Development of the required components of a performance vehicle model to study the suspension system characteristics

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Development of required components of a
performance vehicle model to study the
suspension set-up effectiveness
Kayalarasan – 18028785
Programme: Motorsport Engineering
Module: P04791 – MSc Projects
Submission: Project report
Year: 2018/19
Word count: 10,327 and 30 figures
Supervised by: Collin bell | Senior lecturer | Oxford Brookes university
School of Engineering, Computing and Mathematics

MSc, Dissertation report_ 18028785_Kayalarasan
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Declaration form
Statement of originality
Except for those parts in which it is explicitly stated to the contrary, this project is my own work. It has
not been submitted for any degree at this or any other academic or professional institution.
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Highlights
 This project concept based on ‘Reverse engineering’, the data collected from a race car
processed using the Matlab tool developed in a very efficient and innovative way that assists to
understand the existing level of suspension performance and aid for optimization.
 With the literature review made for this project, the author found that this is the first paper to
research on combining three different approaches consists of tyre modelling, sensor fusion and
rigid body modelling to determine the race car dynamics.
 The Matlab tool created requires input data as minimal as from three-sensors (an Inertial
measurement unit, GPS and potentiometer) from a race car to generate the results that support
a race engineer.
 The behaviour of the chassis and tyres of a race car addressed using mathematical modelling to
analyse the effectiveness of suspension set-up.
 As part of this project, a Matlab model has been created that consists of over 2,000 lines of code.

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Abstract
In four-wheeled motorsport, setting-up a vehicle suspension for a race is very demanding and the
allowed practise period is very limited. Typically, a race engineer would collect data from a race car
during that short-period to analyse and understand the set-up required for that specific race track. In
recent years, using telemetry technology many gigabytes of data acquired using numerous sensors are
transferred to the racing team to precisely monitor and study the vehicle’s performance using data
analyses software. The software and sensors involved in this process are very expensive.
This paper aims to present a simple and an effective Matlab tool, developed from a ten degree of
freedom (DOF) information acquired using just three sensors from a racecar, those data reverse
engineered with an innovative mathematical modelling approach to study the effectiveness of the
suspension set-up by modelling the performance of the components associated with the suspension. The
tool consists of a magic formula 6.1 tyre model to represent the actual tyre force and moment. Followed
by sensor fusion of an inertial measurement unit (IMU) and GPS to extract the driven racing line and the
chassis orientation precisely. Lastly, a six-DOF rigid body model with an inertia tensor to characterise the
dynamics of the chassis by imposing the chassis orientation data.
A quaternion and a Rotation matrix formed to store the chassis orientation using the angular velocity. A
Kalman filter used to fuse the acceleration and GPS sensor to obtain the drift-free velocity and the driven
race line. The inertia tensor combined with the rotation matrix to get the dynamic load transfer and the
forces acting on each wheel. The tyre slip properties obtained by incorporating the tyre model with the
rigid body model. Then, the suspension damping performance computed using the vertical wheel
displacement and the ride heights extracted using the vector theory.
The results generated by the Matlab tool will require further analyses by a race engineer to realize the
optimizing window of the existing suspension set-up in a short amount of time on the circuit. The data
supplied and model validation done using the well-established AVL VSM lap time simulation software, a
good level of agreement obtained even with the limitations and assumptions considered in the tool. The
nature of this concept has the potential to convert into a commercial package software which requires
less computational power and minimal data compare to a conventional method.

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Acknowledgements
I would like to thank everyone who supported me throughout this research work conducted from
January 2019 until September 2019, especially thank you Colin bell at the Oxford Brookes university for
your official guidance. A big thank you to the Oxford Brookes University and all the lecturers who have
supported during the term period.

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Contents
Declaration form...................................................................................................................................................2
Highlights..............................................................................................................................................................3
Abstract ................................................................................................................................................................4
Acknowledgements..............................................................................................................................................5
Contents ...............................................................................................................................................................6
List of figures ........................................................................................................................................................8
List of tables....................................................................................................................................................... 10
List of abbreviations .......................................................................................................................................... 11
1 Introduction............................................................................................................................................... 13
1.1 Background........................................................................................................................................... 13
1.2 Aim and Objectives........................................................................................................................... 15
1.3 Justification....................................................................................................................................... 15
1.4 Structure of the report..................................................................................................................... 16
1.5 Literature review.............................................................................................................................. 17
1.5.1 Tyre modelling........................................................................................................................... 17
1.5.2 IMU and GPS Sensor fusion....................................................................................................... 22
1.5.3 Six-DOF Rigid Body model ......................................................................................................... 25
1.5.4 Summary.................................................................................................................................... 27
2 Methodology ............................................................................................................................................. 28
2.1 Overview........................................................................................................................................... 28
2.2 Tyre Modelling Methodology .......................................................................................................... 29
2.3 IMU and GPS Sensor fusion methodology...................................................................................... 32
2.4 Six-DOF Rigid body model Methodology........................................................................................ 36
2.5 Summary........................................................................................................................................... 41
3 Results and Discussion............................................................................................................................... 43
3.1 Introduction...................................................................................................................................... 43
3.2 Tyre modelling.................................................................................................................................. 43

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3.3 IMU and GPS sensor fusion.............................................................................................................. 47
3.4 Six-DOF Rigid body model................................................................................................................ 50
3.5 Summary........................................................................................................................................... 53
4 Conclusions and Future work.................................................................................................................... 54
4.1 Future work ...................................................................................................................................... 55
4.1.1 Tyre modelling........................................................................................................................... 55
4.1.2 IMU and sensor fusion............................................................................................................... 55
4.1.3 Six-DOF rigid body model .......................................................................................................... 55
5 References................................................................................................................................................. 56
6 Bibliography............................................................................................................................................... 62
Appendix............................................................................................................................................................ 65
Appendix 1: Coefficients used in the Magic formula 6.1 tyre model.......................................................... 65
Appendix 2: IMU and GPS data extracted from the AVL VSM software simulated using the Silverstone
circuit............................................................................................................................................................. 67
Appendix 3: Quaternion computed from the angular velocity of the vehicle .......................................... 67
Appendix 4: Kalman Filter – Position, velocity and direction-wise acceleration ...................................... 68
Appendix 5: Moments from all three motions to study the balance of the car ....................................... 69
Appendix 6: Yaw moment and lateral acceleration to analyse the over and understeer phenomena... 69
Appendix 7: Vehicle corner weight and the respective forces .................................................................. 70
Appendix 8: Tyre Slip angle and the slip ratio............................................................................................. 71
Appendix 9: Chassis and wheel displacement with relevant transmissibility ratio and the input
frequency...................................................................................................................................................... 73
Appendix 10: Suspension system - transmissibility ratio Vs the Input frequency..................................... 75
Appendix 11: Vehicle Coordinate System and dimensions used in the model........................................... 76

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List of figures
Figure 1. Suspension influence on three major performances of an F1 car (Mercedes-f1, 2019) ................. 13
Figure 2. Circuit of The Americas (COTA) turn 1 shows the track elevation on the left, the gap between
Actual and driven racing lines on the right (Formula1, 2019)......................................................................... 14
Figure 3. Indoor tyre testing machine at the calspan tire research facility centre (Calspan, 2019)............. 17
Figure 4. Parameters to develop the Pacejka's Magic Formula Tyre model (Pacejka and Besselink, 2012) 19
Figure 5. The geometry of the TMeasy model with combined tyre forces (Rill, 2013)................................. 19
Figure 6. Brush model - Longitudinal deformation of the rubber between the road and carcass and the
level of flexibility shown in the two regions (Svendenius, and Wittenmark, 2003)..................................... 20
Figure 7. Carcass and tread deformation during combined slip (Xu, et al., 2014)......................................... 20
Figure 8. A 6-axis IMU and the GPS sensor from the Vbox automotive (Ina. 2019) ..................................... 22
Figure 9. State estimation method in the Kalman filter (Understanding Kalman Filters, MATLAB. 2019).. 23
Figure 10. Toyota TS050 Hybrid LMP1 2018 race car (Racecar Engineering. 2019) ....................................... 27
Figure 11. An overview of the project concept................................................................................................ 28
Figure 12. The process flow involved in the different phase of this project.................................................. 28
Figure 13. Comparison between the tyre models with the main aspects to consider (Pacejka and
Besselink, 2012)................................................................................................................................................. 29
Figure 14. An overview of the Sensor fusion methodology........................................................................... 32
Figure 15. The ground rules of vector theory used in the rigid body dynamics ............................................ 36
Figure 16. The interlink between each model in the Matlab tool .................................................................. 41
Figure 17. Pure and combined longitudinal force and slip ratio generated from the MF tyre model.......... 43
Figure 18. Pure and combined lateral force and slip angle generated from the MF tyre model ................. 44
Figure19. Pure and combined Aligning torque and slip angle generated from the MF tyre model ............ 45
Figure 20. Tyre stiffness for pure longitudinal, Lateral and aligning torque from MF tyre model .............. 45
Figure 21. Magic formula 6.1 tyre model developed in Matlab on the left, AVL VSM magic formula 5.2 tyre
model on the right for comparison with different vertical load.................................................................... 46
Figure 22. All 9 terms in the rotation matrix validated against the Matlab toolbox..................................... 47
Figure 23. Vehicle Position, velocity obtained for X-axis using Kalman filter................................................ 48
Figure 24. The driven racing line generated using the sensor fusion method on the left, Image of the
Silverstone circuit cropped from the google map on the right for comparison .......................................... 48
Figure 25. Chassis movement illustrated using the RM and vehicle position with the direction wise
coordinate system............................................................................................................................................ 49
Figure 26. Yaw moment generated by the race car on the Silverstone circuit............................................. 50
Figure 27. The ride height at four locations of the chassis derived using vector theory.............................. 51

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Figure 28. Front left corner weight and the respective forces generated using newton’s second law...... 52
Figure 29. The Slip angle and the slip ratio generated for the forces experienced by the Font left tyre.... 52
Figure 30. Front left suspension transmissibility ratio and the Input frequency to study the damping ..... 53

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List of tables
Table 1. Evolution of Magic formula tyre model............................................................................................. 18
Table 2. The main variables used in the magic formula tyre model............................................................... 19
Table 3. Classification of the inertia tensor..................................................................................................... 25
Table 4. Inputs supplied to the magic formula tyre model............................................................................ 30
Table 5. Matlab coding algorithm and the link between each section in the MF tyre model (OptimumTire
Documentation – OptimumG, 2019)................................................................................................................ 31
Table 6. IMU and GPS data used in the sensor fusion model ........................................................................ 33
Table 7. The vehicle data referred from the AVL VSM software.................................................................... 37
Table 8. Corner locations and the respective inertia tensor with and without aerodynamic loads........... 38
Table 9. Ride height locations vectors from the COG to compute the velocity ........................................... 39

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List of abbreviations
Symbol Unit Description
B - Stiffness factor
C - Shape factor
D - Peak value
E - Curvature
K % Slip ratio
α rad Slip angle
SV - Vertical shit
SH - Horizontal shift
Vx m/s Forward velocity
γ rad Inclination / camber angle
μ - Friction level
Ω rad/s Angular velocity of the tyre
Re M Effective Rolling radius
R0 M Tyre circumference
Fz N Vertical force
Pio Pascal Nominal tyre pressure
Fxo N Pure Longitudinal force
Fx N Combined Longitudinal force
Fyo N Pure Lateral force
Fy N Combined Lateral force
Mxo Nm Pure aligning torque
Mx Nm Combined aligning torque
𝐴𝑐𝑐 𝑋 𝑚/𝑠2 Longitudinal acceleration
𝐴𝑐𝑐 𝑌 𝑚/𝑠2 Lateral acceleration
𝐴𝑐𝑐 𝑍 𝑚/𝑠2 Vertical acceleration
𝐺𝑦𝑟𝑋 rad/s Roll rate
𝐺𝑦𝑟𝑌 rad/s Pitch rate
𝐺𝑦𝑟𝑍 rad/s Yaw rate
𝑃𝑜𝑠 𝑋 m Position in X-axis
𝑃𝑜𝑠 𝑌 m Position in Y-axis
𝑃𝑜𝑠 𝑍 m Position in Z-axis
𝑉𝑒𝑙 𝑋 m/s Velocity X-axis

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𝑉𝑒𝑙 𝑌 m/s Velocity Y-axis
𝑉𝑒𝑙 𝑍 m/s Velocity Z-axis
q - Quaternion
RM - Rotation matrix
∆𝑡 Sec Sampling period
I 𝑘𝑔. 𝑚2 Inertia tensor
F N Force
M Nm Moment
𝜔 rad/s Angular velocity
𝑉𝐹𝐿 𝑅𝐻
m/s Velocity at the front left ride height location
𝑃𝐹𝐿 𝑅𝐻
M Position at the front left ride height location
COG - Center of gravity
IMU - Inertial measurement unit
GPS - Global positioning system
FL - Front left
FR - Front right
RL - Rear left
RR - Rear right
DOF - Degree of freedom

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1 Introduction
Motorsport is a competitive racing event where a group of teams compete with each other with their
vehicles and drivers. In the four-wheel domain, racing circuit changes for every single race within a
session (Federation Internationale de l'Automobile, 2019). As a result, setting up a vehicle for each race is
more challenging within the permitted practising time, that demands information about various factors
such as the environmental condition and track characteristics. To make some essential adjustment in the
vehicle set-up to improve the lap time, a race engineer would get the driver’s feedback and a huge
amount of data analysis recorded from the vehicle using various sensors. Misinterpretation of an issue
due to lack of knowledge on the factors would cost a few seconds in a lap, especially working with less
proficient driver (Giaraffa, 2012).
1.1 Background
Utilizing data obtained from a race car and processed with different concepts to study the effectiveness
of the suspension set-up. The suspension system has several mechanisms such as springs, dampers that
work together to provide optimum ride and handling performance and it is one of the complex and key
aspects to produce a winning show, the significance of a suspension system in a racing vehicle is shown
in Figure 1 (Floyd and Law, 1994),
1. To keep all the tyres grounded to extract the maximum possible grip.
2. To control the chassis movement to have a better aero balance that enhances the aero grip. And
3. To maintain the desired ride height/rake angle that allows the diffuser to work efficiently.
Figure 1. Suspension influence on three major performances of an F1 car (Mercedes-f1, 2019)
To realize the effectiveness of the suspension set-up of a racecar, knowledge and understanding of the
components and its performance that are associated with the suspension system is very crucial
(Haubenreich and Law, 2000),
1. Tyre force and moment ability
2. Chassis dynamics
3. Track characteristics (driven racing line)

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Tyres are the one and only negotiator between the vehicle and the road surface, that handles all the
force and moments. So, it is highly imperative to keep the tyres as good as possible on the track (Lee and
Taheri, 2017). Tyres and suspension work together in series, thus, tuning one will influence another
performance. Tyres capability acknowledged by the force and respective slip properties that dictate the
optimal working range. This information is vital to set-up the suspension accordingly.
The aerodynamics downforces on a race car is another influential factor that pushes the tyres to the
ground, allows creating more accelerations (Imani and Limebeer, 2015). The downforce generating
elements such as front wing work efficiently at certain angles, therefore, controlling the chassis
movement even for a millimetre is important. Also, it affects, the airflow around and underneath the
vehicle due to change in the ride heights/rake angle that disturbs the aero balance and the ground effect
(Berman, 2016). If a vehicle is equipped with a stiffer suspension set-up, that helps to control the load
transfer/body movement (Berman, 2016) but, it worsens the mechanical grip. So, finding a perfect
balance between these two parameters is a demanding goal for a race engineer (F1technical, 2019).
Mechanical grip depends on the suspension set-up that is more important on the low-speed tracks such
as Hungary (Formula1-dictionary, 2019). Each racing circuit has unique characteristics as shown in Figure 2
like racing line, gradients, corners etc. The actual racing line is the shortest path to follow to achieve the
best lap timing. If a driver misses the line then the time spent on that corner will increase from the
desired one (Cardamone, et al., 2010). During the combined conditions (steering input plus braking or
acceleration), high level of load transfer takes place that affects the tyre grip and the aerodynamics
downforce, a clever suspension set-up will produce optimum performance.
Figure 2. Circuit of The Americas (COTA) turn 1 shows the track elevation on the left, the gap between
Actual and driven racing lines on the right (Formula1, 2019)

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1.2 Aim and Objectives
This project aimed to develop a simple and effective Matlab tool that supports a race engineer to process
a Ten-Degree of freedom (DOF) data collected from the components associated with the suspension
system of a race car, to study the effectiveness of the existing suspension set-up.
To achieve this, the following objectives were set:
1. To create the Magic formula 6.1 tyre model in MATLAB that represents the force and moment
ability in a graphical presentation for the given tyre.
2. To create a MATLAB model that allows generating the driven racing line and the chassis
movement as an input of the six-axis Inertial measurement unit (IMU) and GPS data.
3. To create a six-DOF rigid body model in MATLAB that represents the chassis with mass
properties, combines with the previous two models to derive the mass distribution, ride heights,
tyre slip properties and suspension transmissibility for the given input.
1.3 Justification
Before adjusting the suspension set-up, it is vital to understand the performance of the components
associated with the suspension system (Lamers, 2008). In motorsport, understanding a race circuit and
optimizing the set-up is one of the most important steps in improving the performance of the vehicle.
This Matlab tool will allow a race engineer to know the tyres force and moment ability, chassis dynamics
and the driven racing line from a test lap for any given circuit that makes the set-up process quicker and
reduce the development cost. As the motorsport financial regulations are getting stringent that the
racing teams will be allowed to spend equal and limited amount of budget on the vehicle development
process in upcoming years (Adam Cooper, 2019).
The core usage and benefits of this tool:
1. Specifically designed to support a race engineer.
2. Useful for new tracks or where the race engineer lacks knowledge about the racetrack
characteristics.
3. Can be especially useful when working with amateur drivers.
4. Required only ten-DOF information from a race car as an input.
The raw data for this study extracted from the AVL VSM Lap time simulation software and the software
been used by many leading motorsport companies for its advanced simulation technologies (Avl racing,
2012). This paper described a tool developed completely using the Matlab software. The rigid body and

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sensor fusion deal with the matrix and algebra, where the Matlab is the quickest in computing compared
to python and other software (Southhampton, 2019).
1.4 Structure of the report
Because of the complexity of this project, each main section of the report has been divided into three
sub-sections that mirror the models created:
1. Tyre modelling
2. IMU and GPS Sensor fusion
3. Six-DOF Rigid Body model
Initially, the literature review shows some of the similar researches conducted and those were referred
to support this study. Next, the methodologies explained to build the models in the Matlab. Then, the
results obtained from the Matlab tool shown in the graphical form with appropriate comments. From
that, the conclusion and future work presented that highlights the summary of the project and extension
of each model. The appendix section has formed to have the spare figures and the report ends with the
references and bibliography.

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1.5 Literature review
This literature review covers a number of topics that are relevant to the project. It begins by covering
tyre modelling, including the tyre models that are used in the project. It then covers IMU and GPS sensor
fusion. Finally, the review summarises the literature on six-DOF Rigid Body Dynamics.
1.5.1 Tyre modelling
Tyres are one of the crucial and challenging components to fully understand, that creates a major effect
on race car dynamics, the operating temperature on the tread is very important to generate sufficient
level of grip (Milliken and Milliken, 1995) and it is sensitive to factors such as vertical load and camber
angle. Figure 3 shows an Indoor tyre-testing machine, which is one of the methods to acquire tyre force
and moment data. To predict a tyre’s performance, it must have tested in many different setups
according to the usage on the racetrack. As a result, that would raise the testing cost, time and the test
data (Jonson and Olsson, 2016). To address this situation several tyre models have been developed.
Figure 3. Indoor tyre testing machine at the calspan tire research facility centre (Calspan, 2019)
1.5.1.1 Magic formula tyre model
A semi-empirical tyre model industrialised 32 years back by Hans Pacejka which is known as the Magic
Formula (Uil, 2007), widely accepted by the automobile industries and researchers (Schmeitz, et al.,
2005). The Magic formula (MF) tyre model allows computing the steady-state and the transient force and
moment of a tyre that minimizes the testing cost and time drastically (Pacejka and Besselink, 1997). The
MF has gone through many versions to improve the capabilities and the accuracy of the model as shown
in Table 1.

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Table 1. Evolution of Magic formula tyre model
In general, the force and moment data represented in three different ways using a spreadsheet, graphs
and formulas, the last method suits well for theoretical studies and simulations. Singh and
Sivaramakrishnan, (2015) the magic formulas coefficients derived using the least square fitting algorithm
with variables as shown in Figure 4.
Version
names
Released Year Developments from the previous version Reference
Pacejka’87 1987
Pure side force, brake force and self-aligning
torque
Bakker, Nyborg
and Pacejka, (1987)
Pacejka’89 1989
Influences of ply steer, conicity and camber are
included
Bakker, Pacejka
and Lidner, (1989)
Pacejka’94 1993-94 Tyre horizontal force generation at combined slip
Kuiper and Van
Oosten, (2007).
Pacejka’96 1996
The combined lateral and longitudinal response as
well as lateral camber response, Pneumatic trail
and load sensitivity.
OptimumTire
Documentation -
OptimumG. (2019)
Deflt’97 1997 Longitudinal and lateral transient responses
(Pacejka,
Besselink, 1997).
Pac’2002 2002
Includes models for the rolling resistance,
overturning moment and turn slip.
OptimumTire
Documentation -
OptimumG. (2019)
MF 5.2 -
The effect of camber on the longitudinal
coefficient of friction.
Pacejka’06 2006
Includes significant modifications to the pure
lateral, aligning torque models, combined lateral
and longitudinal models to adopt large camber
and turn slip.
MF 6.1 /
MF swift
2010
It contains inflation pressure dependency and has
the Swift model included and used to simulate
higher frequency input 100 Hz and uneven roads.
TASS
International.,
(2019)
MF 7.3
2019
Not released
yet
It can run on real-time platforms and also
temperature sensitive

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Figure 4. Parameters to develop the Pacejka's Magic Formula Tyre model (Pacejka and Besselink, 2012)
The basic equation (equation 1) used to compute the pure and combined force and the moment, Table 2
shows the variables used in the equation.
𝑦 = Dx sin [Cx arctan {Bx. kx − Ex(Bx. kx − arctan(Bx. kx)}] + S𝑣 {1}
Table 2. The main variables used in the magic formula tyre model
B Stiffness factor α Slip angle
C Shape factor SV Vertical shit
D Peak value SH Horizontal shift
E Curvature x Inputs (slip ratio or slip angle)
K Slip ratio y Outputs (force and moment)
1.5.1.2 TMeasy tyre model
Another semi-empirical tyre model is the TMeasy that was developed in 1994 by Georg Rill., (2013). See
Figure 5 the model can generate a 3-dimensional slip and seamlessly shift from standstill to driving
condition makes this suitable for vehicle dynamics simulations (Rill, 2013).
Figure 5. The geometry of the TMeasy model with combined tyre forces (Rill, 2013)

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The longitudinal and lateral force derived from the equations 2 & 3:
𝐹𝑥 = 𝑓𝑆𝑥 𝑁 {2}
𝐹𝑦 = 𝑓𝑆𝑦 𝑁 {3}
Normalized longitudinal and the lateral slips represented by the ‘SN
’ variable. The global derivative factor
‘f’ used to obtain the combined force value.
1.5.1.3 Brush model
Svendenius, and Wittenmark, (2003) studied a method to convert the brush model more flexible to use.
The rubber section between the carcass and the road plays an important role, that considered as two
distinct parts known as adhesion and sliding region used to compute the force and moment as shown in
Figure 6. The bristles at contact patch assumed to be flexible and defined as the slip angle.
Figure 6. Brush model - Longitudinal deformation of the rubber between the road and carcass and the
level of flexibility shown in the two regions (Svendenius, and Wittenmark, 2003)
1.5.1.4 An Analytical tyre model
An Analytical model is another tyre model suitable for theoretical studies shown in Figure 7. The braking
and driving conditions simulated under consideration of a huge number of tyre physical parameters.
Model is mainly concentrated on carcass compliance under combined slip angle (Xu, et al., 2014).
Figure 7. Carcass and tread deformation during combined slip (Xu, et al., 2014).

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Berman, (2016) used Pacejka’89 tyre model to optimise the dampers and three springs of an LMP2 car
and mentioned that the moment of a tyre has a minor contribution to the vehicle dynamics. The tyre
model developed in the Matlab used to regress the lateral force and aligning moment data obtained
from the track testing. Jang and Karnopp (2000) studied the simulation of a vehicle and its power
steering dynamics with different vehicle loads using the Pacejka’87. He established that adjusting the MF
coefficients would help to have a better fit with the experimental data.
Lugner, Pacejka and Plöchl (2005) researched about the tyre models and validated using three different
multi-body simulation softwares. And obtained very good agreement with the experiment data and
found that the implementation of the tyre model requires enough details to achieve accurate results.
Similarly, Jansen, et al., (2005) validated the MF 6.1 swift model accuracy with an experimental data on
parking behaviour and combined slip using the Simulink software. Established that to achieve sufficient
accuracy in the result, all the input parameters relevant to the event must be included in the model.

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1.5.2 IMU and GPS Sensor fusion
A six-axis IMU sensor contains an accelerometer and a gyroscope unit that measures the acceleration
and angular velocity respectively, this sensor typically mounted at the centre of gravity (COG) of a car
figure attached in the appendix (Racing Car Dynamics, 2019). An accelerometer is used to measure the
inertial force acting in the opposite direction and the Gyroscope sensor quantifies the orientation in the
rate of change of angles per second (Starlino, 2009). In vehicles, the IMU and GPS shown in Figure 8 used
to know the position and the orientation (Tim, 2019) and (Racecar Engineering, 2019).
Many pieces of research have done on fusing both the sensors together with data filtering to obtain
collective results to estimate the state of an element. IMU has been used for a variety of applications,
including estimating pitch and roll angles (Oh and Choi, 2013) or even detecting the drowsiness of a driver
(Lawoyin, Fei and Bai, 2014).
Figure 8. A 6-axis IMU and the GPS sensor from the Vbox automotive (Ina. 2019)
Syed, et al., (2007) describes that the rotation matrix obtained by using the gyroscope angular velocity
and then the accelerometer data fused with the rotation matrix, then the result integrated twice to get
the state estimation. The GPS data used to update and correct the drift of the IMU data since the
acceleration and angular data tend to diverge with the time due to the integration. Oh and Choi, (2013)
mentioned that the GPS signal in a vehicle is not robust as it losses some data when the vehicle enters a
weaker signal area such as tunnels, under the trees. Rehbinder and Hu (2004) estimated the attitude of a
robot using the gyroscope and accelerometer unit. Long-term drift-free estimation of attitude from the
IMU achieved by using the state algorithms.
Sanz Díaz, (2015) obtained the state estimation of a quadrotor system by applying the Kalman filter to
linearize the last estimated state. Euler’s angle (equation 4), The Rotation matrix (RM) (equation 5), and

23
Quaternions (equation 6) mathematical approaches were established to represent the orientation. The
Quaternion method is more efficient and less computational effort compared to the other two
approaches.
𝐸𝑢𝑙𝑒𝑟 = [𝑅𝑜𝑙𝑙 (𝜙) 𝑃𝑖𝑡𝑐ℎ (𝜃) 𝑌𝑎𝑤 (𝜓)] {4}
[𝑅𝑀] = [
1 0 0
0 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
0 −𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃
] . [
𝐶𝑜𝑠𝜃 0 −𝑠𝑖𝑛𝜃
0 1 0
𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃
] . [
𝐶𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 0
−𝑠𝑖𝑛𝜃 𝐶𝑜𝑠𝜃 0
0 0 1
] {5}
[𝒒] =
[𝑞0]
[𝒒′
]
{6}
Sensor fusion is the process of integrating the measurement from the GPS and IMU sensor to find the
precise position and orientation (Kok, Hol and Schön, 2017). The drift would be worse in the position
than the orientation due to double integration. The inertial sensors provide high sampling rate data
which are precise in short term and a sensor with low sampling rate is required to compensate the drift.
The Euler’s angle method has a drawback of having the gimbal lock compare to the Quaternion. The
accelerometer does not provide real results, thus, the sensor fusion of both the system is required to
manage the weakness of each sensor (Abyarjoo, et al., 2015).
1.5.2.1 Kalman filter (KF)
Student Dave's Tutorials (2013) explains the Kalman filter (Equations 7 & 8) used to estimate the state
accurately by fusing two signal to removes the drift in the data and it is mainly used in navigation
systems. The filter formed with coefficients those changes over the time to adopt new updated values in
a loop. Figure 9 shows the state estimation computed using the measurement and predicted data.
𝑥 𝑛 = 𝐴𝑥 𝑛−1 + 𝐵𝑥 𝑛 {7}
𝑦𝑛 = 𝐶𝑥 𝑛 + 𝑉𝑛 {8}
Figure 9. State estimation method in the Kalman filter (Understanding Kalman Filters, MATLAB. 2019)

24
The KF consist of two stages, the state prediction using a mathematical state model is the first stage and
the predicted state merged with the measurement state that gives the estimated state (Kalman Filter,
2019). The difference between these two stages is mainly due to the error and the noise presented in the
model and sensors. The process and measurement noise affect the Kalman filter weight between the
prior and residual prediction (Wang, et al., 2011).
1.5.2.2 Complementary filter
The complementary filter (Equations 9) combines the high pass and low pass filter uses the gyroscope
and the acceleration data to find the orientation of an element.
𝐴𝑛𝑔𝑙𝑒 = 0.98. (𝑎𝑛𝑔𝑙𝑒 + 𝑔𝑦𝑟 𝑑𝑎𝑡𝑎. ∆𝑡) + 0.02. (𝑎𝑐𝑐 𝑑𝑎𝑡𝑎) {9}
The acceleration and gyroscope data is reliable only in the long term (low pass) and short term (high
pass) respectively. The weightage introduced based in the equations to obtain accurate angle from
acceleration and gyroscope data. The filter is a simple and easy to use compared to the Kalman filter
(Pieter-Jan.com, 2013).

25
1.5.3 Six-DOF Rigid Body model
Rigid body dynamics is the exploration of the movement of the elements without having physical
deformation. It is consisting of a large number of particles joins together permanently (Landau and
Lifshitz 2013). Sum of all the particles defines the total system performance considering all the particles
have zero-DOF relative to each other. Beer, et al., (2016) the fundamental concept of the mechanics are
the time, mass, force and the space. Space is defined by the coordinate frame of the body to measure
the motion in 3 directions. The motion of an object depends on the mass and the time for the given
force. Pedersen, (2003) states an inertia is defined as the resistance force to a rotational motion of an
object at a particular axis and generally, it is denoted as a scalar with a unit of 𝑘𝑔. 𝑚2
.
Equation 10 shows a 3x3 rank-2 inertia tensor and Table 3 gives information on the tensor group. An
inertia tensor is a function of the position and orientation of a reference frame in a body. Gentile, et al.,
(1995) it is a key factor for dynamics study of an object and the inertia properties differs at various
location and orientation of the object.
𝑅𝑎𝑛𝑘 2 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑡𝑒𝑛𝑠𝑜𝑟 [𝐼] = [
𝐼𝑥𝑥 𝐼𝑥𝑦 𝐼𝑥𝑧
𝐼𝑦𝑥 𝐼𝑦𝑦 𝐼𝑦𝑧
𝐼𝑧𝑥 𝐼𝑧𝑦 𝐼𝑧𝑧
]
{10}
The diagonal terms in the inertia tensor represent the moment of inertia about three-axis and off-
diagonal terms are the product of inertia that indicates the mass distribution (Rigid Body Motion. 2019)
that usually be symmetry (example Ixy = Iyx) (IITG, 2019). The Dyad tensor keeps 3 pieces of information
about all 3 axes the plane, direction and the magnitude in its first, second and third subscripts
respectively (Polymer Nanostructures Lab., 2012).
Table 3. Classification of the inertia tensor
Tensor Terminology Information Components
Rank 0 Scalar Magnitude 1
Rank 1 Vector The magnitude and one direction 3
Rank 2 Dyad The magnitude and two direction 9
Rank 3 Traid The magnitude and three direction 27
Zewari and Quinn (1982) applied the mathematical model approach to study the dynamic behaviour of a
vehicle using vector analysis and Euler’s equation of motion. All the motions measured from the centre of

26
the gravity (COG) of the vehicle. The tire-road interaction and the wheel centre velocity discussed with
the relative motion to the suspension. The vehicle sprung mass considered as a six-DOF to make the
model simple and the wheel centre has 3-DOF. Though the vehicle split into the sprung and the unsprung
masses, the whole vehicle considered as a single mass system. If the inertia tensor is a diagonal matrix
with zero product of inertia (equation 11), then the direction of the angular momentum ‘l’ is always
parallel to the angular velocity ‘ω’ and also results in the principal frame of axis for the rigid body. The
non-diagonal matrix converted into a new set of a diagonal matrix using the eigenvectors (New jersey
institute of technology, 2019).
𝑃𝑟𝑖𝑐𝑖𝑝𝑎𝑙 𝑎𝑥𝑖𝑠 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑡𝑒𝑛𝑠𝑜𝑟 [𝐼] = [
𝐼𝑥𝑥 0 0
0 𝐼𝑦𝑦 0
0 0 𝐼𝑧𝑧
] {11}
Liu, et al., (2019) inertia properties usually predicted by a 3D cad model tend to produce large errors due
to the uncertainty of the geometry dimensions, density etc. Two coordinate systems defined to find the
inertial parameters, one will act as the global and another one is local coordinate fixed on the body.
Antoine cerfon, (2019) explains the mass distribution of the body is the key factor to define the inertia
tensor with an assigned origin. A fixed reference required to define the orientation of an object. The
rotational matrix transfers the object state information from the local to the global origin.
Dolatabadi and Kabganian, (2006) considered the model body as a perfectly rigid body for the simulation
of attitude dynamics and kinematics. The reference frame is fixed and useful to define the position of the
body. Body frame fixed at the COG of the model and the origin is beneficial to know the velocity and
body force. The relation between acceleration, velocity and displacement known as the kinematics of
motion and classified as translation and rotation. Quaternion is more stable to represent rotations
compared to other methods except for singular points. A rigid body has 6-axes of freedom in space
motion.
Ganapathi (2005) highlighted that a body rotates in the space would not continue to rotate forever if the
‘l’ and ‘ω’ not acting in the same direction. The Euler’s equation takes more time to compute as it uses
the trigonometry and the gimbal lock that affects the degree of freedom. By normalizing the Quaternion,
the drift error eliminated so that the points act in the same direction with unit magnitude and Matlab
used to visualize the model movement.

27
1.5.4 Summary
Very limited journals published on the mathematical modelling on the racing vehicles due to the
competitive nature of the motorsport, especially in the vehicle dynamics domain (Berman, 2016).
Considering all the studies presented, majorities only handled a single topic but no research been found
on combining all three domains (Tyre modelling, sensor fusion and rigid body dynamics) together.
In this study, a Matlab tool developed that consists of three models that represent actual tyre
performance, chassis dynamics and the driven racing line. All the models have combined together to
study the efficiency of the suspension set-up by reverse engineering a 10-DOF data (six-axis IMU + GPS+
four-wheel vertical displacement) acquired from a race car.
Initially, a MF tyre model developed to get the force and moment of a tyre. Secondly, sensor fusion of an
IMU and GPS to generate the driven racing line and the chassis orientation. Thirdly, a six-DOF rigid body
model developed with an inertia tensor to impose the chassis orientation data, this model mainly helps
to study the load transfer, tyre force/slip properties and the ride height of the chassis. It was achieved by
using the inertia tensor that holds the vehicle’s mass properties and the rotation matrix. Using the
output from the rigid body model and the wheel displacement the suspension transmissibility
performance computed. All pieces of information are key to analyse a vehicle suspension performance by
a race engineer (Kayalarasan, 2019). Figure 10 shows a Le Mans Prototype LMP1 race car and its
parameters used in this study, simulated using the AVL VSM lap time simulation software.
Figure 10. Toyota TS050 Hybrid LMP1 2018 race car (Racecar Engineering. 2019)
To the level of the author’s knowledge and the research on literatures, this is the first paper of results
produced by handling three different domains.

28
2 Methodology
The key findings from section 1.5 on tyre modelling, sensor fusion, rigid body dynamics and the
justification for the methods handled to develop the Matlab tool described in this section.
2.1 Overview
Figure 11 illustrates an overview of the project concept and the actual usage scenario, starting from the
data acquisition to the major outputs created from the Matlab tool. The detailed process shown in Figure
12 that explains the way of using each model to achieve the overall aim of this project.
Figure 11. An overview of the project concept
Figure 12. The process flow involved in the different phase of this project
MF 6.1 tyre model was developed for the Pure and combined conditions for longitudinal,
Lateral and algining torque. The model was used to calculate the slip angle for all four tyres
for inputs such as vertical force, lateral force, camber angle and tyre pressure.
Using the gyroscope data, A rotation matrix created from the quaternion that tells the
orientation of the chassis around the race track. The acceleration and the GPS data fused
together to obtain a drift free vecloity and the racing line with the help of a kalman filter.
The dynamics of the chasssis calculated using the inertia tensor and rotaiton matrix. The
vertical movement of the chassis and the wheel data used to get the suspenion
transmisibillity and the track bumpiness.
The Rigid body and the tyre model combined together, the acceleration data from the IMU
and the corner weight of the vehicle obtained from the rigid body dynamics used to
calculated the force and slip properties of a tyre.

29
2.2 Tyre Modelling Methodology
Among the tyre models considered in section 1.5.1, the semi-empirical, MF tyre model is being widely
used in the vehicle handling simulations due to its less complexity, low computational effort and accurate
fit of steady-state force and moment with minimal input data shown in Figure 13 (Uil, 2007). The MF
model generates force and moment with respect to the slip properties of a tyre for various inputs such as
vertical load, inflation pressure, inclination angle (Pacejka, 2005). When a force is acting on a tyre, it
tends to deform due to the nature of the rubber and slip from the road surface that affects the grip level.
The slip on the longitudinal and lateral direction is known as slip ratio and slip angle respectively.
Figure 13. Comparison between the tyre models with the main aspects to consider (Pacejka and
Besselink, 2012)
The latest version of the MF tyre model 6.1 developed without the swift factor using the Matlab software
to study only the steady-state performance. This model handles up to 8Hz (Delft tyre, 2019) and the
coefficients have increased from its predecessors and eliminated the number of measurements needed

30
by having new pressure coefficients in the equations that adjusts the force and moment according to the
tyre pressure. A set of scaling coefficients included tuning the performance to the real-time event, for
example, the friction level can be set according to road conditions such as dry and wet asphalt (Pacejka
and Besselink, 2012). The construction defects in the tyres such as the ply-steer and conicity create an
offset in the Lateral and longitudinal forces, these effects represented by the horizontal and vertical shift
function SV and SH.
The limitations of this approach:
1. The coefficients derived using the least square fitting algorithm, therefore, the coefficients used
in the MF model doesn’t have any physical meaning to the tyre properties (Rao, et al., 2006).
2. This model does not have the velocity dependency factor, hence it runs with fixed velocity
coefficients thus the force and moment is not sensitive to vehicle velocity (Uil, 2007).
3. Tyre temperature-sensitive factor is missing, thus the tyre degradation not accounted in this
model (Uil, 2007).
Despite this, the MF model is capable of delivering the best approximation of the steady-state tyre force
and moment compared to other models (Uil, 2007).
MF 6.1 tyre model equations were published in many journals, the Pacejka and Besselink (2012) book was
referred for this study and it includes over 100 coefficients obtained from the tyre testing experiments
(Madsen and Dirr, 2014). The inputs of the MF tyre model shown in Table 4, the Matlab code and the
coefficients used in the magic formula 6.1 tyre model attached in the Appendix section.
Table 4. Inputs supplied to the magic formula tyre model
Vx Forward velocity (m/s) Ω Angular velocity of the tyre (rad/s)
K Longitudinal Slip ratio (%) Re Effective Rolling radius (m)
α Lateral Slip angle (rad) R0 Tyre circumference (m)
γ Inclination / camber angle (rad) Fz Vertical force (N)
μ Friction level Pio Nominal tyre pressure (Pas)
The coefficients were stored in an excel file to alter them more efficiently for any existing data for similar
tyres, but this approach affects the computational effort. The SAE coordinate system and SI units used to
develop the model and the algorithm of MF tyre model’s Matlab code shown in Table 5. Turn slip effect
also considered in the model to represent complete steady-state performance (Pacejka and Besselink,
2012). the stiffness of a tyre computed by taking partial derivation of the pure performance curve and
that varies with the input sample rate such as slip angle and slip ratio, assumed as 100Hz.

31
Table 5. Matlab coding algorithm and the link between each section in the MF tyre model
(OptimumTire Documentation – OptimumG, 2019)
Region Sections Data Requires to build the model
Fxo Pure Longitudinal force -
Fx Combined Longitudinal force Fxo , Fyo
Fyo Pure Lateral force -
Fy Combined Lateral force Fxo , Fyo
Mxo Pure aligning torque Fxo , Fyo, Fx, Fy
Mx Combined aligning torque Fxo , Fyo Fx, Fy
To obtain the force (Fx, Fy) and moment (Mz) and to reduce the computational effort few input
parameters have kept constant such as the velocity, camber angle, friction level, rolling radius and the
tyre pressure. Day and Law (1996) noted that the moments make less impact on the vehicle handing thus
the overturning moment neglected. Pneumatic trail and residual torque equations in the aligning torque
section partially referred from the PAC2002 tyre model presented in (Kuiper and Van, 2007) to make the
model less complex. The Matlab MF 6.1 tyre model validated against the AVL VSM MF 5.2 tyre model.

32
2.3 IMU and GPS Sensor fusion methodology
Precise measurement on the body orientation and path travelled plays an important role in many fields
such as motorsport, aerospace and robotics (Madgwick, et al., 2011). In motorsport, that will influence
the prediction of the aerodynamic effect and the racing line of the vehicle. A six-axis IMU and a GPS
sensor fusion methodology flow shown in Figure 14.
Figure 14. An overview of the Sensor fusion methodology
The sensor fusion process separated in two stages, firstly to find the rotation matrix and the earth frame
acceleration. Secondly, combining that data with the GPS to obtain the racing line. Table 6 shows a six-
axis IMU and the GPS data extracted from the AVL VSM software at the logging rate of 100Hz and a
graph attached in the Appendix.
The chassis angular velocity captured at the COG of the vehicle where the IMU mounted. Quaternion and
rotation matrix approaches handled to represent the chassis orientation and rotation. Hamilton’s
quaternion ’q’ is a four-dimensional hypersphere denoted in vectors with an extension of the complex
numbers that illustrate the orientations and rotations of a rigid body in a three-dimensional space and it
6-axis IMU sensor
Body frame
Acceleration
Gyroscope /
Angular velocity
Direction-wise
Acceleration
GPS
Quaternion
Rotation matrix
Kalman Filter
Drift free velocity
and position
Driven racing line with chassis
orientation plot in Matlab
Earth frame
Acceleration
Sensor fusion

33
is widely used in robotics, quantum mechanics and video games (Visualizing quaternions, 2019) and
(Understanding Quaternions, 2019).
Table 6. IMU and GPS data used in the sensor fusion model
IMU GPS
𝐴𝑐𝑐 𝑋 Longitudinal acceleration (m/s2
) 𝑃𝑜𝑠 𝑋 Position in X-axis (m)
𝐴𝑐𝑐 𝑌 Lateral acceleration (m/s2
) 𝑃𝑜𝑠 𝑌 Position in Y-axis (m)
𝐴𝑐𝑐 𝑍 Vertical acceleration (m/s2
) 𝑃𝑜𝑠 𝑍 Position in Z-axis (m)
𝐺𝑦𝑟𝑋 Roll rate (rad/sec) 𝑉𝑒𝑙 𝑋 Velocity in X-axis (m/s)
𝐺𝑦𝑟𝑌 Pitch rate (rad/sec) 𝑉𝑒𝑙 𝑌 Velocity in Y-axis (m/s)
𝐺𝑦𝑟𝑍 Yaw rate (rad/sec) 𝑉𝑒𝑙 𝑍 Velocity in Z-axis (m/s)
Quaternion is less intuitive compared to DCM and Euler’s angle but it is more efficient and requires less
computational effort relative to the other two approaches. Moreover, it does not have singularities
known as the gimbal lock presented in the Euler’s angle (Sanz Díaz, 2015).
Quaternion is a mix of one real element and three imaginary elements and it defined in several terms
shown in (equation 12).
[𝑞] = [
𝑊
𝑋
𝑌
𝑍
] = [
𝑞0
𝑞1
𝑞2
𝑞3
] = [
cos 0.5 𝜃
𝑖 sin 0.5 𝜃
𝑗 sin 0.5 𝜃
𝑘 sin0.5 𝜃
] {12}
The first component of the quaternion is a scalar that denotes the amount of the rotation about an axis
and the rest of the three imaginary vectors (i, j & k) specify the axis to rotate. The value ‘0.5’ is a double
cover factor that rotates an object twice as the angle value to avoid the singularities (Sarabandi and
Thomas, 2018) and ‘θ’ is the angular velocity (Yun, Bachmann and McGhee, 2008).
Product of quaternions ⊗ works quite distinctively and determined by Hamilton’s rule (equation 13 to
16). The multiplication of the imaginary vectors is not commutative (jk ≠ kj).
𝑖2
= 𝑗2
= 𝑘2
= 𝑖𝑗𝑘 = −1 {13}
𝑖𝑗 = 𝑘 ; 𝑗𝑖 = −𝑘 {14}
𝑗𝑘 = 𝑖 ; 𝑘𝑗 = −𝑖 {15}
𝑘𝑖 = 𝑗 ; 𝑖𝑘 = −𝑗 {16}
The angular velocity about each axis computed individually (equation 17, 18 & 19) and multiplied together
using product quaternion rule (equation 20 & 21). Same process looped for the whole lap time 98 sec.

34
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑋 − 𝑎𝑥𝑖𝑠 = [cos(0.5 ∗ 𝐺𝑦𝑟𝑋) , sin(0.5 ∗ 𝐺𝑦𝑟𝑋) , 0, 0] {17}
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑌 − 𝑎𝑥𝑖𝑠 = [cos(0.5 ∗ 𝐺𝑦𝑟𝑌) , 0, sin(0.5 ∗ 𝐺𝑦𝑟𝑌) , 0] {18}
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑍 − 𝑎𝑥𝑖𝑠 = [cos(0.5 ∗ 𝐺𝑦𝑟𝑍) , 0, 0, sin(0.5 ∗ 𝐺𝑦𝑟𝑍)] {19}
[𝑋 ⊗ 𝑌] = [
𝑞0
𝑞1
𝑞2
𝑞3
] =
[𝑋1. 𝑌1 − 𝑋2. 𝑌2 − 𝑋3. 𝑌3 − 𝑋4. 𝑌4]
[𝑋1. 𝑌2 + 𝑋2. 𝑌1 + 𝑋3. 𝑌4 − 𝑋4. 𝑌3]
[𝑋1. 𝑌3 − 𝑋2. 𝑌4 + 𝑋3. 𝑌1 + 𝑋4. 𝑌2]
[𝑋1. 𝑌4 + 𝑋2. 𝑌3 − 𝑋3. 𝑌2 + 𝑋4. 𝑌1]
{20}
[𝑋𝑌 ⨂ 𝑍] = [
𝑞0
𝑞1
𝑞2
𝑞3
] =
[𝑋1𝑌1 . 𝑍1 − 𝑋2𝑌2. 𝑍2 − 𝑋3𝑌3. 𝑍3 − 𝑋4𝑌4. 𝑍4]
[𝑋1𝑌2. 𝑍2 + 𝑋2𝑌1. 𝑍1 + 𝑋3𝑌4. 𝑍4 − 𝑋4𝑌3. 𝑍3]
[𝑋1𝑌3. 𝑍3 − 𝑋2𝑌4. 𝑍4 + 𝑋3𝑌1. 𝑍1 + 𝑋4𝑌2. 𝑍2]
[𝑋1𝑌4. 𝑍4 + 𝑋2𝑌3. 𝑍3 − 𝑋3𝑌2. 𝑍2 + 𝑋4𝑌1. 𝑍1]
{21}
Quaternion axis defined as the normalized unit vector (equation 22) so that the sum of their square is
one. Following that, a single orientation defined in two distinct quaternions in a three-dimensional space
known as the quaternion conjugate (equation 23).
𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 [𝑞] =
𝑞
√𝑞02 + 𝑞12 + 𝑞22 + 𝑞32 {22}
𝐶𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒 [𝑞] = 𝑞−1
= [
𝑞0
−𝑞1
−𝑞2
−𝑞3
] {23}
Rotation matrix computed using the quaternions (Madgwick, 2011). Craig, (2009) It has three vectors in a
column those are orthogonal to each other and rotates independently thus the off-diagonal terms are
not symmetric rather it computed by the same equation with a different sign show in (equation 24).
Matlab offers quaternion and rotation matrix functions in its built-in robotics toolbox (Robotics System
Toolbox, 2019), the manually computed rotation matrix data validated against the toolbox data.
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 [𝑅𝑀] = [
2𝑞0
2
− 1 + 2𝑞1
2
2 (𝑞1 𝑞2 + 𝑞0 𝑞3) 2 (𝑞1 𝑞3 − 𝑞0 𝑞2)
2 (𝑞1 𝑞2 − 𝑞0 𝑞3) 2𝑞0
2
− 1 + 2𝑞2
2
2 (𝑞2 𝑞3 + 𝑞0 𝑞1)
2 (𝑞1 𝑞3 + 𝑞0 𝑞2) 2 (𝑞2 𝑞3 − 𝑞0 𝑞1) 2𝑞0
2
− 1 + 2𝑞3
2
] {24}
Earth frame acceleration (equation 25) derived by using the RM and the raw acceleration data. This data
further used in the Kalman filter.
[
𝐴𝑐𝑐 𝑁𝑜𝑟𝑡ℎ
𝐴𝑐𝑐 𝐸𝑎𝑠𝑡
𝐴𝑐𝑐 𝐷𝑜𝑤𝑛
] = [𝑅𝑀]−1
. [
𝐴𝑐𝑐 𝑋
𝐴𝑐𝑐 𝑌
𝐴𝑐𝑐 𝑍
] {25}
Typically, the data outputted from an IMU will be noisy and tend to have drift and the GPS sensor
generates low frequency signals. Literature has shown that both sensors data must be fused together to
get the drift-free velocity and position. A Kalman filter (Equations 26 to 40), adapted from Kalman Filter,
(2019) and Kalman Tutorial, (2019) used to fuse both the sensor’s data.

35
𝑆𝑡𝑎𝑡𝑒 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑖𝑜𝑛 (𝑋 𝑝) = 𝐴. 𝑋 𝑛−1 + 𝐵. 𝑥 𝑛 {26}
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑝𝑟𝑒𝑑𝑖𝑐𝑖𝑡𝑜𝑛 (𝑃𝑝) = 𝐴 . 𝑃𝑛−1. 𝐴 𝑇
+ 𝑄 {27}
𝐼𝑛𝑛𝑜𝑣𝑎𝑡𝑖𝑜𝑛 (𝑦̃) = 𝑍 𝑛 − 𝐻. 𝑋 𝑝 {28}
𝐼𝑛𝑛𝑜𝑣𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑆) = 𝐻. 𝑃𝑝. 𝐻 𝑇
+ 𝑅 {29}
𝐾𝑎𝑙𝑚𝑎𝑛 𝑔𝑎𝑖𝑛 (𝐾) = 𝑃𝑃. 𝐻 𝑇
. 𝑆−1 {30}
𝑆𝑡𝑎𝑡𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑖𝑜𝑛 (𝑋 𝑛) = 𝑋 𝑝 + 𝐾. 𝑦̃ {31}
𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑒𝑠𝑡𝑖𝑎𝑚𝑡𝑖𝑜𝑛 (𝑃𝑛) = (𝐼 − 𝐾. 𝐻). 𝑃𝑝 {32}
Inputs to the KF:
𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑒𝑐𝑡𝑜𝑟 (𝑥 𝑛) = 𝐼𝑀𝑈 = [𝐴𝑐𝑐] {33}
𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 (𝑍 𝑛) = GPS = [
𝑃𝑜𝑠
𝑉𝑒𝑙
] {34}
𝐼𝑛𝑖𝑡𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑖𝑜𝑛 (𝑋1) = [
𝑃𝑜𝑠1
𝑉𝑒𝑙1
] {35}
Constant variables in the KF equations:
𝑆𝑡𝑎𝑡𝑒 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 (𝐴) = [
1 ∆𝑡
0 1
] {36}
𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝑚𝑎𝑡𝑟𝑖𝑥 (𝐵) = [0.5 . ∆𝑡2
∆𝑡
] {37}
𝑆𝑐𝑎𝑙𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐻) = [1 1] {38}
𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑒𝑟𝑟𝑜𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑄) = 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑛𝑜𝑖𝑠𝑒2
. [∆𝑡2
0
0 ∆𝑡2] {39}
𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑚𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑅) = 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑛𝑜𝑖𝑠𝑒2
. [∆𝑡2
0
0 ∆𝑡2] {40}
The process noise and measurement noise covariance are the standard deviations of the error produced
by the sensors while at rest, the values assumed as 0.01 (Scott, 2019). The Kalman filter was performed
individually for all three-axis and the gravitational acceleration was neglected from the Z-axis data.
The complementary filter presented in section 1.5.2.2 also considered, as it is relatively simpler and less
computational effort. A Kalman filter was preferred because the complementary filter only fuses the
accelerometer and gyroscope signals to compute accurate Euler’s angle (Wu, et al, 2016).

36
2.4 Six-DOF Rigid body model Methodology
A 3-dimensional nature of this study described using the vector algebra shown in Figure 15, this also used
in several multi-body system (MBS) programs such as MSC Adams and Madymo (Blundell and Harty,
2004).
In addition to that, a new approach has been handled in this section to generate the outputs:
1. Inertia tensor summation at every time step using the chassis orientation [RM].
2. Dynamic mass distribution of the vehicle.
3. Orientation and position of the rigid body to study the ride heights.
4. Force and moment at every time step to generate all four tyres slip properties.
5. All four suspension transmissibility performance.
Figure 15. The ground rules of vector theory used in the rigid body dynamics
In this model, the chassis (sprung mass) assumed as a rigid body with six-DOF and the wheels considered
having only one-DOF the vertical displacement. Input data referred from the AVL VSM software shown in
equation41 and Table 7.
𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑡𝑒𝑛𝑠𝑜𝑟 𝑎𝑡 𝐶𝑂𝐺 [𝐼] = [
168 0 0
0 990 0
0 0 1140
] 𝑘𝑔. 𝑚2 {41}
O
P
Q
𝑃𝑍
𝑃𝑌
𝑃𝑋
𝑅 𝑄
𝑅 𝑃
𝑅 𝑃𝑄
X
Y
Z
(𝜙)
Y’
Z’
P’
(𝛼)
(𝛼)
X’

37
Table 7. The vehicle data referred from the AVL VSM software
Values Code Description Values Code Description
955.50 M Total mass (kg) 53.0 WD Mass distribution front (%)
506.41 M_FA Front axle mass (kg) 0.3546 R0 Tyre circumference (m)
449.08 M_RA Rear axle mass (kg) 185000 Pio Tyre Pressure (Pas)
434.41 M_FS Front sprung mass (kg) - M_FA_DF Front axle downforce (N)
373.08 M_RS Rear sprung mass (kg) - M_RA_DF Rear axle downforce (N)
72.0 M_FUS Front un-sprung mass (kg) - WH_TL Wheel Displacement (m)
76.0 M_RUS Rear un-sprung mass (kg)
The principal axis reference frame (COG) attached to the chassis, so both travel together. The inertia
tensor summation (equation 42) provides variation in inertia properties due to the orientation of the
chassis, that computed using the rotational transformation matrix (Coursera, 2019). For example, Figure
15 shows a rotation about the X-axis converted the XYZ frame into new X’Y’Z’ thus a new inertia property.
𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑡𝑒𝑛𝑠𝑜𝑟 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 [𝐼𝑆] = [𝑅𝑀] . [ 𝐼] . [𝑅𝑀]−1 {42}
The Euler’s equation of motion applied to generate the force and moment at the COG using the inertia
tensor summation, angular acceleration vector (𝛼), angular velocity vector (𝜔) and spin tensor[𝜔].
𝐹𝑜𝑟𝑐𝑒 (𝐹) = 𝑀𝑎𝑠𝑠. 𝐴𝑐𝑐𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑚. 𝑎 {43}
𝑀𝑜𝑚𝑒𝑛𝑡 (𝑀) = [𝐼𝑆]. {𝛼} + [𝜔]. [𝐼𝑆] . {𝜔}
{44}
𝑆𝑝𝑖𝑛 𝑡𝑒𝑛𝑠𝑜𝑟 [𝜔] = [
0 −𝐺𝑦𝑟𝑍 𝐺𝑦𝑟𝑌
𝐺𝑦𝑟𝑍 0 −𝐺𝑦𝑟𝑋
−𝐺𝑦𝑟𝑌 𝐺𝑦𝑟𝑋 0
]
{45}
The parallel axis theorem (equation 46 to 54) used to find the inertia properties at various location of the
chassis by shifting the reference frame to desired locations. Using the COG inertia tensor, a new inertia
tensor summation for all four corners generated to get the dynamic weight distribution of the vehicle.
𝐼 𝑋𝑋_𝐹𝐿 = 𝐼 𝑋𝑋_𝐶𝑂𝐺 + (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1). (𝑌𝐹𝐿
2
+ 𝑍 𝐹𝐿
2
) {46}
𝐼 𝑌𝑌_𝐹𝐿 = 𝐼 𝑌𝑌_𝐶𝑂𝐺 + (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1). (𝑋 𝐹𝐿
2
+ 𝑍 𝐹𝐿
2
) {47}
𝐼 𝑍𝑍_𝐹𝐿 = 𝐼 𝑍𝑍_𝐶𝑂𝐺 + (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1 + 𝑀_𝑅𝐴 + 𝑀_𝑅𝐴_𝐷𝐹1 ). (𝑋 𝐹𝐿
2
+ 𝑌𝐹𝐿
2
) {48}
𝐼 𝑋𝑌_𝐹𝐿 = 𝐼 𝑋𝑌_𝐶𝑂𝐺 − (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1). (𝑋 𝐹𝐿
2
+ 𝑌𝐹𝐿
2
) {49}
𝐼 𝑌𝑋_𝐹𝐿 = 𝐼 𝑋𝑌_𝐹𝐿 {50}
𝐼 𝑌𝑍_𝐹𝐿 = 𝐼 𝑌𝑍_𝐶𝑂𝐺 − (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1). (𝑌𝐹𝐿
2
+ 𝑍 𝐹𝐿
2
) {51}

38
𝐼 𝑍𝑌_𝐹𝐿 = 𝐼 𝑌𝑍_𝐹𝐿 {52}
𝐼 𝑋𝑍_𝐹𝐿 = 𝐼 𝑋𝑍_𝐶𝑂𝐺 − (𝑀_𝐹𝐴 + 𝑀_𝐹𝐴_𝐷𝐹1). (𝑋 𝐹𝐿
2
+ 𝑍 𝐹𝐿
2
) {53}
𝐼 𝑍𝑋_𝐹𝐿 = 𝐼 𝑋𝑍_𝐹𝐿 {54}
During yaw, sprung and un-sprung rotates together thus the 𝐼𝑧𝑧_𝐹𝐿 (equation 48) contains full mass of
the vehicle. The approach handled to compute the vehicle’s corner weight was by splitting the chassis
into 4 regions (FL,FR,RL & RR), the term 𝐼 𝑋𝑋 in the inertia tensor shows the mass about the X-axis for a
𝑚2
. The vectors represented as the corner distance from the COG shown in Table 8, those were tweaked
to match the static corner loads (FL = 253.2 kg) and the aerodynamics load added later.
Table 8. Corner locations and the respective inertia tensor with and without aerodynamic loads
COG to Corner distance vectors (m)
Front left (FL) Front right (FR)
𝑋 𝐹𝐿 1.4110 𝑋 𝐹𝑅 1.4110
𝑌𝐹𝐿 0.5172 𝑌𝐹𝑅 − 0.5172
𝑍 𝐹𝐿 −0.3249 𝑍 𝐹𝑅 −0.3249
Rear left (RL) Rear right (RR)
𝑋 𝑅𝐿 −1.6140 𝑋 𝑅𝑅 −1.6140
𝑌𝑅𝐿 0.4940 𝑌𝑅𝑅 −0.4940
𝑍 𝑅𝐿 −0.3449 𝑍 𝑅𝑅 −0.3449
Inertia tensor [𝐼] without aerodynamics load (𝒌𝒈. 𝒎 𝟐
)
Front left Front right
[
253.2 −179.4 232.2
−179.4 2051.7 41.3
232.2 41.3 3102.5
] [
253.2 179.4 232.2
179.4 2051.7 −41.3
232.2 −41.3 3102.5
]
Rear left Rear right
[
224.5 59.4 −250.0
59.4 2213.3 12.7
−250.0 209.1 3635.5
] [
224.5 −59.4 −250.0
−59.4 2213.3 −12.7
−250.0 −209.1 3635.5
]
Inertia tensor [𝐼] with aerodynamic load (𝒌𝒈. 𝒎 𝟐
)
[
662.0 −966.3 607.1
−966.3 3765.9 222.5
607.1 222.5 6901.1
] [
662.0 966.3 607.1
966.3 3765.9 −222.5
607.1 −222.5 6901.1
]
[
613.4 978.2 −683.1
978.2 990 209.1
−683.1 209.1 3862.3
] [
613.4 −978.2 −683.1
−978.2 990 −209.1
−683.1 −209.1 3862.3
]

39
At higher speeds, race cars generate a huge amount of downforce that reaches up to 3 to 4 times the car
weight (Zhang, et al., 2006) and (Floyd and Law, 1994). It plays a significant role in estimating the corner
loads of the vehicle. The aerodynamics loads at the front and rear axles are vectors that frequently
change over time. The downforce vector in the AVL VSM software given for each axle (not for each
corner). Therefore, it was evenly assigned to each respective corners after conversion from N to kg. This
approach gives a constant value at the straights due to almost negligible chassis orientation, (equation
55) invented to adopt the corner load variation at the straights. With the four corner load and the
lateral/longitudinal acceleration from the IMU, using newton’s second law (equation 46) the force at
each wheel identified for the respective acceleration and used in the tyre model as an input to generate
the tyre slip angles and slip ratio. In the force calculation, the front and rear of the vehicle considered
independently, therefore, the lateral acceleration divided by two before multiplying with the corner
loads. On the other hand, the longitudinal force acts on all four wheels thus the longitudinal acceleration
divided by four.
𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝐹𝑍 𝑎𝑡 𝑡ℎ𝑒 𝐹𝐿 𝑐𝑜𝑟𝑛𝑒𝑟 = (𝐼 𝑥𝑥_𝐹𝐿 𝑓𝑟𝑜𝑚 𝐼𝑠 +
𝑀_𝐹𝐴_𝐷𝐹
2
) −
(𝑀_𝐹𝐴_𝐷𝐹)1
2
(𝑘𝑔) {55}
Using the vector theory (equation 56 & 57), the vertical velocity at the ride height points computed and
integrated to obtain the actual ride heights of the chassis using the spin tensor, distance vector and the
change in COG velocity (Blundell and Harty, 2004). The vectors from the COG given in Table 9.
𝑉𝐹𝐿 𝑅𝐻
= [𝜔]. [
𝑋 𝐹𝐿 𝑅𝐻
𝑌𝐹𝐿 𝑅𝐻
𝑍 𝐹𝐿 𝑅𝐻
] + [
∆𝑉𝑒𝑙 𝑋
∆𝑉𝑒𝑙 𝑌
∆𝑉𝑒𝑙 𝑍
] (𝑚/𝑠) {56}
𝑃𝐹𝐿 𝑅𝐻
= ∫ 𝑉𝐹𝐿_𝑅𝐻 ∆𝑡 (𝑚) {57}
Table 9. Ride height locations vectors from the COG to compute the velocity
COG to Ride height locations vectors (m)
𝑋 𝐹𝐿_𝑅𝐻 1.211 𝑋 𝐹𝑅_𝑅𝐻 1.211
𝑌𝐹𝐿_𝑅𝐻 0.1500 𝑌𝐹𝑅_𝑅𝐻 −0.1500
𝑍 𝐹𝐿_𝑅𝐻 −0.3145 𝑍 𝐹𝑅_𝑅𝐻 −0.3145
𝑋 𝑅𝐿_𝑅𝐻 −1.3140 𝑋 𝑅𝑅_𝑅𝐻 −1.3140
𝑌𝑅𝐿_𝑅𝐻 0.0700 𝑌𝑅𝑅_𝑅𝐻 −0.0700
𝑍 𝑅𝐿_𝑅𝐻 −0.2895 𝑍 𝑅𝑅_𝑅𝐻 −0.2895

40
The suspension transmissibility is the ratio between the input and output amplitude shows the damping
performance (equation 58). It was computed using the vertical displacement of the wheel and chassis
(ride heights) for all four suspensions (Giaraffa, 2012), both the displacement data were converted into
absolute values for better visualization. root, (2019) The suspension components react based on the
input frequencies, therefore, the wheel travel converted from the time domain to the frequency domain
using (equation 59). The natural frequency of all four-suspension system varies due to different corner
loads so does the damping performance.
𝑇𝑟𝑎𝑛𝑠𝑚𝑖𝑠𝑖𝑏𝑖𝑙𝑙𝑖𝑡𝑦 𝑟𝑎𝑡𝑖𝑜 =
𝐶ℎ𝑎𝑠𝑠𝑖𝑠 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝑂𝑢𝑡𝑝𝑢𝑡)
𝑊ℎ𝑒𝑒𝑙 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝐼𝑛𝑝𝑢𝑡)
{58}
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 =
(
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝜋. 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
)
2. 𝜋
(𝐻𝑧)
{59}
The limitations of this approach:
1. The torsional stiffness of the race car chassis is not accounted in the model, therefore the load
transfer, ride heights and suspension transmissibility performance vary from the actual.
2. The road surface assumed to be flat throughout the circuit that reduced the accuracy of the ride
height predictions.
3. The downforce at each corner data was not available that influence the slip properties and the
corners loads calculations.
The nature of mathematical modelling followed in this section has the potential to improve the
performance predictions of race car dynamics by adding extra parameters and coefficients (Zewari and
Quinn, 1982). The limitations considered in this model were to simplify and explain the concept in a
practical way.
AVL VSM software generates over hundreds of data like dynamic ride heights, corner loads, wheel force,
slip angles and slip ratios etc., these data were directly overlapped with the rigid body model results to
validate the accuracy of the Matlab tool.

41
2.5 Summary
All three models developed in the Matlab linked together shown in Figure 16, the output from one model
used as an input to another that supports to generate the complete output of this project.
Figure 16. The interlink between each model in the Matlab tool
A quaternion created to form the rotation matrix, to establish the drift-free racing line a Kalman filter
used to fuse both the IMU and GPS data. RM directed to the six-DOF rigid body model and combined
Magic formula
Tyre model
IMU and GPS sensor
fusion
Six-DOF
Rigid body model
Corner load
(Fz)
Rotation matrix
(RM)
Drift free velocity
and racing line
Lateral force
(Fy)
Longitudinal force
(Fx)
Inertia tensor
(I)
Ride heights
(RH)
Suspension
transmissibility
ratio
Vertical wheel
displacement
Slip ratio
(k)
Slip angle
(α)
Data required to study the efficiency of suspension set-up
Yaw moment
(YM)
Quaternion
(q)

42
with the inertia tensor to know the dynamic of the chassis such as the corner loads, lateral/longitudinal
forces at each wheel and the yaw moment experienced by the vehicle. Then, the forces data used in the
MF tyre model to generate the tyre slip properties. Lastly, the suspension transmissibility ratio acquired.
The Matlab code generated attached in the appendix section.
Assumptions considered in this section:
To reduce the overall computational effort few inputs considered as constant in the MF tyre model such
as vertical load, tyre pressure, camber angle and rolling radius. The maximum lateral force taken as the
constant vertical load. The slip angle and slip ratio are sensitive to the vertical loads thus the accuracy of
the result compromised.

43
3 Results and Discussion
3.1 Introduction
All three models were developed in the Matlab software using the methodology and validation process
presented in section 2. The models have linked together as a single tool to exchange information to
generate each component performance that requires further analysis by a race engineer. In this section,
the result obtained from each model shown and the explanations made based on how a race engineer
could use and interpret the graphs to study the effectiveness of the existing suspension set-up of a race
car. This Matlab tool produced results similar to a conventional Laptime simulation software.
3.2 Tyre modelling
Figure 17 shows the longitudinal force vs slip ratio for pure (Fxo) and combined (Fx) conditions, that
highlights the grip limit of the tyre during pure (𝐴𝑐𝑐 𝑋 ) and combined accelerations (𝐴𝑐𝑐 𝑋 & 𝐴𝑐𝑐 𝑌).
Figure 17. Pure and combined longitudinal force and slip ratio generated from the MF tyre model
The longitudinal force generates the slip ratio if a tyre crosses the force limit that makes the wheel to
spin. This data supports to study the tyres grip limit during acceleration and braking, while acceleration
when load transfer takes place that deforms the tyre and shrinks the grip level (Tsinias, 2014). A tyre
deforms more during the combined condition, thus the gap between the peak performance of pure and
combined curve differs. Slip ratio usually positive for traction and negative for braking.
Stiffness
Peak slip ratio: 0.12
Force: -7587 N, -8631 N

44
The peak lateral force (Fy) reached around 90% of the peak slip ratio (Berman, 2016) as shown in Figure
18. Asymmetric curve and a vertical shift observed and it shows the tyre has construction defect such as
ply-steer (Pacejka and Besselink, 2012) and that offset by the inclination angle. The peak performance gap
between the pure and combined force indicates the tyre deforms more during the lateral combined
condition compare to the longitudinal case.
Figure 18. Pure and combined lateral force and slip angle generated from the MF tyre model
This plot helps to know the maximum force that a tyre can handle while cornering before losing the grip.
Similarly, the lateral force has also a limit and a friction ellipse will help to know the longitudinal and
lateral performance limit of a tyre.
Aligning torque (Mzo, Mz) generates when the vertical load center of the pressure on the footprint is not
in-line with the tyre centreline, that leads to the deformation of the tyre contact patch. The slip angle at
which the peak of the aligning torque is always smaller than the slip angle at the peak lateral force see
Figure19. From the driver’s perspective, they will feel a decrease in steering wheel torque before they
reach the maximum amount of the tyre lateral force (Pacejka, 2005). The Kingpin angle and caster angle
have a major contribution to the Mz performance. Noticeably the combined condition curve (Mz) shifted
down shows a similar amount of steering feedback ability on either turn.
Peak slip angle: 4.3, 6.48
Force: 5749 N, 7586 N
Cornering stiffness
Force: -4691 N, -7058 N

45
Figure19. Pure and combined Aligning torque and slip angle generated from the MF tyre model
Figure 20 shows stiffness of the tyre that outlines the amount of force required generating a unit
deformation of slip angle or the slip ratio and those limits a tyre. The relaxation length is one of the
important aspects for the transient response (Pacejka, 2005) and that varies with the stiffness of the
tyre. The factors such as tyre pressure, vertical load and temperature influences the stiffness.
Figure 20. Tyre stiffness for pure longitudinal, Lateral and aligning torque from MF tyre model
Maximum slip limit
Stiffness
Torque: - 110 Nm, -66 Nm
Torque: 94 Nm, 139 Nm

46
The longitudinal force and slip ratio generated for a range of vertical loads and plotted against the AVL
VSM MF 5.2 tyre model shown in Figure 21. Due to commercial factor, the data from the AVL VSM cannot
be extracted thus a picture shown. A good level of agreement between the models found and the
variation observed due to the limitations and the assumptions considered in section 2.2.
Figure 21. Magic formula 6.1 tyre model developed in Matlab on the left, AVL VSM magic formula 5.2
tyre model on the right for comparison with different vertical load

47
3.3 IMU and GPS sensor fusion
The quaternion converted into the rotation matrix shown in Figure 22, RM defines the axis vector length
from the reference point those values/position changes according to the chassis movement. RM is the
main factor to understand the dynamics of the chassis combined with the inertia properties and this
helps to visualize the chassis rotation in Matlab.
Figure 22. All 9 terms in the rotation matrix validated against the Matlab toolbox
The vehicle position, velocity and earth frame/direction wise acceleration for X-axis shown in Figure 23.
The role of GPS measurement covariance was significant on the error estimation (Quadri and Sidek,
2014). By tuning both the noise covariance, the state estimation changes that help to obtain better
results. Especially having a smaller process noise value would produce accurate results, where the
acceleration noise is smaller and that supports the low-frequency GPS measurement data. The Kalman
gain was indirectly computed using the prediction and estimate errors. The Y and Z-axis data attached in
the appendix.
All three-axis position data and the rotation matrix combined, that gives information about the track
characteristics such as gradient, corners and the chassis orientation shown in Figure 24 and Figure 25.

48
Figure 23. Vehicle Position, velocity obtained for X-axis using Kalman filter
Figure 24. The driven racing line generated using the sensor fusion method on the left, Image of the
Silverstone circuit cropped from the google map on the right for comparison
Start / Finish line

49
Figure 25. Chassis movement illustrated using the RM and vehicle position with the direction wise coordinate system
Chassis movements like roll, pitch and yaw (Gyroscope)
Driven racing line (Acc+ GPS)
North – South
East – West
Up - Down

50
3.4 Six-DOF Rigid body model
Figure 26 shows the yaw moment experienced by the race car at each corner for a threshold of ±250Nm,
the comparison with the AVL VSM data attached in the Appendix. Yaw moment determines the balance
of a car around a corner and this cause by the tyres force and moment ability that varies based on the
dynamic mass distribution. The under and over yaw moment means understeer and oversteer
respectively occurs due to the inertia difference between front and rear (Technical Papers – OptimumG,
2019). Going faster through corners requires higher yaw moment, desired at the slow-speed corner entry
and vice versa at the corner exit. The yaw moment should be equal to zero (steady-state) when the
vehicle is at the apex.
Figure 26. Yaw moment generated by the race car on the Silverstone circuit
Figure 27 shows the dynamic ride heights, the first 10 secs shows good fit with the AVL data later
deviated due to the limitation mentioned in section 2.4, but the trend looks very similar. An aero-map
provides a desired ride height range for all four corners, that has potential to generate a huge amount of
downforce from the diffuser that improves the cornering performance (Zhang, Toet and Zerihan, 2006).
The inertia makes the chassis to move and rotate that leads to variation in the ride heights from the
desired range, which mainly affects the ground effect and the aero balance of the car. The aero balance
defines the amount of front and rear axle downforce and it helps to keep the centre of pressure at the
preferred location, especially important at the corner entry to have a good amount of tyres grip (SEAS,
2019).

51
Figure 27. The ride height at four locations of the chassis derived using vector theory
The front left dynamic corner weight and the respective forces shown in Figure 28, a race engineer could
get the amount of force that acts on the tyre contact patch. Overloading a tyre while cornering would
make the tyre to slip and lose the grip, that leads to oversteer or understeer and at the straights, the
wheel spins make the vehicle slower (Haubenreich and Law, 2000). Using the tyre slip angle and slip ratio
data generated using the tyre model shown in Figure 29 the performance ability of all four tyres were
known. The plot shows a decent fit with the AVL data that was due to the unavailability of the downforce
data at each corner declared in section2.4 and the assumptions made in the tyre model section 2.5.

52
Figure 28. Front left corner weight and the respective forces generated using newton’s second law
Figure 29. The Slip angle and the slip ratio generated for the forces experienced by the Font left tyre

53
The level of suspension damping determines the chassis movement and the stiffness of the suspension
components define the damping performance (Berman, 2016), the damping behaviour varies based on
the input frequencies from the road surface shown in Figure 30. In this case, as the frequency gets
bigger the damping becomes poorer and makes the chassis to displace more.
Figure 30. Front left suspension transmissibility ratio and the Input frequency to study the damping
3.5 Summary
Initially, the MF 6.1 tyre model developed as a single model that shows the pure and combined
performance ability of a tyre and the agreement with the AVL VSM MF5.2 model shows good level of fit.
Then, the project phase shifted to the sensor fusion, where the quaternion and rotation matrix
computed using the angular velocity data. A perfect match observed in the validation against the Matlab
in-built tool box. The whole vehicle movement recreated combining the driven racing line and rotation
matrix.
Lastly the six-DOF rigid body model results presented, the yaw moment highlighted the over and
understeer area on the circuit. The chassis load transfer, ride heights and the tyre slip properties gave an
insight to optimize the existing suspension set-up. Also, the present damping performance of the
suspension shown. The overall match with the AVL VSM data is acceptable considering the assumptions
and limitations. The results from the models would require further analysis as a conventional Laptime
simulator to understand the vehicle suspension performance. In this section some of the plots only show
the front left corner performance, rest of the figures attached in the Appendix.

54
4 Conclusions and Future work
The Matlab tool developed with a ten-DOF data acquired from an LMP1 race car has been presented. The
aim of the study was to develop a simple and an effective tool that supports a race engineer to realise
the effectiveness of the existing suspension set-up by providing the dynamic performance of the
components associated with the suspension.
The three objectives stated were created using the mathematical modelling approach, that addressed
the performance of the tyres, driven racing line and chassis of a race car according to the race track.
Initially, the magic formula 6.1 tyre model created to represent the actual tyre component of a
performance vehicle, that gives the force and moment data based on the given inputs. The model built
with enough complexity to address the real slip angle and slip ratio experienced by the race vehicle. In
this study, few assumptions made to reduce the computational power.
An IMU and GPS data extracted from the AVL VSM software using an LMP1 vehicle simulated at the
Silverstone circuit. The chassis orientation precisely calculated using the quaternion and rotation matrix,
the acceleration and the GPS data used to create the driven racing line.
The rotation matrix used in the six-DOF rigid body model with an inertia tensor, that represented the
actual chassis component with mass properties. Both matrices combined to understand the dynamic
load transfer. The forces at each corner computed using the newton law and the vector theory applied to
obtain the ride heights of the chassis. The tyre force and moment data are determined by combining the
tyre model with the rigid body model. The suspension transmissibility ratio was calculated using the
vertical wheel travel data and the ride heights.
Even with the limitations and assumptions considered in this Matlab tool, the overall accuracy of the
results shows good agreement with the AVL VSM results. The results obtained from the tool require
further analysis by a race engineer to optimise the set-up. It can be predominantly useful to know the
race circuit characteristics, while working with amateur drivers and while using fewer data acquisition
sensors. The results show that the concept used in this tool is reliable and has the potential to develop
with additional parameters.

55
4.1 Future work
4.1.1 Tyre modelling
The presented tyre model can be extended further with the swift factor to represent the higher
frequency inputs that allow the tyre to react for the short wavelength inputs (larger than 0.1m).
Therefore, the Matlab tool can be used to ride comfort and transient studies. To improve the model
accuracy, all the real-time data must be used by replacing the constants considered. The temperature
sensitivity coefficient can be incorporated to address the degradation on the tyre’s force and moment
data over time.
4.1.2 IMU and sensor fusion
The driven racing line can be included with the track width data to clearly see where the vehicle position
is relative to the width of the track. In order to achieve this, a real vehicle has to drive slowly on either
side of the track edges that can be further developed with the same methodology used in section 2.3 to
get a complete 3D profile of the race track.
4.1.3 Six-DOF rigid body model
A coefficient can be applied to represent the actual chassis stiffness, that will change the inertia tensor
properties and can also improve the forces and moment prediction accuracy. The aerodynamics load on
each corner can be assigned correctly to predict the accurate vertical force on the wheels. The complete
3D track profile data from the above section can be used the correct the ride height accuracy. This
Matlab tool can be connected to the real-time telemetry to receive the input data from the car while on a
test lap, this allows generating the output instantly to speed up the analysis process.

56
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65
Appendix
Appendix 1: Coefficients used in the Magic formula 6.1 tyre model
Pure Lateral Pure Longitudinal Pure Aligning torque
pCy1 1.338 pCx1 1.579 qBz1 12.035
pDy1 0.8785 pDx1 1.0422 qBz2 -1.33
pDy2 -0.06452 pDx2 -0.08285 qBz3 0
pDy3 0 pDx3 0 qBz4 0.176
pEy1 -0.8057 pEx1 0.11113 qBz5 -0.14853
pEy2 -0.6046 pEx2 0.3143 qBz9 34.5
pEy3 0.09854 pEx3 0 qBz10 0
pEy4 -6.697 pEx4 0.001719 qCz1 1.2923
pEy5 0 pKx1 21.687 qDz1 0.09068
pKy1 -15.324 pKx2 13.728 qDz2 -0.00565
pKy2 1.715 pKx3 -0.4098 qDz3 0.3778
pKy3 0.3695 pHx1 2.16E-04 qDz4 0
pKy4 2.0005 pHx2 0.0011598 qDz6 0.0017015
pKy5 0 pVx1 2.03E-05 qDz7 -0.002091
pKy6 -0.8987 pVx2 1.06E-04 qDz8 -0.1428
pKy7 -0.23303 pPx1 -0.3485 qDz9 0.00915
pHy1 -0.001806 pPx2 0.37824 qDz10 0
pHy2 0.00352 pPx3 -0.09603 qDz11 0
pHy3 0 pPx4 0.06518 qEz1 -1.7924
pVy1 -0.00661 qEz2 0.8975
pVy2 0.03592 qEz3 0
pVy3 -0.162 qEz4 0.2895
pVy4 -0.4864 qEz5 -0.6786

66
pPy1 -0.6255 qHz1 0.0014333
pPy2 -0.06523 qHz2 0.0024087
pPy3 -0.16666 qHz3 0.24973
pPy4 0.2811 qHz4 -0.21205
pPy5 0 pPz1 -0.4408
pPz2 0
Combined Lateral Combined Longitudinal Combined Aligning torque
rBy1 10.622 rBx1 13.046 sSz1 0.00918
rBy2 7.82 rBx2 9.718 sSz2 0.03869
rBy3 0.002037 rBx3 0 sSz3 0
rBy4 0 rCx1 0.9995 sSz4 0
rCy1 1.0587 rEx1 -0.4403
rEy1 0.31.48 rEx2 -0.4663
rEy2 0.004867 rHx1 -9.97E-05
rHy1 0.009472
rHy2 0.009754
rVy1 0.05187
rVy2 4.85E-04
rVy3 0
rVy4 94.63
rVy5 1.8914
rVy6 23.8

67
Appendix 2: IMU and GPS data extracted from the AVL VSM software simulated
using the Silverstone circuit
Appendix 3: Quaternion computed from the angular velocity of the vehicle

68
Appendix 4: Kalman Filter – Position, velocity and direction-wise acceleration

69
Appendix 5: Moments from all three motions to study the balance of the car
Appendix 6: Yaw moment and lateral acceleration to analyse the over and
understeer phenomena
Over yaw moment /
Oversteer
Under yaw moment /
Understeer
Big oversteer at the vale turn

70
Appendix 7: Vehicle corner weight and the respective forces

71
Appendix 8: Tyre Slip angle and the slip ratio

72

73
Appendix 9: Chassis and wheel displacement with relevant transmissibility ratio
and the input frequency

74

75
Appendix 10: Suspension system - transmissibility ratio Vs the Input frequency

76
Appendix 11: Vehicle Coordinate System and dimensions used in the model
X +
Y +
Z +
GyrZ / Yaw
GyrX / Roll
GyrY / Pitch

77
Code Definition Values (m) Code Definition Values (m)
A
COG to front-
wheel center
1.411 L Overall length 4.650
B
COG to Rear
wheel center
1.614 H Overall height 1.9
C Front overhang 1.0 W Overall width 1.05
D Rear overhang 0.625 FT Front track 1.55
E COG to ground 0.3445 RT Rear track 1.55
COG
a bc d
L
H
e

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