Safety and Reliability of

Distributed Embedded Systems

Technical Report ESL 04-01

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 hardware-in-the-loop (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 Hardware-in-the-Loop Test Facility for Automotive ACC

Implementations”

Forthcoming:

ESL04/04 “Control Technologies For Automotive Drive-By-Wire Applications”

ESL04/05 “10-Node Distributed ACC System: Co-Operative Implementation”

ESL04/06 “10-Node Distributed ACC System: Pre-Emptive 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 Two-Wheel 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 hardware-in-the-loop 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 Two-Wheel 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 single-wheel versions, two-wheel versions, or full four-wheel

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 four-wheel 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 two-wheel 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 X-direction. 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 Two-Wheel Traction Model

1.2 System Block Diagram

The diagram below shows a high-level 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 non-linear. It can also be seen in the model that the vehicle is

front-wheel 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 real-time 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 X-axis, 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: Un-packed 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 co-efficient 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 drive-by-wire, 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, high-end 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

Mercedes-Benz 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

‘look-up-table’ of data values and use interpolation for intermediate values. However, in order to

provide for an effective, efficient simulation, a second-order 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

Higher-order 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 co-efficient 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 0-100 %. 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 shift-map,

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 5-speed

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 servo-valve actuated

system. Although some systems under development allow for a purely servo-actuated 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

servo-actuated 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 0-100 %,

representing the amount of pressure to apply. The constant of 1.5 relates u

b

to the intermediate

pressure in the brake line (0-150 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 quasi-linear 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 0-60 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. 101-113.

Gerdes, J.C., Maciuca, D.B. and Hedrick, J.K. (1993) "Brake System Modeling for IVHS

Longitudinal Control”, Advances in Robust and N Systems, DSC-Vol. 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 377-399.

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. 1098-1110.

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