mathematical modelling for the continuous simulation of a bicycle - research and design

8/3/2019 Mathematical Modelling for the Continuous Simulation of a Bicycle - Research and Design

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USING MATHEMATICAL MODELLING TO CREATE A

CONTINUOUS SIMULATION OF A BICYCLE

EXTENDED RESEARCH AND DESIGNSEAN ROBINSON


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Page 2 of50 Sean Robinson 2012

Abstract This paper documents the necessary mathematics required to solve the individual

stages in creating a limited physics engine for a bicycle. These solutions are tested through a

series of experiments to prove viability and accuracy. The set of solutions is then amalgamated

into a full model to show how the inputs are used within the complete system to return accurate

and appropriate results. Using these solutions, a working system is designed through a series of

increasingly specialised diagrams that culminate to describe the complete structure and mechanics

of the final product.

CONTENTS

1. Introduction ............................................................................ ................................................................ ........ 4

2. Refined Evaluation of Software Languages and Development Platforms ...................................................... 6

2.1. Introduction ..................................................................... ....................................................................... ... 7

2.2. Modularisation .......................................................................................................................................... 7

2.3. Mathematical Formula .............................................................................................................................. 8

2.4. Operating System ...................................................................................................................................... 8

2.5. .DLL Files .................................................................................................................................................... 8

2.6. Conclusions ...................................................................... ....................................................................... ... 8

3. Extended Research ....................................................................................................................................... 10

3.1. Introduction ..................................................................... ....................................................................... . 11

3.2. Defining Constants .................................................................................................................................. 11

3.3. Calculating Velocity ................................................................................................................................. 12

3.3.1. Power .................................................................................................................................................. 12

3.3.2. Surface Friction ................................................................................................................................... 12

3.3.3. Braking ................................................................................................................................................ 12

3.3.4. Slopes .................................................................................................................................................. 13

3.3.5. Watts to Overcome Gravity ................................................................................................................ 13

3.3.6. Watts to Overcome Surface Friction ................................................................................................... 13

3.3.7. Watts to Overcome Air Resistance ..................................................................................................... 13

3.3.8. Final Power Output ............................................................................................................................. 14

3.3.9. Calculating Acceleration .......................................................................... ............................................ 14

3.4. Integrating Velocity ................................................................................................................................. 17

3.5. Calculating Lean ............................................................................... ........................................................ 20

3.6. Calculating Steering Angle .................................................................................................. ..................... 24

3.7. Integrating and Mapping Movement ....................................................................... ............................... 27

3.8. Enhancements ............................................................................... .......................................................... 29

3.9. Conclusions ...................................................................... ....................................................................... . 29

4. User Specification ......................................................................................................................................... 304.1. Introduction ..................................................................... ....................................................................... . 31


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4.1.1. Motivation ................................................................ ......................................................................... .. 31

4.1.2. Scope .......................................................................... ......................................................................... 31

4.1.3. Definitions ................................................................. ......................................................................... . 31

4.1.4. Customer .................................................................. .......................................................................... . 33

4.2. Core Functionality ................................................................................................................................... 33

4.2.1. Overview ............................................................................................................................................. 33

4.2.2. Core Features .................................................................................................................. .................... 33

4.2.2.1. Calculate Acceleration ............................................................................... ................................. 33

4.2.2.1.1. Inputs ..................................................................................................................................... 33

4.2.2.1.1.1. Power ............................................................................................................................. ........ 33

4.2.2.1.1.2. Brake ...................................................................................................................................... 34

4.2.2.1.1.3. Slope ...................................................................................................................................... 34

4.2.2.1.1.4. Watts To Overcome Friction ........................................................................... ....................... 34

4.2.2.1.1.5. Watts to Overcome Gravity ................................................................................................... 34

4.2.2.1.1.6. Watts to Overcome air Resistance ..................................................................... .................... 34

4.2.2.1.2. Output .................................................................... ................................................................ 34

4.2.2.1.2.1. Acceleration ........................................................................................................................... 34

4.2.2.2. Integrate Velocity ....................................................................................................................... 35

4.2.2.3. Calculate Yaw ............................................................................................................................. 35

4.2.2.4. Generate New Position .............................................................................................................. 35

4.3. Constraints............................................................................................................................................... 36

4.3.1. Interaction ................................................................. ......................................................................... . 36

4.3.2. Performance ............................................................................. ........................................................... 36

4.3.3. Structural .................................................................... ......................................................................... 36

4.3.4. Operational ......................................................................................................................................... 37

4.3.5. Legal .................................................................................................................................................... 37

4.4. Testing ..................................................................................................................................................... 37

5. Refined UML Diagram ......................................................................... .......................................................... 386. Use Case Diagram and Specification ................................................................... .......................................... 39

6.1. Use Case Diagram .................................................................................................................................. .. 40

6.2. Derive New State .................................................................................................................. ................... 40

6.1.1. Description ............................................................... .......................................................................... . 40

6.1.2. Actors .................................................................................................................................................. 40

6.1.3. Preconditions ...................................................................................................................................... 40

6.1.4. Flow of Events ..................................................................................................................................... 41

6.1.5. Alternate Flow of Events ........................................................................ ............................................. 41

6.1.6. Post Conditions ................................................................................................................................... 41


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6.3. Get New Coordinates .............................................................................................................................. 42

6.3.1. Description ............................................................... .......................................................................... . 42

6.3.2. Actors .................................................................................................................................................. 42

6.3.4. Preconditions ...................................................................................................................................... 42

6.3.6. Flow of Events ..................................................................................................................................... 42


6.3.8. Post Conditions ................................................................................................................................... 42

6.4. Get Lean................................................................................................................................................... 42

6.4.1. Description ............................................................... .......................................................................... . 42

6.4.2. Actors .................................................................................................................................................. 42

6.4.4. Preconditions ...................................................................................................................................... 42

6.4.6. Flow of Events ..................................................................................................................................... 42


6.4.8. Post Conditions ................................................................................................................................... 43

6.5. Get Pedal Difficulty Coefficient ............................................................................................................... 43

6.5.1. Description ............................................................... .......................................................................... . 43

6.5.2. Actors .................................................................................................................................................. 43

6.5.4. Preconditions ...................................................................................................................................... 43

6.5.6. Flow of Events ..................................................................................................................................... 43


6.5.8. Post Conditions ................................................................................................................................... 43

7. Sequence Diagram ................................................................................ ........................................................ 44

8. Activity Diagrams .......................................................................................................................................... 45

8.1. Acceleration ........................................................................................................................................ 45

8.2. Integration........................................................................................................................................... 46

8.3. Distance ...................................................................... ......................................................................... 46

8.4. Position Plotting ............................................................................................. ..................................... 47

8.5. Lean Calculations ................................................................................................................................ 479. Conclusions ............................................................ .......................................................................... ............. 48

Bibliography ........................................................................ .................................................................... .............. 49

1.INTRODUCTIONFollowing on from research previously carried out (1), this paper seeks to accumulate the gathered

information and refine the individual solutions before presenting an amalgamated solution to the

overall problem.


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The initial stage of this paper will be comprised of the methodology used to develop the distinct

solutions into a completed system. Each of these stages will be carefully explained such that their

individual use and the way in which they fit into the general solution will be made clear.

Once the method has been successfully demonstrated, extensive testing and experimentation to

ensure accuracy throughout each stage will be completed and documented. The aim of this testing,

in addition to providing preliminary results, will be to form the basis of unit testing carried out on

the completed project in the development stage. By testing each component, this will translate into

either specific methods within classes or slightly higher level activities at a later stage.

The second part of this paper will serve as the completed design for the final system. Design will

take the form of a series of increasingly specialised documents and diagrams in order to provide a

full view of the completed system, from abstract concepts to program flow minutia.

The first document to be presented will be a user Specification; this will detail the exact nature of

the project including specific mechanics and more general constraints pertaining to client andsystem requirements.

Beginning with a high level of abstraction, a Use case diagram will be developed and documented in

the form of a use case specification. As this project will have very limited user interaction, particular

emphasis will be placed on the activities taking place within each use case. Although these are not

strictly initiated by the user or another system, documenting a brief flow of events will be important

when specialising the design further.

Using the use case and specification, a sequence diagram will be developed that shows the

initialisation and interaction between classes throughout the chain of events described in the use

case specification. This will provide a graphical representation of events as they transpire within the

system. A sequence diagram is suited well to this task as the flow of events follow a linear path

through the different classes.

The lowest level of graphical description to be developed will be activity diagrams. Each activity

diagram will represent a distinct but significant section of calculations in the chain.

Throughout this design, a UML diagram will be built that represents a prototype of the system.

Testing scenarios will not be included as they will need to be developed as the build progresses in

the development phase. The UML will present the logic used in the engine and detail the

relationships that will be constructed between the different classes.

This paper will conclude with a detailed section reporting on the methods and experiments carried

out in the course of this design phase. Individual solutions, in addition to the general system, will be

examined and compared against other potential mechanisms. Any difficulties encountered in the

design of this solution will be documented here with solutions that will need to be incorporated into

the design before development.


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2.REFINED EVALUATION OF SOFTWARE LANGUAGES AND DEVELOPMENTPLATFORMS

PLATFORM AND LANGUAGE EVALUATION FOR USE IN

THE DEVELOPMENT OF A CONTINUOUS SIMULATION

SEAN ROBINSON


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Abstract This document outlines to

necessary qualities and technical

requirements for a project aimed at creating

a continuous simulation of a bicycle. The

programming language C++ and

development platform Visual Studio 2010 is

evaluated with these items in consideration.

This document concludes by indicating how

feasible it is to create this project using the

tools described and presents alternatives

should there be a more viable solution.

CONTENTS

1. Introduction .................................................. 7

2. Modularisation .............................................. 7

3. Mathematical Formula ................................. 8

4. Operating System.......................................... 8

5. .DLL Files ....................................................... 8

6. Conclusions ................................................... 8

Bibliography .............Error! Bookmark not defined.

2.1. INTRODUCTIONThe C++ language (2) is under consideration as

it is one of the most widely used and powerful

tools available. While it can be used on a

variety of platforms, the dedicated IDE Visual

Studio (3) is the usual way in which to develop

with this language.

The criteria under consideration range from

fundamental technical requirements to moreabstract concepts that would benefit this

project. Details such as modularisation,

support for mathematical formula, operating

system compatibility and the ability to

develop the solution as a .DLL file will all be

considered.

The research undertaken will evaluate the

way in which the language and platform

indicated support the different criteria. Adiscussion regarding the effectiveness of this

support will conclude with the most feasible

solution.

Should these choices be deemed insufficient

for the needs, then other candidates will be

considered and subjected to a similar

feasibility test.

2.2. MODULARISATIONThe most critical factor in this project, with

regard to structure, is modularisation. Many

of the equations or solutions are highly

generalised and can be used for a variety of

different situations. In order to promote code

reuse within this project and also forcomponent reuse externally, it is important to

ensure that solutions remain distinct and

independent from the overall solution.

Through this, solutions can be reused for

different scenarios.

An example of this structure is integration.

When plotting the course of objects through a

simulation, integration is used to derive new

positions based on factors such as velocityand current position. There are several

situations where integration can be used,

such as integrating the velocity based on

acceleration and position based on velocity.

To promote reuse, a general solution can be

developed and then used for both problems.

C++ supports modularisation in several ways.

The most fundamental structural advantages

here are methods and classes. By

encapsulating general solutions in classes and

general equations into functions, either

internal or external to those classes, a library

of abstract solutions can be developed to

support both this application and others (4).

Modularisation also lends itself well to unit

testing. By decomposing overarching

problems into individual and general tasks,

unit tests can easily be developed and applied

(5). This decomposition supports the project


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methodology being applied elsewhere in the

application.

2.3. MATHEMATICALFORMULA

It is important to note that this application

amounts to a limited scope physics engine

simply concerned with the propulsion and

steering of a bicycle. This in turn, indicates

significant use of mathematical and physical

mechanics. The language used to create this

application must be able to allow a developer

to make use of mathematical formula in order

to successfully model these equations.

It should be noted that this is an evaluation of

built in functionality and not simply the ability

to create the appropriate functionality using

the language.

Almost any language will support basic

arithmetic functions and it is unknown which

specific equations will need to be used until

further research has been undertaken. The

most basic requirements that can be

extrapolated initially include the use of

trigonometric functions for calculating the

gradient of a slope and possibly equations of a

circle for use with the steering methods.

Pre-existing libraries bundled with the C++

language provide extensive mathematical

ability that can be utilised throughout this

project (6). This reduces the volume of code

to be written and therefor, reduces elements

that require testing.

2.4. OPERATING SYSTEMThe system that will be running this project is

Windows 7; the output of this project will

need to be in a format that this operating

system can use. In addition to running the

output, it is advisable to develop the solution

in the operating system that it is being built

for.

As Visual Studio 2010 is a Microsoft product,

there is no difficulty in it running on their

standard operating system (7). It is also clear

that the output .DLL file will run with the

existing system, this is covered in more detail

under the following .DLL Files section.

2.5. .DLLFILESThis project will generate a suite of classes

that provide the functionality specified.

Although they could simply be generated as

separate .CPP files, in order to provide a highlevel of modularisation and protection for the

software, the solution will be built as a .DLL

file that can be used in the existing simulation

software.

Visual Studio 2010 provides functionality to

build a solution that outputs a .DLL file by

default (8). As long as the relevant access

points in the .DLL are properly exported and

the .DLL file is referenced within the existingsystem, there is no difficulty and only a very

small difference in building the project in this

manner.

2.6. CONCLUSIONSIt has been shown that the Visual Studio 2010

package in conjunction with the C++ language

is an ideal choice for building this project as it

satisfies all of the criteria indicated.

Modularisation was an important criterion to

consider so that the output of this project will

remain a protected, general solution with

extended elements pertinent to this specific

case. It has been found that C++ meets these

needs very well.

Support for mathematical formula has been

found to be extensive and more than capablefor the needs of this project.


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The development criteria required of the

Visual Studio 2010 package indicated that an

output be compatible with Microsoft

Windows 7 and have the ability to develop

the solution as a .DLL file. This paper has

documented that the chosen platform does

indeed conform to these standards.

Based on the satisfactory way in which both

the language and platform chosen fulfil the

criteria indicated, this project will be

developed in C++ using the Visual Studio 2010

package. The solution will be built as a .DLL

file and then referenced in the existing

simulation package.


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3.EXTENDED RESEARCH

CREATING A CONTINUOUS AND INTERACTIVE

SIMULATION USING BICYCLE DYNAMICS AND

ACCURATELY MODELLED ENVIRONMENTAL FACTORS

SEAN ROBINSON


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Abstract Following on from the work

completed in a previous paper, where the

specific dynamics and mechanics of

particular problems were investigated, this

paper sets out to compile these results and

demonstrate the significant steps needed tobe taken in order to develop a coherent

model. Each stage required will be

introduced and explained before a

demonstration that shows the solution in

practice is given. These demonstrations and

experiments show how it is possible to

model a bicycles dynamics to a high degree

of accuracy in a limited scope and then

interact with the model using accurately

modelled environmental factors. This paper

will document the logical enhancements thatcould be taken in order to extend the project

and then conclude with a summary of the

work.

TABLE OF CONTENTS

1. Introduction ................................................ 11

2. Defining Constants ...................................... 11

3. Calculating Velocity ..................................... 12

I. Power...................................................... 12

II. Surface Friction ....................................... 12

III. Braking .................................................... 12

IV. Slopes ..................................................... 13

V. Watts to Overcome Gravity .................... 13

VI. Watts to Overcome Surface Friction ...... 13

VII. Watts to Overcome Air Resistance .... 13

VIII. Final Power Output ............................ 14

IX. Calculating Acceleration ......................... 14

4. Integrating Velocity ..................................... 17

5. Calculating Lean .......................................... 20

6. Calculating Steering Angle .......................... 24

7. Integrating and Mapping Movement .......... 27

8. Enhancements ............................................ 29

9. Conclusions ................................................. 29

Bibliography .............Error! Bookmark not defined.

3.1. INTRODUCTIONWhereas the previous report was focussed

almost exclusively on finding solutions to

problems, this paper documents the timeline

that was taken in order to pipe thosesolutions together. In some respects it is

more difficult to concatenate solutions

together as there is not usually an existing

perfect solution, the solution needs to be

developed dynamically for the particular

problem.

This paper takes a linear format as that is the

way in which the experiments were

completed. As each experiment is reliant onprevious work completed, it is necessary that

problems are accurately solved before moving

on. Some issues that were solved were found

to be simply too complicated or costly and it is

recommended that these are avoided in

favour of more simple solutions.

The experiments were simply completed in a

Microsoft Excel Workbook as this provided

immediate and clear results that could be

plotted to give a visual representation of

exactly what was happening. This workbook

can also form a core component of any testing

modules for work that expands on the basic

demonstrations given here.

Each experiment takes place over 30 steps

using a randomised delta based on 0.1

seconds. Using a randomised delta allows the

experiment to mimic to difference in screen

refresh rates if the model was developed for a

computer.

3.2. DEFINING CONSTANTSTo provide a more interactive and dynamic set

of experiments, a single page in the workbook

was created to house the constants used

throughout the work. These constants range

from fundamental constants such asacceleration due to gravity on earth, to more


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technical data such as the bicycles wheelbase

length, to more arbitrary factors such as an air

resistance coefficient.

As this paper progresses, it will becomes clear

what each of these constants represent.

It is also important to note that by having a

single cell that is used throughout, it is easy to

modify certain factors to see immediate

changes in results. It would be the work of

seconds to see how doubling the riders

weight affects acceleration on a slope or how

much wind resistance would be required to

bring the rider to a complete stop.

Wheelbase 115 cm

Caster Angle 18 Degrees

Gravity 9.82 m/s/s

Centre of mass height 66 cm

Tyre Thickness 2.5 cm

Velocity 10 m/s

Dt 0.1 s

Randomise 1 Boolean

Weight of Object 80 kg

MaximumDeceleration 0.5 G

Air Resistance 0.7 Coefficient

3.3. CALCULATINGVELOCITY

3.3.1.POWERThe first item to consider when modelling the

propulsion of an object is the amount of

power that is being fed into it. On a bicycle

this is the amount of watts generated by the

cyclist pedalling. It would easy to simply forgo

this and use an arbitrary value for

acceleration but in order to accurately model

and allow for hardware integration, a power

output should first be defined. Based on

research by Coggan (9), a value of 100 watts

foran experienced cyclist has been used. This

is achieved through 25 watt increases initially

to approximate a build up as pedalling

commences.

3.3.2.SURFACE FRICTIONSurface friction is a difficult thing to model in

a limited example as each surface will have a

very different coefficient for the friction

applied. In this limited experiment, a value of

0.1 has been used for a flat road based on

work by May (10).

In a more developed project, this value can

easily be changed to reflect the surface

currently being used. This could either be anarbitrary value that seems to give an

appropriate feel or something more accurate

that has been researched.

3.3.3.BRAKINGBraking can be modelled in many different

ways. In bicycles, the brake force applied to a

machine is the result of friction generated

between the brake pad and wheel. In a

simplistic model, the velocity could simply be

reduced by a fraction of a certain value

defined as maximum brake.

In a more complex model, the brake force

could be modelled in a specific measure

relative to the actual physics, such a G

(gravity) or Newtons. In this model, the brake

force if modelled in gravities.

Should a user be allowed to apply excessive

braking then it will be quite likely that a crash

will occur at some point as the bicycle will

decelerate too quickly. As this is not a

concern of the basic model described here, a

maximum safe brake value has been defined

as 0.5g has been used based on a report by

Whitt and Wilson (11). Any brake measure

applied will be considered a fractional value of

its maximum, which is then applied to the

0.5g value in order to determine a brake force


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in gravities that need to be used to reduce the

velocity.

3.3.4.SLOPESIn this experiment, a slope was defined simplyby specifying a percentage; this percentage is

a fraction of angle defined on a road ranging

from 0 with a horizontal plane to 100 with a

fully vertical surface. Conversely, the value

can also be defined as a negative which is

indicative of a slope that is being travelled in a

downward direction. In order to output a

visual representation, this slope was tallied in

a separate column that provided a general

overview of how the landscape changed

within the experiment.

3.3.5.WATTS TO OVERCOMEGRAVITY

Following on from the slope definition is

something that directly affects the velocity

because of it. Gravity is apparent from any

cycling situation but it begins to affect the

machine and rider depending on the

steepness of the hill currently being traversed.

Travelling up a hill will increased the

gravitational pull, travelling downwards will

do the inverse and the rider will find himself

being pulled forwards. In this model, the

force isnt actually calculated but a value of

watts is given in the calculation below. This is

the amount of watts that should be

subtracted from the initial power in order to

compensate for the effect of gravity. Positive

and negative values are both used. The

formula, given by May (10) is:

= Where:

G: Standard acceleration due to gravity

P: Weight of object is kg

: Slope Percentage

V: Current velocity in m/s

3.3.6.WATTS TO OVERCOMESURFACE FRICTION

Surface friction has already been discussedand explained. Similarly to gravity, a formula

has been used to identify the amount of watts

required to overcome this friction. Provided

again by May, the formula is:

= . Where:

0.1: Friction coefficient

V: Current velocity in m/sP: Weight of object is kg

3.3.7.WATTS TO OVERCOME AIRRESISTANCE

The final subtraction of power in the model

comes in the form of air resistance. Air

resistance plays a huge role in cycling and is

an important factor to consider when

attempting to model accurately. It would beoutside the scope of this project to determine

a truly valid coefficient for wind resistance so

0.7, a value based on work by May (10) has

been used in addition to the rest of the

formula for calculating the watts required to

overcome the resultant force. This formula

has been simplified as wind is not being

considered in this project, the referenced

website has further information that can be

used for an expanded model.

= ()Where:

: Air resistance coefficientV: Current velocity in m/s


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3.3.8.FINAL POWER OUTPUTThe final power if what remains once all

subtractions for environmental factors havebeen applied. The formula is simply:

= Where:

w: Power output in watts

p: Initial power output from pedalling

: Watts to overcome gravitywf: Watts to overcome friction

wa: Watts to overcome air resistance

3.3.9.CALCULATINGACCELERATION

Acceleration was the goal of this section of

experimentation, a value that could be used

throughout further experiments that allowed

for the propulsion of an object. The

acceleration is calculated using the mass of an

object and a force acting upon it. Using the

following formula (12), this can be calculated:

=

Where:

f: Force

m: Mass

With regard to propulsion of an object, this is

the final step of the section. If a basic

integration method was being used, then this

acceleration could be simply added to the

existing velocity, which then could be used to

calculate the new position using an Euler

integration (13). As a runge-Kutta4 method is

going to be used, there is much more work to

be done in order to fully utilise the results so

far so this will be covered in the next section.

Below is the final datasheet from the excel

workbook that shows how user modified data

in the left columns propagate throughout the

rest.

wer Friction Brake Slope% Veloc (m/s) Gravity w Friction w Air Res w Final P (w) Accel (m/s)

0 0.1 0 0 0 0 0 0 0 0

25 0.1 0 0 0 0 0 0 25 0.3125

50 0.1 0 0 0.3125 0 2.5 0.021362305 47.4786377 0.593482971

75 0.1 0 0 0.905982971 0 7.24786377 0.520544838 67.23159139 0.840394892

100 0.1 0 2 1.746377864 27.43908899 13.97102291 3.728315818 54.86157228 0.685769654

100 0.1 0 2 2.432147517 38.21390179 19.45718014 10.07088828 32.2580298 0.403225372

100 0.1 0 2 2.83537289 44.54937884 22.68298312 15.95616754 16.81147051 0.210143381

100 0.1 0 2 3.045516271 47.85115165 24.36413017 19.77337543 8.01134275 0.100141784100 0.1 0 2 3.145658055 49.42457936 25.16526444 21.78876296 3.621393228 0.045267415

100 0.1 0 2 3.190925471 50.13582099 25.52740377 22.74301417 1.59376107 0.019922013

100 0.1 0 2 3.210847484 50.44883567 25.68677987 23.17165592 0.692728541 0.008659107

100 0.1 0 6 3.219506591 151.7546627 25.75605273 23.35963193 -100.8703473 -1.260879342

100 0.1 0 6 1.958627249 92.32185402 15.66901799 5.25960848 -13.2504805 -0.165631006

100 0.1 0 6 1.792996243 84.51467091 14.34396994 4.034931616 -2.893572474 -0.036169656

100 0.1 0 6 1.756826587 82.80977801 14.0546127 3.795637475 -0.660028183 -0.008250352

100 0.1 0 6 1.748576235 82.42088941 13.98860988 3.742413358 -0.151912643 -0.001898908

100 0.1 0 6 1.746677327 82.33138248 13.97341861 3.730234105 -0.035035196 -0.00043794

100 0.1 0 6 1.746239387 82.31073974 13.96991509 3.727428992 -0.008083826 -0.000101048100 0.1 0 0 1.746138339 0 13.96910671 3.726781956 82.30411133 1.028801392


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100 0.1 0 -6 2.774939731 -130.7995591 22.19951785 14.95748945 193.6425519 2.420531898

100 0.1 25 -6 5.070471629 -239.0017507 40.56377303 91.25215113 207.1858265 2.589822832

100 0.1 25 -6 7.535294461 -355.1836397 60.28235568 299.5013087 95.39997532 1.192499692

100 0.1 25 -6 8.602794152 -405.5013052 68.82235322 445.6733175 -8.994365601 -0.11242957

100 0.1 50 -6 8.240364582 -388.4178249 65.92291666 391.685343 30.80956527 0.385119566

100 0.1 50 -6 8.375484148 -394.7868208 67.00387318 411.2707309 16.51221671 0.206402709

100 0.1 25 -6 8.456886857 -398.6238189 67.65509485 423.3792808 7.589443211 0.09486804

100 0.1 0 0 8.551754897 0 68.41403918 437.7879212 -406.2019604 -5.077524505

100 0.1 0 0 3.474230392 0 27.79384313 29.35444578 42.85171108 0.535646389

100 0.1 0 0 4.00987678 0 32.07901424 45.13267992 22.78830584 0.284853823

100 0.1 0 0 4.294730603 0 34.35784483 55.45054522 10.19160995 0.127395124

100 0.1 0 0 4.422125728 0 35.37700582 60.53287459 4.090119592 0.051126495

This data was mapped to several graphs in order to obtain an overview of the output. Some of the

more significant show the correlation between velocity and the height of a slope or the amount of

acceleration given by certain amounts of power in watts.

0

10

20

30

40

50

60

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Height

Height


16/50

0

1

2

3

4

5

6

78

9

10

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Velocity (m/s)

Velocity (m/s)

-500

-400

-300

-200

-100

0

100

200

300

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Final Power (w)

Final Power (w)


17/50

3.4. INTEGRATING VELOCITYUsing a base velocity calculated by adding the

acceleration to the previous velocity and

subtracting the deceleration due to braking,

the distance an object moves can be

calculated by integrating the velocity using

the Runge-Kutta 4 method developed by

Runge (14) and then modified by Kutta (15) .

RK4 is usually used to integrate a position but

can become very complex when using

multiple variables; a good balance can be

achieved by using RK4 to integrate the

velocity based on a differing delta time and by

then using a simple Euler to calculate the

distance moved.

The RK4 method works by taking several

derivatives of a value in a given time step and

then summing them using weights. This then

gives a good approximation of what the

velocity will be (16).

The first step is to calculate the derivatives.

Each derivative has a slightly different formula

but the one used here is as follows:

D1:

(2 + )

D2:

(2( + 0.51) + (.))D3:

(2( + 0.52) + (.))D4:

(2( + 2) + (.))

-6

-5

-4

-3

-2

-1

0

1

2

3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Acceleration (m/s)

Acceleration (m/s)


18/50


Where:

t: Delta TimeT: Current Time

V: Current velocity in m/s

D1: Derivative 1

D2: Derivative 2

D3: Derivative 3

D4: Derivative 4

e: Constant

The derived velocity can now be calculated

using the following formula

= +16 + 23 +

33 +

46

This integrated velocity is now used in a basic

Euler by multiplying the current distance

moved by the derived velocity squared.

In this manner, thought slightly convoluted in

parts, it is possible to obtain the accuracy of

using RK4, keep the cost of the program down

by using more basic approaches where

possible and obtain a solid value of distance

moved that can be used to accurately indicate

where the object is going to be at the next

step.

Below is the datasheet used to calculate the

integration during the experiment.

dt Time Position Velocity d1 d2 d3 d4 dv

0.119469 0 1 0 0.119469 0.099016 0.099283 0.097513 0.102263

0.015962 0.119469 1.010457734 0 0.011154 0.010889 0.010889 0.010886 0.010933

0.075284 0.135431 1.010577262 0.3125 0.003095 0.029943 0.028646 0.02727 0.337091

0.037452 0.210715 1.124207402 0.905983 -0.04796 -0.03945 -0.04002 -0.03735 0.865274

0.146287 0.248167 1.872906372 1.746378 -0.44146 -0.6252 -0.54566 -0.36602 1.221511

0.139759 0.394454 3.364996098 2.432148 -0.63703 -1.21403 -0.8964 -0.62454 1.51841

0.079715 0.534213 5.670564296 2.835373 -0.43599 -1.07795 -0.8265 -0.62915 2.023033

0.076205 0.613928 9.7632255 3.045516 -0.45209 -1.2008 -0.90042 -0.69053 2.1546720.176485 0.690133 14.40583602 3.145658 -1.08806 -2.37198 -1.33843 -1.13575 1.538221

0.16561 0.866618 16.77196004 3.190925 -1.0446 -2.34921 -1.33699 -1.12883 1.599952

0.181669 1.032229 19.33180658 3.210847 -1.15841 -2.51006 -1.3836 -1.20688 1.518747

0.103211 1.213898 21.63840049 3.219507 -0.66187 -1.72004 -1.14687 -0.88443 2.006153

0.136179 1.317109 25.6630501 1.958627 -0.53083 -0.77871 -0.66858 -0.45113 1.312539

0.121356 1.453288 27.38580863 1.792996 -0.43363 -0.60169 -0.53911 -0.38031 1.277074

0.115821 1.574643 29.01672769 1.756827 -0.40593 -0.55843 -0.50489 -0.3622 1.274365

0.12158 1.690464 30.64073508 1.748576 -0.42442 -0.57332 -0.51904 -0.36696 1.252556

0.157394 1.812044 32.20963235 1.746677 -0.54915 -0.68163 -0.62162 -0.3979 1.154419

0.082432 1.969438 33.54231542 1.746239 -0.28767 -0.42312 -0.38809 -0.3039 1.3772380.086196 2.05187 35.43910112 1.746138 -0.30084 -0.4388 -0.40167 -0.31145 1.363931

0.172084 2.138066 37.29940996 2.77494 -0.95476 -1.81656 -1.199 -0.85455 1.468199

0.093859 2.31015 39.45501925 5.070472 -0.95173 -3.96273 -1.79124 -2.01853 2.657441

0.047556 2.404009 46.51701066 7.535294 -0.71666 -4.89909 -2.46003 -2.4499 4.554494

0.140591 2.451565 67.2604288 8.602794 -2.41886 -15.3699 -0.23682 -19.6798 -0.28254

0.053788 2.592156 67.34025614 8.240365 -0.88644 -6.54006 -2.65754 -3.35287 4.467948

0.042478 2.645943 87.30281102 8.375484 -0.71153 -5.46399 -2.70574 -2.73098 5.078489

0.114501 2.688421 113.0938629 8.456887 -1.9366 -12.8421 -0.94909 -12.9081 1.385701

0.035488 2.802922 115.0140314 8.551755 -0.60696 -4.82873 -2.67346 -2.4525 5.541115

0.188078 2.83841 145.7179848 3.47423 -1.30681 -2.99306 -1.47123 -1.50911 1.5168130.183596 3.026488 148.0187054 4.009877 -1.47238 -3.9352 -1.5315 -2.2554 1.566344


19/50


0.195959 3.210084 150.4721404 4.294731 -1.68317 -4.6733 -1.50263 -3.05531 1.446341

0.142212 3.406043 152.5640436 4.422126 -1.25775 -4.0925 -1.60551 -2.25642 1.937094

Graphing this data can give a clear indication

of what is happening over time. The followinggraphs indicate the movement of an object

over the steps based on the velocity shown.

The derived velocity is also given for

comparison. A correlation can be mapped

here that shows a higher velocity will push theposition of an object further from the origin at

a greater speed.

0

20

40

60

80

100

120

140

160

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Position

Position

0

12

3

4

5

6

7

8

9

10

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Velocity

Velocity


20/50


3.5. CALCULATING LEANThe lean, or Yaw, of an object is the amount

of angle by which it is rotated about a virtual

line drawn along the Z axis at the lowest part

of the plane. For humans, this would

represent the angle our bodies make to the

perpendicular when swaying to either side.

The lean of a bicycle can be thought of in the

same way. As a machine goes around acorner, a certain degree of leaning is

inevitable and this can be calculated.

To provide some level of interactivity and

dynamism, functionality allowing users to

alter the angle of steering was provided. This

meant that at each step, or row, a certain

amount of steering could be applied either

left or right. These angles referred to a

change in steering, not the actual angle. From

here, the new angle was calculated. The

workbook that details the steering is quite

large so each section will be displayed

individually in order to provide easier viewing.

Left

Turn

Right

Turn

Change in

Steering

Steering

Angle0 0 0 0

5 0 -5 -5

5 0 -5 -10

0 3 3 -7

15 0 -15 -22

5 0 -5 -27

8 0 -8 -35

0 14 14 -21

0 10 10 -11

0 10 10 -10 5 -3 4

-4

-3

-2

-1

0

1

2

34

5

6

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

dv

dv


21/50


0 5 -5 9

0 10 -10 19

0 3 -3 22

6 0 6 16

8 0 8 8

11 0 5 -3

5 0 -5 -8

9 0 -9 -17

6 0 -6 -23

0 9 9 -14

0 0 0 -14

0 4 4 -10

0 0 0 -10

0 9 9 -1

0 2 0 1

0 0 0 1

0 3 -3 4

0 0 0 4

0 6 -6 10

0 2 -2 12

In order to calculate the lean, something

called the radius of the turn needed to be

known. The radius of the turn is the radius of

an imaginary circle drawn based on the angle

of the steering. With the steering at a certain

angle and with a flat plane to experiment

with, riding a bicycle while maintaining the

angle will eventually lead the rider back to the

origin. This can visualised by imagining the

rider leaving a mark from the tyres wherever

they go. When the radius of a circle is

mentioned in relation to steering in this

paper, metres is the measurement being

considered.

The upright turn radius of a circle is the radius

that would be made based on a bicycle with

no lean. This is simply a starting point and

gives a base to work with for refinement. The

formula, given by Sharp (17) is:

=

()

Where:

r: Radiusw: Wheelbase in cm

: Steering Angle: Caster Angle

The ideal lean can now be calculated. Ideal

here means ideal tyres and centre of mass,

infinitely slim and low respectively. As this is

never the case, a more developed solution is

required. This is a modification of the answer

given by the ideal lean formula given by

Fajans (18).

= ()Where:

: Leanv: Velocity

g: Acceleration due to gravity

r: Upright turn radius

The modification mentioned previous is a way

of giving the increase the lean should take asa result of tyre and centre of mass


22/50


considerations. The output of this formula is

added to the ideal lean in order to give a more

accurate representation. The formula, given

by Cossalter (19), is:

(. ().)Where:

: Caster Anglet: Tyre thickness in cm

h: Centre of mass height

For the sake of this demonstration, the tyre

thickness has been given as 2.5cm. This was

simply the output from a brief averaging of

several tyres and types. The centre of mass,

given by h, was decided upon using a similar

measure.

The value now given is the amount of lean, or

yaw, which the object being modelled will

need to be rotated by in order to map closely

to real world dynamics. A final value was

calculated that was to be an experiment in

how far it is realistically possible to pursue

accuracy in a limited model such as this. The

lean of a bicycle affects the radius of its

turning circle by a small amount. The new

radius was calculated using a formula given by

Cossalter (19).

= ()() : Caster Anglew: Wheelbase in cm

: Steering Anglea: Lean

The relevant sections of the data sheet are

included below:

Upright Turn Radius Ideal Lean Lean Increase Lean Final Radius

0 0 0 0 034.83177044 0 0 0 34.83177044

17.41588522 0.000571009 -2.3402E-05 0.000547607 17.41588261

24.87983603 0.003359537 -0.000137686 0.003221851 24.8797069

7.916311464 0.039212037 -0.00160664 0.037605398 7.910714635

6.450327859 0.093116958 -0.003810765 0.089306193 6.424622346

4.975967206 0.163063977 -0.006653422 0.156410555 4.915224511

8.293278676 0.113400991 -0.004637643 0.108763348 8.244274485

15.83262293 0.063558452 -0.002603104 0.060955348 15.80321855

174.1588522 0.005953483 -0.000243994 0.00570949 174.1560136

43.53971305 0.024107832 -0.00098793 0.023119902 43.5280769319.35098358 0.05449215 -0.002232182 0.052259968 19.32456481

9.166255379 0.042592919 -0.001745085 0.040847835 9.158609285

7.916311464 0.041331104 -0.001693416 0.039637688 7.910093436

10.88492826 0.028866895 -0.001182905 0.02768399 10.88075741

21.76985653 0.0143012 -0.000586095 0.013715105 21.76780906

58.05295074 0.005351622 -0.000219328 0.005132294 58.05218617

21.76985653 0.014263006 -0.00058453 0.013678476 21.76781998

10.24463837 0.03029816 -0.001241538 0.029056622 10.24031396

7.572124009 0.103188813 -0.004221561 0.098967252 7.535071618

12.43991801 0.207431862 -0.008440571 0.198991291 12.1944346312.43991801 0.435097909 -0.017275416 0.417822493 11.36977004


23/50


17.41588522 0.408403976 -0.016277156 0.392126819 16.09399126

17.41588522 0.377953275 -0.015124305 0.36282897 16.28205025

174.1588522 0.040993918 -0.001679609 0.039314309 174.0242783

174.1588522 0.04179372 -0.00171236 0.040081361 174.0189765

174.1588522 0.042735516 -0.001750923 0.040984592 174.0126022

43.53971305 0.028223109 -0.001156531 0.027066577 43.52376544

43.53971305 0.037588967 -0.001540169 0.036048798 43.51142584

17.41588522 0.107433427 -0.004394558 0.103038869 17.32351469

14.51323768 0.136358722 -0.005571199 0.130787524 14.38928732

0

20

40

60

80

100

120

140

160

180

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Upright Turn Radius

Final Radius

00.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031

Ideal Lean

Lean


24/50


From the visual representation, it is very clearthat the difference between ideal and actual

lean is very small, as is the difference between

the initial radius and the final radius. These

changes are very small and will go unnoticed

in almost every application using these

equations. Unless a very high degree of

accuracy is required, there is little point in

making use of the full set.

3.6. CALCULATINGSTEERING ANGLE

As the radius of the turn changed slightly due

to the lean of the bicycle, it is apparent that

the angle of rotation will also be different. In

order to calculate this new angle, an

experiment was devised. The aim of the

experiment was to derive a new point on the

virtual circle based on the distance travelledin the step. By taking the angle between this

new point and the origin, which would be the

top point of the circle at X = 0, Y = Radius, it

would be possible to accurately model the

slight change.

Despite several days of experimentation, this

method was never conclusively proved.

Discrepancies between the new derived angle

and the original steering angle were too largeto ignore. This coupled with the vast amount

of time being tunnelled into the solution andwith the possibility of a very small return but

huge costs associated meant that the pursuit

of the solution was abandoned. The method

used to arrive at the results shown later will

not be detailed.

The first step in completing this was to derive

a new point based on the distance travelled;

the new point would be a new point on the

circumference of the virtual circle. The new

point was derived using formula from

MathWarehouse (20).

= : Arc lengthr: Radius

: Inner angle between both points in radians

With this inner angle, the new point could bederived using the parametric equation of a

circle. An offset was then applied in order to

ensure the point had moved in the correct

direction around the circumference. The two

points now represented a coord across the

virtual circle and a tangent was given by the

original point. The coord tangent equation

given by Warendorff (21), states that the

angle between a coord and an angle is half of

the inscribed angle between the points. This

-150

-100

-50

0

50

100

150

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Change In Radius

Change in Steering


25/50


means that by halving the angle already

calculated, the angle between the points

could be obtained.

At this point, the new angle showed large

differences to the original and could not be

consolidated despite attempting several

tweaks and rebuilds. There is no doubt a way

to realise this solution, but considering the

extra cost and complexity for such a small

reward in a small project indicates that the

matter should not be pursued further.

The relevant data is given below.

x1 y1 Distance In Angle x2 y2 Angle Diff

0 0 0 0 0 0 0 0

0 34.83177044 2.72365E-06 4.48249E-06 -2.7237E-06 34.83177044 2.24124E-06 4.999998

0 17.41588261 0.007671051 0.025249469 -0.00767105 17.41588092 0.012624735 9.987375

0 24.8797069 0.129164651 0.297605718 -0.12916523 24.87937162 0.148802859 6.851197

0 7.910714635 0.748529241 5.424202687 -0.74964572 7.875327307 2.712101343 19.2879

0 6.424622346 3.007668784 26.8364621 -3.11633219 5.733370743 13.41823105 13.58177

0 4.915224511 1.903198808 22.19645682 -1.95040058 4.55134123 11.09822841 23.901770 8.244274485 3.097967283 21.54106847 -3.17036212 7.669026873 10.77053424 10.22947

0 15.80321855 2.505435053 9.08825408 -2.51591747 15.60502893 4.54412704 6.455873

0 174.1560136 8.917113432 2.935139 -8.92100915 173.927777 1.4675695 0.46757

0 43.52807693 7.996715297 10.53137339 7.951808532 42.79558648 5.265686694 1.265687

0 19.32456481 5.326940935 15.8019621 5.259734204 18.59499937 7.900981052 1.099019

0 9.158609285 4.016425364 25.13929105 3.888919047 8.291949872 12.56964553 6.430354

0 7.910093436 1.383989933 10.02984391 1.376939439 7.789327054 5.014921957 16.98508

0 10.88075741 1.854653751 9.771170056 1.845685906 10.72307443 4.885585028 11.11441

0 21.76780906 2.169375502 5.71298218 2.165786218 21.65979874 2.85649109 5.143509

0 58.05218617 1.236323776 1.220833672 -1.23641723 58.03902182 0.610416836 2.3895830 21.76781998 1.393239209 3.669049812 -1.39419027 21.72324838 1.834524906 6.165475

0 10.24031396 2.500530828 13.99786492 -2.52530642 9.93653189 6.998932459 10.00107

0 7.535071618 1.140764954 8.678639387 -1.14511771 7.448883917 4.339319693 18.66068

0 12.19443463 5.988721 28.15238985 -6.22656322 10.75321483 14.07619493 0.076195

0 11.36977004 2.865699468 14.44846862 -2.89594476 11.01053452 7.224234311 6.775766

0 16.09399126 0.189352234 0.674449643 -0.1893566 16.09287737 0.337224821 9.662775

0 16.28205025 1.803351402 6.349128663 -1.80703613 16.18228539 3.174564332 6.825436

0 174.0242783 17.67808631 5.823287949 -17.7084749 173.1271446 2.911643975 1.911644

0 174.0189765 6.62304149 2.181743661 6.621442682 173.8929575 1.09087183 0.090872

0 174.0126022 0.762456583 0.251175499 0.762454144 174.0109318 0.12558775 0.8744120 43.52376544 1.493802479 1.967476581 1.493509221 43.49813316 0.983738291 3.016262

0 43.51142584 11.89277748 15.66833452 11.74525113 41.89622005 7.83416726 3.834167

0 17.32351469 12.81166186 42.39477319 11.67532036 12.79808797 21.19738659 11.19739

0 14.38928732 4.241694814 16.89829901 4.180530049 13.76861497 8.449149507 3.55085


26/50


Despite there being a very close correlation

between the derived Y values and a broad

-30

-25

-20

-15

-10

-5

0

5

10

15

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31x1

x2

0

20

40

60

80

100

120

140

160

180

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

y1

y2

0

5

10

15

20

25

30

35

40

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Angle

Original


27/50


correlation between the original angle and

the angle considering lean, it is recommended

to use the angle of steering for these

calculations.

3.7. INTEGRATING ANDMAPPING MOVEMENT

The final stage in this process is to actually

plot some points to demonstrate the

movement. Movement is considered to be

the distance and direction travelled to the

next point. The angle of the turn and distance

travelled in the current dt is already known so

the new point can be calculated using the

parametric equation of a circle and then

adding the previous position, which will act as

the origin.

dt Time Angle Movement X Y

0.068843953 0 0 0 0 0

0.074739419 0.068843953 -5 0.003972623 -0.000346062 0.0039575210.022569843 0.143583373 -10 0.003054153 -0.000876144 0.006965322

0.021356104 0.166153216 -7 0.102812872 -0.013399561 0.109012618

0.076438383 0.18750932 -22 0.781158696 -0.305885767 0.833347296

0.014518368 0.263947703 -27 2.045595522 -1.234131248 2.65620806

0.025705863 0.278466071 -35 5.255263349 -4.247093188 6.962001044

0.114520799 0.304171934 -21 6.339302551 -6.517796338 12.88067184

0.06889548 0.418692733 -11 3.544323525 -7.193746519 16.35994197

0.127438314 0.487588213 -1 5.182484641 -7.284147499 21.54163809

0.023090648 0.615026526 4 3.371261219 -7.04909923 24.90469541

0.180005126 0.638117175 9 8.06788396 -5.78763868 32.873350810.029316446 0.818122301 19 2.354997378 -5.021300878 35.10017344

0.118324639 0.847438747 22 3.146233466 -3.843268943 38.01753968

0.025785655 0.965763386 16 1.667161222 -3.383963912 39.62018294

0.10360612 0.991549041 8 2.630018933 -3.018120376 42.22463262

0.100718349 1.09515516 -3 1.717853055 -3.107980321 43.9401338

0.002857348 1.19587351 -8 1.736843071 -3.349580411 45.66009114

0.044747185 1.198730858 -17 2.993869404 -4.22447245 48.52327432

0.188248492 1.243478042 -23 2.327484513 -5.133457079 50.66592014

0.065680894 1.431726534 -14 1.9700843 -5.609826812 52.57754354

0.169819225 1.497407428 -14 9.998422682 -8.027462416 62.279269890.120381681 1.667226654 -10 0.161225934 -8.055444957 62.43804892

0.107851001 1.787608334 -10 0.852591764 -8.203421671 63.27770099

0.126320329 1.895459335 -1 3.554743207 -8.265429046 66.83190334

0.077535861 2.021779665 1 0.617145321 -8.254663835 67.44895477

0.087319371 2.099315526 1 11.20717209 -8.059170859 78.65442168

0.19686197 2.186634897 4 8.015844019 -7.500296855 86.65075929

0.051299809 2.383496867 4 2.160636998 -7.349654721 88.80613842

0.064542392 2.434796677 10 8.62022407 -5.853519662 97.29553433


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-40

-30

-20

-10

0

10

20

30

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Angle

Angle

-6

-5

-4

-3

-2

-1

0

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

X

X

0

20

40

60

80

100

120

140

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Y

Y


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The graphs above clearly show the movement

between data points based on the criteria

previously explained. By comparing the X and

Angle graphs, it becomes apparent that

altering the angle of the steering has an effect

on movement, but even a sudden alteration

still describes a smooth curvature. This is an

ideal result as it satisfies many criteria with

regard to accuracy of dynamics but also

outputs a model that is highly indicative of

what would be expected.

3.8. ENHANCEMENTSDespite overall success in the experiments

detailed here, there have been some sections

that presented substantial difficulty. A

complete solution would be providing proof

of the improved turning equations. Despite

the differences being too large, there is still

something to be gained for truly accurate

models if the problem can be solved or

conclusively proved.

Although the transition of the object can be

clearly tracked over the graphs, there are

issues that limit the model. The Z axis has

been completely ignored in this paper as the

values for it have no bearing on the research.

Converting to three dimensions requires the

alteration of the current Y axis becoming the

new Z component and a new Y set being

established to hold vertical data. At every

point in this model, the vertical component

will just be the height of the terrain currentlybeing used. As the equations are calculated

using the slope before any integration occurs,

the propulsion of the object has already

factored the modifications required by the

terrain. The object is then moved by the

specified amount in the indicated direction.

Improving on this would mean a substantial

overhaul to much of the system but would

allow the bicycle to potentially leave the

ground in moments of severe terrain by

utilising equations of inertia and momentum.

A final point to make is that a second run of

RK4 could be used to integrate the actual

movement more accurately. Although the

velocity is already integrated, as

demonstrated by the smooth curvature,

further refinement could be achieved with

little added cost.

3.9. CONCLUSIONSThis paper detailed the pipeline of equations

and calculations necessary to model the core

features of a bicycle. Power is the first thing

needing to be calculated, this is a value

determined by the power being put in to

something minus the power being taken out.

These factors are usually the riders pedal

power and any environmental factors.

It was then shown how this power gave an

acceleration, which could then be integrated

using RK4 to give a measure of distance that

would be traversed during the current delta

time step.

The third stage was steering. The lean of thebicycle was demonstrated to be easily

calculable but the addition of further

equations and complexity meant that choices

should be made with regard to specific

problems. This paper recommends that

unless a very high level of accuracy to real

world dynamics is paramount, more basic

measures will easily suffice. The Yaw of a

bicycle should be calculated based on an

upright position and any small increase can be

safely ignored as the difference is infinitesimal

and will be unseen by any users.

Furthermore, any change in radius from lean

should be avoided in all but the most

important cases. The closest approximation

to a solution shown here clearly demonstrates

a very high cost and minimal return. Finally,

the movement of the object was plotted in

order to prove the accuracy and workings ofthe solutions provided.


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4.USER SPECIFICATION

MATHEMATICAL MODELLING FOR THE CONTINUOUS

SIMULATION OF A BICYCLE

SPECIFICATION

SEAN ROBINSON


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Abstract- This document is a detailed specification of a limited physics engine currently in

development. The overview of the project is first explained so that the premise and overarching

aims are made clear. The functionality of the application will be listed with explanations of any

design choices made. In addition to the core system, there are certain project constraints that

need to be adhered to, these will be reported on with explanations. Project constraints range from

technical issues concerning interaction with the application to legal notes concerning the

intellectual property of the project. The testing that will take place following the completion of

the project will be documented in order to demonstrate completeness.

4.1. INTRODUCTION4.1.1.MOTIVATION

This application is the output of a level 3 project in Computer Games Development. The application

will exist as a suite of libraries tasked with dealing with the physics of the overall application, which

is a virtual bicycle simulator. The bicycle simulator, being developed by the Glamorgan Games

Company, will be a demonstration of the collaboration that exists between students and teaching

staff. The simulator itself works using a training bicycle providing feedback to a computer; this data

will be processed using the physics engine detailed here. The processed data will then be passed to

the front end of the application, which will use it to correctly output a graphical display indicating

the current state of the simulation.

4.1.2.SCOPEThe potential scope for an application of this type could be enormous; one of the more esoteric

tasks needing to be tackled is the evaluation of methodology and solutions in order to arrive at a

final product that is realistic within the time frame. Core functionality will be provided in order to

develop a working prototype that is fully tested. This prototype can then form the base of a more

detailed project. This document will detail only those aspects regarding an enhanced prototype, not

a complete solution.

Rationale: A complete solution would take much more time than is available for this project;

decisions will be made in order to ensure that a working product is delivered that achieves the core

functionality criteria detailed later in this document. Any remaining time can then be used to

enhance the application.

4.1.3.DEFINITIONSSimulation A way in which a scene or group of objects can dynamically change over time and be

output in a relevant graphical format.

Mathematical Modelling A system of using mathematical equations to determine the relationships

between objects and factors and how both sides respond to alterations in their linked counterparts.

Prototype A basic model of the final solution that has been proved. This acts as the base for future

work.


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Yaw/Lean The angle at which an object is rotated about an invisible line drawn along the base of

the Z axis in the current plane.

Turn Radius The radius of a virtual circle that would be drawn if an object completed a full circle by

travelling forward without change in the steering angle.

State The data currently stored in the environment variables that defines the way in which the

scene and all objects currently exist.

Acceleration The amount that the velocity is increasing or decreasing.

Velocity The rate at which the object is being propelled forwards.

Friction Refers to the friction of the floor surface currently acting to slow the object down.

Dt Delta time, the amount of time elapsed between game cycles.

Runge-Kutta/RK A method of integration that utilises weighted summations of derivatives in order

to generate a smooth curvature of movement.

4th

Order A scale on the Taylor Series of Expansion

Axis An imaginary line drawn to indicate a direction in the current plane.

Euler A basic integration method that is very quick to calculate but lacks accuracy.

Wheelbase The distance between the centres of both tyres.

Caster Angle The angle at which the head tube is reclined from the perpendicular.

Head Tube The tube connecting the handlebars to the front fork of a bicycle.

Steering Angle The angle at which the handlebars are rotated about the Y axis.

FPS Frames Per Second, the rate at which a screen is redrawn.

Exponential A value raised to the power of another.

Cost In this circumstance, cost indicates computational costs.

Encapsulation The way in which certain functionality is removed and stored independently to

promote privacy and modularisation.

Modularisation By promoting encapsulation, the reuse of functionality rises.

Coefficient A multiplicative factor

IDE Integrated Development Environment, a platform used in the development of software.

DLL Dynamic Link Library A Microsoft implementation of a shared library concept.

C++ - A programming language.


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4.1.4.CUSTOMERThe outcome of this project is to deliver an enhanced prototype of the engine detailed to the

Glamorgan Games Company. This document, in addition to any research papers and design

documents, will be delivered. Full testing reports and the commented source code will form the

core deliverable.

4.2. CORE FUNCTIONALITY4.2.1.OVERVIEW

The premise behind this application is simply the ability to move a bicycle across landscape using

propulsion based on the users output. The environment itself will modify the trajectory of the

machine based on factors such as friction and gravity. Steering the bicycle is accomplished by

turning the handlebars on the training machine; the value of turn is used to calculate rotational

details such as Yaw.

4.2.2.CORE FEATURES4.2.2.1. CALCULATE ACCELERATION

Acceleration is one of the most important considerations when modelling movement. This is

the amount that the velocity of an object must increase or decrease in order to reflect

changes made to the current state.

Acceleration will be calculated by modifying the force put into an object by the rider by the

force removed through environmental factors. These factors are gravity on slopes, friction

from the surface and air resistance. Acceleration will be measured in ms/s.

The acceleration value will be added to the current velocity to give a current speed; this will

then be integrated to calculate the distance travelled.

Rationale: By calculating the acceleration implicitly using the environmental factors, an axis

of calculation is effectively removed from the pipeline. By modelling a trajectory that utilises

all affecting factors, it is known that the height position of any point modelled will simply be

relative to its position on the bicycle and height of the ground. While this may lack accuracy

to a real world scenario, the lower computational and time cost is something that is

important to consider within the limited scope of the project.

4.2.2.1.1. INPUTS4.2.2.1.1.1. POWER

The inputs required calculating acceleration will be a power indicating the pedalling

currently taking place; this will be delivered in Watts. A default value of 100 will be

used in the absence of external input.

Rationale: The energy input into a bicycle when cycling is often measured in Watts.

Furthermore, studies have shown that 100 watts of output is a good average for

amateur cyclists.


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4.2.2.1.1.2. BRAKEThe brake value will create a deceleration value that will be subtracted from the

final acceleration value. This will be measured in ms/s and will be the output of a

function taking a percentage of the maximum safe deceleration defined as 0.5g. The

fraction of this used will be defined by the percentage of brake applied.

Rationale: Research in this project has shown 0.5g to be a safe maximum

deceleration constant. By taking a fraction of this as defined by the percentage of

brake used, it can be ensured that safe braking will always take place and will

remove the requirement for calculating scenarios where the machine becomes out of

the control of the rider.

4.2.2.1.1.3. SLOPEThe current slope will be defined using the Pythagorean Theorem to find the

hypotenusebetween two points in 3D space defined by the centre of the tyres. This

value will be converted to a percentage that defines a slope. The percentage will

range from 0% on a fully horizontal plane, to 100% for a fully vertical plane.

Rationale: Calculations involving gravity that are detailed later in this document use

the slope value as a percentage. This can also improve integration with other

systems as many roads and hills in the real world are measured using percentages.

Should an environment be loaded that already has these values mapped, calculating

the slope becomes a simplermatter of returning a value instead of calculating the

hypotenuse between two points.

4.2.2.1.1.4. WATTS TO OVERCOME FRICTIONThe watts required to overcome gravity will be subtracted from the input force. This

is calculated by multiplying the velocity in ms/s by the weight of the object in kg by

the friction coefficient defaulted to 0.1. The formula and coefficient value are both

outputs from previous research.

4.2.2.1.1.5. WATTS TO OVERCOME GRAVITYThe same format as friction, although the formula changes so that standard

acceleration from gravity is multiplied by the weight of the object in kg, the

percentage of the slope and the velocity in ms/s.

4.2.2.1.1.6. WATTS TO OVERCOME AIR RESISTANCEAs previously, but achieved by multiplying the air resistance coefficient determined

by research by the velocity in ms/s cubed.

4.2.2.1.2. OUTPUT4.2.2.1.2.1. ACCELERATION

After subtracting all environmental factors from the initial power, the acceleration is

achieved by dividing the force by the mass of the object. The mass in this case will

simply be the weight of the rider and bike in kg.


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4.2.2.2. INTEGRATE VELOCITYThe velocity will be integrated over the current dt using the Runge-Kutta 4

thorder method.

This will then be used in an Euler method to output the distance travelled in the current dt.

Rationale: Integrating the velocity instead of the position using RK4 is as effective as the

inverse but there is an added bonus of then being able to use integration for the position too.

While the cost may be a little too high to consider integrating both attributes using RK4, a

basic Euler method for position will provide a smooth integration curvature based on

derivatives of Velocity.

4.2.2.3. CALCULATE YAWThe yaw of the object is calculated by following a series of equations that were developed or

researched previously in the project. The initial calculation generates a turn radius using the

following:

= ()

Where:

r: Radiusw: Wheelbase in cm

: Steering Angle: Caster Angle

The lean is then calculated by the following:

= ()Where:

: Leanv: Velocity

g: Acceleration due to gravity

r: Upright turn radius

Rationale: Increase in lean due to bicycle components and changes in turn radius due to leanare not considered as research has shown the differences to be almost undetectable and

carry a high added cost.

4.2.2.4. GENERATE NEW POSITIONThe new position will be calculated using the parametric equation of a circle with the radius

set as the distance to travel, degree as the steering angle and the origin based on the

previous position.

Rationale: The equations have already been used and it is the obvious choice based on

information already known. With a radius and degree of turn and wishing to calculate a


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coordinate (A point on a virtual circumference defined by the distance to move), the

immediate choice is this equation.

4.3. CONSTRAINTS4.3.1.INTERACTION

Interaction with this library will take place purely through the passing of variables within the

system. The bicycle will monitor the values required; this data is then fed into the library. The

important considerations are the power and brake factors for acceleration, friction and air

resistance coefficient if differing from the default provided and the steering angle. The slope of

the terrain is defined by the hypotenuse so the position of the centres of both wheels must be

supplied in order for the calculations to work.

The output of all this will be very small, simply new values for the centre of the wheels in 3D

space. From these values, the graphical end of the application can build the remainder of the

bicycle. These values will be given as floating point coordinate values in their own data type

consisting of an X, Y and Z component. In order to provide feedback to the bicycle, the

percentage of slope calculated will also be returned so that the servo controlling the difficulty in

pedalling can be adjusted appropriately, this will be returned as the percentage itself.

This cycle of processing will take place every time it is called, this will usually be done once per

game cycle as defined by the refresh rate. The structural considerations are detailed later in this

document.

4.3.2.PERFORMANCEPerformance has been an important area to consider when researching this application. As the

processing cycle is likely to take place very frequently, it is important to ensure that the pipeline

is as streamlined as possible. While accuracy is a notable goal

mathematical modelling for the continuous simulation of a bicycle - research and design

Documents