autonomous vehicle control

Autonomous Vehicle ControlAn investigation into the application of Model Predictive Control

for active vehicle safety

Presented by:James Gillard

Prepared for:M. S. Tsoeu

Dept. of Electrical and Electronics EngineeringUniversity of Cape Town

Submitted to the Department of Electrical Engineering at the University of Cape Townin fulfilment of the academic requirements for a Master of Science degree in Control

Systems Engineering

November 4, 2015

Declaration

1. I know that plagiarism is wrong. Plagiarism is to use anothers work and pretend

that it is ones own.

2. I have used the IEEE convention for citation and referencing. Each contribution to,

and quotation in, this report from the work(s) of other people has been attributed,

and has been cited and referenced.

3. This report is my own work.

4. I have not allowed, and will not allow, anyone to copy my work with the intention

of passing it off as their own work or part thereof.

Signature:. . . . . . . . . . . . . . . . . . . . . . . . . . .

M. S. Tsoeu

Date:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgements

i

Abstract

ii

Contents

1 Introduction 1

1.1 Background to the study . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Overview of Vehicle Stability . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Purpose of the study . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Problems to be Investigated . . . . . . . . . . . . . . . . . . . . . 4

1.3 Scope and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Plan of development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature Review 6

2.1 Vehicle Safety Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Vehicle Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Path Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Scaled Vehicles as Testbeds . . . . . . . . . . . . . . . . . . . . . 12

3 Vehicles and Models 14

iii

3.1 Pneumatic Tyre Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Tyre Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Tyre Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Vehicle Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Results 24

5 Discussion 25

6 Conclusions 26

7 Recommendations 27

A Additional Files and Schematics 31

B Addenda 32

B.1 Ethics Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iv

List of Figures

1.1 ISO Standard 8855 vehicle coordinate system . . . . . . . . . . . . . . . 3

2.1 Timeline showing various safety measures and road accident fatalities . . 7

2.2 Diagram showing different catagories of Vehicle Safety Systems . . . . . . 8

3.1 Slip angle, Force and Moment positive directions. . . . . . . . . . . . . . 15

3.2 Bottom view of the tyre contact patch subjected to a lateral slip angle. . 16

3.3 Measured tyre data with Pacejka Magic Formula fit for a range of operating

Normal Forces. Adapted from http://www.optimumg.com . . . . . . . . 18

3.4 Left: View of driven and side-slipping tyre. Right: The tyre undeer

different slip conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Left: The tyre at pure side slip, from small to large slip angle. Right: The

resultig side force generated for each case presented. . . . . . . . . . . . . 20

3.6 Linear and Brush Tyre models versus lateral slip angle . . . . . . . . . 21

3.7 Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

v

List of Tables

vi

Chapter 1

Introduction

1.1 Background to the study

With over 89.7 million vehicles produced worldwide in 2014 [1] the automobile clearly

forms a foundation on which many aspects of our daily lives are built. Cars provide us

with true freedom of mobility as well as the ability to perform countless tasks that would

otherwise be impossible. However, the cost of this freedom is also evident. With over

30 thousand deaths relating to motor vehicle accidents in 2013, motor vehicle crashes

have become the leading cause of death among those under the age of 55 in the United

States [2]. Often these crashes are brought about through the actions of the driver or the

vehicles interaction with the environment. Excessive speed in combination with a loss

of traction due to changes in road surface friction are key components to many loss of

control type accidents. In these conditions the vehicle is operating at the very limits of

its dynamic capabilities, far from the typical conditions most drivers are used to. Without

the necessary skill or reaction speed the driver is often unable to avoid losing control,

resulting in potentially fatal crashes.

In recent years we have seen control systems introduced into almost every facet of our

everyday lives. Our desire to have higher rates of efficiency and reduced rates of failure

have resulted in a surge in the field. It should therefore come as no surprise that

automotive manufacturers have used these techniques in order to address the problems

of vehicle safety. These systems, including Anti-Lock Braking (ABS), Traction Control

(TCS), Electronic Stability Control (ESC), and Roll Stability Control (RSC) which utilize

control over the drive and brake systems of the vehicle to augment the drivers actions and

improve the handling response of the vehicle. These control systems have been shown to

1

1.2. BACKGROUND

greatly reduce the number of single vehicle crashes. In 2012 it is estimated that in the U.S.

electronic stability control (ESC) alone saved an estimated 446 lives among passenger car

(PC) occupants, and 698 lives among light truck and van (LTV) occupants, for a total

of 1,144 lives saved among passenger vehicle (PV) occupants [3].

Despite the efforts made by automotive manufacturers to ensure passenger safety accidents

still happen far too often and therefore the need for increasingly advanced control systems

is necessary. These systems need to go further in assisting the driver to not only avoid

potentially dangerous situations, but also help to maintain passenger safety even during

an extreme event. This is being done by researchers by using new technologies to provide

information about the environment, including road friction conditions, obstacles and

even pedestrians or other vehicles. The aim of this dissertation is therefore to attempt

to use low cost sensors in order to leverage the necessary measurements and determine

approximate state boundaries for the vehicle. This information can then be used to

implement control strategies which will enable the vehicle to maintain a level of dynamic

stability, even under extreme circumstances.

1.2 Background

1.2.1 Overview of Vehicle Stability

In order to develop the controllers that will be discussed in this dissertation it is first

necessary to define the basic dynamics that are relevant to maintaining control of a

vehicle. Figure 1.1 below shows a vehicle with a coordinate system as defined by the ISO

Standard 8855.

In the figure it can be seen that the positive x-axis is in the forward direction of travel,

the positive y-axis is to the drivers left and the positive z-axis is pointing up from the

ground. The angular rotations about the x,y and z-axes are termed as the vehicles

pitch, roll and yaw respectively and importantly a global coordinate system has also

been defined. Finally, the angle shown in the figure represent the steering angle of the

front wheels.

2

1.2. BACKGROUND

Figure 1.1: ISO Standard 8855 vehicle coordinate system

When discussing vehicle stability one of the most important quantities is the vehicles

sideslip angle. This angle is defined to be the difference between the vehicles heading

angle and its velocity vector and is a measure of how sideways the vehicle is travelling.

Its obvious that at normal highway speeds a sideslip angle of zero is typical, but it

would be rather distressing to the driver if it were 90. It is therefore logical to put some

constraint on the sideslip angle in order to maintain stability.

The vehicles yaw rate, which is the rate at which the vehicle rotates about the z-axis,

is also of particular interest to us when dealing with stability. If the yaw rate becomes

too large and is uncontrolled it can lead to the vehicle effectively spinning out, another

undesirable result.

Finally, the roll angle of the vehicle will be of particular interest in this paper. Many

production ESC systems are developed to control the first two quantities, side-slip and

yaw rate, but there are limited control systems which focus on minimizing the vehicles roll

angle at the same time. Vehicle rollover accidents are becoming more and more prevalent

with many people now driving sport utility vehicles (SUVs). The high centre of gravity

causes the vehicle to behave in a similar manner to an inverted pendulum, with its centre

of mass located well above the tire contact points on the ground. The inverted pendulum

is a well documented and researched classical control problem due to its high level of

instability and under-actuated architecture. It is therefore critical to maintain relatively

small roll angles in order to prevent the vehicle from reaching an uncontrollable level of

instability.

3

1.2. BACKGROUND

It is important to note that these quantities, the sideslip angle, yaw-rate and roll angle,

are all generated by tire forces as this is the only medium through which the vehicle

interacts with the external environment. It therefore becomes essential to have a good

understanding of the tire forces and their effects on the above quantities when the vehicle

is at the limits of its handling capabilities.

Other aspects that play a role in the vehicles handling properties are governed by the

very structure of the vehicle itself. Properties such as mass distribution, tyre structure

and suspension set up all form part of the cars handling. These factors will all be taken

into account by the dynamic modelling of the vehicle in coming chapters.

1.2.2 Purpose of the study

The purpose of this study is therefore to develop controllers capable of constraining the

yaw-rate, sideslip and roll angles to maintain dynamic safety. This work aims to allow

a vehicle to operate safely at a maximum possible speed while tracking a desired path.

Meeting these goals ensures that the control system will be capable of not only sensing a

possible emergency situation, but also reacting to it in order to maintain driver safety.

1.2.3 Problems to be Investigated

To fully explore this project the following are identified as key problems which will need

to be investigated in order to develop a working system:

Will the use of real time friction calculation enhance the control of an autonomousvehicle

Which control methods are best suited to maintaining vehicle safety

Can the controllers be designed to work in conjunction with a driver

Given an optimal line can vehicle safety be maintained at high speed

Can these control system be used to optimize the vehicles energy efficiency

Will the introduction of friction estimation improve controller performance

4

1.3. SCOPE AND LIMITATIONS

1.3 Scope and Limitations

A fully autonomous vehicle system would require the use of a number of different control

systems in order to function properly. These systems include, but are not limited

to, a vision system for obstacle detection and recognition, vehicle steering, braking

and cornering control systems, path generation algorithms and finally localization and

mapping algorithms.

The scope of this project, however, is confined to the control system which will be used

to drive a remote controlled (RC) model vehicle at high speed while following a given

trajectory or path. The only available inputs that will be used are those typically available

to the everyday driver, namely: steering, braking and throttle control. The vehicle used

will be modified to resemble the dynamics of a full scale passenger vehicle as closely as

possible, with the exception to its drive system. The proposed test bed is a 1/10th scale

RC car driven by a brushless DC motor. Therefore the project will focus mainly on the

steering, braking and cornering control techniques which allow the vehicle to drive at the

limits of friction, while maintaining dynamic safety. In order to test this the vehicle will

be made to maneuver at maximum speed while avoiding excessive vehicle roll, yaw and

sideslip.

If the project time and budget allows, a simple camera system will be developed in order

to provide the robot with a visual path input, allowing it to follow a line or stay within

the bounds of a track, but there will be no work done on the recognition of obstacles.

Finally in order to push the car to the limits of friction, it is also important to know

exactly how much friction is available. Therefore, if time allows, a friction estimation

system will be designed and integrated into the controller architecture.

1.4 Plan of development

5

Chapter 2

Literature Review

With vehicles playing such an important role in our daily lives it should come as no

surprise that there has been a tremendous amount of research done in the field of vehicle

safety and control systems, even though it is still fairly new. The following chapter will

provide the reader with a broad overview of the current research being done, while paying

special attention to the works which closely relate to this dissertation.

2.1 Vehicle Safety Systems

This section will provide the reader with an overview of some of the safety systems

already employed by vehicle manufactures and discuss some of the prominent production

technologies currently available. This section will address the strengths and weaknesses

of particular designs.

The time-line on the following page shows a number of vehicle safety systems which

have become standard in modern light motor vehicles (LMVs) in the U.S. and compares

this to the number of vehicle accident fatalities per year. As can be seen below since

the introduction of electronic stability control (ESC) there has been a steep decrease in

fatalities per annum.

6

2.1. VEHICLE SAFETY SYSTEMS

Figure 2.1: Timeline showing various safety measures and road accident fatalities

The decrease in fatalities with the increase in technology has sparked further interest

in vehicle safety systems, with the new aim of making passenger vehicles completely

autonomous. This means that the safety systems will not only have to localize themselves

in their environment, but also be able to operate in constantly changing situations and

react to emergency situations.

Figure 2.2 below shows the current trends in vehicle safety research as well as the

respective categories under which they fall. At the top level the systems are separated

into vehicle stabilization and path tracking. These two subsections form the basis on

which almost all vehicle safety systems are built, either on their own or as a combination

of sub-categories. The vehicle stabilization category shows systems which do not take

any path information into account, but rather rely on particular vehicle states as their

input. The second category is a fairly new type of vehicle control which takes into account

the vehicle states as well as a desired trajectory when computing the controller output.

While most of the path tracking systems are still in the research phase, manufacturers are

slowly integrating camera and/or LIDAR systems into their vehicles in order to leverage

the extra information needed to determine the vehicles position in relation to the rest of

its environment.

7


Figure 2.2: Diagram showing different catagories of Vehicle Safety Systems

2.1.1 Vehicle Stabilization

i) Yaw Stabilization

The first category in Fig.2.2 represents vehicle stabilization which does not take any path

information into account. Some of the most prominent and successful production systems

in this category are Electronic Stability Control or ESC, Anti-lock Braking System (ABS)

and Traction Control systems.

ESP helps to stabilize the vehicle yaw rate and slip angle by using differential braking,

meaning that the brakes are applied to individual wheels in order to mitigate any unwanted

yaw motion. A detailed explanation can be found in [4]. Anti-lock braking and traction

control systems also help the driver remain in control by preventing excessive wheel slip,

which allows for the maximum static friction to be used to steer or slow the vehicle instead

of allowing the tires to slide and begin using the lower kinetic friction.

Besides these systems there are a number of other stabilization techniques which have

been thoroughly researched, however, the sensors and actuators required are not yet

available in production vehicles. One of the most commonly researched topic is that of

an active suspension system. This strategy uses actuators in the vehicles chassis to apply

a force to individual wheel suspension, which leads to better road adhesion or mitigates

vehicle pitching or rolling. The results have shown great promise by limiting the vehicle

roll angle in high C.G vehicles as well as maintaining a smoother travel over uneven

surfaces [5, 6, 7].

Other systems, which are still in the research stages of their development, include more

global stabilization techniques such as the envelope control proposed by J. Gerdes et

al. [8]. Their proposed strategy uses a sliding surface technique in order to maintain

a defined envelope of safe operating conditions. Their work has shown that a control

system can be used in conjunction with a driver in order to maintain vehicle safety, where

the control system only intervenes if it has determined that the driver command results

8


in the vehicle leaving a safe operating region.

ii) Rollover Prevention

As mentioned in the introduction there are extremely high numbers of fatalities on the

roads each year due to vehicle accidents. With this dissertations focus on vehicle safety

it should be noted that an important aspect of this is rollover prevention. Of the nearly

32,000 fatalities on U.S. roads in 2004 the National Highway Traffic Safety Administration

(NHTSA) estimated that nearly a third of all road fatalities involve vehicle rollover while

accounting for only 2.4% of all highway accidents[9].

During cornering, the vehicles roll moment causes a lateral load transfer from the inner

wheels to the outer wheels. This load transfer strongly influences the lateral dynamics

of the vehicle. Due to the nonlinear properties of pneumatic tyres, the total force

capability of the front or rear axle decreases as a result of this load transfer. To overcome

this problem some researchers have employed an active roll control system in order to

reduce the total load transfer during cornering. An active roll control system enables the

modulation of the normal force experienced at each corner of the vehicle body and hence

has the ability to limit the roll motion of the vehicle chassis.

One common method of implementing this active roll control is achieved through the use

of active suspension. Active suspension was initially introduced as a means to provide

a solution between the conflict of vehicle ride comfort and handling. Although active

suspension systems have been around for a few years, although not in production vehicles,

most of the research is based on passenger ride comfort. Safarudin et al. [10] showed how

an active suspension system can be used to mitigate vehicle rollover.

The methods investigated in this dissertation, however, will use methods more similar to

those used by [11] where an active steering controller was designed in order improve lateral

stability and manoeuvrability. The same approach will be taken in this dissertation where

only steering and throttle/braking commands will be used to mitigate rollover. By using

this approach it means that this rollover prevention system can be easily implemented

on any production vehicle without the need for any modification.

9


2.1.2 Path Tracking

Recent developments and substantial reduction in the cost of positioning and sensing

technologies such as differential global positioning systems (DGPS), Light Detection and

Ranging (LIDAR) and even stereo camera vision systems have opened up a number of

opportunities to gather more information about a vehicles surrounding environment. This

information can then be used to generate path information which can be used to either

simply stay on a given path or generate an optimal line on which to travel. This section

will briefly discuss some aspects of path planning, but the main focus will fall on the path

tracking problem.

i) Path Planning

When generating an optimal path there have been two approaches made by researchers:

a) the path is generated oine using a priori knowledge of the desired route

or

b) the path is generated online using current information to determine the optimal path

between two points.

Recursive optimisation techniques have shown that a path generated oine can even

mimic the path taken by professional race drivers by using a function which optimizes

the length of the path travelled versus the speed at which the vehicle can maintain that

path. This method nicely separates the objectives of path planning and path tracking,

which allows for a focus on one or the other. The problem with this approach, however,

is that its hardly a realistic approach to the problem.

Online path generation on the other hand only has access to current information and needs

to be able to execute quickly for the vehicle to maintain high speed. This technique is

commonly used for object avoidance where there is no way of knowing when an obstacle

may appear and is a far more realistic, although more difficult, approach. The combined

path planning and tracking approach requires that the vehicle control inputs and path

information are combined in a single optimization problem. A good example of this was

shown by Gerdes et al. [12] where a unified controller was used to simulate a race drivers

approach to selecting a line, tracking the desired path and maintaining vehicle control.

This race driver approach is a common theme when it comes to controlling vehicles at

high speed because race drivers are a perfect example of how one can control a vehicle

at the very limits of its capabilities.

10


ii) Path Tracking

When it comes to path tracking there are several methods which have successfully been

employed. Until fairly recently, however, these techniques have required a relatively

slow and constant vehicle speed which severely limited their usefulness. Again, with

recent improvements in processing power and available technologies there has been much

improvement in the speeds at which these controllers can be used.

At low speeds kinematic or linear dynamics are often used to model the vehicle due

to the lack of dynamic excitement in the vehicle itself. This approach has been shown

to be fairly effective in the classical nonholonomic vehicle control problem where speeds

dont venture much higher than 0.5m/s, but as speeds increase a more thorough dynamic

model is needed often containing a nonlinear tyre model. This is because the dynamic

characteristics of a car vary strongly with speed, especially when the performance limits

are extended by aerodynamically generated downforce, increasing the frictional coupling

between the tyres and the ground. Within an operating range around the straight

running state, a linear representation of a (good) car can be expected to be accurate. As

manoeuvring severity increases, tyre shear forces saturate in a smooth and progressive

manner. Optimal controls derived assuming linearity can be expected to work well for

gentle manoeuvring and not so well for limit operation therefore.[13]

Although a nonlinear tyre model would more accurately describe the vehicle dynamics

there have been a number of implementations using a linearised tyre model for situations

where the slip is kept fairly low. In 2003 Rosetter [14] developed a potential field

lanekeeping system which actively assisted the driver to stay in the centre of the lane.

In his work he used a linear approximation for the forces generated by the tires thereby

simplifying the overall vehicle model. Because the goal in his paper was not to have a fully

autonomous system, where the primary goal is accurate and high performance tracking,

but rather a driver assistance system it follows that a simplified model is sufficient for his

design. This control scheme was shown to work well in conjunction with a driver providing

good local stability, but is not sufficient for a fully autonomous implementation.

A more relevant example of path tracking using linearised models can be seen in the works

done by Thommyppillai et al. [15] and Beal [16]. Both of these authors used a Nonlinear

Model Predictive Control (NMPC) approach to path tracking. This method proves to

be very accurate even when performing high slip manoeuvres due to the structure of the

Model Predictive Control, or MPC, framework. MPC has been shown in a number of

cases to be a considerably robust method of control which, in this case, caters for the

neglected nonlinearity in the tyre modelling. It is for this reason, among others, that

11


model predictive control has been chosen as the control strategy for this dissertation.

As mentioned above the control schemes used for path tracking can often be improved

through the use of a nonlinear tyre model. Because the tyre is the only contact point

between the vehicle and the environment it is the only source of force generation. Falcone

et al. [17] include the tire nonlinearity in the formulation of their MPC controller,

however, their work focusses mainly on the linear operating region of the tyre. The

controller is tested using the standard double-lane change maneuver and proves to work

well in simulation, but its overall complexity means that computation time becomes a

problem for real-time implementation.

2.1.3 Scaled Vehicles as Testbeds

In this section we will briefly discuss using the proposed scaled vehicle, a 1/10th scale radio

controlled (RC) car, as a test bed and how it relates to full size vehicles. As mentioned

above an important part of vehicle safety is rollover prevention, however, the rollover

testing of full scale vehicles is not only an expensive endeavour, but a fairly dangerous

one too. Because this project is aimed at real vehicle safety systems it is important to

convince ourselves that the test bed closely relates to a full scale vehicle, thereby making

the controller architecture fully transferable to a larger platform. Therefore if results

from scaled vehicles tested in a controlled environment can be shown to closely relate to

the dynamic behaviour of full size vehicles, then such an approach can be an effective

means of investigating vehicle rollover.

Scaled vehicles have proven to be reliable test beds for a number of different applications

[18, 19, 20]. The main advantages of using a scaled vehicle (like the Radio Controlled

car used for this dissertation) are vehicle costs, ease of modifications and the relative

size of the testing facility required to perform various standard manoeuvres. Since the

testing area can be reduced in size it is also easier to maintain more accurate control over

the testing conditions such as road surface and obstacle placement etc. Finally a scaled

vehicle is also more easily pushed to its dynamic limits allowing researchers to observe

what happens when the vehicle reaches the nonlinear region of operation.

In the works of Lapapong et al. [21] an RC car chassis was used on a scaled rolling road

to test the validity of their dynamic model of the vehicle. Using a simple bicycle model,

which will be discussed in greater detail later in this report, they were able to closely

mimic the dynamic responses of a full scale vehicle. By comparing the frequency responses

12


of both the scaled vehicle and a full scale test vehicle they found that at low frequency

the scaled vehicle closely matched the full-sized car, but concluded that due to tyre-lag

effects there was a noticeable error in their responses. By including a better model of the

scaled tyres, however, Andrew Liburdi [22] found that he was able to accurately replicate

simulation results of double-lane change and sine sweep manoeuvres which also used a

bicycle model.

The works done in testing and validating scaled vehicles for use in dynamic vehicle control

therefore shows great promise and thus allows this dissertation the freedom to use such

a vehicle provided it is accurately modelled.

13

Chapter 3

Vehicles and Models

The estimation and control systems which will later be presented in this paper take

advantage of several different models of the dynamics of the vehicle. While it is possible

to derive a model containing all the degrees of freedom as described in [23] by Shim

et al., the models presented here represent only the dynamics which are critical to the

tasks laid out in the scope and limitations section of this report. These models, while

still accurate, reduce the overall complexity and allow for a more transparent model for

analytical use. This reduction in complexity will also help to reduce the computational

burden on the processors which will be used to implement the controllers in real time.

The models presented here can be adapted to a range of vehicles, but for the work in this

dissertation the parameters used will be those of the RC test platform. These parameters

can be found in the appendix of this report.

3.1 Pneumatic Tyre Modelling

Tyre characteristics are of crucial importance for the dynamic behaviour of the road

vehicle [24]. Since the tyres are the only contact point between the vehicle and the outside

environment it is of critical importance to have a clear understanding of their operation

and the forces they generate. The problem, however, is that modern automobile tyres are

extremely complicated products to say the least; they are a composite of various rubber

compounds, steel and often kevlar and the designs have been refined through decades of

practical development. This section will begin by defining some of the terminology used

when dealing with tyre dynamics before presenting some of the tyre models which will

be used in this dissertation.

14

3.1. PNEUMATIC TYRE MODELLING

3.1.1 Tyre Quantities

This section will briefly outline some of the important quantities and definitions with

regards to tyre modelling. The definitions used here will follow those set out by his book:

Tyre and Vehicle Dynamics [24] which was written by Hans B. Pacejka, who is regarded

as the foremost authority on tyre modelling.

i) Pure Longitudinal Slip

The upright wheel rolling freely, that is without applying a driving torque, over a flat

level surface along a straight line (ie. with no side slip) may be defined as the starting

situation with all slip equal to zero. When a driving torque is applied, however, the

condition of zero slip is no longer fulfilled and a build-up of additional tyre deformation

and possibly partial sliding at the contact patch may occur. As a result horizontal forces

are developed. This slip quantity will serve as an input into the tyre system with the

resulting output being the forces developed.

Figure 3.1: Slip angle, Force and Moment positive directions.

Figure 3.1 shows a freely rolling wheel, with some forward speed Vx (the longitudinal

component of the total velocity vector V of the wheel centre) and angular speed of

rotation can be taken from measurements. By dividing these two quantities, the so-

called effective rolling radius re is defined:

re =Vx

(3.1)

When a torque is then applied to the wheel spin axis, a longitudinal slip arises and is

defined as follows:

= Vx reVx

(3.2)

15


The sign of is taken such that for a positive slip value, a positive longitudinal force

Fx arises, that is, a driving force. It can noted that at wheel lock up (ie. when = 0)

= 1, but for very large values of in comparison to Vx, can become very large. Inorder to limit this we rather define as follows:

=

VxreVx if Vx rereVxre

if Vx < re(3.3)

This approach is not taken by Pacejka in his definition, however, for computational

purposes it becomes necessary for this further definition.

ii) Pure Lateral Slip

Lateral slip, similarly to the longitudinal slip above, is defined by the ratio of lateral

and forward components of the velocity vector V . As in Pacejka, this corresponds to the

tangent of the slip angel . Again the sign convention results in a positive longitudinal

force generated by a positive slip angle.

tan = VyVx

(3.4)

Figure 3.2 below shows the distortion of the tyre structure when subjected to a lateral

slip angle. This deformation is what leads to the lateral force generation which is then

used to steer the vehicle.

Figure 3.2: Bottom view of the tyre contact patch subjected to a lateral slip angle.

This lateral slip, , combined with the longitudinal slip, , and vertical load, Fz, (which

16


may be considered a given quantity that results from the normal deflection of the tire)

are all that are needed as inputs to our tyre models. These models will be described in

greater detail in the following section, however, for now we will define the output forces

as follows:

Fx = Fx(, , Fz) Fy = Fy(, , Fz) (3.5)

iii) Combined Slip

The notion of combined slip comes about due to the limited static frictional force available

to the tyre at any given time. ETC ETC

3.1.2 Tyre Modelling

There are a large number of factors which influence the way in which tyre force is

generated. Tread patterns specially designed to remove water from between the tyre

and the road, the rubber itself may be softer for more grip or harder for longevity, the

inflation pressure, the tyre wear and even the asymmetry of the tyre are all factors which

may change the magnitude or distribution of the force generated. Whilst all of these

factors play a role in vehicle handling it would be prohibitively complex to include all

of them. Therefore the models used in this dissertation seek to capture only the vital

characteristics of tyre-force generation, while maintaining a sufficient level of accuracy.

i) Pacejka Magic Formula Tyre Model

The Pacejka Magic Formula tyre model, named after its creator Hans B. Pacejka, is

a semi-empirical tyre model which uses a combination of data curve-fitting as well as

physical tyre properties in order to approximate the forces generated by the tyre. Pacejka

has developed a series of models over the last 25 years, the first of which was presented

in [25] where it was shown that the proposed formula was not only extremely accurate in

describing the measured data, but also characterized some of the typifying quantities such

as slip stiffness and peak values which permitted the calculation of forces and torques in

conditions which deviate from those imposed during the actual measurements. Since its

inception in 1987 it has become the most widely used tyre model and can be found in

most vehicle modelling software.

17


The formula was termed magic because there is no particular physical basis for the

structure of the equations chosen, yet they fit a wide variety of constructions and operating

conditions as seen in [25]. Using the formula each tyre is characterised by 10-20 coefficients

for each important force it can produce at the ground contact patch, typically lateral and

longitudinal force, and self-aligning torque, as a best fit between experimental data and

the model. These coefficients are then used to generate equations showing how much

force is generated for a given vertical load on the tire, camber angle and slip angle [24].

The general form of the equation as found in [24] is as follows:

Fy = D sin[C arctanB E(B arctan(B))]With stiffness factor:

B =CF

CD

peak factor:

D = Fz

and cornering stiffness:

CF = BCD

(3.6)

The shape factors C and E as well as the friction coefficient may be estimated or

determined through regression techniques.

The following figure shows the approximate force generation vs measured tyre data:

Figure 3.3: Measured tyre data with Pacejka Magic Formula fit for a range of operatingNormal Forces. Adapted from http://www.optimumg.com

18


Although the Magic Formula is the most accurate model of vehicle tyres it is computationally

intensive and highly nonlinear. It will therefore not be used in the controller design for

this project, but rather as a tool in simulation to verify that the proposed controllers

perform as intended. The required test data needed for this empirical model will be

taken from [26] where scaled pneumatic tyres were tested and compared to their full scale

counterparts. In his paper Polley found that the force producing characteristics of scaled

DuBro pneumatic tyres exhibited similitude with full-scale tyres. He then went on to

solve for the various constants to be used in the Magic Formula; these constants will

therefore be used in this dissertation.

ii) Brush Tyre Model

The brush tyre model, see [24], describes the structure of a tyre using a row of elastic

bristles which touch the road plane and can deflect in a direction parallel to the road

surface. The assumption is that the slip (both longitudinal and lateral) relies on the

compliance of bristles or tread elements of the tyre which represents the elasticity of

the real tyre carcass and tread. As can be seen in the figure below, when the wheel speed

vector V shows an angle with respect to the wheel plane, side slip occurs. When the

wheel velocity of revolution multiplied with the effective rolling radius re is not equal

to the forward component of the wheel speed Vx = V cos , we have fore-and-aft slip.

Under these conditions slip occurs and the corresponding forces are developed.

Figure 3.4: Left: View of driven and side-slipping tyre. Right: The tyre undeer differentslip conditions.

As depicted in Fig. 3.4 the tread elements move from the leading edge (right hand

19


side of figure) to the trailing edge. The tip of each element will remain adhered to the

ground as long as static friction allows, that is, it will not slide across the road surface.

Simultaneously the base point, which is fixed to the tyre carcass, moves backward with

the linear speed of rolling (re) with respect to the wheel axis located at the contact

patch centre labelled C in the pictures.

The resulting deflection of the tyre bristles varies linearly with distance from the leading

edge and the tips form a straight contact line which is parallel to the wheels velocity

vector V . The maximum force generated bby this tyre model is dependent on 3 variables

namely: The constant coefficient of friction , the vertical (normal) force distribution Fz

and the stiffness of the tread elements cpy. The pressure distribution, and consequently

the maximum deflection, are assumed to have a parabolic distribution and therefore as

soon as the straight contact line intersects this parabolic distribution sliding will start.

This intersection can more easily be seen in figure 3.5. The remaining part of the contact

Figure 3.5: Left: The tyre at pure side slip, from small to large slip angle. Right: Theresultig side force generated for each case presented.

line will coincide with the parabola for the maximum possible deflection. At increasing

slip angle, the side force that is generated will increase. The distance of its line of

action behind the contact centre is termed the pneumatic trail t. As the slip increases,

the deformation shape becomes more symmetric and, as a result, the trail gets smaller.

This is because the point of intersection moves forward, thereby increasing the sliding

range and decreasing the range of adhesion. This continues until the wheel speed vector

runs parallel to the tangent to the parabola at the foremost point. Then, the point of

intersection has reached the leading edge and full sliding starts to occur. The shape has

now become fully symmetric and the side force attains its maximum.

20

3.2. VEHICLE MODELLING

iii) Linear Tyre Model

As can be seen in figure 3.5 and 3.3 there is a range of values for slip over which tyre force

is generated fairly linearly. It can therefore be assumed that for small values of slip the

force can be approximated by a constant C or C multiplied by the respective slip value.

Despite the complex design of tyres and the influence stick-slip friction, their behaviour

at low levels of lateral force is dominated by the elastic nature of the rubber. This model

will therefore yield good results up to approximately half the maximum force of the tyre

as shown in the figure below:

Figure 3.6: Linear and Brush Tyre models versus lateral slip angle

This model will be used in the design of the MPC controller as the linear approximation

of the force can easily be included in the state-space representation of the vehicle model.

3.2 Vehicle Modelling

In this section we will define the two vehicle models which have been used in this

dissertation for controller design. The first is a linear bicycle model which has been used

extensively in vehicle handling studies and the second is a two track roll model which

reduces the vehicle roll dynamics to a simple inverted pendulum. While these models do

not capture all of the dynamics of a real vehicle, they do provide us with enough detail

to design the required controllers as well as maintaining a minimum complexity.

21


i) Bicycle Model

The bicycle model shown in fig. 3.7 is a simplified chassis model for a four wheel vehicle.

Since the model only focuses on planar dynamics of the vehicle the four wheels are

combined into just a single track, thereby neglecting roll dynamics. This model is a

three state model only and is used to describe the rotational (yaw) motion as well as the

longitudinal and lateral velocities of the vehicle.

Beyond neglecting the roll dynamics, there are a few other key assumptions made. The

mass of the vehicle is considered to be located entirely in the rigid base of the vehicle

chassis. Furthermore, the slip and steering angles for the left and right wheels are

considered to be the same; this assumption is made based on the vehicle travelling at

typical driving speeds and traversing corners of moderate radius. The forces can therefore

be lumped together as a single equivalent wheel as shown in the figure. For the purposes

Figure 3.7: Bicycle Model

of friction estimation and lateral force generation a constant longitudinal velocity is often

used [16], however, we have seen in the previous section that the longitudinal tyre forces

play a large role in how the lateral forces are generated and therefore unlike [16] we will

include these effects. The equations of motion of the bicycle model are derived from first

principles by writing force balance equations in the vehicle x and y - coordinate frames

as well as a moment balance equation about the z - axis.

max = Fxf + Fxr, may = Fyf + Fyr, Izz = aFyf + bFyr (3.7)

22


The vehicle sideslip angle is calculated as

tan1UyUx

(3.8)

By taking into account the vehicle steering angle we can rewrite the acceleration and

moment equations which define the vehicle states

ax + Vy =1

m[Fxr Fyf sin ] (3.9)

ay + Vx =1

m[Fyf cos + Fyr] (3.10)

=1

Izz[aFyf cos bFyr] (3.11)

where the parameters m , Izz and are the mass of the vehicle, the yaw moment of inertia

and the steering angle of the front wheels respectively. The forces Fxr, Fyr and Fyf are

the forces generated by the tyres at the front and rear axles and are highly nonlinear

at elevated levels of lateral acceleration. Therefore either a nonlinear brush model or

Pacejka model may be used to describe the force generation characteristics. However, for

simplicity of design and to keep this bicycle model linear, we will consider the case where

the slip angles remain fairly small and we can therefore use a linear model to describe

the force generation. In either case we require the front and rear slip angles of the vehicle

tyres. These are given as follows:

f = tan1 Vy + a

Vx r = tan1 Vy b

Vxr =

r VxVx

(3.12)

It can be noted at this point that since the vehicle is rear wheel drive only there is no

effect longitudinal forces developed at the front of the vehicle. The steering input, , is

also included in these expressions

ii) Two-Track Yaw-Roll Model

23

Chapter 4

Results

24

Chapter 5

Discussion

25

Chapter 6

Conclusions

26

Chapter 7

Recommendations

27

Bibliography

[1] Statista. Motor vehicle production of the United States and worldwide from 1999 to

2014, 2015.

[2] Statista. Number of road traffic fatalities in the United States 2009 to 2013, 2015.

[3] Nhtsa. Estimating Lives Saved by Electronic Stability Control, 20082012, 2014.

[4] Rajesh Rajamani. Electronic stability control. In Vehicle Dynamics and Control,

pages 201240. Springer, 2012.

[5] Haiping Du, Kam Yim Sze, and James Lam. Semi-active H control of vehicle

suspension with magneto-rheological dampers, 2005.

[6] a Trachtler. Integrated vehicle dynamics control using active brake, steering and

suspension systems, 2004.

[7] Bart L J Gysen, Johannes J H Paulides, Jeroen L G Janssen, and Elena Lomonova.

Active electromagnetic suspension system for improved vehicle dynamics. Vehicular

Technology, IEEE Transactions on, 59(3):11561163, 2010.

[8] Stephen Erlien, Susumu Fujita, and J Christian Gerdes. Safe driving envelopes for

shared control of ground vehicles. In Advances in Automotive Control, volume 7,

pages 831836, 2013.

[9] Alexander Strashny. An Analysis of Motor Vehicle Rollover Crashes and Injury

Outcomes, 2007.

[10] Mochamad Safarudin, Amrik Singh, and Haery Sihombing. Rollover Prevention

System for Passenger Vehicles, 2010.

[11] S Solmaz, M Corless, and R Shorten. A methodology for the design of robust rollover

prevention controllers for automotive vehicles with active steering. International

Journal of Control, 80(11):17631779, November 2007.

28

BIBLIOGRAPHY

[12] Kirstin L R Talvala, Krisada Kritayakirana, and J Christian Gerdes. Pushing

the limits: From lanekeeping to autonomous racing. Annual Reviews in Control,

35(1):137148, April 2011.

[13] M Thommyppillai, S Evangelou, and R S Sharp. Advances in the development of a

virtual car driver. Multibody System Dynamics, 22(3):245267, 2009.

[14] Eric J. Rossetter. A potential field framework for active vehicle lanekeeping

assistance, 2003.

[15] M Thommyppillai, S Evangelou, and R S Sharp. Car driving at the limit by

adaptive linear optimal preview control. Vehicle System Dynamics, 47(12):1535

1550, December 2009.

[16] Craig E. Beal. Applications of Model Predictive Control to Vehicle Dynamics for

Active Safety and Stability, 2011.

[17] Paolo Falcone, H Eric Tseng, Francesco Borrelli, Jahan Asgari, and Davor Hrovat.

MPC-based yaw and lateral stabilisation via active front steering and braking.

Vehicle System Dynamics, 46(sup1):611628, September 2008.

[18] Matthew Polley, Andrew Alleyne, and Edwin De Vries. Scaled vehicle tire

characteristics: dimensionless analysis. Vehicle System Dynamics, 44(2):87105,

February 2006.

[19] Alexander Liniger, Alexander Domahidi, and Manfred Morari. Optimization-based

autonomous racing of 1:43 scale RC cars. Optimal Control Applications and Methods,

pages n/an/a, 2014.

[20] W E Travis, R J Whitehead, D M Bevly, and G T Flowers. Using scaled vehicles to

investigate the influence of various properties on rollover propensity, 2004.

[21] S Lapapong, V Gupta, E Callejas, and S Brennan. Fidelity of using scaled vehicles

for chassis dynamic studies. Vehicle System Dynamics, 47(11):14011437, November

2009.

[22] Andrew Liburdi. Development of a Scale Vehicle Dynamics Test Bed, 2010.

[23] Taehyun Shim and Chinar Ghike. Understanding the limitations of different vehicle

models for roll dynamics studies. Vehicle System Dynamics, 45(3):191216, March

2007.

[24] Hans B. Pacejka. Tire and Vehicle Dynamics, 2012.

29

BIBLIOGRAPHY

[25] E Bakker, L Nyborg, and H B Pacejka. Tyre Modelling for Use in Vehicle Dynamics

Studies, 1987.

[26] Matthew Polley and Ag Alleyne. Dimensionless analysis of tire characteristics for

vehicle dynamics studies, 2004.

30

Appendix A

Additional Files and Schematics

31

Appendix B

Addenda

B.1 Ethics Forms

32

IntroductionBackground to the studyBackgroundOverview of Vehicle StabilityPurpose of the studyProblems to be Investigated

Scope and LimitationsPlan of development

Literature ReviewVehicle Safety SystemsVehicle StabilizationPath TrackingScaled Vehicles as Testbeds

Vehicles and ModelsPneumatic Tyre ModellingTyre QuantitiesTyre Modelling

Vehicle Modelling

ResultsDiscussionConclusionsRecommendationsAdditional Files and SchematicsAddendaEthics Forms

autonomous vehicle control

Documents