Department of Engineering Science
Energy and Power Group
Traction Control for an Electric Vehicle
Transfer Report
Nathan Ewin
Balliol College
Supervisor: Dr. MD McCulloch
December 2011
Abstract
This report explores the tools required to develop traction control for an electric vehicle. A starting
point for this is to consider slip control of an individual wheel. A model of a vehicle is needed to
ensure the fidelity of the control. As friction between the tyre and the road is the main force that
acts on the vehicle, this is studied in detail. The Dahl model (PR Dahl 1968) for friction is
investigated as it is a dynamic model that captures some key friction phenomena with only two
parameters. Simulation of the model is conducted in the Simulink environment. The model was
validated against results from the literature on the LuGre model (Canudas de Wit et al. 1995) as
well an experiment on micro-slip (Olofsson 1995). It was found that for the Dahl model displayed
drift for a constant applied force less than the Coulomb friction force. The addition of a damping
term produced a presliding displacement curve that fit the results of the more complex LuGre
model. The Dahl model was shown to capture the general presliding behaviour shown by the
experimental data. A quarter vehicle model was built in Simulink to enable simulation of tyre and
suspension dynamics. The response of this model was validated against a similar model found in the
literature. The Dahl model and quarter vehicle model were combined to model vehicle motion in
the longitudinal direction. Simulations of acceleration tests for two different electric vehicles were
validated against test data from the vehicle manufacturers and car reviewers. The model showed
reasonable agreement with the data given that torque and power had to be estimated.
A review of the literature on slip control showed that many techniques relied on measuring the
vehicle speed which is problematic. Most also assume a static relationship between slip and friction
based on the Magic Formula (H. B. Pacejka & Bakker 1992), where changes in road condition and
vehicle speed are not taken into account.
A control that relies on neither of these is Maximum
Transmissible Torque Estimation (MTTE) (Yin et al. 2009). In this research an MTTE controller
has been modelled in Simulink to verify its performance. The initial simulations showed that for a
stationary vehicle even a gentle ramp in torque demand would result in the controller severely
limiting the motor torque. This was due to the zero initial speed causing errors in the wheel speed
derivative calculation. A solution was to delay the controller turning on until the vehicle was in
motion.
Future work, will look at extending the single wheel slip control into a full vehicle traction control.
A full vehicle model will be implemented using the Dymola physical modelling package. This will
help verify the traction controller, through simulation of the changes in vehicle dynamics and road
condition. Vehicle testing using Westfield's I-Racer is planned to verify the performance of the
controller experimentally. Novelty will be sort through adapting the controller to predict the
vehicle's behaviour by incorporating real time vehicle modelling and vehicle sensors signals.
Table of Contents
Nomenclature .................................................................................................................................... 3
1 Introduction ...................................................................................................................................... 4
1.1 Background ............................................................................................................................... 4
1.2 Report Structure ........................................................................................................................ 5
1.3 Project Structure ....................................................................................................................... 6
2 Literature Review ............................................................................................................................. 7
2.1 Friction ......................................................................................................................................7
2.1.1 Static Friction Phenomena ................................................................................................ 7
2.1.2 Dynamic Friction Phenomena ...........................................................................................9
2.2 Friction Models ....................................................................................................................... 11
2.2.1 Static Models ...................................................................................................................11
2.2.2 Dynamic Models ............................................................................................................. 12
2.2.3 Summary ......................................................................................................................... 15
2.3 Tyre Models ............................................................................................................................ 16
2.4 Vehicle Models ........................................................................................................................17
2.4.1 Quarter Vehicle ................................................................................................................17
2.4.2 Half Vehicle .....................................................................................................................18
2.4.3 Full Vehicle ..................................................................................................................... 19
2.4.4 Summary ......................................................................................................................... 20
2.5 Slip Control .............................................................................................................................20
2.5.1 Vehicle Speed Estimation ................................................................................................22
2.5.2 Slip Ratio Estimation ...................................................................................................... 23
2.5.3 Sliding-mode Control ......................................................................................................24
2.5.4 Model Based Control ...................................................................................................... 25
2.5.5 Summary ......................................................................................................................... 27
3 Work to Date ...................................................................................................................................28
3.1 Dahl Friction Model ...............................................................................................................28
3.1.1 Analytical model ............................................................................................................. 28
3.1.2 Simulation with Displacement Excitation ......................................................................29
3.1.3 Simulation with Force Excitation ................................................................................... 31
3.1.4 Comparison to the LuGre Model .................................................................................... 33
3.1.5 Comparison to Experimental Data .................................................................................. 37
3.2 Quarter Vehicle Model ............................................................................................................41
3.2.1 Description of Model ...................................................................................................... 41
3.2.2 Suspension frequency response ...................................................................................... 43
3.2.3 Longitudinal Acceleration Test ....................................................................................... 44
3.2.4 Summary ......................................................................................................................... 48
3.3 MTTE Controller .................................................................................................................... 48
3.3.1 Modifications to Controller .............................................................................................49
3.3.2 Comparison of Model Simulation to Literature .............................................................. 50
3.3.3 Summary ......................................................................................................................... 50
4 Future Work .................................................................................................................................... 51
4.1 Simulation .............................................................................................................................. 51
4.2 Testing .....................................................................................................................................52
4.3 Research Timing Plan ............................................................................................................. 53
5 Bibliography ................................................................................................................................... 54
2
Nomenclature
Symbol
Units Description
Symbol Units Description
A
m2
Frontal area
RR
/
Ride rate
a
m/s2
acceleration
T
Nm
Torque
Cd
/
Drag coefficient
v
m/s
Velocity
CS
N-s/m Suspension damping
vS
m/s
Stribeck velocity
Da
N
Aerodynamic drag force
VW
m/s
Wheel velocity
F, Fd
N
Friction Force
VX
m/s
Vehicle longitudinal velocity
FC
N
Coulomb Friction force
x
m
Displacement
FV
N
Viscous Friction force
α
/
Relaxation factor
Fe
N
Excitation force
δS
/
Striebeck form factor
Fs
N
Static friction (stiction) force
ζ
/
Damping ratio
i
/
Dahl exponent parameter
λ
/
Longitudinal slip ratio
JW
kg.m2 Wheel inertia
μ
/
Krr
/
KS
Rolling resistance coefficient
Coefficient of friction
3
ρ
kg/m Air density
kN/m Suspension stiffness
σ0
N/m
Dahl stiffness parameter
KU
kN/m Tyre stiffness
σ1
Ns/m
Microscopic damping
parameter
M
kg
Vehicle mass
σ2
Ns/m
Viscous friction parameter
MS
kg
Sprung mass
τ
s
Time constant
MU
kg
Unsprung mass
ω
rad/s Wheel angular velocity
N
N
Normal Force
ωd
rad/s Damped natural frequency
P
W
Apparent power
ωn
rad/s Undamped natural frequency
r
m
Wheel rolling radius
ωtyre
rad/s Tyre natural frequency
Rr
N
Rolling resistance force
Table 1 Symbols and meanings
3
1 Introduction
This report explores the tools required to develop a traction controller of an electric vehicle. A
dynamic friction model is investigated for its effectiveness at modelling the force between the tyre
and the road. This is combined with a vehicle dynamics model to simulate a quarter of the vehicle,
providing a building block for a real time full vehicle model. The numerical models are validated
against results from simulation and experimental work found in the literature. Finally a numerical
model of a slip controller is adapted from the literature.
1.1 Background
The concerns about climate change and energy security have lead to an urgent need to move away
from the use of fossil fuels. In transport, which is responsible for 17% of global emissions (IEA
2009), this means the electrification of road based vehicles. There is much debate about what is the
best zero emissions energy storage medium for a vehicle and how best to make the transition to this
from conventional fuels. What is clear is that whether it is a battery power electric vehicle (EV) like
the Nissan Leaf or a parallel hybrid such as the Toyota Prius, almost all rely on electric motors to
provide improvements in efficiency. What has not been recognised until more recently is that
electric motors offer advantages that enable EVs to be safer, more responsive, and more enjoyable
to ride than their Internal Combustion Vehicle (ICV) counterparts. Utilising these advantages will
likely speed up the adoption of electric motors in the transport sector.
Three key advantages of an EV over an ICV for traction control are precise measurable torque, fast
response, and simple actuation (Y. Hori et al. 1998).
The near linear relationship between the
phase current and the torque of an electric motor mean that the torque can be measured accurately
from the current sensor signals. The current and therefore the torque can be controlled precisely
using Pulsed Width Modulation of the inverter. An accurate torque measurement together with the
speed measured from the rotor encoder can give knowledge of the state of traction of the wheel.
The response time of an electric motor is under 10ms compared to over 200ms for an ICE. Longer
system delays can reduce the stability of the control as well as the performance. Finally traction
control for EV can be provided at no additional hardware cost as the electric motor provides the
actuation. In addition distributing the propulsion by having two or four electric motors has the
advantages of: added redundancy in terms of safety (He & Y. Hori 2006) ; distributed mass which
improves vehicle dynamics; and independently controlled wheels which allow advanced traction
control.
4
A starting point for developing traction control is to consider slip control of an individual wheel
(Yin et al. 2009). The purpose of slip control is to prevent wheel lock during braking and wheel
spin while accelerating.
This will constrain this study to longitudinal vehicle dynamics. Slip
control consists of sensing the state of the wheel and feeding this information back to modulating
the torque applied to the wheel.
Slip control methods vary depending on the sensor signals used and the assumptions on the
conditions that the vehicle operates under. Conventionally slip control has required the vehicle
speed to estimate the slip ratio, λ:
λ=
V w−V x
Vw
Equation
1.1
Where 'Vw' is wheel velocity and 'V' is vehicle velocity. The desired slip ratio is obtained from the
Magic Formula (H. B. Pacejka & Bakker 1992), a well known empirical relationship between
friction and slip. The controller uses the error between the estimated and desired slip ratio as
feedback compensation. This is due in part to slip control evolving with ICVs that cannot measure
the engine torque precisely. It is possible to obtain the vehicle speed from encoders on the none
driven wheels, and accelerometers, optical sensors, magnetic marker sensors (Yin et al. 2009), and
GPS (Bevly et al. 2003), on the chassis. A more accurate vehicle speed can be estimated by fusing
multiple sensor signals using advanced filters (Gustafsson et al. 2001). For slip control developed
specifically for EVs the torque measured from the electric motor means that slip ratio can be
estimated without measuring the vehicle speed (Fujii & Hiroshi Fujimoto 2007). To deal with
uncertain road conditions sliding-mode controllers can be used (E. Kayacan et al. 2009), (Amodeo
et al. 2010). Alternatively model based approaches exists that only consider the dynamics of the
wheel and therefore do not depend on a static slip-friction relationship (Y. Hori 2004), (Yin et al.
2009).
1.2 Report Structure
In chapter two a review of the relevant literature is carried out. First the behaviour of friction is
studied as this is the primary force acting on a vehicle. A distinction is made between static and
dynamic phenomena as this has an effect on the simulation and control of friction. Next the models
of friction are considered for how well they capture the different phenomena and how suitable they
are for numerical simulation. Secondly vehicle models are reviewed for the purpose of verifying a
slip controller by simulation. The range of models considered is intentionally kept broad to keep in
mind the wider context of developing a traction controller. Tyre models are also considered for this
reason although are outside the scope of this research. Finally the literature on slip control is
5
reviewed. Distinctions are made between whether or not the vehicle speed is measured and whether
or not static relationship between friction and slip ratio is assumed.
Chapter three details the simulation work to date. This has focused on how well the Dahl model
captures friction. In the first section a numerical Dahl model is taken through a validation process.
This involves comparison to: an analytical solution; a more complex friction model, and to
experimental data from the literature. In the second section the friction model is combined with a
quarter vehicle model to validate it against real vehicle data. Finally the third section details the
modelling of a slip controller and the modifications required to verify it with results from the
literature.
Chapter four lays out a plan for future work to the end of the project. This describes how a slip
controller can be developed into a traction controller. The type of simulations required to verify the
performance of the controller are discussed. Finally plans for testing the controller on a real vehicle
are laid out.
1.3 Project Structure
This research forms part of a Technology Strategy Board (TSB) project on the YASA Motor. Due to
there being several changes in consortium partners and project scope, the research has unavoidably
been divided into two parts. This report only describes the second part of this research. The first
part, which studies pressure contacts for integrating an electric motor with an inverter, is detailed in
a separate report (Ewin 2011). This report describes the modelling of an electrical contact in PSpice
software and verification of the model through experimentation.
6
2 Literature Review
“The critical control forces that determine how a vehicle turns, brakes and accelerates are
developed in four contact patches no bigger than a man's hand.”
- Anon, after (Gillespie 1992)
To study slip control an understanding of the vehicle dynamics is necessary. As friction is the
primary force acting on a vehicle, this makes a logical starting point. First the known friction
phenomena will be discussed. Following on from this friction models are reviewed based on how
well they capture these phenomena and their usefulness for slip control. Tyre and vehicle models
are then considered based on their degree of complexity and therefore their applications. Finally,
the effectiveness of existing slip controls methods is reviewed in terms of their ability to improve
performance while maintaining stability.
2.1 Friction
Friction between two bodies exhibits a complex range of behaviour. This can be divided into static
or dynamic phenomena depending on whether the friction force is solely dependent on the relative
velocity between the bodies or not. That is to say for static phenomena there is a single friction
force per velocity value. For dynamic phenomena there exist multiple solutions. Friction models
are described as static or dynamic depending on the phenomena they capture. No friction model
exists that completely and succinctly describes friction.
The following review of friction phenomena and models relied on three survey papers on this topic
(Olsson et al. 1998) , (Iurian et al. 2005), (Armstrong-Helouvry et al. 1994). The original sources
are cited where appropriate although the author acknowledges the work in these papers in
summarising this research topic.
2.1.1 Static Friction Phenomena
The classic works on friction only considered steady-state velocity and applied force. Figure 2.1
shows the relationship of force (F) and velocity (v) for combinations of the steady-state friction
phenomena.
7
Figure 2.1: Static Friction Phenomena, (Iurian et al.
2005)
Coulomb
Work by Da Vinci (1519), Amontons (1699) and Coulomb (1785) led to a definition of friction as a
force between two bodies in contact, that opposes motion, is proportional to normal force (N) and
independent of contact area. Coulomb friction describes the characteristic behaviour of two bodies
in contact with relative motion, under dry conditions. This relationship is depicted in Figure 2.1
and is defined as:
F =−F C. sgn (v )
Equation 2.1
Where 'FC ' is the Coulomb friction force and 'v' is the relative velocity between the bodies.
Viscous
As this report is concerned with friction between a tyre and road surface, the effects of lubricants
such as water must be taken into account. Reynolds (1886) was first to study viscous friction. The
simplest relation gives friction linearly proportional to the relative velocity between the two bodies,
Equation 2.2. The combined Coulomb-viscous friction relationship is shown Figure 2.1.
F =−F V . v
Equation 2.2
Where 'FV ' is the viscous coefficient of friction which it typically much smaller than FC.
8
Stiction
Static friction or stiction describes the friction force between two bodies at rest that is independent
of velocity (Armstrong-Helouvry et al. 1994). When a force is applied to a body, below the brakeaway force, according to Newton's third law an equal and opposite force is generated. Therefore the
friction force is a function of applied force. The first to show that the static force is higher than the
Coulomb friction force was Morin (1833). The combined Coulomb-static-viscous behaviour is
shown in Figure 2.1c).
Stribeck effect
At the beginning of the 20th century it was found for conditions where static friction is greater than
Coulomb friction the friction force initial decreases with velocity (Stribeck 1902), and therefore is
continuous and not as previously thought. This is described as being in regime of partial fluid
lubrication (Armstrong-Helouvry et al. 1994). The Stribeck curve is depicted in Figure 2.1. The
friction force due to the Stribeck effect can be described by the following equation:
F =( F S − F C ). exp(∣v / v S∣δ )
S
Equation 2.3
Where vS is the Stribeck velocity and δS is the form factor. The Stribeck effect is responsible for
stick-slip motion exhibited by a body under friction (Canudas de Wit et al. 1995). This is not
necessarily a separate phenomenon as suggested by (Iurian et al. 2005).
2.1.2 Dynamic Friction Phenomena
In the 20th century experiments of greater sensitivity have shown that friction exhibits rate
dependent phenomena (Johannes et al. 1973), (Hess & Soom 1990) and the behaviour of static
friction is closely linked with the microscopic properties of the contact between bodies (Rabinowicz
1951).
These dynamic friction phenomena are shown in Figure 2.2.
Understanding these
phenomena will be necessary to control a wheel of a vehicle as it begins to slip (lose traction) on the
road.
9
Figure 2.2:Clockwise from top left: Friction-velocity hysteresis loop, after
(Olsson et al. 1998); varying break-away force, after (Iurian et al. 2005);
and presliding displacement, after (Olsson et al. 1998).
Frictional Lag
Friction force varies depending on whether the velocity is increasing or decreasing due to a lag
effect when the acceleration reverses (Hess & Soom 1990). This relationship is shown by the graph
in the top left of Figure 2.2. The friction force is greater for increasing velocity. Friction is said to
exhibit hysteresis. The size of the hysteresis loop is dependent on normal load, viscosity and
frequency of velocity reversal. This is significant for controlling a wheel as the delay could cause
instability.
Varying Break-Away Force
The break-away force, i.e. the force at which the object subject to friction begins to slide, is
dependent on the rate at which the external tangential force is applied (Johannes et al. 1973). This
behaviour is shown in the graph in the top right of Figure 2.2. The break-away force decreases with
a higher rate of applied force. This is important as during a braking event a higher tractive force
will be available if the rate of applied force is modulated.
10
Presliding Displacement
For an applied tangential force less than breakaway force there is a small presliding displacement
between the two bodies in contact (Courtney-Pratt & Eisner 1957). This is due to the contact
asperities between the two surfaces acting in a spring like manner. Presliding displacement is also
known as the Dahl effect. The characteristic behaviour is shown in the graph in the bottom left of
Figure 2.2. Before sliding occurs friction is a function of displacement rather than velocity. This
accounts for the behaviour during velocity reversal which was previously thought of as a
discontinuity, see Figure 2.1. This is helpful for control purposes as it allows uniqueness of
solutions.
2.2 Friction Models
Now that the behaviour of friction has been described, models of friction will be looked at for how
well they capture this behaviour and how suitable they are for real time modelling and control.
2.2.1 Static Models
The static models of friction are summarised below. These models only to include the dynamic
friction phenomena by including time dependency or time delay (Iurian et al. 2005).
Classic Model
The classical model of friction combines the effects of Coulomb and viscous friction, stiction, and
the Stribeck effect. The friction force can be described by the switching function in Equation 2.4
and the expression to describe friction during relative motion in Equation 2.5, after Olsson et al.
(1998).
{
F ( v)
if v ≠0
F= F e
if v=0 ∧∣F e∣<F s
F s . sgn ( F e )
otherwise
Equation 2.4
}
F (v)= F C +( F S −F C ) . exp (∣v / v S∣δ )+ F V v
S
Equation 2.5
The major drawback of this model is the non-uniqueness of solutions at zero velocity. This
discontinuity causes problem when using such a friction model for control purposes due to the
difficulty in detecting zero velocity (Olsson et al. 1998).
11
Karnopp
A solution to the above problem that avoids switching between different states for sticking and
sliding is to implement a dead zone around zero velocity (Karnopp 1985) . This means that the
velocity is held at zero while below a predefined level. The friction force exactly matches the
applied force while below the break-away force. The model proves efficient for simulations
(Olsson et al. 1998), and has been adapted into dynamic models (Haessig & Friedland 1990). The
down side for simulation purposes is that the complexity of the model generally increases with the
complexity of the system (Haessig & Friedland 1990). The model does not capture the dynamic
phenomena and the dead zone is not a true representation of friction (Olsson et al. 1998).
Armstrong
The classical model has been extended to a seven parameter model that includes some of the
dynamic effects of friction, (Armstrong-Helouvry et al. 1994). The model consists of two separate
equations, one for sticking and one for sliding. The equation for sticking gives the friction force as
proportional to presliding displacement. From the friction phenomena mentioned earlier, this is a
rather simplistic representation as the relationship is shown to be exponential due to plastic
deformation of asperities. The equation for sliding incorporates the Stribeck effect along with
frictional lag and varying break-away force. A switching function, most likely requiring an
additional parameter, will be needed to change between the two equations (Olsson et al. 1998).
This causes difficulties during simulation (Eborn & Olsson 1995).
2.2.2 Dynamic Models
The dynamic friction models are summarised below. Dynamic models include additional time
varying state variable, other than velocity, that determine the friction force.
Dahl
The Dahl model (PR Dahl 1968) is a dynamic representation of sliding and rolling friction that
captures the presliding displacement phenomena. It is based on the hypothesis that friction is
caused by the forming and breaking of quasi-static contact bonds. The analogue of the two surfaces
being connected by a stiff spring is often used to describe the phenomena. The Dahl model is
described by the follow relationship between friction force and displacement (P Dahl 1977):
i
dF
F
=σ 0 .∣1− sgn(v )∣ . S
dx
Fc
Equation 2.6
(
)
F
sgn(v )
Fc
Equation 2.7
S=sgn 1−
12
Where 'σ0' is the (rest) stiffness parameter, 'i' is the exponent parameter, 'F C' is the Coulomb friction
force, and 'S' is the stabilising factor.
The dynamic equation can be obtained as follows:
i
dF dF dx
F
= ⋅ =σ 0 .∣1− sgn ( v )∣ . S.v
dt dx dt
Fc
Equation 2.8:
When i=1, S=1 and Equation 2.8 simplifies to:
dF
F
=σ 0 ( v− .∣v∣)
dt
Fc
Equation 2.9:
The steady-state Dahl Model simplifies to Coulomb friction:
F =F C sgn(v)
Equation 2.10
The Dahl model does not include the Stribeck effect as it is rate independent. The Dahl model is a
simple dynamic model of friction that captures some of the key phenomena and only requires two
parameters to be defined. The Dahl model represents a simplified LuGre model, described below.
Bristle
A model of friction is presented based on the bristles-like nature of how the asperities deform on
the surface of a body under friction (Haessig & Friedland 1990). When a tangential force is applied
to the bristles they deflect like a spring. If the force is large enough the bristles will slip in a
random manor. The complexity resulting from modelling each individual asperity means that the
model is of little use numerically. However, it does provide a useful analogy to visualise friction.
Figure 2.3: Visualisation of bristle interpretation of friction,
(Magallan et al. 2011)
13
Rest-Integrator
A computationally more efficient model than the Bristle model is proposed (Haessig & Friedland
1990).
This models the stiction and sliding friction as two separate functions. It therefore relies
on a switching function to transition between the two modes, and so is discontinuous with respect to
position. This is likely to cause numerical problems during simulation, (Iurian et al. 2005).
Bliman-Sorine
A second-order Dahl model has been developed in an attempt to include the Stribeck effect (Bliman
& Sorine 1995). The model replaces the time variable with a space variable. It effectively consists
of a fast Dahl model and slow Dahl model connected in parallel. This gives rises to the Stribeck
effect after a change in direction of motion, but not during steady state (Canudas de Wit et al. 1995).
The model does not capture the effects of frictional lag or varying break-away force. This model
provides a slightly more realistic representation of friction than the Dahl model, but at the cost of
higher computational complexity.
LuGre
The LuGre friction model as proposed by Canudas de Wit et al. (1995) is based on the bristle
interpretation of friction (Haessig & Friedland 1990).
The LuGre model can be seen as an
extension of the Dahl model that includes the effects of stiction and the Stribeck effect. The LuGre
is rate-dependent and so also includes the effects of frictional lag and varying brake-away force. In
the LuGre model the average bristle displacement, z, is defined by the following equation:
ż =v−
0
∣v∣
g  v
Equation 2.11
The functions 'g(v)' is used to model the Stribeck effect, given as:
S
g  v=F C  F S −F C e∣v/ v ∣
S
Equation 2.12
The friction force can then be defined as follows:
F = 0 z 1 ż  2 v
Equation 2.13
where σ0 is the bristle stiffness, σ1 represents microscopic damping, and σ 2 is the viscous friction
coefficient. This is known as the standard parameterisation of the model as it uses linear viscous
14
friction and constant microscopic damping. An alternative is to model the viscous friction as in
Equation 2.14. This models lubricant being forced into the interface as velocity increases. It also
ensures that the model is dissipative.
2
− v/ v d 
 1 v = 1 e
Equation 2.14
Canudas de Wit & P. Tsiotras (1999) developed the LuGre friction model to better represent a tyre.
This considers the friction force distributed across a contact patch rather than applied at a single
point. This is to more accurately model the tyre as it deforms under load, as shown in Figure 2.4.
Figure 2.4: a) point contact, lumped model. b) contact patch,
distributed model, after (Canudas de Wit & P. Tsiotras 1999)
Although the distributed model is more realistic it introduces PDEs that are computationally slow to
solve. Canudas de Wit & P. Tsiotras (1999) introduce the concept of average bristle displacement to
transform the PDEs to ODEs.
Leuven
It has been shown that the presliding hysteresis behaviour of the LuGre model does not completely
agree with experimentation (Swevers et al. 2000). The Leuven model is proposed which includes
hysteresis behaviour with nonlocal memory (Swevers et al. 2000).
This is shown to fit
experimental data with more accuracy than the LuGre model. The greater complexity of the model
is noted as a draw back for use in control (Iurian et al. 2005).
2.2.3 Summary
Early works on friction identified the static phenomena of Coulomb, viscous, stiction and the
Stribeck effect. This behaviour is captured by the classical model of friction. The discontinuity of
zero velocity presented problems for using this model for control purposes. Armstrong and
15
Karnopp present models that overcome this. More recently the dynamic behaviour of friction has
been investigated, which shows friction exhibits memory, hysteresis and varying break-away force
behaviour. A number of dynamic friction models have been developed that aim to capture these
phenomena. The majority of these have been extended from the Dahl model, including the widely
used LuGre model. This factor combined with its simplicity makes the Dahl model a good starting
point for modelling friction and so it will be using in the simulation work of the subsequent chapter.
2.3 Tyre Models
The tyre forms the only connection to the ground for a vehicle. Therefore its behaviour has a strong
influence on the vehicle dynamics. Having said this complex tyre modelling is outside the scope of
this report. As this research has focused on longitudinal motion on a flat road the focus has been on
friction force generation. This is largely covered by friction models. In addition most of the
friction models considered have been of a point contact type, therefore the deformation of the tyre
has not been of interest. The discussion is included for completeness and to introduce the reader to
the Magic Formula tyre model (H. B. Pacejka & Bakker 1992), which in its simplest form is a static
friction model.
Figure 2.5: Typical tyre/road friction profiles for varied road condition and vehicle velocity
after (Harned & Johnston 1969)
Tyre modelling can be divided into four categories (H. Pacejka 2006). These broadly define the
level of complexity, the accuracy and the insight into the behaviour of the tyre. The first consists of
the empirical models that are curve-fits to experimental data. These provide a high level of
accuracy but little understanding of the tyre. The most well known of these is the Magic Formula
(MF), (H. B. Pacejka & Bakker 1992), which has become the standard for comparing friction
models to. The MF fits a curve to steady state experimental data such as the curves in Figure 2.5.
This data is generated under strictly controlled test conditions that mean the velocity of the vehicle
16
and the angular velocity of the wheel are independent (Canudas-de-Wit et al. 2003). In reality these
two velocities are closely coupled. The static maps in Figure 2.5 provide no understanding of the
transition between different conditions which will be continuously varying in the real world.
The second category are the semi-empirical models. These are still based on empirical data but
contain structures related to physical models. An example of this is the similarity method, (H.
Pacejka 2006). The third category consists of simplified physical models. These provides a good
representation of major observed characteristics while simplifying the mathematics to allow
efficient simulation. An example of this is the brush type tyre model, (Dugoff et al. 1970) . The
fourth category is the complex physical models. This includes finite element models that can model
individual tread elements. These are used for detailed analysis of the tyre and can be computational
intensive. The first and fourth categories provide the greatest accuracy while the second and third
categories are less complex and therefore more useful for simulation.
2.4 Vehicle Models
A vehicle model consists of a set of differential equations that represent a simplified mathematical
model of a vehicle. Vehicle models are used to simulate real vehicles and therefore save on
expensive time consuming testing. A model will never completely capture the behaviour of a
vehicle as assumptions will always have to be made. It is therefore important to select the correct
model for the required application and level of accuracy. The complexity of a vehicle model is
determined by the number of bodies it includes and the degrees of freedom they have.
Vehicle models in the literature can be broadly split into three categories: quarter vehicle; half
vehicle; and full vehicle models. This will help divide up the discussion, although there are varied
applications and levels of complexity within each.
2.4.1 Quarter Vehicle
Quarter vehicle (QV) models consist of a single wheel and single vehicle mass. They typically only
consider either vertical motion or longitudinal motion. The vertical QV model consists of a wheel
mass and vehicle mass, each with a single degree of freedom. The masses are connected by a
suspension model, Figure 2.6. This consists of a spring-damper, and possibly an active component.
The tyre is connected to the road by a spring, and sometimes a damper as well, which represent the
tyre dynamics. The vertical QV is useful for simulating the motion of an individual suspension unit
and to analyse its performance relative to a varying road profile (Alyaqout et al. 2007). The model
does not include a suspension stop or a mechanism for wheel lift-off. Therefore it may not be able
to simulate the vehicle response from a sudden road displacement such as a speed bump or pot hole,
17
particularly at high speed.
Figure 2.7: QV longitudinal
Model, (Yin et al. 2009)
Figure 2.6: QV Suspension
Model, (Alyaqout et al. 2007)
A longitudinal QV model also consists of a vehicle mass and wheel mass, as in Figure 2.7. These
are rigidly connected to each other, and the tyre is rigidly connected to the road. The vehicle can
move in the longitudinal axis and the tyre can rotate about the lateral axis. This model has been
used widely to look at slip/skid control of a wheel, (Cai et al. 2010), (Yin et al. 2009), (Fujii &
Hiroshi Fujimoto 2007). In these papers only the longitudinal motion of the car is of interest, and
as the road is considered flat the suspension dynamics can be ignored. Several of the papers that
use the longitudinal QV model ignore the drag force and most ignore the rolling resistance force.
Due to this the longitudinal QV will over perform as there are no losses. This will result in a
reduction in slip controller performance (Yin et al. 2009). It is predicted that the errors due to
ignoring external forces will be minor compared to other modelling assumptions.
2.4.2 Half Vehicle
The Half Vehicle (HV) models also fall into two categories: steering models and suspension
models. Both make the assumption that right and left sides of the vehicle behave identically and
therefore ignore roll dynamics. Steering models, also referred to as a bicycle or single track model,
only consider motion in the plane of the road (longitudinal and lateral) and therefore ignores pitch
and vertical dynamics. It is often used to analyse the yaw dynamics for developing active steering
control, (Falcone & Borrelli 2007), (Marino & Cinili 2009). The HV steering model is represented
by a single vehicle mass with three degrees of freedom: longitudinal; lateral; and yaw. A diagram of
the model is shown in Figure 2.9. For the purposes of traction control this is of little use, as during
cornering the controller would need to differentiate between the left and right side wheels.
18
The HV suspension model ignores lateral motion and therefore only considers longitudinal, vertical
and pitch dynamics. The model consists of a vehicle sprung mass and two unsprung masses
representing the wheels, as in Figure 2.8. The suspension is represented by two separate springdampers for front and rear. Each wheels is connected to the road by a spring, representing the tyre
stiffness. A HV suspension model generally has five or seven degrees of freedom depending on
whether the wheel rotational dynamics are modelled. The model has been used to simulate the
vehicle dynamics during longitudinal braking and accelerating to develop active suspension (Cao et
al. 2007), and to analyse suspension performance with varying road profile (Roy & Z. Liu 2008).
Inclusion of wheel rotational dynamics allows study of torque distribution between front and rear
wheels (Yamakawa et al. 2007).
Figure 2.9: HV Steering Model, (Iurian et al.
2005)
Figure 2.8: Half vehicle model, after (Cao et
al. 2007)
2.4.3 Full Vehicle
Full vehicle models provide the most physically similar models to a real vehicle. A basic full
vehicle model has 14 degrees of freedom, such as the vehicle model proposed by Shim & Ghike
(2007). This allows full 6-axis motion of the chassis plus vertical and rotational motion of the
wheels. This still makes the assumption of lumped mass and a rigid vehicle body. More advanced
models, such as the commercially available ADAMS/Car and CarSim, have over one hundred
degrees of freedom and models the individual components of a vehicle. The downside to many
commercial packages is some of the functionality of the model is often hidden. In addition to the
simulations that can be performed by QV and HV models, a full vehicle model can simulate the roll
dynamics that occur during cornering and therefore can be used to determine wheel lift-off (Shim &
Ghike 2007).
19
Figure 2.10: 14DOF Full Vehicle Model, after (Shim & Ghike 2007)
2.4.4 Summary
In this section a review of the vehicle models and their applications has been carried out. The
longitudinal QV model is used extensively by research on slip control. The vertical QV model
provides a simple model of the suspension which is used within many of the half-vehicle and fullvehicle models. The combination of these two models with an appropriate friction model will
provide a useful building block for developing a full vehicle model necessary for developing
traction control.
2.5 Slip Control
Slip is the difference in vehicle and wheel velocities. The longitudinal slip ratio can be defined as
(Fujii & Hiroshi Fujimoto 2007) :
r ω−V
max(r ω , V ,ϵ)
Equation 2.15
λ=
Where 'V' is the longitudinal vehicle velocity, 'r' is the wheel radius, 'ω' is the angular wheel
velocity, and ε<<1 is included to ensure that the denominator is never zero. Slip occurs whenever
a torque is applied to the wheel as it is essential for generating friction at the tyre-road interface
(Gillespie 1992). The steady state relationship between slip ratio and friction coefficient can be
seen in Figure 2.11. This shows how peak the friction force depends upon slip ratio and road
condition. Slip above this optimum value will lead to reduced friction force and therefore reduced
20
controllability of the vehicle. As a result, torque applied to the vehicle should be modulated to
maintain the maximum friction force at the wheel.
Figure 2.11: Magic formula μ-λ relationship, after
(Magallan et al. 2011)
In the absence of any control, during sudden braking wheel lock can occur which will reduce
controllability of a vehicle and increase its stopping distance. When a vehicle accelerates without
slip control, particularly in icy conditions, wheel spin is likely. On a positive gradient this may
prevent the vehicle moving at all.
The major advantage of implementing slip control using an electric machine compared to hydraulic
brakes is the much quicker response of the system. The response time of an electric motor is of the
order of 1ms whereas the response time of a hydraulic ABS is between 55-130ms, (Sakai & Yoichi
Hori 2001), (Yesim Oniz et al. 2009) . This is due to a combination of the dead time of the solenoid
valve and the delay of the hydraulic circuitry. Not only will this lead to a slower response but will
also lead to greater instability within the control loop (H. Fujimoto et al. 2004). An electric
machine therefore provides a useful starting point for investigating slip control as it provides fast
actuation and accurate measurements of the torque and angular speed.
The criteria for a slip controller can now be drawn up. This will provide a means of comparing the
slip control strategies that have been proposed to date. As mentioned earlier the primary aim of a
slip controller should be to allow the maximum friction force between the wheel and road. In
normal operating conditions, where the torque applied to the wheel is less than the limit of traction,
the controller should not impede the performance of the vehicle. During sudden acceleration and
deceleration of the wheel the controller should limit the applied torque to the maximum determined
by the tyre-road interface. The controller should be robust to changes in external conditions. This
is apparent from Figure 2.5, Figure 2.11, where different road conditions and vehicle speed give rise
21
to different peaks in friction force. Other factors such as tyre wear and normal load will also effect
the μ-λ relationship (Gillespie 1992).
2.5.1 Vehicle Speed Estimation
Conventional slip control has taken the approach of estimating the slip ratio to maintain it at an
optimum value. The first problem with this is obtaining an accurate value for vehicle velocity to
calculated slip ratio. The second problem is determining an optimum value as this varies as
previously described. The simplest method to estimate the vehicle speed is to use position encoders
on the non-driven wheels. Although this is not possible when considering 4WD or when braking
with all four wheels. Also changes in tyre rolling radius cause additional errors (Daid & Kiencke
1995).
The vehicle speed can be estimated directly from sensors mounted on the vehicle chassis. This is
generally achieved using an accelerometer, although (Yin et al. 2009) notes several others such as
optical sensors and magnetic marker sensors. Both are dismissed, as optical sensor robustness
would deteriorate over time due to dirt and oil ingress, and magnetic markers would require a large
change to road infrastructure. GPS is another method that can be used to estimate vehicle speed,
(Bevly et al. 2003). However, loss of signal, particularly in built up areas means it is not a reliable
option. Several papers, (Fujii & Hiroshi Fujimoto 2007) and (Yin et al. 2009), note that the
accuracy of accelerometers is affected by a speed offset due to the long-time integral calculation.
This suggests that the accelerometer signal on its own is not accurate enough to be usefu l. The
single accelerometer has also been rejected as a valid vehicle velocity measurement due to the
gradient of the slope producing an undetectable error, (Yin & Yoichi Hori 2010). As accelerometers
are now readily available in three axis packages this would no longer be a problem.
Sensor Fusion
Sensor fusion involves using the signals from multiple sensors and a filter to compute the
parameters of vehicle dynamics with a higher accuracy than an individual sensor can achieve. One
such method uses a Recursive Least Squares (RLS) algorithm to estimate the vehicle speed from the
wheel speed and an accelerometer signal (Zhang et al. 2007) . An experiment of a vehicle
decelerating from 12m/s to 2m/s over 2s was used to validate the method. A fifth wheel instrument
was used to provide a reference for the vehicle speed. The results showed an accuracy of +/-0.3m/s
between estimation and reference value. A better measure of performance would be to show how
well the estimated slip ratio tracks the desired value, then the significance of any error would be
easier to determine. In Figure 2.12 at time 1.8s there is considerable slip, although this could be
22
attributed to slow response of the hydraulic ABS used in the test.
Figure 2.12: Vehicle velocity estimation
using RLS, after (Zhang et al. 2007)
A similar method using a Kalman filter instead of an RLS filter has also been considered
(Gustafsson et al. 2001).
The experimental results show the vehicle speed error is +/- 0.25m/s.
This can be improved to +/- 0.05m/s if the tyre radius is estimated rather than using a constant
value. Both filter methods account for the change in tyre radius. The RLS method uses the wheel
speed signal from a driven wheel and accounts for the error due to slip whereas the Kalman method
only notes that this is possible. These papers show how sensor fusion can be used to obtain a better
estimation of vehicle speed than using an accelerometer on its own. What they do not show is the
improvement this will have on a slip controller. A control strategy relying on this method will also
have the problem of selecting a suitable slip ratio.
2.5.2 Slip Ratio Estimation
The slip ratio can be estimated without knowledge of vehicle velocity (Fujii & Hiroshi Fujimoto
2007). The method is based on the following differential equation for slip ratio, λ:
J
−ω̇
ω̇
T
λ+(1+ 2 w ) − 2
ω
r M ω r Mω
Equation 2.16
λ̇=
Where 'ω' is the wheel speed, 'r' is the radius of the wheel, 'J W' is the inertia of the wheel, 'M' is the
mass of the vehicle and 'T' is the torque applied to the wheel by the motor. With the exception of λ
all the values on the right of the equation are assumed to be known or can be calculated from
sensors signals. This Slip Ratio Estimation (SRE) has the advantage of integrating the signals from
the wheel speed encoder to calculate the slip and therefore reduces noise.
compared with a Disturbance Observer (DOB).
The performance is
The methods are simulated and then tested
23
experimentally on a low-μ to high-μ road. The results show that the DOB fails to estimate the slip.
The SRE tracks the slip accurately apart from after re-adhesion of the tyre where there is an offset.
The DOB fails due to noise from the low resolution encoder which is not attenuated by its pseudoderivative high pass filter. The SRE has an offset due the error not converging for ώ < 0.
The SRE is developed into a non-linear slip controller based on feedback linearisation. This
assumes a linearised relationship between slip and friction. The controller is tested experimentally
during acceleration (Fujii & Hiroshi Fujimoto 2007) and braking events (Suzuki & H. Fujimoto
2010), Figure 2.13. The results show that although the slip ratio estimation is fast and accurate the
slip controller only maintains the slip ratio with +/-0.3 of the reference value. This means that the
friction force will not be at its maximum value.
Figure 2.13: Experimental results of brake turn event with SRE control, after (Suzuki
& H. Fujimoto 2010)
The main drawback of this method along with the sensor fusion approach is that the road conditions
will vary and therefore it is difficult to select the optimal reference slip ratio.
2.5.3 Sliding-mode Control
In reality there will always be uncertainty in estimating friction force due to varying road
conditions. An alternative to choosing a reference slip ratio based on the peak friction predicted by
Magic Formula is to use a sliding mode control (Schinkel & Hunt 2002), (Harifi et al. 2008), (K. Xu
et al. 2011). The desired slip ratio is determined in real time. A sliding surface is defined based on
the dynamics of the system.
The torque to the wheel is controlled to satisfy this solution.
Simulations show that a sliding mode controller can maintain a desired slip ratio when the friction
coefficient between the tyre and road is altered. The drawback of this control method is that it
requires the vehicle velocity to be measured (Amodeo et al. 2010). Chattering is often caused by a
discontinuous control action, the reduction of which is often the focus of the research effort
(Amodeo et al. 2010).
Noise from sensors are likely to amplify this problem when tested
24
experimentally, of which there is little evidence of in the literature.
2.5.4 Model Based Control
Model Following Control (MFC), (Y. Hori 2004) uses a simple model of the vehicle to estimate the
wheel speed. The error between the estimated speed and the actual wheel speed is used as negative
feedback to the demand signal. A block diagram of MFC is shown in Figure 2.14. MFC is based
on the idea that the inertia seen by the wheel depends on the slip ratio (Y. Hori 2004) :
M actual ≃M wheel+M body (1− λ )
Equation 2.17
where Mactual is the total equivalent mass, Mw is the wheel mass, and Mbody is the equivalent vehicle
mass. A rapid increase in wheel velocity that causes slip to occur will be observed as a decrease in
wheel inertia.
Figure 2.14: MFC block diagram (Y. Hori 2004)
MFC is tested by accelerating a vehicle across a low-μ surface. The results showed that with a gain
of Kp = 20 and time constant of τ = 0.1 the slip ratio λ ≤ 0.4. This is significantly greater than the
optimal slip ratio range 0.1<λ<0.2 stated in the paper.
A deceleration test on track with μ≤0.5
showed that the braking torque could be controlled to maintain the slip ratio between λ = +/-0.2.
This is compared to no MFC where the wheel locks.
A load follow control termed Maximum Transmissible Torque Estimation (MTTE) is proposed by
(Yin et al. 2009). This uses the feedback from the model to provide a saturation limit for the torque
demand. The method is based on estimating the friction force acting on wheel from:
25
T J V˙
Fd= − w 2 w
r
r
Equation 2.18:
A relaxation factor is used to define the ratio of chassis acceleration to wheel acceleration:
(F d −F dr )/M
V̇
=
V˙ w (T max−r.F d )r / J w
Equation 2.19:
α=
As severe slip will occur when the difference between the velocities of the chassis and wheel
increase α should be set close to 1. The maximum transmissible torque is then estimated as:
Jw
1 r.F d
 M r2
Equation 2.20:
T max=
Figure 2.15 shows a diagram of the controller. For simulation purposes the controller uses a one
wheel vehicle model based on the Magic Formula, to determine the wheel velocity. For the
experimental controller the wheel speed is calculated from the wheel encoder signal. A low pass
filter (LPF) is required to smooth the signal from the encoder. The low pass filter on the torque
signal is need to keep the two in phase. Gain compensation proportional to the rate of torque
demand is added to Tmax to compensate for system delays that would otherwise unduly restrain the
torque. This is not shown in Figure 2.15.
Figure 2.15: MTTE Controller, after (Yin et al. 2009)
The theoretical analysis shows that for the system to be stable α > 1, although this will constrain
Tmax to lower than the maximum torque available from the road. It also shows that for stability the
time constant of the LPF should typically be greater than that of the electro-mechanical system. Yin
et al. (2009) tests these parameters through simulation and experimentation. The test consists of a
vehicle driving at a constant low speed and the road surface friction coefficient suddenly
26
decreasing. Both show that for α < 1 the slip ratio is actually relatively stable compared to the noncontrolled case. The torque output is under damped which may affect driver comfort, although this
is not mentioned by the authors. There appears to be little difference in the performance when α =
0.9 and α = 0.5, although the authors state otherwise. Both show the importance of the time
constant of the LPF being longer than that of the electro-mechanical system to improve stability.
The performance of the MTTE Control is compared to Model Following Control (MFC) strategy.
The results show that MFC is only able to control slip with a gain that results an unstable torque.
Yin et al. (2009) notes that the drag force is ignored which will reduce the controller performance,
particularly at high velocities. The rolling resistance is also ignored. The wheel radius is assumed
to be constant, although this will vary according to the vehicle dynamics. This will result in an
error in Tmax , which could reduce performance or stability.
2.5.5 Summary
Conventional slip control that uses single accelerometers to estimate the vehicle speed has been
noted as having poor accuracy due to offset errors and long time integral calculation. To overcome
this, sensor fusion approaches have been adopted using multiple sensors and advanced filters. An
alternative approach to slip control has been to estimate the slip from its rate of change without
sensing vehicle speed. Both of these methods have the draw back of using a static relationship
between slip and friction coefficient to choose a desired slip ratio, based on a particular road
condition. Sliding model control presents a solution to the uncertainty in the road condition,
although the occurrence of chatter and the lack of experimental work leave cause for concern.
Model based control only considers the dynamics of the system avoiding the need to estimate slip or
measure vehicle speed. Although feedback compensation has shown limited performance a method
using feedback saturation has shown promising results.
27
3 Work to Date
This report explores the tools required to develop a traction controller of an electric vehicle. A
dynamic friction model is investigated for its effectiveness at modelling the force between the tyre
and the road. This is combined with a vehicle dynamics model to simulate a quarter of the vehicle,
providing a building block for higher complexity vehicle models. The numerical models are
validated against results from models and experimental work found in the literature.
Finally a
numerical model of a slip controller is adapted from the literature.
3.1 Dahl Friction Model
The literature shows that no complete model of friction exists. A range of dynamic models that can
be seen as extensions of the Dahl model have been developed that capture friction phenomena to
varying degrees, and with varying degrees of complexity. The Dahl model is a dynamic model of
friction that captures the behaviour of presliding displacement with only two parameters. The
following section investigates the effectiveness of the Dahl model within a simple dynamic system.
This will help determine its usefulness in modelling friction for a tyre-road interface.
A numerical model of the Dahl friction model has been constructed in Simulink. A process of
validation is then undertaken to determine the fidelity of the model. The first stage of validation is
to check the numerical model is behaving as expected by comparing it to an analytical solution.
Next the model can be compared to other models in the literature, in this case the LuGre model is
used. This stage will only show how well the model agrees with the simulation work of other
researchers. Therefore the third stage is to validate the model against experimental work found in
the literature. A final stage would be to validate the model against the author's own experimental
work. This has not yet been undertaken.
3.1.1 Analytical model
In this section an analytical solution to the Dahl equation is derived. For simplicity only the case of
the exponent parameter, i = 1 is considered. The simplified Dahl friction-displacement equation is:
dF
F
=σ(1− . sign(v ))
dx
Fc
Equation 3.1:
The solution can be derived in parts depending on the 'sign(v)' term. For v = 0 it is straight forward
to see that F = σ.x . This shows that σ defines the initial slope of the friction-displacement curve
when an object is at rest. For v>0 Equation 3.1 becomes:
28
Ḟ +
σ
F =σ
Fc
Equation 3.2
Given the initial condition of F=0 at x=0 the Laplace transform of Equation 3.2 is:
F (x)=F c (1−e
−σ
x
Fc
)
Equation 3.3
This equation defines the friction force for any displacement, given that the velocity remains
positive. If the direction of the velocity reverses (v<0) then 'σ.F/FC' in the Dahl equation will take a
negative sign. A new solution will exist with initial conditions (x1, F1) taken from final state of the
previous solution:
F ( x )=( F 1+F c )e
σ
( x− x1 )
Fc
Equation 3.4
−F c
Likewise, if the velocity reverses a second time the solution will be:
−
F (x)=(F 2−F c )e
σ
(x−x 2 )
Fc
+F c
Equation 3.5
For Equation 3.3 and Equation 3.5, as displacement increases the friction force will tend toward the
Coulomb friction asymptote. For Equation 3.4 the friction force will tend toward the negative
Coulomb friction asymptote. These equations fully describe the presliding displacement shown in
Figure 2.2 of the Literature Review chapter. Now that an analytical solution has been found, the
numerical model can be compared to it to check for errors produced by the numerical solver.
3.1.2 Simulation with Displacement Excitation
The simple case of a fixed linear displacement with two velocity reversals is used for the first
simulation.
This allows an initial comparison to the analytical solution without the added
complexity of the friction driving the displacement, even though this is more realistic.
The
Simulink model is shown in Figure 3.1. The Dahl differential equation is contained within the
MATLAB Function block. The Dahl parameters are set to the arbitrary values of i = 1, σ = 1, and
Fc = 1.
29
1
2
x
v
D is p la c e m e n t
R am p3
V e o lc it y
d u /d t
In te rp re te d
M A T L A B Fcn F'
D e riv a tiv e
D a h l e q u a tio n
F r ic t io n F o r c e
R am p1
R am p2
F
3
1
s
In te g ra to r2
Figure 3.1: Simulink Dahl model with displacement excitation
The excitation displacement is shown in Figure 3.2. The Solver used was the ode23s (stiff/Mod.
Rosenbrock) with a relative tolerance of 1x10-3. A plot of the resultant friction against displacement
is shown in Figure 3.3. The analytical solution is shown on the same graph for comparison. The
plot shows a very good match between the numerical and analytical solutions. The maximum error
between points on the two curves is 0.91%. Reducing the relative tolerance of the solver to 1x10 -4,
reduced the maximum error to 0.19%. Changing the parameters to σ =190E6 N/m, Fc = 3060N, to
represent a stiff contact, had little effect on the accuracy (max error 0.22%).
Figure 3.2: Excitation displacement profile
30
Figure 3.3 Force-displacement curves for analytical and numerical
models
3.1.3 Simulation with Force Excitation
In the previous section the numerical model was shown to be almost identical to the analytical
model for a displacement excitation. In this section the slightly more realistic case of a unit mass
excited by a external force subject to friction is simulated. The Simulink model was adapted from
the one used by Chou (2004). The model is shown in Figure 3.4. The model is governed by the
Dahl Equation 2.8 and Newton's second law:
1
(F −F )
m e
Equation 3.6
ẍ=
Where Fe is the excitation force.
3
4
R am p3
A ccel
F _ a p p lie d
G
x ''
G a in
1
s
In te g ra to r
x'
1
s
In te g ra to r1
1
x
R am p1
R am p2
5
F _ fric tio n
1
s
F'
In te g ra to r2
In te rp re te d
M A T L A B Fcn
2
v
M A T L A B Fcn
Figure 3.4: Simulink Model, adapted from (Chou 2004)
31
The values of parameters used in the model are shown in Table 3.1. The model used the ode23s
solver with a relative tolerance of 1E-4. The excitation force profile is shown in Figure 3.5. A plot
of the friction force against displacement is shown in Figure 3.6, with the analytical solution shown
for comparison. The Simulink Dahl model shows a very good fit to the analytical solution.
Parameter
Symbol
Value
Units
Mass
m
1
kg
Gravitational acceleration
g
10
m s-2
Coefficient of Friction
μ (mu)
0.1
Gain
G
1/m
kg-1
Coulomb Friction Force
FC
μ.m.g
N
Dahl stiffness
σ (s)
1
N/m
Dahl shape factor
i
1
Simulation time
t
6.3
s
Table 3.1: Model Parameters
Figure 3.5 Excitation force and resultant friction force
32
Figure 3.6 Friction force against displacement for the numerical
and analytical models
3.1.4 Comparison to the LuGre Model
It has been shown that the Dahl model behaves as predicted analytically for a displacement
excitation and for a force excitation. The model will now be validated against another model from
the literature. The model used for comparison is the LuGre model (Canudas de Wit et al. 1995).
This model is viewed as one of the more advanced friction models.
The details of a presliding displacement simulation using the LuGre model are readily available
(Canudas de Wit et al. 1995). The simulation involved applying a trapezoidal shaped excitation
force to a unit mass subject to friction. The magnitude was limited to 95% of the stiction force to
look at the behaviour before slip occurs.
The Simulink Dahl model in Figure 3.4 was adapted to match as closely as possible the simulation
described above. The parameters were altered from Table 3.1 to σ = 105N/m and FC = 1.5N, taking
FC = FS as the Dahl model doesn't contain a term for stiction. The excitation force is shown in
Figure 3.7. The duration of the original simulation was not given. The only information available
is a qualitative description of the excitation force being slowly ramped (Canudas de Wit et al. 1995).
Based on this the excitation waveform period was set to 60 seconds.
33
Figure 3.7 Excitation force applied to model
The results of the simulation are shown in Figure 3.8, Figure 3.9. Figure 3.8 shows a plot of the
friction force against the presliding displacement for the Simulink Dahl model and the LuGre model
(Canudas de Wit et al. 1995). The models initially show a good match as the excitation force is
ramped from 0 N to 1.425 N (t ≤ 10s). When the excitation force is then held at 1.425 N (10 < t <
15s) the displacement should remain constant, although from Figure 3.9 it can be seen that the
displacement continues to increase. The sudden change in the excitation force at t =10s causes the
velocity to oscillate rather than returning to zero resulting in the unexpected displacement.
Figure 3.8 Friction verses presliding displacement for the
Simulink Dahl model and the LuGre model
34
Figure 3.9: Force, displacement, velocity and acceleration profiles for the Dahl
model
The equations that govern the response of the system are:
Ḟ =σ 0 (v−
Equation 3.7:
F
.∣v∣)
Fc
v̇=
1
.( F e −F )
m
Equation 3.8:
Equation 3.7 tells us that the friction force, 'F', will only reach steady steady when it is equal to the
Coulomb friction force, 'FC'. On the other hand Equation 3.8 shows that the velocity, 'v', will only
reach steady state when the friction force equals the excitation force 'F e'. As 'Fe' and 'FC' are not
equal the the friction force will oscillate about the excitation force as shown Figure 3.10. Drift,
where an applied force less than that of the break-away force produces an unbounded displacement,
is found to also be a property of the LuGre model for an oscillating excitation force (Pierre Dupont
et al. 2002). Drift is more pronounced in the Dahl model due to there being no damping term.
35
Figure 3.10 Friction force-velocity oscillations
An additional viscous damping term is added to the system model to minimise this oscillation. The
differential equation now becomes:
v̇=
1
.(F e −F −σ 1 . v )
m
Equation 3.9
Where σ1 is the damping coefficient. The damping coefficient was set to σ1 = 100 Ns/m. The
results of the model simulated with damping are shown in Figure 3.11 and Figure 3.12. Figure 3.11
shows that the presliding behaviour of the model has a very close match with the LuGre model.
This is due to the velocity oscillations being almost completely removed by the damping term, as
can be seen in Figure 3.12. A damping coefficient between 100 < σ1 < 1000 Ns/m was found to
give very similar results.
It is also worth noting that the simulation of the model with damping ran much faster than the
original, at 1.2s compared to 16.2s. This is an important consideration when simulating more
complex models.
36
Figure 3.11: Friction force versus presliding displacement for
the modified Dahl model and the LuGre model
Figure 3.12: Velocity profile for modified Dahl model
It has been shown that for a simple dynamics system the Dahl model only capture the presliding
behaviour of the more complex LuGre model by including a damping term. Without the damping
term the two parameter Dahl model exhibits drift for a constant force under the Coulomb friction
force.
3.1.5 Comparison to Experimental Data
Having compared the Simulink Dahl Model to the simulation results of another friction model in
the literature the next stage in validation is to compare the model to experimental data from the
literature. The LuGre model is qualitative compared to experimental work between metal contacts
(Canudas de Wit et al. 1995). Here we go one step further and attempt to match the parameters of
the model to experimental data to make a quantitative comparison.
Olofsson (1995) presents experimental work determining the presliding displacement of a steel37
brass contact. The experiment consisted of a flat plate and a hollow cylinder pressed together by a
normal force of 6kN. The cylindrical contact was then rotated by an applied torque and the angular
displacement was measured. Plots of the applied torque profile, the angular displacement profile
and the force versus displacement are given (Olofsson 1995).
Model Input
As the experimental set up was not fully detailed it was necessary to calculate the radius, r, of
cylindrical contact. This allows the displacement and force excitations to be calculated from the
torque and angular displacement profiles given. The radius can be calculated from the ratio of
maximum torque and maximum force given by the plots. This gave r = 11.6mm. The same value
was obtained from the ratio of the minimum torque to minimum force. The ratio of the normal load
over the pressure (given as 39.1MPa) showed that this is a reasonable value for the radius of the
cylinder.
Model Parameters
To simulate the experiment the model parameters need to be defined so that they match the
experimental set up. The parameters that need to be defined are: the normal force; the Coulomb
friction force; the Dahl stiffness parameter; and the Dahl exponent parameter. The normal force is
6kN (Olofsson 1995). The Coulomb friction force is the asymptote that the friction-displacement
curve tends to. This can be estimated by visual inspection. The stiffness coefficient 'σ' can be
estimated from the initial gradient of the friction-displacement graph. The exponent parameter of
between 1 and 1.5 has been found by (P Dahl 1977) to give the best fit to experimental data. The
simulation parameters are given in Table 3.2.
Parameter
Symbol Value
Units
Normal Force
N
6000
N
Gravitational acceleration g
9.81
m s-2
Gain
G
g/N
kg-1
Coulomb Friction Force
FC
1400
N
Dahl stiffness parameter
σ
130
kN/mm
Dahl exponent parameter
i
1
kN-s/mm
Damping Coefficient
σ1
11.4
Simulation time
t
177.8
s
Table 3.2: Dahl model parameters
38
Simulation
Now that the inputs and parameters have been determined the simulation can be run. The Dahl
model was first simulated with the displacement excitation, as in Figure 3.1. The displacement
profile was extracted from the experimental data (Olofsson 1995). The resulting friction force is
plotted against displacement in Figure 3.13. These results show that there is a reasonable match
between the model and the experimental data. In the first part of the curve, starting at the origin, the
model under-predicts the friction force.
Figure 3.13: Comparison of Dahl model simulated with a
displacement excitation to experimental data
A second comparison was done using the Dahl model with force excitation. The force excitation
profile is given in Figure 3.14. The force excitation input to the model was low pass filtered to
remove the high frequency transients that would distort the response. The results are shown in
39
Figure 3.14: Excitation force applied to model along with applied
force from experiment (Olofsson 1995)
Figure 3.15 and Figure 3.16. It should be noted that the model displacement is offset by 12μm to
align the two curves. This was done as the model could not capture the highly irregular behaviour
of the rate of friction initially increasing (from the origin). The model shows a good fit to the
majority of the experiment. The additional displacement at the end of the simulation can be
attributed to the choppy nature of the excitation force which causes the velocity to fluctuate.
Figure 3.15: Displacement, velocity and acceleration profiles from simulation
40
Figure 3.16: Presliding displacement hysteresis behaviour for simulation and
for experimental data after (Olofsson 1995)
3.2 Quarter Vehicle Model
This section looks at a QV model that has been built in Simulink. The model combines the
longitudinal and vertical QV models that were described in the literature review chapter together
with the Dahl friction model. The literature showed that for simulating slip control it is necessary to
have a QV longitudinal model combined with a friction model. The QV suspension model is
included to develop an understanding of the vehicle dynamics.
3.2.1 Description of Model
Figure 3.17: Quarter Vehicle Model
A diagram of the QV Model is shown in Figure 3.17. 'Ms' is the mass of the vehicle chassis or
sprung mass. 'Mu' is the mass of the wheel or unsprung mass. 'K s' and 'Cs' are the spring constant
41
and damping coefficient of the suspension, respectively. 'K u' and 'Jω' are the spring stiffness and
rotational inertia of the wheel, respectively. The model has four degrees of freedom: vertical
motion of the sprung mass; vertical motion of the unsprung mass; longitudinal motion of the
vehicle; and rotational motion of the wheel. Its is assumed that the suspension rigidly connects the
longitudinal motion of the wheel centre to that of the sprung mass. Although the tyre deforms under
load according to its stiffness, the forces that act at the tyre contact patch are all considered to act at
a single point below the wheel centre. The equation of motion of the QV model are:
M S Z¨ S =−K S Z S −Z U −C S  Ż S −Z˙U −M S . g
Equation 3.10
M U Z¨U =−K U  Z U −Z O −K S Z U −Z S −C S  Z˙U −Z˙ S −M U . g
Equation 3.11
M t Ẍ =F − R r −D a
Equation 3.12
J ω ω̇=T −r.F
Equation 3.13
Symbol
Unit
Description
Value
MS
kg
Sprung mass
400
MU
kg
Unsprung mass
40
r
m
Wheel rolling radius
0.25
Wheel inertia
2.5
J W = M U . R2
2
Source
JW
kg.m
KS
kN/m
Suspension stiffness
20
(Shim & Ghike 2007)
CS
N-s/m
Suspension damping
1131
Gives 0.2 damping ratio (0.20.4 typical (Gillespie 1992))
KU
Krr
kN/m
Tyre stiffness
200
(Shim & Ghike 2007)
Rolling resistance coefficient
0.015
(Gillespie 1992)
Table 3.3: Quarter Vehicle Model Parameters
42
3.2.2 Suspension frequency response
In this section the suspension response of the QV model is validated against a similar model in the
literature, (Gillespie 1992). It is useful to define the key parameters that describe the model.
Gillespie (1992) defines the ride rate as:
Ks.Ku
K sK u
Equation 3.14
RR=
Ride rate is the effective stiffness of the QV model. The damped natural frequency of the system is
defined as:
 d = n  1− 2
Equation 3.15
Where the undamped natural frequency, ωn , and damping ratio, ζ, are defined as:

RR
Ms
Equation
3.16
 n=
Cs
 4.K s M s
Equation 3.17
=
Gillespie (1992) states that a damping ratio between 0.2 and 0.4 is typical for a modern passenger
car. It is also worth noting the natural frequency of the tyre:

Ku
Mu
Equation
3.18
tyre =
Given the above equations and the values for spring and damping coefficients in Table 3.3 the
damped natural frequency for a damping ratio of 0.2, should be 1.05Hz. Simulations were run on
quarter vehicle model with a sinusoidal road input for frequencies between 0Hz and 20Hz. For the
period between 0-2Hz the interval between frequencies has 0.05Hz, and for 2-20Hz the interval was
1Hz. This was done to capture the more transient behaviour at lower frequencies. The results are
show in Figure 3.18. It can be clearly seen there is a resonant peak at 1.05Hz. Also of note is the
bump on the graph at 11.24Hz which is due to the tyre resonance frequency. This shows that the
Simulink model is behaving correctly according to theory and is in agreement with the results of a
similar model (Gillespie 1992).
43
Figure 3.18: Quarter Vehicle Frequency Response of sprung mass displacement
to road displacement
3.2.3 Longitudinal Acceleration Test
In this section the quarter vehicle model is used to simulate real world EVs during a 0-60mph
acceleration test. Ideally a vehicle test would be carried out where the wheel torque, wheel
rotational velocity, and vehicle velocity are logged, and the road conditions are known. As this has
not been possible to date, the wheel torque will need to be estimated based on the data provided by
the manufacturer and car reviewer. The model will be validated by comparing the 0-60mph time to
the time quoted by the manufacturer and car reviewer.
It is assumed that the acceleration test is conducted on dry asphalt. The relationship between the
coefficient of friction, μ, and slip ratio, λ, is given by Y. Hori (2004) . For dry asphalt and moderate
slip, 0.03< λ<0.2, the coefficient of friction can be taken to be μ max = 0.9.
Tesla Roadster
The vehicle parameters and 0-60mph time were taken from the vehicle manufacturer website
(TeslaMotors 2011). A second 0-60mph time from a car review was found to provide a measure of
experimental error (Robinson 2009). The simulations consist of applying a torque signal to the
quarter vehicle model and measuring the time it takes to reach 26.8m/s. The simulation process is
iterative, in that each step aims to increase the realism and therefore the accuracy of the model. The
external forces applied in each simulation are listed Table 3.4. The aerodynamic drag and rolling
resistance are calculated using the following equations (Gillespie 1992),:
44
D a =½ . ρ.C d. A. v 2
Equation 3.19
R r =K rr . m. g
Equation 3.20
Sim
Torque
Drag
Roll Resistance
1
Step to 3000Nm @ t=0s
No
No
2
Step to 2750Nm @ t=0s
No
No
3
Step to 2750Nm @ t=0s
Yes
Yes
4
Ramp from 0 (t=0s) to 2750Nm (t=1s)
Yes
Yes
Table 3.4
The results of the simulations are shown in Figure 3.19. In the first simulation the maximum torque
available from the motors is applied instantaneously and the resistive forces are ignored. The
torque exceeds the traction limit of the tyre-road interface and therefore produces excessive slip
(0.43 @ t =3.03s). In reality this would result in a reduced friction coefficient between the tyre and
road. Hence this simulation is a poor reflection of reality, which is shown by the results.
Figure 3.19: 0-60mph times from simulations, Car reviewer,
and Manufacturer
The second simulation reduces the torque step to just below the traction limit. This results in much
lower slip (0.06 @ t =3.03s)although the 0-60mph time is exactly the same as the first simulation.
This is to be expected as the Dahl friction model is rate independent.
The third simulation includes the effects of drag and rolling resistance. These have a minor effect
on the 0-60mph time and provides evidence for why they are often ignored in simple longitudinal
models (Yin et al. 2009) (Cai et al. 2010).
The fourth simulation ramps the torque up over a second and then holds at a constant value
thereafter. This was chosen as a more realistic input of a driver steadily increasing pressure on the
accelerator pedal. This has a major influence of the 0-60mph time, which comes within 1.3% of
45
time given by the manufacturer. Given that there is a 5% difference between the results of the
vehicle manufacturer and reviewer this is an acceptable degree of accuracy.
Nissan Leaf
A second set of simulations were done based on the vehicle parameters of the Nissan Leaf (Nissan
2011). The Nissan Leaf is heavier and has less power than the Tesla Roadster. Simulating EVs
with different performance levels helps to improve the fidelity of the model. The external forces
and power limits for the simulations are shown in Table 3.5. The 0-60mph times for the simulation
are shown in Figure 3.20, along with the times given by the manufacturer (Nissan 2011) and car
reviewer (Austin 2010).
Sim Torque
Drag
Roll Resistance
1
Ramp from 0 (t=0s) to 2220Nm (t=1s)
Yes
Yes
2
Ramp as above, power limit Pmax = 80kW
Yes
Yes
3
Ramp as above, Pmax = 1.7*Vw+31 (kW)
Yes
Yes
Table 3.5:
The iterative process to improve the model accuracy continues from the previous set of simulation
on the Tesla Roadster. Therefore the first Leaf simulation is the same as the fourth Roadster
simulation, although the torque is limited to the rated torque of the Leaf. Excessive slip is not a
concern as the rated torque of the Leaf is much lower than the traction limit. From the results in
Figure 3.20 it can be seen that the 0-60mph time is half that of the experimental data. The
difference in error compared with the Roadster simulation is due to the power limit of the Leaf not
being taken into account. The Roadster power limit did not need to be considered as it does not
have a significant effect until 100mph (Robinson 2009).
Figure 3.20: 0-60mph times from simulations, Car reviewer
(Austin 2010), and Manufacturer (Nissan 2011)
46
The second simulation used the same torque demand but also limited the product of torque and
angular wheel speed to the rated power of the motor. This makes a significant improvement on first
simulation but still has an error of 15% compared with the experimental data. To understand the
cause of this error the experimental velocity data (Austin 2010) is used to estimate the vehicle
acceleration. This is done by fitting a power regression curve to the velocity data, Figure 3.21. The
derivative of this gives the acceleration.
The curves do not necessarily provide an accurate
depiction of reality for t < 2s, due to the large time step between the first two data points. This is of
little importance though as within this time the vehicle is likely to be limited by the available torque
Velocity (m/s), Accel*10 (m/s2)
and not the available power.
Velocity (m/s)
Power Regression for Velocity (m/s)
Acceleration*10 (m/s2)
Power Regression for
Acceleration*10 (m/s2)
35
30
25
20
15
10
5
0
0
2
4
6
8
10
12
14
Time (s)
Figure 3.21: Velocity and acceleration profiles for Nissan Leaf acceleration test
From the velocity and estimated acceleration and knowing the mass of the vehicle (1525kg) the
apparent power of the vehicle can be estimated from:
P=m.a. v
Equation 3.21
The data shows that the acceleration test had an apparent power of approximately 56kW throughout
the speed range. This corresponds to the power driving the vehicle forward after losses are taken
into account. The power output at the wheel will be higher as it has to overcome the drag force and
rolling resistance. By adding the power losses due to these external forces an output power limit
related to vehicle velocity can be determined Figure 2.13. This is obviously only applicable up to
the rated power. The third simulation used this power limit instead of the rated power given by the
manufacturer. This gives a 0-60mph time within 1.7% of the time given by the manufacturer.
47
120.0
100.0
f(x) = 1.71x + 30.91
Power (kW)
80.0
60.0
output power (kW)
@ wheel
Linear Regression
for output power
(kW) @ wheel
40.0
20.0
0.0
10
15
20
25
30
35
40
45
Velocity (m/s)
Figure 3.22: Nissan Leaf estimated output power during
acceleration test
3.2.4 Summary
The combined QV model and Dahl friction model, provides reasonably accurate match to
experimental data given that only the rated values for torque and power were available. The
simulation results showed that the torque and power profiles had the greatest influence on the
fidelity of the model. For an EV, the torque and power output of the motor would be easily
obtainable from the current and position sensors.
The losses due to drag and rolling resistance
would be harder to measure although only had a minor affect.
3.3 MTTE Controller
A model of a MTTE Controller (Yin et al. 2009) was built in Simulink. The controller description
and diagram are included in the literature review of this report. The vehicle model that is used is
shown in Figure 3.23. Only the peak friction coefficients are give for the Magic Formula friction
model that is used.
Therefore the other three parameters are estimated based on the parameters
given by Cai et al. (2010). It was found that the the block for calculating slip (λ) produced a NaN
error. The solution was to add a small constant to the denominator (Fujii & Hiroshi Fujimoto
2007).
48
Figure 3.23: one-wheel vehicle model with Magic
Formula μ-λ function, after (Yin et al. 2009)
3.3.1 Modifications to Controller
It was found that if the controller is turned on at the same time as the torque demand is initiated then
the controller will limit the torque to several orders of magnitude lower than the maximum
transmissible torque. To mitigate this a switch was included that turned on the controller at a set
time after the start of the simulation.
Figure 3.24 Slip ratio convergence for varied controller turn on
The model was run with a torque ramp 100Nm/s and controller turned on a t = 1s, 5s and 10s. The
slip ratio over time is plotted in Figure 3.23. This is not a realistic simulation as such due to the
extremely high torque but it does show that the MTTE converges to an optimum slip ratio,
regardless of switch on time. However, a delay that produces significance slip will take much
longer to converge. The optimum slip ratio was defined by the Magic Formula model of friction,
within the vehicle model, which had a peak value at λ = 0.1.
49
3.3.2 Comparison of Model Simulation to Literature
With the modifications made to the controller model as mentioned previously a comparison is made
to the results from the literature. The simulation consists of a step torque demand, T = 100Nm,
applied to the the vehicle at t = 0s. At t = 2s the road surface changes from high-μ (μmax = 1) to lowμ (μmax = 0.3). The results from the simulation are shown on the right of Figure 3.25, with the
results from the literature for comparison (with severe slip). The behaviour at the change of friction
coefficient is very similar between the two. The friction and torque curves tend to different
asymptotes as only the peak parameter for the friction model were given.
Figure 3.25: Comparison of Simulink MTTE model with results for literature,(Yin et al. 2009)
3.3.3 Summary
In this report a model of the MTTE slip controller has been reproduced in Simulink. The model has
been shown to be a reasonably faithful representation of the original work, given that only one of
the parameters of the Magic Formula friction model is known. This report has shown for a linear
increase in torque demand the controller severely restricts the output torque. The model has been
adapted to include a delay in turning on the controller, which has mitigated the problem.
50
4 Future Work
The aim of this research is to develop a robust traction controller for 2WD and 4WD electric
vehicles.
The literature shows that there exists a variety of ways to control slip, which is
fundamental to traction control.
An attractive solution to slip control is the Maximum
Transmissible Torque Estimation (MTTE) proposed by Yin et al. (2009). This does not require the
vehicle speed to be sensed, nor does it assume a static relationship between friction force and slip
ratio. Most of the research on slip control has only considered a single wheel (quarter vehicle
model).
To develop effective traction control all the driven wheels need to be considered.
Therefore to verify a traction controller by simulation a full vehicle model will be required.
A slip controller should be able to maintain traction of a wheel through changes in road conditions,
vehicle dynamics and tyre forces. Road surface conditions can change suddenly due to rain and ice
reducing the friction coefficient and the road profile can change due to pot holes and speed bumps.
The vehicle dynamics determine the normal load on the wheels, which will vary due to pitch
moments caused by acceleration and roll moments caused by turning manoeuvres. The friction
force is proportional to the normal force, although during a turn not all of the friction force will be
available for driving the vehicle forward. A proportion of the friction force will be required to
generate the lateral force to enable the vehicle to turn, after the theory of friction circles (Gillespie
1992). A traction controller will need to do all of the above while ensuring that the combined
torque to all the driven wheels does not lead to vehicle instability.
4.1 Simulation
To verify a traction controller by simulation the need for a full vehicle model has been highlighted
above. Due to the time constraints of the project a commercial package has been sought. Dymola
is a physical modelling and simulation tool that is used widely in the automotive industry. Its key
advantage is that it is based on the open Modelica modelling language. Therefore the fundamental
equations that govern the components are open to view and edit. In addition Dymola can be
interfaced with a Simulink controller. As the I-Racer test vehicle, discussed in the next section, also
uses Simulink to compile it's ECU this provides a close approximation to reality and an efficient
route to implementation. An evaluation licence of Dymola has been acquired to trial the software.
If this proves satisfactory a full academic licence will be purchased with the available TSB funding.
The controller will be developed in Simulink. It will consist of individual MTTE slip controllers
for the torque to each of the driven wheels. Novelty will be sort in developing a traction controller
to manage the interaction between the individual slip controllers. The research will look at what
information can be gained from including a real time vehicle dynamics model within the controller.
51
For example estimating the normal loads on the wheel from how fast the vehicle is accelerating.
This will draw on the work to date on vehicle and friction modelling.
To verify the performance of the traction control a number of simulations will by run. In each case
the controller simulation will be compared to a no control case to give a measure of the controller
performance. First the launch control will be tested.
For this the vehicle will be simulated
accelerating from stationary on a flat asphalt road. A torque demand will be applied that is high
enough to produce slip in the absence of the control. This will test the basic performance of the slip
controllers. Secondly, steering manoeuvres will be considered. This will include fixed steering
angle cornering and slalom manoeuvres. These will verify the performance of the controller for
variations in the normal load on the wheels caused by the pitch and roll dynamics of the vehicle.
Thirdly, sudden changes in the road condition will be simulated. This corresponds to patches of ice
causing the tyre-road friction coefficient to drop. Simulation tests will consist of the vehicle driven
in a straight line from a high-μ to low-μ surface, and on a split-μ surface where the coefficient of
friction is different for right and left tyres. Finally, a combination of these tests will be simulated to
verify that these coupled conditions do not lead to instability. The research to date will help with
developing and validating this model of higher complexity.
4.2 Testing
Once confidence has been built through the simulation tests the controller will be implemented and
tested on a real vehicle. Vehicle testing will be carried out in collaboration with TSB project partner
Westfield Sportscars Ltd. Westfield are able to provide access to their I-Racer electric vehicle, and
support in integrating the controller. The I-Racer is a high performance battery electric race vehicle
that contains two independent electric motors that directly drive the rear wheels. The I-Racer has a
3-axis accelerometer built into the chassis. Westfield also have a full CAD model of the I-Racer
which will enable the Dymola vehicle model components to be tuned to match those of the I-Racer.
The same set of tests will be carried out as described in the above section. The launch control and
cornering test will be conducted on a dry asphalt track.
For launch control, accurate
displacement/time measurements will be taken with OU's FinePix HS10 high speed video camera.
For cornering the I-Racer's built in accelerometer will be used to calculate the vehicle velocity as
well as the dynamics of the chassis. There is also the possibility of using a speed encoder from one
of the non-driven wheel to provide a second vehicle speed reference. For the change in road
condition tests the low-μ road can be constructed from acrylic sheet (Yin et al. 2009) or aluminium
plate (Y. Hori 2004) lubricated with water. To aid with the vehicle testing it will be possible to draw
on the supported of the TSB project partners: YASA Motors (motor manufacturer); Sevcon (motor
52
controller manufacturer); TRW Connekt (test engineering consultancy).
Further work will look at what improvements can be made to the traction controller by
incorporating signals from the chassis accelerometers and steering wheel position. Simulation of
the vehicle and controller under braking conditions will also be considered.
4.3 Research Timing Plan
2012
Q1
Activity
Q2
2013
Q3
Q4
Q5
J F M A M J J A S
Model I-Racer in Dymola
Implement Traction Controller in Simulink
and Interface with Dymola
Simulation, verification and adaption of
controller
Integrate controller into I-Racer ECU
Conduct validation tests of I-Racer with
controller
Modify Controller
Repeat specific validation tests/ conduct
additional tests
Write up Thesis
Table 4.1
53
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