creating a continuous and interactive simulation using bicycle dynamics and real world factors

8/3/2019 Creating a Continuous and Interactive Simulation Using Bicycle Dynamics and Real World Factors

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CREATING A CONTINUOUS AND INTERACTIVE

SIMULATION USING BICYCLE DYNAMICS AND

ACCURATELY MODELLED ENVIRONMENTAL FACTORS

SEAN ROBINSON


2/21

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 .................................................. 2

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

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

I. Power........................................................ 3

II. Surface Friction ......................................... 3

III. Braking ...................................................... 3

IV. Slopes ....................................................... 4

V. Watts to Overcome Gravity ...................... 4

VI. Watts to Overcome Surface Friction ........ 4

VII. Watts to Overcome Air Resistance ...... 4

VIII. Final Power Output .............................. 5

IX. Calculating Acceleration ........................... 5

4. Integrating Velocity ....................................... 7

5. Calculating Lean .......................................... 10

6. Calculating Steering Angle .......................... 14

7. Integrating and Mapping Movement .......... 17

8. Enhancements ............................................ 19

9. Conclusions ................................................. 19

Bibliography ......................................................... 20

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.

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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Page 3 of21 Sean Robinson 2012

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.CALCULATING VELOCITYI. POWER

The 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 (1), 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.

II. SURFACE FRICTIONSurface friction is a difficult thing to model ina 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 (2).

In a more developed project, this value can

easily be changed to reflect the surface

currently being used. This could either be an

arbitrary value that seems to give anappropriate feel or something more accurate

that has been researched.

III. 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 tyre of the 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 (3). 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.

IV. 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.

V. WATTS TO OVERCOME GRAVITYFollowing 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 (2) is:

= Where:

G: Standard acceleration due to gravity

P: Weight of object is kg

: Slope PercentageV: Current velocity in m/s

VI. WATTS TO OVERCOME SURFACEFRICTION

Surface friction has already been discussed

and 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/s

P: Weight of object is kg

VII. 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 be

outside the scope of this project to determine

a truly valid coefficient for wind resistance so

0.7, a value based on work by May (2) 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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VIII. FINAL POWER OUTPUTThe final power if what remains once all

subtractions for environmental factors have

been 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

IX. CALCULATING ACCELERATIONAcceleration 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 (4), 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 (5). As a runge-Kutta4 method is

going to be used, there is much more work tobe 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 0.100141784

100 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 -0.000101048

100 0.1 0 0 1.746138339 0 13.96910671 3.726781956 82.30411133 1.028801392

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


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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

0

1

2

3

4

5

6

7

89

10

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

Velocity (m/s)

Velocity (m/s)


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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 (6) and then modified by Kutta (7) .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 andthen summing them using weights. This then

gives a good approximation of what the

velocity will be (8).

-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)

-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)


8/21


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) + (.))Where:

t: Delta TimeT: Current Time

V: Current velocity in m/s

D1: Derivative 1D2: 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 2.0230330.076205 0.613928 9.7632255 3.045516 -0.45209 -1.2008 -0.90042 -0.69053 2.154672

0.176485 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 1.377238


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0.086196 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 1.516813

0.183596 3.026488 148.0187054 4.009877 -1.47238 -3.9352 -1.5315 -2.2554 1.566344

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 indicationof what is happening over time. The following

graphs 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 mappedhere that shows a higher velocity will push the

position 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


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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 a

corner, 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.

0

1

2

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

-4

-3

-2

-1

0

1

2

3

4

5

6

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

dv

dv


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Left

Turn

Right

Turn

Change in

Steering

Steering

Angle

0 0 0 0

5 0 -5 -55 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 -1

0 5 -3 4

0 5 -5 90 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 -140 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 therider 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 (9) is:


12/21


= ()

Where:

r: Radius

w: 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 byFajans (10).

= ()Where:

: Caster Anglev: 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 as

a result of tyre and centre of mass

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 (11), 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 closelyto 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 (11).

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

: Steering Anglea: Lean

The relevant sections of the data sheet areincluded below:

Upright Turn Radius Ideal Lean Lean Increase Lean Final Radius

0 0 0 0 0

34.83177044 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 4.9152245118.293278676 0.113400991 -0.004637643 0.108763348 8.244274485


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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 43.52807693

19.35098358 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 12.19443463

12.43991801 0.435097909 -0.017275416 0.417822493 11.36977004

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

180200

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

Upright Turn Radius

Final Radius


14/21


From the visual representation, it is very clear

that 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.

6.CALCULATING STEERINGANGLE

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 travelled

in 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 large

to ignore. This coupled with the vast amount

of time being tunnelled into the solution and

with the possibility of a very small return but

huge costs associated meant that the pursuit

of the solution was abandoned. The method

0

0.05

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

-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


15/21


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 (12).

= : Arc lengthr: Radius

: Inner angle between both points in radians

With this inner angle, the new point could be

derived 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 (13), states that the

angle between a coord and an angle is half of

the inscribed angle between the points. This

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 theextra 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 23.90177

0 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 2.389583

0 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


16/21


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 0.874412

0 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

-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

140160

180

200

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

y1

y2


17/21


Despite there being a very close correlation

between the derived Y values and a broad

correlation between the original angle and

the angle considering lean, it is recommended

to use the angle of steering for these

calculations.

7.INTEGRATING AND MAPPINGMOVEMENT

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 thenadding 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 0.003957521

0.022569843 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 32.87335081

0.029316446 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 42.224632620.100718349 1.09515516 -3 1.717853055 -3.107980321 43.9401338

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


18/21


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 62.27926989

0.120381681 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

-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


19/21


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.

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 currently being

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.

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.

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


20/21


The third stage was steering. The lean of the

bicycle 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 ofaccuracy 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

two dimensional space in order to prove the

accuracy and workings of the solutions provided.

BIBLIOGRAPHY

1. Coggan A. Training Peaks. [Online]. [cited 2012 01 05. Available from:

http://home.trainingpeaks.com/articles/cycling/power-profiling.aspx.

2. May Q. MayQ. [Online].; 2010 [cited 2012 01 05. Available from:

http://www.mayq.com/Best_european_trips/Cycling_speed_math.htm.

3. Whitt FR, G WD. Bicycling Science. 2nd ed. Massachusetts: Massachusetts Institute of technology; 1982.

4. Newton I. Philosophiae Naturalis Principia Mathematica. 1st ed. London; 1687.

5. Euler L. Institutionum calculi integralis volumen secundum St. Petersburg: Imperial Academy of Science;

1769.

6. Encyclopedia.com. Encyclop. [Online].; 2008 [cited 2011 11 02. Available from:

http://www.encyclopedia.com/doc/1G2-2830903781.html.

7. GAP. http://www-gap.dcs.st-and.ac.uk. [Online]. [cited 2011 11 02. Available from: http://www-gap.dcs.st-

and.ac.uk/~history/Biographies/Kutta.html.

8. Fiedler G. gafferongames.com. [Online].; 2006 [cited 2011 11 02. Available from:

http://gafferongames.com/.

9. Sharp RS. Motorcycle Steering Control by Road Preview. Journal of Dynamic Systems, Measurement andControl (ASME). 2007 July;(129).

10. Fajans J. Steering in Bicycles and Motorcycles. American Journal of Physics. 2000 July; 68(7).

11. Cossalter V. Motorcycle Dynamics. 2nd ed.: Lulu; 2006.

12. Math Warehouse. Math Warehouse. [Online]. [cited 2012 01 05. Available from:

http://www.mathwarehouse.com/trigonometry/radians/s=r-theta-formula-equation.php#.

13. Warendorff J. Wolfram Demonstrations project. [Online].; 2012 [cited 2012 01 05. Available from:

http://demonstrations.wolfram.com/TangentChordAngle/.


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