi-static approach and a much more complex

transient model. They do note the following:

“However the transient solution, although more complicated, takes into

account vehicle factors that are not accounted for in the other solution

techniques. A fully transient solution to lap time simulation could thus be

sought which would allow more accurate tuning of a greater number of

vehicle parameters.”

Allen et al. (1994) also note that gross handling characteristics and vehicle stability can

be managed with relatively simple models, but load transfer and tyre force response are

critical. The simulation developed contains models of the engine, drive-train, brake

system, aerodynamics, wheel loading, suspension geometry, and tyres. These are briefly

discussed in the following sections (10.2.2 to 10.2.9).


202

The vehicle design parameters and the intended path are both inputs to the simulation. A

path is created by breaking a racing line into a number of small sections each with a

length and a curvature. This path definition file is either generated by the path

optimisation routine detailed in section 5.1 or from onboard vehicle data acquisition such

as presented by Cassanova et al. (2001). The simulation algorithm was based on a

forwards and then backwards sweep of the data. The algorithm is outlined below:

The first sweep calculates vehicle acceleration using the following steps:

1. The track index, n, is set to 1.

2. The velocity of the vehicle, v, is set the initial vehicle velocity v0.

3. The curvature and length of the track definition are taken from the track

definition file at position n.

4. The maximum lateral acceleration, and resultant velocity vmax, of the vehicle is

calculated for the given curvature.

5. If this velocity is greater than v then the amount of acceleration available to the

vehicle is calculated. Using this acceleration value and the length of the track

segment v is reset according to the equation of motion: v2 = vo2 + 2as

6. Otherwise v is set to vmax

7. n is increased by one

8. Steps 3-7 are repeated until all the track points have been analysed (n is equal to

the number of points in the track definition file)

The second sweep accounts for the braking of the vehicle using the following steps:

9. The curvature and length of the track definition are taken from the track

definition file at position n.

10. vmax for each point is taken from the velocity trace generated from the

acceleration sweep.

11. If v is less than vmax then the amount of deceleration available to the vehicle is

calculated. Using this deceleration value and the length of the track segment v is

reset according to the equation of motion: v2 = vo2 + 2as

12. Otherwise v is set to vmax.

13. n is decreased by 1


14. Steps 9-13 are repeated until all the track points have been analysed

Once this is completed the times for each of the track segments are calculated using the

segment length and the velocity of that segment. Th

compute the lap time.

10.2.2 Engine Model

The engine torque is determined by interpolating a torque curve

setup file. This setup file is created by the program user.

defined in a table for uniform steps in engine RPM. This table may be calculated from

external engine simulations, semi

dynamometer testing.

ignored. The torque from the engine is multiplied by a throttle posit

Equation 10-1. tp is the throttle position and is a

interpolated torque.


203

are repeated until all the track points have been analysed


segment length and the velocity of that segment. The segment times are summed to

Engine Model

Figure 10-1 – Throttle Dependent Engine Curve

The engine torque is determined by interpolating a torque curve

This setup file is created by the program user. The torque of the engine is


external engine simulations, semi-empirical from analysis of test data, or

ometer testing. Given that the simulation is quasi-static, the inertia of the engine is

The torque from the engine is multiplied by a throttle posit

is the throttle position and is a real value between 0 and 1,


are repeated until all the track points have been analysed (n=0)


e segment times are summed to

Curve

The engine torque is determined by interpolating a torque curve defined in the vehicle

The torque of the engine is


empirical from analysis of test data, or by engine

static, the inertia of the engine is

The torque from the engine is multiplied by a throttle position value as shown in

real value between 0 and 1, is the maxτ


204

Equation 10-1

A simple clutch model is used. During the launch phase of a vehicle the clutch model is

activated. A specified launch RPM is maintained until the torque available from the

engine is less than the torque that can be resisted by the tyres. At that point the clutch

model is made inactive and RPM is linked to the slip ratio of the rear wheels and the

gearing between the engine and the wheels.

10.2.3 Drive-train

The drive-train model is designed around the setup for a typical Formula SAE vehicle

with a motorcycle engine, which includes a gearbox, front sprocket, rear sprocket, chain,

differential, and CV or tripod joints. It is assumed that engine torque is only transmitted

to the rear wheels. The drive-train model includes driveline efficiency, torque

multiplication, gear change procedure, and the torque split between the two wheels.

The driveline efficiency affects the available torque from the engine by applying a

multiplier to the engine torque as follows:

Equation 10-2

Where

Equation 10-3

This torque is then multiplied by the drive ratios to determine the torque available at the

wheels. This is done as follows:

maxττ ×= tpengine

enginedrivetraindrive τητ ×=

cvaldifferentichaingearboxdrivetrain ηηηηη ×××=


205

Equation 10-4

: Final drive ratio

Equation 10-5

Gear change points are defined for both an up change and a down change. An up change

is initiated if the engine RPM is above the up change point and the car is not already in

the highest available gear. Similarly a down change is initiated if the engine RPM is

below the down change point given and the car is not already in the lowest available gear.

The time taken for a gear change is a set value. During a gear change it is assumed that

the current gear ratio is set to zero (indicating neutral). This results in no available thrust

to the driving wheels.

Torque is split between the wheels according to the differential type. The simulation

accounts for 3 simple types of differentials:

• Open differential

• Locked Differential

• Limited Slip Differential

For an open differential the torque split is even between the two wheels, while the wheels

travel at different velocities. This is given by the two following equations:

drivefinaltyres R ττ ×=

finalR

TeetharSprocket

ketTeethFrontSprocrRatioCurrentGeaductionimaryfinal n

nRRR

ReRePr ××=


206

Equation 10-6

Equation 10-7

Where is the outside wheel radius and is the inside wheel radius

Since torque is split evenly it is only possible to transmit double the torque determined by

the wheel with the lowest torque capacity. Since the thrust applied by both wheels is even

there will be no contribution towards a vehicle’s yaw moment and as such the open

differential affects the handling of a vehicle the least.

For a locked differential the two wheels are linked together and their velocities are

identical, hence:

Equation 10-8

The torque on each tyre is split based on tyre loading conditions, tyre geometry and the

radius of the turn. Since the torque applied by both wheels is different a yaw moment is

placed on the vehicle that must be resisted by the front tyres.

A torque sensing limited slip differential (LSD) is modelled as a mix between both the

open and a locked differential and comes in a variety of forms. The LSD will allow a

torque differential between the wheels to exist up to a certain bias ratio such that:

Equation 10-9

RightLeft ττ =

1−=−=i

o

i

iodiff r

r

r

rrv

0r ir

0=diffv

minmax ττ ×= BiasR


RBias is the torque bias ratio

differential, for low bias ratios it will behave

Figure

10.2.4 Brakes

Allen et al. (1994) not

performance direction

all four wheels with a bias setting between the front and rear of the vehicle. Brak

assumed even side-to-


207

the torque bias ratio. For high bias ratios the LSD performs like a locked

differential, for low bias ratios it will behave more like an open differential.

Figure 10-2 – Locking Effect vs. Torque Bias Ratio

note that front-to-rear proportioning plays a significant role in limit

performance directional stability. To allow for this it is assumed that braking occurs on

all four wheels with a bias setting between the front and rear of the vehicle. Brak

-side.


For high bias ratios the LSD performs like a locked

more like an open differential.

Locking Effect vs. Torque Bias Ratio

rear proportioning plays a significant role in limit

To allow for this it is assumed that braking occurs on

all four wheels with a bias setting between the front and rear of the vehicle. Braking is


208

Equation 10-10

Equation 10-11

The amount of braking torque is limited by the end that has the least available braking

capacity. The available braking capacity is determined using the tyre model described in

section 10.2.9.

10.2.5 Aerodynamics

The aerodynamics model is a simple one involving constant coefficients for lift and drag.

The model determines that both lift and drag operate on a single point named the centre of

pressure (COP). The COP has a fore-aft location to determine the lift split front to rear

and a height to determine the pitch moment generated by the drag force. The centre of

pressure is assumed to lie along the centre line of the vehicle. Lift and drag are calculated

according to the following functions:

Equation 10-12

Equation 10-13

The coefficients of lift and drag may be determined computationally or through data

acquisition of physical experiments. Failing this, appropriate estimations may be used.

The aerodynamic lift and drag producing elements are assumed to be mounted to the

sprung mass of the vehicle. They will then affect the resultant wheel loads for any given

speed. The wheel loading due to the pitch moment created by drag is as follows:

arBrakesBiassFrontBrake R Reττ ×=

sRightBrakeLeftBrakes ττ =

2

2vACF L

LIFT

ρ=

2

2vACF D

DRAG

ρ=


209

Equation 10-14

Equation 10-15

The wheel loading due to the lift is as follows:

Equation 10-16

Equation 10-17

: The percentage of lift at the front of the vehicle.

This model does not take into account the nonlinear characteristics of down force which

can dramatically alter the vertical dynamics of a vehicle. (Scott Floyd & Harry Law,

1994) These effects are primarily due to altering wing pitch angles and ground effects.

However in order to provide an effective non-linear aerodynamics model it would be

necessary to invest in wind tunnel testing or complicated fluid dynamics simulations. This

is beyond the scope of this thesis but may be worth studying at a later opportunity.

10.2.6 Wheel Loading

Given that the simulation is quasi-steady state it is assumed that all of the forces of the

wheel are reacted by suspension springs. The transient forces of the damper are ignored.

The wheel loads are calculated through the following equations:

w

hFF COPDRAG

FRONT

×−=

w

hFF COPDRAG

REAR

×=

FRONTLIFTFRONT LFF ×−=

)1( FRONTLIFTREAR LFF −×−=

FRONTL


210

Equation 10-18

Equation 10-19

Equation 10-20

Equation 10-21

is the roll force distribution applied to the front wheels and is calculated as a function

of the springs and anti-roll bars. The roll force distribution is the number which defines

how the lateral load transfer is distributed between the front and rear axles. This load

transfer affects vehicle directional stability because of the influence of normal load on

tyre-force response. (Allen et al. 1994)

10.2.7 Suspension Geometry

There are a number of suspension force mechanisms that may be included in the model,

these include: compliance, damping, bump stops, auxiliary roll stiffness, squat / lift

effects, roll steer (toe effects) and tyre camber angle. (Allen et al. 1994) A simple

suspension geometry model has been included in the model to allow for studies of basic

camber, toe, Ackermann, caster and KPI (king pin inclination) settings. In order to

maintain fast computation speeds these will be treated as linear functions of wheel travel

and steering angle where appropriate. Compliance is ignored as suspension connection

points in race cars are generally spherical joints and components are generally designed

for high stiffness.

442

20

20)(0)(0

0

AvCL

L

hAvC

T

hmaR

L

hmaFF LFcpD

F

latFlongFLFL

ρρ−−+−=

442

20

20)(0)(0

0

AvCL

L

hAvC

T

hmaR

L

hmaFF LFcpD

F

latFlongFRFR

ρρ−−−−=

44

)1(

2

20

20)(0)(0

0

AvCL

L

hAvC

T

hmaR

L

hmaFF LRcpD

R

latFlongRLRL

ρρ−+

−++=

44

)1(

2

20

20)(0)(0

0

AvCL

L

hAvC

T

hmaR

L

hmaFF LRcpD

R

latFlongRLRL

ρρ−+

−−+=

FR


211

The deflection of the wheels is dependent on wheel loads, and is calculated as follows:

Equation 10-22

The toe and camber angles are calculated from the wheel deflections as follows:

Camber:

Equation 10-23

Toe:

Equation 10-24

The camber angle of the front tyres, are also affected by the angle of the wheels due to

steering and the effects of caster and KPI. The caster is approximated as a linear change

to camber dependent on steered angle, while the KPI is assumed as linear also it always

adds camber to a wheel. This changes the camber equations for the front wheels to:

Camber:

K

FF 0−=δ

0γδγ γ += C

0αδα α += C


212

Equation 10-25

Equation 10-26

The Ackermann effect will cause the inside wheel to be turned through a different angle

to the outside wheel. Typically the inside wheel will turn more than the outside wheel

consistent with the tighter radius it is running on. However it is not uncommon for race

cars to exhibit anti-Ackermann. This is shown in the following equation:

Equation 10-27

Values of greater than 1 indicate positive Ackermann where values less than one

indicate anti-Ackermann.

10.2.8 Tyre Modelling

“Racing cars operate at the peak of the tyre force curves, where the force is

the greatest. The tyres therefore need to be modelled as closely as possible,

to allow accurate prediction of the race car” (Siegler et al. 2000)

Allen et al. (1994) also state that tyre forces represent a significant proportion (probably

a majority) of vehicle dynamics behaviour. It is not unreasonable to expect that the

accuracy of the lap time simulation will be largely based on the accuracy of the tyre

model.

A common tyre model used for simulations is the Pacejka “magic formula model” for

combined longitudinal and lateral loading conditions. The model features load and

camber dependent friction tyre characteristics and computes lateral force as a function of

KFRFLFFFLFL CCC θθγδγ θγ +++= 0

KFRFRFFFRFR CCC θθγδγ θγ +++= 0

OUTSIDEAINSIDE C θθ ×=

AC


slip angle and longitudinal force as a function of slip percentage

temperature effects are not included in the model. The Pa

characterised in Equation

The coefficients of the model are determined by curve fitting to results from tyre testing.

These tyre tests are generally conducted under steady state conditions. The image below

shows an example of th

different loads, pressures, speeds, cambers, and slip angles. Forces in three directions and

three moments are measured on the tire as well as the temperature during the test. The

measurements are made once the tire has rea

An example of the results obtained during such a test is shown below

9 Note these results show the data in a filtered form suitable for use in a lap time simulation.

F


213

slip angle and longitudinal force as a function of slip percentage

temperature effects are not included in the model. The Pacej

Equation 10-28. (Pacejka & Bakker, 1991)

Equation 10-28

fficients of the model are determined by curve fitting to results from tyre testing.


shows an example of the equipment used to test tires (Figure 10-3



measurements are made once the tire has reached steady state.

Figure 10-3 - Calspan Tire Testing

An example of the results obtained during such a test is shown below

Note these results show the data in a filtered form suitable for use in a lap time simulation.

))}]arctan((arctan{sin[ ααα BBEBCDF −−=


slip angle and longitudinal force as a function of slip percentage. Tyre wear and

cejka magic formula is

fficients of the model are determined by curve fitting to results from tyre testing.


3). Tires are tested with



An example of the results obtained during such a test is shown below (Figure 10-4).9

Note these results show the data in a filtered form suitable for use in a lap time simulation.

))}]


214

Figure 10-4 - Calspan Test Results (Rouelle 2007)

There are a few assumptions that are made in this testing that provide differences between

the results and what is seen in a high performance application such as racing. To start

with there is a delay before full side force generation. A wheel is required to roll a certain

amount in order for the contact patch of the tyre to deform to the slip angle. (Allen et al.

1994) This effect is not implemented in the model as it introduces increased complexity

to the calculation procedure. An assumption is made that the tyre is always able to

produce maximum grip for a given load. However on a real race track surface grip

variations would not allow this. (Crolla & Deakin, 2000) The steady state conditions also

eliminate the effects of temperature on tire grip. This is a large part of the mechanics in a

race car. For example the graph below shows the relationship between temperature of the

tires and the resultant lateral acceleration, as measured by infrared tire temperature

sensors mounted on a racing vehicle.10

10 Figure 10-5 was created using data obtained by the author while working for a race team (Vehicle and

team details withheld)


215

Figure 10-5 - Lateral Acceleration vs. Tire Tempera ture

Implementing these transient tire properties would add complexity and running time to

the lap time simulation. An empirical curve fitting tyre model like the Pacejka model was

the most appropriate type of model to apply to this solution. One problem with using the

Pacejka model computationally is that the ideal slip angle for the tyre is not a given

property and in order to find it, numerical solving techniques must be used. In order to

avoid this situation and create a faster model for the simulation the author decided to

design a custom tyre model for the program.

10.2.9 Tyre Model

A simplified model for a tire was developed for the simulation. Lateral and longitudinal

forces were created for the tyre, while the moments were ignored. The model had the

following features:

• Load dependent friction

• Camber dependent friction

• Defined ideal slip angle


216

The following image shows the shape of a typical tyre curve. (Haney, 2003) There are

three main regions, the linear region, transitional region, and the frictional region. These

regions can clearly be seen by the slope of the curve. This slope is termed the cornering

stiffness of a tire.

Figure 10-6 - Typical Tire Curve

The model developed for the lap time simulation is based on numerically integrating a

given cornering stiffness. The general equation for the cornering stiffness is shown below.

Equation 10-29

K is the cornering stiffness and x is the slip angle. Desired properties of the tires are used

to calculate the coefficients. The cornering stiffness is scaled such that the initial

cornering stiffness is set at 1. Hence A+C = 1. Since approaches 0 as x increase, C

will define a linear drop in tire performance into the frictional range of the tires. Hence

this parameter is set dependent on the desired drop of rate in the frictional region of the

curve. This differs from the Pacejka model which will tend towards zero cornering

stiffness as the slip angle increases. This is of little concern as the majority of the

calculations in a lap time simulation will focus on the transitional portion of the tire

curve, where maximum force is found. The ideal slip angle is found when:

CAeKnBx += −

nBxAe−


217

Equation 10-30

Simple definition of the ideal slip angle was the main purpose for creating this new

model. To achieve this, the ideal slip angle is set at 1 for the coefficients and the equation

is scaled after a table has been created. Hence the coefficient for B can be easily found:

Equation 10-31

This leaves n as a shape factor for the cornering stiffness. A low value will limit the initial

linear region, while a high value will make for a sharper transition. Hence the only two

independent variables in the equation that control the shape of the cornering stiffness

curve are C and n. The equation can be rewritten as:

Equation 10-32

The effect of C can be seen below. Clearly C has a dominant effect after the ideal slip

angle has been reached.

CAenBx −=−

)ln(A

CB −=

CeCKnx

C

C

+−= −)

)1(ln(

)1(


218

Figure 10-7 - Cornering Stiffness for Different C V alues

The effect of n can be seen below. n has most effect on the cornering stiffness before the

ideal slip angle has been reached.

Cornering Stiffness vs. Slip AngleDifferent C Values, n=3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Normalised Slip Angle

Nor

mal

ised

Cor

nerin

g S

tiffn

ess

C (-0.05)

C (-0.1)

C (-0.2)


219

Figure 10-8 - Cornering Stiffness for Different n V alues

In order to go from the cornering stiffness equations to an equation of friction vs. slip

angle it is first necessary to shift the cornering stiffness according to the desired ideal slip

angle. The ideal slip angle is defined as a quadratic function of load as follows.

Equation 10-33

Three points are used to define the quadratic. The first is the ideal slip angle at zero load

to give the point (0, ), the second point is a reference slip angle for a given

reference load to give the point . The solutions to the parameters are as

follows:

Cornering Stiffness vs. Slip AngleDifferent n Values, C=-0.1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Normalised Slip Angle

Nor

mal

ised

Cor

nerin

g S

tiffn

ess

n (2)

n (3)

n (6)

refzrefIdeal baFsign γγγγ +−−= 20 ))((

0γ 0γ

),( refrefzF γ−


This gives a relationship like the following graph. (

Figure 10

The friction is load dependent by the same equation

b =


220

Equation 10-34

Equation 10-35

This gives a relationship like the following graph. ( , ,

10-9 - Ideal Slip Angle vs. Load

by the same equation as the Pacejka Equations:

Equation 10-36

refzF

ba

−

=

))(( 00 refrefsign γγγγ −−

10=refγ 80 =γ =−refzF

baFz )1( −=µ

Kevin Hayward

)

3


b defines as the maximum friction obtainable by the tire.

capability of the tire due to load.

seen in the following graph. In this example the lateral capability dropped off by 10% for

each kilo-Newton of load. The maximum friction was 1.3.

The effect of this loading factor is clearly shown in the graphs for lateral force against slip

angle for different tyre loads.(

angle changes for a given load. In

angle is also increasing.


221

defines as the maximum friction obtainable by the tire. a defines the drop

capability of the tire due to load. is the vertical load on the tire


of load. The maximum friction was 1.3.

Figure 10-10 - Lateral Force vs. Ver tical Load


angle for different tyre loads.( Figure 10-11) This figure also shows how the ideal slip

angle changes for a given load. In this case as the normal load increases the idea slip

angle is also increasing.

zF


defines the drop-off in lateral

is the vertical load on the tire. This relationship is


tical Load


) This figure also shows how the ideal slip

this case as the normal load increases the idea slip


222

Figure 10-11 - Tyre Model for Different Loads

The effects of camber on the tire include camber thrust that increases the directional

lateral capability of the tire and degradation in the maximum lateral force. This usually

results in the outside tires of a car producing increased grip due to camber thrust being

greater than the lateral capability degradation, while the inside tires suffer due to both

camber thrust and grip degradation. This results in an ideal camber for a pair of wheels.

These effects are defined in two parameters in the equation below.

Tyre Model for Different Loads

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1 2 3 4 5 6 7 8 9 10

Slip Angle (deg)

Late

ral F

orce

(kN

)

1.0 kN

2.0 kN

3.0 kN

Ideal Slip Angle


- Friction with camber effects included

- Friction without camber effects included

- Camber Angle

- Shift in lateral capability due to camber thrust

- Degradation in lateral capability due to camber

Figure 10-12 shows th

camber angle of the tyre

Figure

Here we can see that the gain due to camber is not as great as the loss

Hence if both sides of the car had the same load, the ideal camber would be 0. However

as the outside wheel in a turn has a larger load it usually helps to have some camber in the

suspension system.

γµ /w

γµ /wo

γ

c

d


223

Equation 10-37

iction with camber effects included

Friction without camber effects included

Camber Angle

Shift in lateral capability due to camber thrust

Degradation in lateral capability due to camber

shows the relationship between the lateral friction coefficient and the

camber angle of the tyre.

Figure 10-12 - Lateral Friction vs. Camber Angle

Here we can see that the gain due to camber is not as great as the loss



)1)(( // γγµµ γγ dcwow −+=


between the lateral friction coefficient and the

Lateral Friction vs. Camber Angle

Here we can see that the gain due to camber is not as great as the loss to negative camber.




224

Figure 10-13 shows the effect of different cambers on the final tyre model. It should be

noted that there is no evidence of camber thrust for slip angles near zero. This is a

fundamental difference with this model and what happens in the real situation. However

the lap-time simulation is designed such lateral grip calculations will almost always be

performed at close to ideal slip angles for grip.

Figure 10-13 - Tyre Model for different cambers

In order to combine both the lateral and longitudinal grip of the tyre a simple ellipse

model is used. The maximum longitudinal and lateral grips are both calculated for each

point of the simulation. This is shown in Figure 10-14. Longitudinal and lateral grip is

calculated such that it always lies on an ellipse described by the equation below:

Tyre Model for Different Cambers

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 1 2 3 4 5 6 7 8 9 10

Slip Angle (deg)

Late

ral F

orce

(kN

)

0 Deg

-2 Deg

-4 Deg


225

1

22

=

+

MaxLong

Long

MaxLat

Lat

g

g

g

g

Equation 10-38

Lateral Grip

Longitudinal GripMaximumLongitudinalGrip

MaximumLateralGrip

1

22

=

+

MaxLong

Long

MaxLat

Lat

g

g

g

g

Figure 10-14 - Friction Ellipse


226

11 Appendix B: Chapter 7 Non-Conceptual Results

Time s

Front Sprocket - Mass kg Rear Sprocket - COG Height m Final Drive - Weight Dist. (F-R) - Front Initial Camber deg K Front N/m Rear Initial Camber deg K Rear N/m Front Camber Coeff. deg/m Front Spring Prog. m-1 Rear Camber Coeff. deg/m Rear Spring Prog. m-1 Caster Angle deg Front ARB Nm/deg Engine Power Mult. - Rear ARB Nm/deg Upper Change RPM rpm Front ARB Prog. deg-1 Lower Change RPM rpm Rear ARB Prog. deg-1 Frontal Area m2

Front RC Height m Coeff.of Drag - Rear RC Height m Coeff.of Lift - Brake Balance (F-R) - COP Height m Torque Bias Ratio - COP Long. (F-R) -

11-1: Units


227

1 2 3 4 5 Avg Range % Rng

Time 5.069 5.050 5.051 5.056 5.041 5.053 0.028 Mass 306.0 306.0 306.0 306.1 306.0 306.0 0.1 0.00 COG Height 0.200 0.200 0.200 0.200 0.201 0.200 0.001 0.00 Weight Dist. (F-R) 0.607 0.500 0.551 0.567 0.512 0.547 0.107 0.71 K Front 28112 23785 30560 20386 23556 25280 10174 0.34 K Rear 21888 26215 19440 29614 26444 24720 10174 0.34 Front Spring Prog. 0.154 0.159 0.192 0.262 0.045 0.162 0.217 0.43 Rear Spring Prog. 0.195 0.000 0.062 0.453 0.337 0.209 0.453 0.91 Front ARB 3562 2557 3409 4000 2743 3254 1443 0.36 Rear ARB 1490 3912 1226 1149 3961 2348 2812 0.70 Front ARB Prog. 0.316 0.000 0.376 0.007 0.156 0.171 0.376 0.75 Rear ARB Prog. 0.029 0.000 0.000 0.212 0.011 0.050 0.212 0.42 Front RC Height -0.041 -0.036 0.017 0.044 0.018 0.000 0.086 0.86 Rear RC Height 0.020 0.000 0.054 0.038 0.055 0.033 0.055 0.55 Brake Balance (F-R) 1.59 1.90 1.51 0.31 0.47 1.16 1.59 0.80 Torque Bias Ratio 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 Front Sprocket 12 12 15 17 17 14.6 5 0.81 Rear Sprocket 45 40 54 40 40 43.8 14 0.98 Final Drive 3.75 3.33 3.60 2.35 2.35 3.08 1.40 0.56 Front Initial Camber 0.000 -0.895 -0.851 -1.290 -0.528 -0.713 1.290 0.22 Rear Initial Camber -0.137 -1.333 -0.041 -0.063 -0.160 -0.347 1.292 0.22 Front Camber Coeff. -15.4 0.0 -85.9 -79.2 -82.5 -52.6 85.9 0.90 Rear Camber Coeff. -70.9 -7.9 -30.0 -41.1 -40.8 -38.2 63.0 0.66 Caster Angle 0.21 0.00 8.30 9.34 3.78 4.33 9.34 0.93 Engine Power Mult. 0.96 1.03 1.10 0.96 0.99 1.01 0.14 0.72 Upper Change RPM 12126 12158 11425 12494 11135 11868 1359 0.91 Lower Change RPM 6104 5897 5305 5436 7130 5974 1825 0.61 Frontal Area 1.49 1.56 1.71 1.50 1.45 1.54 0.26 0.66 Coeff.of Drag 0.508 0.586 0.500 0.503 0.698 0.559 0.198 0.99 Coeff.of Lift -0.650 -0.650 -0.650 -0.650 -0.639 -0.648 0.011 0.02 COP Height 0.203 0.395 0.212 0.220 0.382 0.282 0.192 0.96 COP Long. (F-R) 0.664 0.500 0.515 0.628 0.502 0.562 0.164 0.41

Table 11-2 : Skidpan Results


228


Time 4.333 4.337 4.347 4.346 4.333 4.339 0.014 Mass 306.0 306.0 306.1 306.0 306.0 306.0 0.1 0.00 COG Height 0.400 0.400 0.363 0.400 0.400 0.392 0.037 0.19 Weight Dist. (F-R) 0.650 0.650 0.650 0.650 0.650 0.650 0.000 0.00 K Front 12133 37202 33456 10000 10000 20558 27202 0.91 K Rear 37867 12798 16544 40000 40000 29442 27202 0.91 Front Spring Prog. 0.034 0.321 0.000 0.485 0.442 0.256 0.485 0.97 Rear Spring Prog. 0.006 0.005 0.500 0.496 0.500 0.301 0.495 0.99 Front ARB 32 981 752 4000 1811 1515 3968 0.99 Rear ARB 1424 1170 4000 2515 1647 2151 2830 0.71 Front ARB Prog. 0.500 0.264 0.223 0.382 0.269 0.328 0.277 0.55 Rear ARB Prog. 0.081 0.095 0.042 0.138 0.479 0.167 0.437 0.87 Front RC Height 0.049 0.041 -0.050 -0.031 0.050 0.012 0.100 1.00 Rear RC Height 0.082 0.000 0.071 0.057 0.000 0.042 0.082 0.82 Brake Balance (F-R) 0.72 2.05 0.46 0.48 0.05 0.75 2.00 1.00 Torque Bias Ratio 9.60 9.06 1.74 1.16 8.00 5.91 8.44 0.94 Front Sprocket 12 12 12 12 13 12.2 1 0.24 Rear Sprocket 50 49 48 45 54 49.2 9 0.64 Final Drive 4.17 4.08 4.00 3.75 4.15 4.03 0.42 0.17 Front Initial Camber -4.268 0.000 -6.000 -3.906 -2.578 -3.350 6.000 1.00 Rear Initial Camber -0.025 0.000 -0.025 -0.042 0.000 -0.018 0.042 0.01 Front Camber Coeff. -4.4 -21.2 -79.6 -13.3 -95.0 -42.7 90.6 0.95 Rear Camber Coeff. -0.9 -0.8 -11.2 -2.5 -0.1 -3.1 11.1 0.12 Caster Angle 1.17 10.00 0.29 3.43 7.65 4.51 9.71 0.97 Engine Power Mult. 1.10 1.10 1.10 1.10 1.10 1.10 0.00 0.00 Upper Change RPM 11016 11028 11302 11596 11021 11193 581 0.39 Lower Change RPM 7344 8000 8000 5241 7863 7290 2759 0.92 Frontal Area 1.40 1.41 1.40 1.40 1.40 1.40 0.01 0.02 Coeff.of Drag 0.501 0.500 0.500 0.500 0.500 0.500 0.001 0.01 Coeff.of Lift -0.632 -0.650 -0.650 -0.524 -0.631 -0.617 0.126 0.19 COP Height 0.400 0.400 0.201 0.209 0.312 0.304 0.199 0.99 COP Long. (F-R) 0.892 0.821 0.514 0.592 0.756 0.715 0.378 0.94

Table 11-3 : Acceleration Results


229


Time 21.377 21.376 21.399 21.371 21.419 21.388 0.048 Mass 306.0 306.0 306.3 306.1 306.2 306.1 0.3 0.01 COG Height 0.201 0.200 0.200 0.200 0.200 0.200 0.001 0.01 Weight Dist. (F-R) 0.596 0.588 0.630 0.602 0.545 0.592 0.085 0.57 K Front 25936 29639 37576 28127 24536 29163 13040 0.43 K Rear 24064 20361 12424 21873 25464 20837 13040 0.43 Front Spring Prog. 0.500 0.472 0.062 0.398 0.468 0.380 0.438 0.88 Rear Spring Prog. 0.500 0.223 0.019 0.113 0.295 0.230 0.481 0.96 Front ARB 3993 3662 3896 2775 3424 3550 1219 0.30 Rear ARB 2148 2089 1934 3161 1820 2230 1341 0.34 Front ARB Prog. 0.474 0.500 0.373 0.498 0.094 0.388 0.406 0.81 Rear ARB Prog. 0.226 0.212 0.241 0.173 0.466 0.264 0.293 0.59 Front RC Height 0.035 -0.048 0.002 0.015 0.025 0.006 0.084 0.84 Rear RC Height 0.051 0.026 0.100 0.006 0.012 0.039 0.094 0.94 Brake Balance (F-R) 1.02 1.08 0.79 0.93 1.36 1.04 0.57 0.28 Torque Bias Ratio 7.37 9.00 6.96 6.06 5.92 7.06 3.08 0.34 Front Sprocket 16 18 16 15 14 15.8 4 0.65 Rear Sprocket 44 54 43 41 40 44.4 14 0.97 Final Drive 2.75 3.00 2.69 2.73 2.86 2.81 0.31 0.13 Front Initial Camber -0.531 -0.517 -0.477 -0.560 -0.738 -0.565 0.261 0.04 Rear Initial Camber -0.470 -0.700 -1.318 -0.764 -0.670 -0.784 0.847 0.14 Front Camber Coeff. -93.6 -89.3 -89.0 -93.6 -95.0 -92.1 6.0 0.06 Rear Camber Coeff. -94.9 -92.2 -87.0 -94.9 -94.5 -92.7 7.9 0.08 Caster Angle 5.66 4.87 6.10 5.56 4.48 5.33 1.62 0.16 Engine Power Mult. 1.09 1.10 1.10 1.10 1.09 1.09 0.01 0.07 Upper Change RPM 12444 11866 12323 12500 12500 12327 634 0.42 Lower Change RPM 5181 5017 6971 7003 5549 5944 1986 0.66 Frontal Area 1.41 1.40 1.40 1.68 1.44 1.46 0.27 0.69 Coeff.of Drag 0.501 0.501 0.501 0.502 0.578 0.516 0.077 0.38 Coeff.of Lift -0.650 -0.650 -0.650 -0.650 -0.650 -0.650 0.000 0.00 COP Height 0.241 0.257 0.278 0.239 0.345 0.272 0.106 0.53 COP Long. (F-R) 0.681 0.656 0.715 0.649 0.591 0.658 0.125 0.31

Table 11-4 : Autocross Results


230


Time 18.534 18.587 18.569 18.536 18.556 18.556 0.054 Mass 306.0 306.3 306.0 306.4 306.0 306.2 0.4 0.01 COG Height 0.200 0.200 0.200 0.200 0.200 0.200 0.000 0.00 Weight Dist. (F-R) 0.621 0.642 0.597 0.583 0.649 0.619 0.066 0.44 K Front 26481 38286 34657 30975 26831 31446 11805 0.39 K Rear 23519 11714 15343 19025 23169 18554 11805 0.39 Front Spring Prog. 0.486 0.383 0.027 0.385 0.486 0.353 0.459 0.92 Rear Spring Prog. 0.025 0.374 0.500 0.314 0.500 0.343 0.475 0.95 Front ARB 2722 3786 4000 3739 3894 3628 1278 0.32 Rear ARB 1607 1645 0 340 3439 1406 3439 0.86 Front ARB Prog. 0.047 0.021 0.500 0.109 0.470 0.229 0.479 0.96 Rear ARB Prog. 0.126 0.492 0.462 0.409 0.097 0.317 0.395 0.79 Front RC Height 0.044 -0.032 -0.030 -0.014 0.041 0.001 0.076 0.76 Rear RC Height 0.023 0.059 0.097 0.003 0.021 0.040 0.094 0.94 Brake Balance (F-R) 0.85 0.78 1.00 1.03 0.75 0.88 0.27 0.14 Torque Bias Ratio 9.73 9.22 9.28 3.26 4.62 7.22 6.47 0.72 Front Sprocket 18 18 18 15 16 17 3 0.52 Rear Sprocket 49 51 48 44 42 46.8 9 0.64 Final Drive 2.72 2.83 2.67 2.93 2.63 2.76 0.31 0.12 Front Initial Camber -0.877 -0.367 -2.064 -0.331 -0.881 -0.904 1.733 0.29 Rear Initial Camber -0.678 -0.820 -0.923 -1.137 -0.387 -0.789 0.750 0.13 Front Camber Coeff. -93.0 -85.6 -48.0 -95.0 -95.0 -83.3 47.0 0.50 Rear Camber Coeff. -95.0 -91.9 -95.0 -95.0 -94.9 -94.4 3.1 0.03 Caster Angle 6.53 5.09 3.38 7.47 5.87 5.67 4.09 0.41 Engine Power Mult. 1.10 1.06 1.10 1.10 1.10 1.09 0.04 0.22 Upper Change RPM 11630 11577 12156 12401 11954 11944 825 0.55 Lower Change RPM 5116 5975 7603 5561 7465 6344 2487 0.83 Frontal Area 1.40 1.40 1.40 1.41 1.40 1.40 0.01 0.03 Coeff.of Drag 0.509 0.501 0.505 0.500 0.500 0.503 0.009 0.04 Coeff.of Lift -0.647 -0.650 -0.650 -0.650 -0.644 -0.648 0.006 0.01 COP Height 0.289 0.237 0.347 0.200 0.200 0.255 0.147 0.73 COP Long. (F-R) 0.697 0.694 0.607 0.613 0.693 0.661 0.091 0.23

Table 11-5 : Endurance Results


231


Score 649.9 657.6 649.9 656.0 653.9 653.5 7.7 Mass 306.0 306.0 306.0 306.4 306.2 306.1 0.4 0.01 COG Height 0.200 0.200 0.200 0.200 0.200 0.200 0.000 0.00 Weight Dist. (F-R) 0.565 0.607 0.560 0.583 0.624 0.588 0.064 0.43 K Front 36579 32273 39842 25718 34731 33828 14124 0.47 K Rear 13421 17727 10158 24282 15269 16172 14124 0.47 Front Spring Prog. 0.312 0.180 0.117 0.195 0.405 0.242 0.288 0.58 Rear Spring Prog. 0.500 0.082 0.243 0.338 0.086 0.250 0.418 0.84 Front ARB 3321 3860 3942 2730 726 2916 3216 0.80 Rear ARB 2752 2725 1349 184 415 1485 2569 0.64 Front ARB Prog. 0.353 0.152 0.213 0.500 0.500 0.344 0.348 0.70 Rear ARB Prog. 0.019 0.184 0.349 0.388 0.058 0.199 0.369 0.74 Front RC Height 0.046 0.031 -0.008 0.050 0.041 0.032 0.058 0.58 Rear RC Height 0.019 0.051 0.100 0.000 0.066 0.047 0.099 0.99 Brake Balance (F-R) 1.18 0.93 1.18 1.02 0.96 1.05 0.26 0.13 Torque Bias Ratio 2.11 6.01 7.09 8.08 1.47 4.95 6.61 0.73 Front Sprocket 16 17 18 15 12 15.6 6 0.96 Rear Sprocket 48 53 54 46 40 48.2 14 0.95 Final Drive 3.00 3.12 3.00 3.07 3.33 3.10 0.33 0.13 Front Initial Camber -0.595 -0.940 -0.748 -1.218 -1.063 -0.913 0.624 0.10 Rear Initial Camber -0.497 -0.898 -0.305 -0.602 -1.050 -0.670 0.744 0.12 Front Camber Coeff. -95.0 -89.9 -81.3 -84.9 -43.9 -79.0 51.1 0.54 Rear Camber Coeff. -83.8 -93.3 -94.6 -95.0 -92.9 -91.9 11.2 0.12 Caster Angle 5.06 4.22 7.84 6.68 7.34 6.23 3.62 0.36 Engine Power Mult. 1.09 1.10 1.10 1.10 1.10 1.10 0.01 0.07 Upper Change RPM 11879 11463 11311 12471 11000 11625 1471 0.98 Lower Change RPM 5221 5060 6052 5000 5628 5392 1052 0.35 Frontal Area 1.68 1.40 1.67 1.47 1.41 1.53 0.28 0.69 Coeff.of Drag 0.500 0.501 0.589 0.500 0.500 0.518 0.089 0.45 Coeff.of Lift -0.650 -0.650 -0.650 -0.650 -0.643 -0.649 0.007 0.01 COP Height 0.292 0.335 0.278 0.392 0.220 0.303 0.173 0.86 COP Long. (F-R) 0.510 0.659 0.584 0.561 0.689 0.600 0.178 0.45

Table 11-6 : Competition Results


232

12 Appendix C: Chapter 7 Conceptual Results

Time s

Front Sprocket - Mass kg Rear Sprocket - COG Height m Final Drive - Weight Dist. (F-R) - Front Initial Camber deg K Front N/m Rear Initial Camber deg K Rear N/m Front Camber Coeff. deg/m Front Spring Prog. m-1 Rear Camber Coeff. deg/m Rear Spring Prog. m-1 Caster Angle deg Front ARB Nm/deg Engine Power Mult. - Rear ARB Nm/deg Upper Change RPM rpm Front ARB Prog. deg-1 Lower Change RPM rpm Rear ARB Prog. deg-1 Frontal Area m2

Front RC Height m Coeff.of Drag - Rear RC Height m Coeff.of Lift - Brake Balance (F-R) - COP Height m Torque Bias Ratio - COP Long. (F-R) -

12-1: Units


233


Time 5.055 5.056 5.098 5.057 5.063 5.066 0.043

Smaller Engine 1 1 1 1 1

Wing Package 1 1 1 1 1

New Tyres 1 1 1 1 1

Mass 296.0 296.0 296.0 296.0 296.1 296.0 0.1 0.002 COG Height 0.250 0.250 0.255 0.250 0.251 0.251 0.005 0.024

Weight Dist. (F-R) 0.519 0.504 0.590 0.512 0.532 0.531 0.086 0.576

K Front 18034 24576 29498 19151 23004 22852 11464 0.382

K Rear 31966 25424 20502 30849 26996 27148 11464 0.382

Front Spring Prog. 0.364 0.489 0.062 0.300 0.499 0.343 0.437 0.873

Rear Spring Prog. 0.496 0.471 0.500 0.500 0.356 0.465 0.144 0.288

Front ARB 84 549 691 1799 309 686 1715 0.429

Rear ARB 1770 3224 3733 353 1203 2057 3381 0.845

Front ARB Prog. 0.164 0.500 0.201 0.222 0.478 0.313 0.336 0.672

Rear ARB Prog. 0.253 0.343 0.500 0.006 0.099 0.240 0.494 0.988

Front RC Height 0.021 0.048 0.036 -0.015 -0.004 0.017 0.063 0.632

Rear RC Height 0.000 0.096 0.093 0.068 0.055 0.062 0.096 0.959

Brake Balance (F-R) 1.33 1.67 1.32 1.76 0.06 1.23 1.71 0.853

Torque Bias Ratio 1.03 1.00 1.00 1.03 1.00 1.01 0.03 0.004

Front Sprocket 16 18 18 16 12 16 6 0.918

Rear Sprocket 40 50 41 48 40 43.8 10 0.674

Final Drive 2.50 2.78 2.28 3.00 3.33 2.78 1.06 0.426

Front Initial Camber -0.181 0.000 -1.901 -0.214 -0.396 -0.538 1.901 0.317

Rear Initial Camber -1.596 -1.728 -1.227 -2.125 -2.136 -1.762 0.909 0.152

Front Camber Coeff. -92.7 -76.3 -91.2 -94.8 -88.0 -88.6 18.6 0.195

Rear Camber Coeff. -94.8 -95.0 -91.8 -91.3 -95.0 -93.6 3.7 0.039

Caster Angle 9.30 8.77 5.86 9.99 9.49 8.68 4.13 0.413

Engine Power Mult. 0.49 0.50 0.49 0.54 0.53 0.51 0.06 0.554

Upper Change RPM 11432 11000 12494 11684 12278 11778 1494 0.996

Lower Change RPM 6570 6522 6906 6499 5901 6479 1005 0.335

Frontal Area 1.67 1.72 1.41 1.57 1.64 1.60 0.30 0.760

Coeff.of Drag 0.500 0.500 0.572 0.500 0.540 0.522 0.072 0.361

Coeff.of Lift -0.650 -0.649 -0.648 -0.650 -0.649 -0.649 0.002 0.003

COP Height 0.268 0.429 0.251 0.449 0.251 0.330 0.198 0.989

COP Long. (F-R) 0.718 0.611 0.746 0.706 0.757 0.708 0.146 0.365

Table 12-2 : Skidpan Results (Conceptual)


234


Time 4.347 4.351 4.337 4.351 4.340 4.345 0.014



New Tyres 1 0 1 1 0

Mass 306.0 306.0 306.0 306.0 306.0 306.0 0.0 0.000 COG Height 0.388 0.400 0.400 0.397 0.400 0.397 0.012 0.060

Weight Dist. (F-R) 0.650 0.623 0.650 0.650 0.650 0.645 0.027 0.179

K Front 10000 34795 18108 39950 10000 22571 29950 0.998

K Rear 40000 15205 31892 10050 40000 27429 29950 0.998


Rear Spring Prog. 0.500 0.168 0.498 0.290 0.000 0.291 0.500 1.000

Front ARB 1364 0 2146 2378 3740 1926 3740 0.935

Rear ARB 4000 1868 3012 3854 0 2547 4000 1.000

Front ARB Prog. 0.000 0.045 0.412 0.064 0.466 0.197 0.466 0.932

Rear ARB Prog. 0.028 0.090 0.000 0.483 0.028 0.126 0.483 0.967

Front RC Height 0.048 0.050 -0.043 -0.027 -0.050 -0.004 0.100 0.999

Rear RC Height 0.097 0.006 0.074 0.087 0.026 0.058 0.091 0.912

Brake Balance (F-R) 2.05 0.38 2.05 0.21 0.05 0.95 2.00 0.999


Front Sprocket 14 12 12 15 12 13 3 0.467

Rear Sprocket 53 47 49 54 49 50.4 7 0.436

Final Drive 3.79 3.92 4.08 3.60 4.08 3.89 0.48 0.195

Front Initial Camber -0.793 -0.436 -0.717 -5.183 -1.228 -1.671 4.747 0.791

Rear Initial Camber 0.000 0.000 -0.013 0.000 0.000 -0.003 0.013 0.002

Front Camber Coeff. -83.4 -23.3 -4.0 -87.5 0.0 -39.6 87.5 0.921

Rear Camber Coeff. 0.0 -0.6 -1.3 -79.1 -0.5 -16.3 79.1 0.832

Caster Angle 0.06 4.80 0.23 5.01 5.52 3.13 5.46 0.546


Upper Change RPM 11674 11111 11021 11190 11234 11246 653 0.435

Lower Change RPM 5094 5000 6538 6657 7799 6218 2799 0.933

Frontal Area 1.40 1.43 1.40 1.40 1.40 1.41 0.03 0.079

Coeff.of Drag 0.500 0.501 0.500 0.500 0.500 0.500 0.001 0.003

Coeff.of Lift 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

COP Height 0.366 0.379 0.220 0.345 0.201 0.302 0.178 0.892

COP Long. (F-R) 0.879 0.771 0.500 0.758 0.523 0.686 0.379 0.947

Table 12-3 : Acceleration Results (Conceptual)


235


Time 21.822 21.834 21.859 21.790 22.119 21.885 0.329



New Tyres 1 1 1 1 1

Mass 336.0 336.1 336.3 336.0 306.0 330.1 30.3 0.005 COG Height 0.250 0.250 0.251 0.251 0.200 0.240 0.051 0.007

Weight Dist. (F-R) 0.603 0.561 0.529 0.574 0.604 0.574 0.075 0.501

K Front 30181 27764 26920 26283 25086 27247 5094 0.170

K Rear 19819 22236 23080 23717 24914 22753 5094 0.170


Rear Spring Prog. 0.138 0.023 0.011 0.027 0.125 0.065 0.127 0.255

Front ARB 1272 724 3986 1003 2431 1883 3263 0.816

Rear ARB 436 1256 1448 2144 51 1067 2094 0.523

Front ARB Prog. 0.136 0.004 0.241 0.392 0.411 0.237 0.407 0.814

Rear ARB Prog. 0.173 0.381 0.500 0.000 0.442 0.299 0.500 1.000

Front RC Height 0.005 0.035 0.039 0.038 0.047 0.033 0.042 0.418

Rear RC Height 0.014 0.050 0.081 0.012 0.001 0.032 0.080 0.797

Brake Balance (F-R) 1.07 1.42 1.60 1.29 0.96 1.27 0.64 0.322


Front Sprocket 16 17 18 17 18 17.2 2 0.337

Rear Sprocket 44 49 54 48 52 49.4 10 0.702

Final Drive 2.75 2.88 3.00 2.82 2.89 2.87 0.25 0.101





Caster Angle 3.52 8.27 6.78 6.17 4.51 5.85 4.75 0.475


Upper Change RPM 11789 12396 12500 12500 12207 12278 711 0.474

Lower Change RPM 6305 6190 7574 6732 7998 6960 1808 0.603

Frontal Area 1.60 1.40 1.42 1.44 1.40 1.45 0.20 0.509

Coeff.of Drag 0.500 0.500 0.503 0.505 0.500 0.502 0.005 0.023

Coeff.of Lift -0.650 -0.650 -0.650 -0.650 0.000 -0.520 0.650 0.748

COP Height 0.371 0.365 0.384 0.312 0.380 0.362 0.071 0.586

COP Long. (F-R) 0.621 0.678 0.630 0.615 0.871 0.683 0.256 0.639

Table 12-4 : Autocross Results (Conceptual)


236


Time 18.961 19.050 19.028 18.964 18.947 18.990 0.103



New Tyres 1 1 1 1 1

Mass 336.0 336.1 336.0 336.0 336.1 336.0 0.1 0.002 COG Height 0.250 0.250 0.250 0.250 0.250 0.250 0.000 0.001

Weight Dist. (F-R) 0.579 0.612 0.586 0.585 0.580 0.589 0.033 0.217

K Front 28365 15387 22818 25004 36605 25636 21219 0.707

K Rear 21635 34613 27182 24996 13395 24364 21219 0.707


Rear Spring Prog. 0.486 0.425 0.297 0.376 0.104 0.338 0.382 0.763

Front ARB 3874 4000 3999 2183 2362 3284 1818 0.454

Rear ARB 2029 333 2376 978 105 1164 2271 0.568

Front ARB Prog. 0.019 0.409 0.025 0.398 0.405 0.251 0.390 0.780

Rear ARB Prog. 0.045 0.284 0.381 0.416 0.166 0.258 0.371 0.742

Front RC Height -0.024 -0.019 -0.006 0.037 -0.003 -0.003 0.060 0.605

Rear RC Height 0.000 0.030 0.025 0.026 0.063 0.029 0.062 0.623

Brake Balance (F-R) 1.23 0.95 1.09 1.29 1.12 1.14 0.34 0.168


Front Sprocket 16 18 15 14 16 15.8 4 0.662

Rear Sprocket 43 47 42 40 46 43.6 7 0.429

Final Drive 2.69 2.61 2.80 2.86 2.88 2.77 0.26 0.106





Caster Angle 7.05 6.50 8.90 6.02 5.47 6.79 3.44 0.344


Upper Change RPM 12120 11846 12411 11620 11825 11964 791 0.528

Lower Change RPM 7901 8000 7795 6777 5006 7096 2994 0.998

Frontal Area 1.75 1.43 1.40 1.41 1.40 1.48 0.35 0.866

Coeff.of Drag 0.501 0.546 0.500 0.500 0.500 0.509 0.046 0.228

Coeff.of Lift -0.650 -0.650 -0.650 -0.646 -0.650 -0.649 0.004 0.006

COP Height 0.430 0.389 0.251 0.322 0.299 0.339 0.179 0.896

COP Long. (F-R) 0.578 0.805 0.798 0.636 0.535 0.670 0.270 0.675

Table 12-5 : Endurance Results (Conceptual)


237


Score 619.2 616.4 620.2 591.5 621.1 613.7 29.6



New Tyres 1 1 1 1 1

Mass 336.0 336.0 336.0 306.0 336.2 330.1 30.2 0.005 COG Height 0.250 0.250 0.250 0.200 0.250 0.240 0.050 0.001

Weight Dist. (F-R) 0.566 0.597 0.537 0.584 0.588 0.575 0.060 0.400

K Front 27380 39442 27648 32272 34671 32283 12062 0.402

K Rear 22620 10558 22352 17728 15329 17717 12062 0.402


Rear Spring Prog. 0.457 0.002 0.000 0.000 0.325 0.157 0.457 0.914

Front ARB 3891 996 3698 3844 2580 3002 2895 0.724

Rear ARB 1010 1604 208 3469 278 1314 3262 0.815

Front ARB Prog. 0.245 0.226 0.048 0.481 0.405 0.281 0.433 0.867

Rear ARB Prog. 0.036 0.168 0.200 0.346 0.329 0.216 0.309 0.619

Front RC Height -0.001 -0.044 0.049 -0.032 -0.011 -0.008 0.094 0.936

Rear RC Height 0.000 0.091 0.055 0.001 0.091 0.048 0.091 0.910

Brake Balance (F-R) 1.39 1.11 1.65 1.17 1.20 1.30 0.54 0.270


Front Sprocket 13 15 17 16 16 15.4 4 0.557

Rear Sprocket 41 48 53 50 51 48.6 12 0.766

Final Drive 3.15 3.20 3.12 3.13 3.19 3.16 0.08 0.033





Caster Angle 6.27 4.06 7.07 5.27 5.90 5.71 3.00 0.300


Upper Change RPM 11853 12500 11905 11782 12410 12090 718 0.479

Lower Change RPM 5648 7469 6138 5708 7074 6407 1821 0.607

Frontal Area 1.75 1.79 1.40 1.40 1.42 1.55 0.39 0.981

Coeff.of Drag 0.517 0.501 0.500 0.500 0.503 0.504 0.017 0.083

Coeff.of Lift -0.650 -0.650 -0.649 0.000 -0.650 -0.520 0.650 0.020

COP Height 0.418 0.326 0.441 0.393 0.281 0.372 0.160 0.809

COP Long. (F-R) 0.572 0.608 0.548 0.812 0.610 0.630 0.264 0.661

Table 12-6 : Competition Results (Conceptual)

Application of Evolutionary Algorithms to Engineering Design

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