Safety and Reliability of
Distributed Embedded Systems
Technical Report ESL 0401
Simulation of Vehicle Longitudinal Dynamics
Michael Short
Michael J. Pont
and
Qiang Huang
Embedded Systems Laboratory
University of Leicester
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Project summary
This technical report is one of a series (listed in full below). Together these reports describe a
complete hardwareintheloop (HIL) simulation that reproduces the behaviour of a passenger car
travelling down a motorway. In the simulation, the speed and position of the car are determined
by an adaptive cruise control system implemented using one or more embedded microcontrollers.
The test bed is intended to be used to assess and compare different software architectures for use
in distributed embedded systems, particularly those for which high reliability is a key design
consideration.
Full list of reports in this series
Available now:
ESL04/01 “Simulation of Vehicle Longitudinal Dynamics”
ESL04/02 “Simulation of Motorway Traffic Flows”
ESL04/03 “Development of a HardwareintheLoop Test Facility for Automotive ACC
Implementations”
Forthcoming:
ESL04/04 “Control Technologies For Automotive DriveByWire Applications”
ESL04/05 “10Node Distributed ACC System: CoOperative Implementation”
ESL04/06 “10Node Distributed ACC System: PreEmptive Implementation”
Acknowledgements
The work described in this report was supported by the Leverhulme Trust (F / 00212 / D)
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Contents
1.
Introduction
............................................................................................................1
1.1
The TwoWheel Traction Model
....................................................................1
1.2
System Block Diagram
...................................................................................2
2.
Equations Of motion
..............................................................................................3
2.1
Vehicle load distribution
................................................................................3
2.2
Drag forces
.....................................................................................................4
2.3
Tractive properties of the tyre/road interface
.................................................4
2.4
Wheel dynamic equations
..............................................................................6
3.
Powertrain Model
...................................................................................................8
3.1
Engine torque curve
.......................................................................................8
3.2
Engine dynamics
..........................................................................................10
3.3
Gearbox model
.............................................................................................11
4.
Brake System Model
............................................................................................12
4.1
Servo actuated brake system
........................................................................12
4.2
Brake friction characteristic
.........................................................................13
5.
Parameter Selection
..............................................................................................14
6.
Simulation Results
................................................................................................15
6.1
Acceleration Performance
............................................................................15
6.2
Braking Performance
....................................................................................16
7.
Conclusions
..........................................................................................................17
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1. Introduction
This document describes the development of a model of the longitudinal dynamics of a
passenger car. The model described here provides a suitably detailed description of a ‘host
vehicle’, which is controlled by a distributed embedded system in a hardwareintheloop real
time test facility. The motorway simulation environment in which the testing is to take place is
detailed in an associated technical report (ESL04/02), while the development of the interface to
the embedded system and the overall system integration can be found in the report ESL04/03.
1.1 The TwoWheel Traction Model
For a simulation of vehicle dynamic performance, the gross vehicle dynamics and the
tyre/wheel dynamics must both be considered. These can both be captured by simplified lumped
mass models, and may consist of singlewheel versions, twowheel versions, or full fourwheel
models for cornering as well as acceleration/braking analysis [Gillespie 1992]. In order to
simulate the dynamics of a passenger vehicle whilst driving down a motorway, the lateral forces
acting upon the vehicle may be, in general, neglected. This is because the motorway systems in
many countries worldwide have been designed to be as straight as possible, and the forces
involved during steering to change lanes have a relatively small effect, and act for only small
periods of time, when compared to the much larger longitudinal forces involved when cruising at
high speed. This is in stark comparison to, for example, a racecar simulation where the
acceleration, braking and cornering dynamics are much more coupled and essential to produce a
realistic model. For this reason, a full fourwheel model is considered unnecessary for the
purposes of this simulation. Passenger vehicles of the type under simulation here are generally
built to be as stable as possible with a centre of gravity (COG) as close to the centre of the car as
possible. However, the effects of longitudinal load transfer are still present, and for this reason a
simplified twowheel dynamic model is best suited to describe the dynamics. Figure 1 shows a
schematic of the model.
It can be seen from the figure that car is represented by a lumped mass m, which has a
forward velocity V in the Xdirection. The vehicle’s wheels have a rotation rate ω
i
, a rolling
radius of R
i
, and a polar moment of inertia J
i
, where the subscript i=f,r describes either the front
or rear wheel. A coefficient of friction µ
i
exists between the wheel’s tyre and the contact surface.
The dimensions B and C represent the distance between the vehicles centre of mass and the front
and rear axles respectively, with the wheelbase represented by the dimension L. The road
gradient θ indicates the road’s inclination from the normal.
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Figure 1: Schematic Of The TwoWheel Traction Model
1.2 System Block Diagram
The diagram below shows a highlevel system block diagram outlining the main elements of
the vehicle model, and the system variable dependencies between them. The main inputs to the
model are the throttle and brake settings, and the main outputs are the vehicles velocity, and the
front and rear wheel velocities. Additionally, there are three other parameters of interest: the road
gradient, road conditions and wind speed.
Figure 2: Vehicle System Block Diagram
It can be seen from the model that the many system elements are tightly coupled and highly
interactive, and many are also nonlinear. It can also be seen in the model that the vehicle is
frontwheel drive. This report will first consider the gross vehicle dynamics that govern the
generation of vehicle speed, and consider each block in turn backward toward the systems inputs.
Once each element had been described, suitable values are then determined for each system
parameter. Following this, a suitable methodology for solving the equations in realtime is then
developed.
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2. Equations Of motion
The primary forces of interest in the simulation of vehicle longitudinal dynamics are the
forces acting about the Xaxis, acting on the vehicle through the tyre/road interface. These forces
act to both accelerate and brake the car, and are proportional to the normal force Z at the
tyre/road contact points:
friFF
ii
ZiX
,,
=
=
µ
Equation 04/01/A
The total force acting upon the vehicle body due to these forces is simply the summation of
the reaction forces at the front and rear contact points:
r
XXX
FFF
fT
⋅
+
⋅
=
22
Equation 04/01/B
It is straightforward to choose the forward velocity V and tyre rotational velocities ω
r
and ω
f
as the main dynamic states to be simulated. Considering the forward velocity V, summing the
forces acting on the car’s mass m:
m
VFgF
V
dX
T
)()sin(
.
−
+
−
=
θ
Equation 04/01/C
Where F
XT
is obtained from equation B, g sin(θ ) is the gradient term and F
d
(V) is the force
due to drag, discussed in section 2.2. In order to evaluate equation B, we must apply equation A
and calculate the normal forces acting at each wheel; in order to do so we require knowledge of
the load distribution at each wheel.
2.1 Vehicle load distribution
The static load distribution is simply a function of vehicle geometry and the grade angle, and
is obtained by summing the forces at each contact point. There is, however, a dynamic load
distribution that can transfer load between the front and rear wheels as the vehicle accelerates and
brakes. Examining the geometry, the following two equations may be formulated to describe the
front and rear normal forces:
( )
( )
L
H
Vm
L
H
L
C
mgF
f
Z
.
sincos −
⎟
⎠
⎞
⎜
⎝
⎛
+= θθ
Equation 04/01/D
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( )
( )
L
H
Vm
L
H
L
B
mgF
r
Z
.
sincos +
⎟
⎠
⎞
⎜
⎝
⎛
−= θθ
Equation 04/01/E
The first terms on the right of these equations are the static load terms, whilst the second,
acceleration dependent terms, are the dynamic loading terms. In agreement with intuition, whilst
the vehicle is accelerating the load is transferred to the rear wheels, and during braking it is
transferred to the front wheels.
2.2 Drag forces
The equations of motion contain a drag term F
d
(V) that is both velocity dependent and acts to
limit the vehicle linear maximum speed. The drag term is a combination of both aerodynamic
resistance forces and ‘rolling’ resistance forces, which are both functions of the forward velocity
V [Gillespie 1992]:
)(),(
2
1
2
VmgCFVACF
rrda
== ρ
rad
FFF
+
=
Equation 04/01/F
Where C
r
is the rolling resistance coefficient, A is the frontal area of the vehicle and C
d
is the
aerodynamic drag coefficient. The prevailing wind speed may be added to the vehicle speed
before evaluation of the aerodynamic drag. The density of air, ρ, can be taken to be equal to 1.23
Kg/m
3
. Now that the gross forces acting on the vehicle body have been examined, the interaction
between the vehicle tyre and the road surface that produces the tractive forces must be
considered.
2.3 Tractive properties of the tyre/road interface
Evidence has shown that the effective coefficient of force transfer µ
i
is a consequence of the
difference between the forward velocity of the car V and the rolling speed of the tyre, ωR
i
.
Although many different parameters can and do affect the friction coupling, for a model such as
this it is best described in terms of the wheel slip ratio λ, which is defined below [Olsen et al.
2003]:
fri
RV
RV
ii
ii
i
,,
),max(
=
−
=
ω
ω
λ
Equation 04/01/G
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The maximum function in the denominator of the above equation allows its use for both
acceleration and braking models. A slip ratio equal to zero means that the forward velocity and
tyre rolling speed are equal, which implies an absence of either engine or brake torque. A
positive slip ratio implies that the tyre has a positive finite rolling velocity, and the car posses a
greater finite forward velocity. A negative slip ratio implies that the car has a finite forward
velocity and the tyre has a greater equivalent positive rolling velocity. At each extreme, i.e. +1
and –1, the wheel is either ‘locked’ at zero speed, or ‘spinning’ with the vehicle at zero speed.
When both tyre and car velocity are equal to zero, the slip is mathematically undefined, and is
taken to be zero for simulation purposes.
Experimental studies have produced several clearly defined friction/slip characteristics
between the tyre and road surface for a variety of different driving surfaces and conditions
[Stichin 1984]. For the purposes of simulation, four types of road condition are to be modelled:
• Normal: The road is dry and maximum traction is theoretically possible.
• Wet/Raining: Overall traction is reduced by about 20 %.
• Snow: Unpacked snow lies on the road surface. Maximum traction reduced by 65 %.
• Ice: Packed frozen snow and black ice lie on the road surface. Highly dangerous –
maximum traction reduced by 85 %.
Graphically, these four conditions are shown in figure 3 below:
Slip/Friction Graph
1
0.5
0
0.5
1
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
Slip Ratio
Tyre Friction (u)
Normal
Wet
Snow
Ice
Figure 3: Plot Of Friction/Slip Characteristics
For a given set of tyre test data, several analytical models exist to analyse and simulate these
relationships, the most popular of which is the Pacejka ‘magic’ model [Bakker et al. 1989]. The
Pacejka model is defined mathematically as follows:
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friBBEBCD
iiiii
,)))),arctan((arctan(sin()( =
−
−
=
λ
λ
λ
λ
µ
Equation 04/01/H
The model defines in excess of 40 constants that are determined from the given set of
experimental data, and the overall model coefficients B, C, D and E are then calculated from a
combination of these constants. The required model coefficients to produce the slip/friction
relationships as shown in the graph above were determined to be as shown in Figure 4:
Surface B C D E
Dry Tarmac 10 1.9 1 0.97
Wet Tarmac 12 2.3 0.82 1
Snow 5 2 0.3 1
Ice 4 2 0.1 1
Pacejka Coefficients
Figure 4: Equivalent Model Parameters
In general, the model produces a good approximation of the tyre/road friction interface, with
the only notable anomaly occurring when the road surface is covered in snow. Under these
conditions, it has been noted that the peak friction coefficient occurs when the slip is at either
extreme of the range (i.e. –1,1). This is due to the ‘snowplough’ effect and is not well captured
by this model [Olsen et al. 2003]. The computational complexity involved in using a more
accurate snow model is not required for the purposes of this simulation. In order to calculate the
slip, and thus determine the effective tractive forces, the equations of motion that allow
calculation of wheel velocity will be examined.
2.4 Wheel dynamic equations
In a similar manner to developing the gross equations of motion for the vehicle body,
summing the torques about each wheel enables the following equation for wheel acceleration to
be written:
fri
J
i
dbre
i
iiiii
,,
)(
.
=
−
−
+
=
ω
τ
τ
τ
τ
ω
Equation 04/01/I
Where τ
e
is the torque delivered by the engine to each wheel and τ
b
is the torque applied to
each wheel due to the brakes. τ
r
is the reaction torque on each wheel due to the tyre tractive
force:
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friFR
ii
Xir
,,
=
=
τ
Equation 04/01/J
If C
fi
is the viscous friction coefficient of the i’th wheel, the viscous friction torque τ
di(ωi)
can
be written as:
friC
iii
fid
,,
)(
=
=
ω
τ
ω
Equation 04/01/K
Solving these equations and integrating them wrt time allows a simulation of the gross
longitudinal motion of the vehicle and each of its wheels. In order to determine the torque
delivered to each wheel in equation I, by the engine and the braking system, it is necessary to
develop further model equations that describe the vehicle’s Powertrain and braking systems.
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3. Powertrain Model
The vehicle Powertrain includes the engine itself, the torque converter, gearbox and final
drive differential to deliver the torque to the front wheels. When the car is in motion, it is
assumed that the engine speed is equal to the wheel speed scaled by the current gear and final
axle drive transmission system. However, when the car is stationary or at very low speeds in 1
st
gear, it must be assumed that the torque converter is in operation and a finite amount of fluid slip
is occurring. Under these conditions the actual torque delivered is determined by a combination
of the converter slip, the throttle setting and wheel velocity. Since the simulation of the vehicle at
low speed is generally not required, it is assumed that the torque converter is locked at all times
and no fluid coupling takes place, and the engine RPM is saturated at some minimum value. In
addition, since we are assuming the vehicle is drivebywire, it is assumed that small servomotors
activate both the throttle and brakes. Figure 5 shows a schematic block diagram of the Powertrain
model.
Figure 5: Powertrain Block Diagram
Before considering the engine and servo dynamics, the amount of torque that is available for
a given engine RPM will be considered.
3.1 Engine torque curve
A typical modern, highend passenger vehicle such as that which is under simulation will
have a large capacity engine with a maximum RPM of around 6000. For example, a typical
MercedesBenz V8 engine is capable of delivering approximately 800 NM of torque at around
3700 RPM. A torque/RPM plot of a V8 engine such as this is shown in figure 6 below:
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Typical V8 Torque Data
0
100
200
300
400
500
600
700
800
900
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Engine Speed (RPM)
Engine Torque (Nm
)
Typical
Figure 6: Typical Engine Torque Data
The standard method of modelling a torque/RPM relationship such as this is to employ a
‘lookuptable’ of data values and use interpolation for intermediate values. However, in order to
provide for an effective, efficient simulation, a secondorder polynomial equation of the form
shown below was employed to capture the relationship:
2
cxbxay ++=
Equation 04/01/L
The use of a polynomial also has the advantage that a finite torque is available at low RPM
without the torque converter model. After applying curvilinear regression on the available data to
determine suitable coefficients, the following equation can be used to provide a good
approximation of the maximum available engine torque (T
Max
) at a given RPM (R) for a typical
V8 engine:
2
0000217.0152.07.528 RRT
Max
−+=
Equation 04/01/M
Higherorder polynomials may be used to produce an improved fit, yet as shown in figure 7
this is not explicitly necessary. If the current gear ratio is η
g
and the final drive ratio is η
f
, the
engine RPM R is determined as follows:
ffg
R
ω
η
η
=
Equation 04/01/N
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Typical V8 Torque Data
0
100
200
300
400
500
600
700
800
900
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Engine Speed (RPM)
Engine Torque (Nm
)
Typical
Model
Figure 7: Model vs. Typical Torque Data.
This torque/RPM characteristic defines the maximum torque that the engine can deliver at a
given RPM; however the actual torque developed at the crankshaft is a function of the chemical
energy delivered into the engine by the current throttle setting, and the dynamics of the engine
itself.
3.2 Engine dynamics
If we assume that the conversion of chemical energy into output torque by the engine may be
described by a first order time lag, and the throttle is actuated by a servo with an associated time
lag, these lags may be lumped together into a single equivalent lag τ
es
. Defining an energy
transfer coefficient µ
e
governing the actual amount of torque developed as a function of the
maximum, T
Max
, the following equations may be used to describe the wheel torque τ
ef
to the
throttle setting u
t
:
.
01.0
eeste
u µτµ −=
Equation 04/01/O
fgMaxee
T
f
η
η
µ
τ
=
Equation 04/01/P
Where η
g
and η
f
are the current gear ratio and final drive ratio respectively, and the input
throttle setting u
t
is constrained to a value between 0100 %. The final element of the Powertrain
that requires a description is the automatic gearbox, which decides the current gear and
consequently the value of η
g
.
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3.3 Gearbox model
In practice, many commercially available passenger vehicles now have automatic gearboxes
that are controlled by an electronic control unit (ECU). Since this element of the Powertrain is
not under consideration in the development of the embedded control system, it is assumed that
the automatic gearbox is explicitly modelled as part of the simulation. Many of the modern
gearbox ECU’s are connected to the vehicle communications network, and run sophisticated
adaptive algorithms involving many instrumented variables such as RPM, vehicle speed, driving
wheel speed and throttle setting. An automatic gearbox is best described in terms of a shiftmap,
that relates the threshold for changing each gear up or down as a function of throttle setting and
wheel speed [Gillespie 1992]. Figure 8 below shows the simplified shift map for the 5speed
gearbox that is to be used in this simulation:
Figure 8: Automatic Gearbox Shift Map
It can be seen in figure that there are two corner points in the shift profile of each gear – one
at 30% throttle and the other at 80%. This enables the engine to operate in the desirable region of
its torque curve at most times. In order to allow for better acceleration performance, if the throttle
is increased the engine RPM is allowed to increase in proportion before changing up a gear. In
addition, the shift map allows the gearbox to drop into a lower gear if sudden acceleration is
desired by sharply increasing the throttle. The following section describes the final element of the
model, the braking system.
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4. Brake System Model
The brake system model used in this simulation is based around a servovalve actuated
system. Although some systems under development allow for a purely servoactuated brake for
each wheel, a more common current implementation is to be used. In this system, an engine
mounted hydraulic pump generates a hydraulic pressure of 150 Bar, which is fed to one of four
servoactuated proportional valves to modulate the brake line pressure delivered to each wheel.
Since the model under development here is only a two wheel traction model, only two valves and
braking systems are under consideration: the front and rear. Figure 9 below shows a schematic of
this braking system:
Figure 9: Brake System Block Diagram
4.1 Servo actuated brake system
Although somewhat of a simplification, the dynamics of the servo valve and the hydraulic
systems can be modelled as simple lags in the time domain [Gerdes et al. 1993]. These two lags
can be incorporated into a single lag τ
bs
for the purposes of our simulation. If the front and rear
callipers are modelled as a simple pressure gain K
c
, then the pressure applied to the brake disk p
b
can be modelled as follows:
fripuKp
i
iii
b
bsbcb
,,5.1
.
=−= τ
Equation 04/01/Q
In this equation, the input brake setting u
b
is constrained to a value between 0100 %,
representing the amount of pressure to apply. The constant of 1.5 relates u
b
to the intermediate
pressure in the brake line (0150 Bar).
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4.2 Brake friction characteristic
This brake line pressure is then converted into a braking torque via a friction relationship that
varies with vehicle speed, temperature and several other parameters [Gerdes et al. 1993].
However, a quasilinear relationship may be assumed and a simplified friction characteristic is
utilised for this simulation:
friKp
i
i
bbb
iii
,),,1min( ==
α
ω
τ
Equation 04/01/R
Where K
b
is a pressure/torque conversion constant for each brake system. It can be seen that
as the wheel velocity ω approaches zero, the effective brake torque also approaches zero. Above
a wheel velocity of α, the steady state brake torque will be equal to p
b
multiplied by K
b
. In order
to simplify things further, in the simulation K
c
will be taken to be unity and its value lumped into
K
b
. In a typical passenger vehicle, the front brakes tend to have a larger capacity than the rear
brakes due to the effects of dynamic load transfer and suspension. Now that suitable equations
have been developed to describe each element of the system, some parameters that represent a
modern passenger vehicle will be suggested.
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5. Parameter Selection
This section shall suggest suitable values for the remaining model parameters to achieve
performance similar to a modern passenger vehicle. The values are not intended to explicitly
model a single make or model of car, but were determined as average values for this type of
vehicle. Parameters with the subscript i should be taken as values for both the front and rear
wheels.
Detailed information on specific makes and models of motor vehicles is freely available
either direct from a particular manufacturer, or via the World Wide Web.
s
s
radsNmC
C
C
mR
mKgJ
Kgm
mH
mC
mB
mL
bs
es
f
r
d
i
i
i
2.0
2.0
/1.0
01.0
29.0
3.0
/5.4
1626
6.0
5.1
5.1
0.3
1
2
=
=
=
=
=
=
=
=
=
=
=
=
−
τ
τ
01.0
/666.6
/33.13
1:83.0
1:1
1:41.1
1:19.2
1:56.3
1:82.2
5
4
3
2
1
=
=
=
=
=
=
=
=
=
i
b
b
f
BarNmK
BarNmK
r
f
α
η
η
η
η
η
η
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6. Simulation Results
This section presents some simple simulation results that were generated using the models
that have been outlined. Some simple measures of performance are gleaned from the simulation
data, in order to verify the simulation parameters give a realistic performance. The following
section describes the acceleration performance of the simulated vehicle.
6.1 Acceleration Performance
The acceleration performance of a passenger vehicle such as this is normally quoted as the
time taken, in seconds, to reach 60 MPH from a standing start. Figure 10 shows the simulation
car velocity in a similar test, with tarmac as the road surface. It can be seen that the 060 MPH
time is 6.8 seconds, which is comparable to figures quoted for this measure of performance in
manufacturers’ data. The top speed reached was 147.8 MPH, again which is a realistic figure for
a vehicle with a V8 engine.
Acceleration Performance
20
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (s)
Velocity (MPH)
Car Velocity
Figure 10: Acceleration Performance
In addition to observing the car velocity, a plot of each wheel velocity (front and rear) is
given in figure 11. It can be noted in this figure that the front wheels are rotating somewhat faster
than the rear: this indicates that there is slip present, creating the driving tractive force. The rear
wheels rotate at a value slightly less than the equivalent rotating circumference of the vehicle,
due to the presence of wheel friction.
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Acceleration Performance
50
0
50
100
150
200
250
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (s)
Velocity (rads/s)
Front Wheel
Velocity
Rear Wheel
Velocity
Figure 11: Wheel Velocity
6.2 Braking Performance
Another common measure of vehicle performance is the braking distance at various speeds.
Figure 12 shows the braking performance of the vehicle, when braked from 60 MPH to standstill.
It can be seen that the braking distance is 47.2m, which is again typical of figures quoted for this
measure of performance in manufacturers’ data
Braking Performance
0
10
20
30
40
50
60
70
0 12 22 31 38 43 46 47
Distance (M)
Velocity (MPH)
Car Velocity
Figure 12: Braking Performance
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7. Conclusions
This document has described the development of a suitable model of passenger vehicle
longitudinal dynamics. In order to create a suitable test environment in which the project aims
may be investigated, a realistic motorway simulation environment is required. This simulation
environment is described in detail in technical report ESL 04/02.
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References / Bibliography
Bakker, E., Pacejka, H. and Lidner, L. (1989) “A new tire model with application in vehicle
dynamic studies”, SAE Paper No. 890087, pp. 101113.
Gerdes, J.C., Maciuca, D.B. and Hedrick, J.K. (1993) "Brake System Modeling for IVHS
Longitudinal Control”, Advances in Robust and N Systems, DSCVol. 53, ASME Winter Annual
Meeting, New Orleans, LA.
Gillespie, T. (1992) “Fundamentals of Vehicle Dynamics”, Society of Automotive Engineers
(SAE), Inc.
Olsen, B., Shaw, S.W., and Stepan, G. (2003) “Nonlinear Dynamics of Vehicle Traction”,
Vehicle System Dynamics, Vol. 40, No.6, pp 377399.
Stichin, A. (1984) “Acquisition of transient tire force and moment data for dynamic vehicle
handling simulations”, Society of Automotive Engineers, Vol. 4, No. 831790, pp. 10981110.
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