Document [original]

A Predictive Potential Field Concept for

Shared Vehicle Guidance

zur Erlangung des akademischen Grades eines

DOKTORS DER INGENIEURWISSENSCHAFTEN (Dr.-Ing.)

der Fakultät für Maschinenbau

der Universität Paderborn

genehmigte

DISSERTATION

von

Dipl.-Ing. Thorsten Brandt, M. S.

aus Euskirchen

Tag des Kolloquiums: 12.11.2007

Referent: Prof. Dr.-Ing. habil. Jörg Wallaschek

Korreferent: Prof. Dr. rer. nat. habil. Michael Dellnitz

Acknowledgements

This thesis owes its existence to the influence and support of many institutions and

individuals. First of all, I would like to express my sincere thanks to my advisors and com-

mittee members Prof. J¨

org Wallaschek, Prof. Michael Dellnitz, Prof. Ansgar Tr¨

achtler,

and Prof. Rolf Mahnken.

In particular, I would like to acknowledge Prof. Wallaschek. During my time as a

scholar of the International North Rhine-Westphalia Graduate School of Dynamic Intel-

ligent Systems he was very active in continuously providing me with excellent research

conditions. As a result both his research group at the Heinz Nixdorf Institute (HNI) as

well as the L-LAB, a public private partnership between the University of Paderborn and

the Hella KGaA Hueck & Co., were very enjoyable and stimulating places.

Therefore, I would like to say thanks to all colleagues and students within the HNI and

the L-LAB. Among many, special thanks go to Silke Hanisch, diploma student of the first

hour, and to Tobias Hesse, Michael Kluge, Christian L¨

oper, and Christoph Sondermann-

W¨

olke, whose contributions had a great impact on the synthesis of this thesis. A key role

was also played by Jun.-Prof. Thomas Sattel: together we started to discover the fields

of driver assistance and many ideas evolved during some intensive discussions, which I

appreciate very much.

Special thanks go to the ”human-machine group” at the L-LAB. The interdisciplinary

ensemble of engineers and psychologists often led to vivid discussions, which made work

real fun. Among these, I am particularly indebted to Michael B¨

ohm for his help in

the experimental design of the simulator study. Also, I would like to thank Dr. Frank

Flemisch from the DLR in Braunschweig. With his visions, such as the h-metaphor, he

had a significant influence on this thesis.

A personal highlight for me was the opportunity to spend 3 months as a visiting re-

searcher at the group of Prof. J. Christian Gerdes at the Dynamic Design Lab at Stanford

University. Therefore I thank him and his entire group - particularly Joshua Switkes. This

was made possible by the financial support of the Hans-Lenze-Stiftung and the great en-

gagement of the graduate school team headed by Dr. Eckhard Steffen as well as the

administration team at the HNI and the L-LAB, who did a great job. Most notably were

Kerstin Hille and Marina Kass¨

uhlke, who always kept the administrative burdens as low

as possible.

My sincere thanks also go to Christoph Sondermann-W¨

olke, Christian L¨

oper, Sabine

Raphael, and Karina Hirsch for their great engagement in proofreading this document.

Last but not least a special mention goes to my family. Thanks - in particular to my

favorite women Andrea and Emma - for all their support and their understanding, which

made it possible to complete this thesis in the first place.

Abstract

Modern driver assistance systems are gradually engaging in vehicle guidance. Examples

are systems for lane-keeping and for lane departure warning. For future assistance sys-

tems, e.g. for lane-change or evasion maneuvers, collision-free motion planning has to

be addressed. This work combines motion planning and trajectory tracking into a uni-

fied framework of potential field methods for shared vehicle guidance between driver and

assistance system. Therein, a so-called elastic band, originally introduced in robotics,

acts like a virtual antenna, similar to the antennae of an insect, sensing trajectories of

low hazards in the environment. Specifically for the automotive application, the mo-

tions of other traffic participants are also anticipated by extrapolation methods. The

tracking algorithms consist of curvature based feedforward control in combination with

a potential field based guidance controller, which is shown to be stable in the sense of

Lyapunov. Besides that, a Lyapunov function provides bounds for the tracking error de-

pending on the initial conditions and the parameters of the controller. The steering angle,

proposed by the guidance system, is communicated to the driver via an assistance-torque

at the steering wheel. In parallel, the driver’s steering intension is incorporated to shape

the planned trajectories. This interactive guidance concept is experimentally tested in a

driving simulator. Therein, different configurations of the assistance torques are analyzed.

Kurzfassung

Moderne Fahrerassistenzsysteme unterst¨

utzen den Fahrer bereits heute in der Fahrzeug-

f¨

uhrung. Beispiele hierf¨

ur sind Systeme zur Spurhalteassistenz und zur Spurverlassenswar-

nung. Zuk¨

unftige Systeme wie Spurwechsel- oder Ausweichassistent machen Methoden zur

kollisionsfreien Bewegungsplanung im Verkehrsraum erforderlich. Die vorliegende Arbeit

stellt einen durchg¨

angigen Ansatz zur Bewegungsplanung und Bahnfolgeregelung auf Ba-

sis von Potentialfeldmethoden vor. Die Umgebung des Fahrzeugs wird dabei zun¨

achst in

einer Gefahrenkarte abgebildet. Ein elastisches Band, welches ¨

ahnlich den F¨

uhlern von

Insekten vor dem Fahrzeug her geschoben wird, detektiert Bereiche des Verkehrsraums

mit besonders niedrigem Gefahrenniveau. Hierbei werden, erg¨

anzend zu Ans¨

atzen aus

der Robotik, auch die Bewegungen anderer Verkehrsteilnehmer ber¨

ucksichtigt. Die Bahn-

folgeregelung besteht aus einer Kombination von kr¨

ummungsbasierter Vorsteuerung und

potentialfeldbasiertem Regler. F¨

ur das potentialfeldbasierte Regelkonzept wird Stabilit¨

im Sinne Ljapunov’s nachgewiesen. ¨

Uber eine entsprechende Ljapunov-Funktion k¨

on-

nen abh¨

angig von Anfangsbedingungen und Reglerparametern Schranken f¨

ur Quer- und

Winkelabweichung von der geplanten Trajektorie angegeben werden. Der vom Assistenz-

system vorgeschlagene Lenkwinkel wird dem Fahrer ¨

uber ein entsprechendes Moment

am Lenkrad kommuniziert. Der resultierende, vom Fahrer gew¨

ahlte Lenkwinkel, wird

wiederum in der Trajektorienplanung ber¨

ucksichtigt. Diese Interaktion zwischen Fahrer

und Assistenzsystem wird anhand verschiedener Auspr¨

agungen des Assistenzmoments in

einem Fahrsimulator experimentell untersucht.

Contents
Abstract III
Kurzfassung V
Nomenclature IX
Introduction 1
1 StateoftheArt................................. 1
1.1 Driver Assistance Systems . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Environmental Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Human Machine Interaction . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Vehicle Guidance Control . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 MotionPlanning ............................ 8
2 Thesis Contributions and Outline . . . . . . . . . . . . . . . . . . . . . . . 14
Lateral Vehicle Dynamics 17
1 Reference Frames and Geometry . . . . . . . . . . . . . . . . . . . . . . . . 17
2 TireModeling.................................. 19
2.1 TireKinematics............................. 19
2.2 Tire Forces: The DugoffModel .................... 21
3 Double-TrackModel .............................. 26
3.1 Kinematics ............................... 26
3.2 Dynamics ................................ 27
4 Single-TrackModel............................... 29
4.1 Kinematics ............................... 30
4.2 Dynamics ................................ 31
5 Linear Single-Track Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 Vehicle-Fixed Formulation . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Trajectory-Fixed Formulation . . . . . . . . . . . . . . . . . . . . . 34
6 Steering Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.1 Ackermann Steering . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.2 Self Steering Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 37
7 Model Validation with Experimental Data . . . . . . . . . . . . . . . . . . 41
7.1 Steady State Cornering Maneuver . . . . . . . . . . . . . . . . . . . 43
7.2 Unsteady Cornering Maneuver . . . . . . . . . . . . . . . . . . . . . 45
Potential Field based Motion Planning 49
1 HazardMap................................... 52
1.1 Road................................... 52
1.2 Obstacles ................................ 56
1.3 Hazard Map Composition . . . . . . . . . . . . . . . . . . . . . . . 60

VIII CONTENTS
2 ElasticBand................................... 62
2.1 Equilibrium Configuration . . . . . . . . . . . . . . . . . . . . . . . 62
2.2 NodePlacement............................. 65
2.3 Constraining Longitudinal Displacements . . . . . . . . . . . . . . . 65
3 Cooperative Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1 Driver’s Steering Intention . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Vehicle State Extrapolation . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Driver’s Maneuver Strategy . . . . . . . . . . . . . . . . . . . . . . 68
4 Algorithm and Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 Distance Computation . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 InitialSolution ............................. 70
4.3 Equilibrium Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Simulations ................................... 76
5.1 Scenario I: Entering Traffic....................... 76
5.2 Scenario II: Crossing Animal . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Scenario III: Passing Maneuver with Oncoming Traffic ....... 80
5.4 Pseudo Code of Motion Planning Algorithm . . . . . . . . . . . . . 81
Potential Field based Vehicle Guidance Control 83
1 General Concept of Potential Field Guidance . . . . . . . . . . . . . . . . . 83
2 Mapping a Virtual Guidance Force on Control Inputs . . . . . . . . . . . . 85
2.1 NonlinearMapping........................... 85
2.2 LinearMapping............................. 86
3 PathTrackingError .............................. 87
4 GuidanceKinematics.............................. 89
5 GuidanceDynamics............................... 92
6 Stability Analysis and Controller Design . . . . . . . . . . . . . . . . . . . 93
6.1 Lyapunov’s Direct Method . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 StabilityAnalysis............................ 94
6.3 Steady State Tracking Error . . . . . . . . . . . . . . . . . . . . . . 97
6.4 A Bound on the Tracking Error - Collision Avoidance . . . . . . . . 97
6.5 Sample Controller Design . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Simulations ...................................103
8 Comment on Human Vehicle Guidance . . . . . . . . . . . . . . . . . . . . 106
Shared Vehicle Guidance between Driver and Assistance System 107
1 Vehicle Guidance Control Loop . . . . . . . . . . . . . . . . . . . . . . . . 107
2 Driving Simulator Exploration . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.1 Haptic Human Machine Interface . . . . . . . . . . . . . . . . . . . 109
2.2 Driving Simulator Setup . . . . . . . . . . . . . . . . . . . . . . . . 111
2.3 Lane-Keeping and Collision Avoidance Experiments . . . . . . . . . 113
2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 118
Conclusion 123
1 Summary ....................................123
2 FutureWork...................................124
Appendix 125
Bibliography 129

Nomenclature

Reference frames

E : (E∗,Eex,Eey,Eez)Earth-fixed frame

R : (R∗,Rex,Rey,Rez)Road-fixed frame

G : (G∗,Gex,Gey,Gez)Guidance-frame (trajectory-fixed)

T : (T∗,Tet,Ten,Teb)Trajectory-fixed (Frenet) frame

V : (V∗,Vex,Vey,Vez)Vehicle-fixed frame

FL : (FL∗,FLex,FLey,FLez) wheel-fixed frame Front Left

FR : (FR∗,FRex,FRey,FRez) wheel-fixed frame Front Right

RL : (RL∗,RLex,RLey,RLez) wheel-fixed frame Rear Left

RR : (RR∗,RRex,RRey,RRez) wheel-fixed frame Rear Right

Notation of matrices and vectors

ErE∗,P position vector of a point Pin !

E (vector from the origin E∗of !

E to

P, projected on the unit vectors Eex,Eey,Eez)

VvPvelocity of a point Pwith respect to !

E projected on !

VvP

xVex-component of E

VvP

VaPacceleration of a point Pwith respect to !

E projected on !

VaP

xVex-component of E

VaP

CE,V transformation matrix from !

V to !

E , e.g. E

EvP=CE,V ·E

VvP

dt (·) time derivative with respect to reference frame !

Objects involved in motion planning

Rcenter of the road

∂Bl,∂Brleft and right border of the road

Ojobstacle

Vhost-vehicle

Pplanned path

XNOMENCLATURE

Vehicle parameter

a,bfront and rear axle distance to the Center of Gravity CG

dtrack width

mmass

Izyaw moment of inertia

x,Cq

αlongitudinal stiffness and cornering stiffness of tire q

SG self Steering Gradient

Vehicle dynamics

βside slip angle

ψyaw angle

δ,δSW steering angle at the wheels and at the steering wheel

δAAckermann steering angle

xlongitudinal slip at tire q

αqcornering angle at tire q

Ux,U

yVex- and Vey-component of EvCG,Ux:= E

VvCG

x,Uy:= E

VvCG

Ut,U

nTet- and Ten-component of EvCG,Ut:= E

TvCG

t,Un:= E

TvCG

xlongitudinal force at tire q

ylateral force at tire q

M∆Fxyaw moment caused by differential braking

Potential field motion planning and guidance

Pinode of elastic band

riposition vector of Piwith respect to !

RVBl

i,RVBr

ihazard potential of the left and the right border of the road at Piin !

RVOj

ihazard potential of obstacle Ojat Piin !

RVhaz

iresulting hazard potential at Piin !

RVint

iinternal potential of elastic band at Piin !

κ,κGcurvature of the driven and of the proposed path

ICC Instantaneous Center of Curvature of the driven path

(instantaneous center of rotation)

ICCGInstantaneous Center of Curvature of the proposed path

Rqgeneralized coordinates of the host-vehicle Vwith respect to !

R∆qpath tracking error (longitudinal, lateral, and angular deviation)

R∆qlat lateral path tracking error (lateral and angular deviation)

Chapter 1

Introduction

Driving is a complex task: wrong decisions taken by drivers contributed to approximately

86% of all accidents on German roads in 2005 (2006, [2]). About 440,000 injured and more

than 5,000 killed people in the same year tragically prove that the safety on our roads

needs to be increased. A very promising approach consists in driver assistance systems

that provide support in the different subtasks of driving.

1.1 State of the Art

According to Donges (1982, [28]) driving can be divided into navigation,guidance, and

stabilization. Therein, navigation refers to the global planning process of a route from a

starting location to a desired destination. On the guidance level, trajectories are planned

locally through the traffic and appropriate control inputs such as the steering angle are

predefined. However, depending on the states of the vehicle and on the environmental

conditions it might be necessary to adapt the control inputs in a way that the vehicle

behaves stable along the planned trajectories. This task is performed on the stabilization

level.

1.1.1 Driver Assistance Systems

Navigation systems basically provide comfort functionalities, whereas safety is affected

by systems that act on the guidance or the stabilization level. Among these systems,

Van Zanten (2002, [126]) reports the longest tradition for stabilizing systems as for

example the Anti-lock-Braking System (ABS) or the Electronic Stability Program (ESP).

Thurner (1998, [113]) expects the remaining safety potential of these accident avoiding

active safety systems much higher than the almost exploited potential of passive safety

systems aiming at crashworthiness, see Figure 1.1.

Today’s safety systems work mostly on the stability level and therefore rely on sensors

that measure the states of the vehicle such as the yaw rate. Future developments will

focus on active guidance systems that link the vehicle to its environment. This trend

mainly originates from the rapid development of environmental sensor systems such as

RAdio Detection And Ranging (RADAR), LIght Detection And Ranging (LIDAR),

or video and effective algorithms for sensor fusion, object detection and tracking. This

21INTRODUCTION

development is already reflected in first systems on the market. Examples are the Adaptive

Cruise Control (ACC) for longitudinal guidance (Winner et al. (1996, [123])) and Lane

Departure Warning (LDW) for lateral guidance, see Suzuki and Jansson (2003, [112]).

Figure 1.1: Roadmap of driver assistance and safety systems by Thurner (1998, [113])

Driver assistance for lateral vehicle guidance is also discussed by Feng, Tan and

Tomizuka (2000, [32]) and a general overview of research fields affected by driver assis-

tance systems on the guidance level is given by Vahidi and Eskandarian (2003, [117]).

1.1.2 Environmental Sensors

Arrangements of state of the art environmental sensors are already capable to achieve

circumferential perceptibility as exemplary sketched in Figure 1.2.

Figure 1.2: Principle sketch of environmental perception

1.1 STATE OF THE ART 3
Therein, each of the sensor systems indicated in Figure 1.2 has specific properties con-
cerning range, precision, and robustness with respect to environmental conditions. Some
representative characteristic data of state of the art environmental sensors are collected
in Table 1.1.
Table 1.1: Environmental sensor specifications
Sensor type Range Opening Longitudinal Velocity Robustness
angle precision acquisition (weather)
LIDAR 150 m up to 16◦<0.5 m derivative good
77 GHz RADAR 150 m 8◦...12◦<1 m direct very good
24 GHz RADAR 50 m 70◦...120◦<0.5 m direct very good
Video 50 m up to 45◦<1 m derivative neutral
IR-scanner 30 m >120◦<0.1 m derivative good
Similar data for RADAR, LIDAR, and camera systems are also compiled by Reichart
(2000, [89]). Each of these single-sensor systems has already been applied in automotive
applications: early algorithms for image processing at high speeds for autonomous auto-
motive guidance were developed and experimentally tested by Dickmanns (1990, [26]),
while RADAR and LIDAR systems have primarily been used in longitudinal guidance
systems such as ACC. A comprehensive review of environmental recognition, especially
on vision systems, and future trends is given by Dickmanns (2002, [24]).
Beyond the further development of single-sensor systems, as collected in Table 1.1,
another innovation thrust is generated by sensor fusion. Sensor fusion aims at combining
different sensor technologies in order to increase range, precision, and - most importantly
- robustness compared to single sensor solutions. Developments in this active field of
research are listed in the contributions of Vukotich and Kirchner (2001, [121]), of
K¨
ampchen, F¨
urstenberg and Dietmayer (2004, [54]), and of Darms and Winner
(2006, [23]).
In general, the processing of environmental sensor data can be divided into two tasks.
First, static elements of the environment have to be identified. Therein, lane detection
is of special interest for vehicle guidance systems. Nowadays, lane detection algorithms
are sufficiently robust and already in production with systems such as LDW. A sample
algorithm is given in Zheng and Cheng (2004, [71]). The second task involves the
detection and tracking of other traffic participants. Classical tracking approaches rely on
a single dynamic model to predict the motion of detected objects. Modern approaches
take different parallel models into account, weighted accordingly to the previous motion of
the tracked object. Therefore, the object state estimation, especially for highly dynamic
driving maneuvers, is an important issue. One recent approach was tested by K¨
ampchen,
Weiss, Schaefer and Dietmayer (2004, [55]), where data, based on sensor fusion
incorporating an Inertial Navigation System (INS), a Global Positioning System (GPS)
receiver, two infrared laser scanner, a 77 GHz RADAR, and a monocular vision system
were processed. Alternative research deals with the prediction of trajectories of objects
by means of learning algorithms as the one presented by Vasquez et al. (2004, [118]).
A further stimulus for guidance systems can be expected by using differential GPS in
combination with INS and digital maps. Therein, modern GPS systems allow the precise

41INTRODUCTION
measurement of the location of the host-vehicle as well as the measurement of the dynamic
states of the vehicle, see Simon (2003, [104]) and Ryu (2004, [94]). In addition, digital
maps can be used to precondition driver assistance systems by predicting static elements
of the environment as proposed in the work of Schraut (2000, [99]).
Last but not least, the proceeding deployment of wireless network applications promises
a further improvement of environmental perception by car to car communication, see
L¨
ubke (2004, [72]); and therefore a further stimulus for the development of vehicle guid-
ance systems can be expected.
1.1.3 Human Machine Interaction
The question how to assist the driver properly in order to improve his decisions is very
difficult. Up to now most systems are restricted to inform the driver, see for example
Buld and Kr¨
uger (2002, [13]). In the near future the other end of the spectrum might
be marked by systems, such as the automatic emergency brake or evasion systems as the
one proposed by St¨
ahlin, Schorn and Isermann (2006, [108]), that guide the vehicle
fully autonomously if an accident seems unavoidable. Though, the degree of cooperation
between driver and assistance system is still very low in both variants. Furthermore, both
approaches have significant drawbacks. A system, that is limited to provide information,
can be quite ineffective if the driver is inattentive. On the other hand systems, that take
over the driving task completely, hold the risk of failures that cannot be corrected by
the driver. Beyond that, the legal situation of autonomous systems, when involved in
accidents, is a precarious topic.
Goodrich and Boer (2000, [43]) consider the process of sharing responsibility be-
tween a human driver and a collision avoidance system as a trade offbetween safety
enhancement and human autonomy. Therein, the predicted safety enhancement must be
compelling enough to justify the cost to the drivers autonomy.
The h-metaphor of Flemisch et al. (2003, [35] and 2004, [34]) provides a guideline
for shared vehicle guidance between driver and assistance system. Thereby, the hstands
for horse, comparing the interaction between driver and vehicle to the interaction between
rider and horse. The main idea is to have a continuous shift of the driving task between
driver and vehicle. The metaphor refers to the extremes of interaction as loose rein and
tight rein. Loose rein characterizes a mode where the rider/driver might be inattentive and
the horse/vehicle is in charge to move safely through its environment and for example not
to collide with obstacles. When the rider indicates, by tight reins, that he is aware of the
situation, he can force the horse to initiate actions, as for example to jump over a trench,
that the horse would not initiate on its own. In the same way the driver can dominate
over vehicle guidance and for example force the vehicle to ”collide” with an beverage can
lying on the road, that is wrongly identified as an obstacle by the environmental sensors.
The coordination of vehicle guidance between driver and assistance system has to be
established via an appropriate human machine interface. Due to the gradually increasing
number of systems that interact with the driver, human machine interfaces in vehicles
experience a continuous change. In general, the communication with the driver is based
on visual,acoustic, and haptic information. As the driving task is mainly based on visual
information, the visual channel usually denotes the highest demand. In spite of this,

1.1 STATE OF THE ART 5
many assistance and information systems also communicate via visual information with
the driver. Bielaczek (1999, [7]) notes that this congestion may lead to a reduction
of the driving performance. Schattenberg (2002, [97]) approves this observation by a
collection of surveys cited in his work.
One approach to resolve the conflict between a rising amount of information on the one
hand and the demand on the visual channel on the other, is to distribute the communica-
tion with the driver on the visual, the acoustic and the haptic channels. In this regard the
work of Zeilinger (2005, [127]) focuses on haptic communication and gives an overview
of active haptic interfaces used in automotive research projects. Figure 1.3 shows for
example a vehicle equipped with active side sticks. Huang (2004, [51]) tries to include
the properties of active haptic interfaces into a description of the overall vehicle guidance
control loop including driver, driver assistance system, and human machine interface.
Figure 1.3: DaimlerChrysler ”SideStick” according to Zeilinger (2005, [127])
In terms of the h-metaphor haptic interfaces are best suited to transmit bidirectional
information between driver and assistance system, thereby keeping the driver continuously
involved in vehicle guidance. Penka (2001, [84]) conducted surveys, which showed that
haptic interfaces such as active side sticks can lead to a higher lane-keeping precision and
to a reduction of the visual demand of the driver compared to force feedback steering
wheels. These results were supported by surveys of Buld and Kr¨
uger (2002, [13]) in
the scope of the project EMPHASIS (Effort Management and Performance HAndling in
SIcherheitskritischen Situationen).
However, drivers are trained to guide their vehicles via steering wheels and will therefore
hardly accept new interfaces such as active side sticks in the near future. For this reason
force feedback steering wheels are favored as haptic human machine interfaces. From
a technological point of view, force feedback steering wheels are further promoted by
the development of active steering systems: assuming steer-by-wire, there is no direct
mechanic link between the deflection of the steering wheel and the steering angles at the
wheels. Thus, a force feedback actuator can apply a torque at the steering wheel to display
information from the assistance system, while the steering angles at the wheels can be set
according to the chosen guidance strategy and vehicle stability requirements. Examples
of activities in the field of steer-by-wire systems are the contributions of Kaltenbach
et al. (2000, [53]), Winner et al. (2004, [122]), and Bajcinca (2006, [5]).
State of the art steering systems are not yet based on ”by-wire technology”. Neverthe-
less, different active steering systems, keeping a mechanical link between steering wheel
and wheels, have evolved. The first class of active steering systems aims at changing the

61INTRODUCTION
steering ratio between the deflection of the steering wheel and the steering angle at the
wheels. In order to do so, a planetary gear is used to add an additional steering angle to
the one commanded by the driver. This technology is for example in production at BMW
(2005, [100]) to adapt the steering ratio to the driving speed.
Electro-mechanical power steering systems that apply an additional steering torque to
the one actuated by the driver are another class of active steering systems. The electrical
actuator, for example in production at Volkswagen (2005, [101]), can also be used to
display haptic information to the driver. One of the first systems applying an assisting
torque for lane-keeping was presented by Naab and Reichart (1994, [77]) in the scope
of PROMETHEUS (PROgramMe for an European Traffic with Highest Efficieny and
Unprecedented Saftey). Other examples are listed in the contributions of Pohl and
Ekmark (2003, [85]) and Consano and Murdocco (2005, [20]).
The different steering technologies are compared by Holle (2004, [50]). Noteworthy
is, that the combination of a planetary gear to add an additional steering angle and
of an actuator to provide an additional torque at the steering wheel, yields the same
functionality as a steer-by-wire system. The torque can then be used to communicate
with the driver, while the additional steering angle could be used for vehicle stability
enhancement.
Following the guidelines of the h-metaphor and establishing shared vehicle guidance
without further increasing the visual demands of the driver, force feedback steering seems
to be a promising solution. In doing so, haptic interaction can be achieved based on
today’s active steering systems.
1.1.4 Vehicle Guidance Control
In order to guide the vehicle cooperatively with the driver, the assistance system has to be
able to generate adequate guidance strategies. Hence, maneuvers have to be planned and
appropriate control inputs have to be determined. The process of planning collision-free
maneuvers through the traffic is referred to as motion planning, while the generation of
the according control inputs is denoted as vehicle guidance control. Noteworthy is, that
vehicle guidance control provides quantities like the steering angle, which finally have
to be coordinated with the driver. In general, longitudinal and lateral vehicle guidance
can be distinguished. For lane-keeping and collision avoidance lateral guidance control is
predominant and is therefore emphasized in the following.
The first lateral vehicle dynamics models and the analysis of the vehicle’s response to
steering input already date back to the 1940s and 1950s. Significant contributions are the
works of Riekert and Schunck (1940, [91]) and Segel (1956, [102]). However, the
question about how to set the control inputs in order to track a particular trajectory only
gained interest in combination with modern sensors and controller hardware. Since the
early 1990s various approaches about how to realize previously planned driving maneuvers,
ranging from lane-keeping up to autonomous driving, were proposed.
One category of vehicle guidance control strategies are model based approaches. Koe-
pele and Starkey (1990, [61]) for example compute the steering angle based on an
inverse linear vehicle model. Model deficiencies due to nonlinearities are compensated by
an additional feedback controller that regulates the lateral positional as well as the yaw

1.1 STATE OF THE ART 7
rate deviation. Analogously, Peng and Tomizuka (1990, [83]) combine a feedback with
a feedforward controller, which are both subjected to gain scheduling. A further example
of model based control of the steering angle for trajectory tracking is given by Consolini,
Piazzi and Tosques (2001, [21]).
In parallel to model based approaches alternative control strategies evolved trying to
imitate the driver’s behavior. Neusser, Nijhuis and Spaanenburg (1992, [78]) pro-
pose a neural network controller for this purpose. Hattori, Hosaka, Taniguchi and
Nakano (1992, [47]) record human drivers behavior in following winding roads in order
to create a rule base to guide vehicles along lane markings. Also Kov´
acs and K´
oczy
(1999, [64]) formulate a fuzzy logic rule base for path tracking. Bulirsch, V¨
ogel, Von
Stryk, Chucholowski and Wolter (2003, [14]) formulate a driver model on two lev-
els: the anticipation and the stabilization level. On the anticipation level optimal control
problems are solved repeatedly on a moving prediction horizon to generate set points. On
the stabilization level these set points are tracked by a position controller.
A control concept applicable at high lateral accelerations is given by Freund and
Mayr (1997, [37]). The presented controller for trajectory tracking includes longitudi-
nal as well as lateral guidance. However, the applied pole placement procedure requires
knowledge of all vehicle’s states, which in particular requires observation of the side slip
angle. Other concepts for trajectory tracking at high curvatures are discussed by S¨
onitz
(2001, [106]). In particular, the guidance control concept based on feedback linearization
was recently applied in related form by St¨
ahlin, Schorn and Isermann (2006, [108])
within the scope of the project PRORETA and by K¨
onig, Kretschmer, Neubeck
and Wiedemann (2006, [63]) for maneuvers at the stability limit of lateral dynamics.
It is important to note, that especially for vehicle guidance control systems at high(way)
speeds a look-ahead distance has to be incorporated in order to achieve stability, see for
example Park and Nikravesh (1996, [82]) or Guldner, Tan and Patwardhan
(1996, [46]), who review different control concepts in the scope of the PATH (California
Partners for Advanced Transit and Highways) program. The advantage of incorporating
a spatial preview was also theoretically shown by Peng and Tomizuka (1990, [83]).
For lane-keeping an unique approach to vehicle guidance control is given by Rossetter
(2003, [93]): lateral and yaw deviation from the lane are combined into an error potential.
This potential, in Figure 1.4 symbolized by a spring, causes a virtual guidance force applied
to the vehicle. The effect of this virtual guidance force is synthesized by appropriate
settings of the control inputs. Look-ahead can be realized by evaluating the potential at
positions ahead of the vehicle. A more detailed discussion on potential field methods will
be given within the scope of motion planning.
Figure 1.4: Potential field based vehicle guidance control

81INTRODUCTION

1.1.5 Motion Planning

Motion planning in presence of moving obstacles and therefore the need for collision

avoidance occurs in a variety of disciplines. Examples are collision avoidance in nautics,

in aviation, and in robotics. Among these, collision avoidance at sea has probably the

longest tradition. Depending on the distances between the involved ships, Vincent

(1977, [120]) distinguishes between micro and macro collision avoidance and formulates

the problem as differential game. Other examples are the works of Djouani and Hamam

(1993, [27]), who address motion planning for ships by minimizing a multi criteria cost

function with respect to time, energy, and reduction of rudder wearing. The contribution

of Zwierzewicz (2001, [129]) also formulates the problem of collision avoidance of two

ships as nonlinear optimal control problem.

However, the proposed strategies are very specific to nautics due to the underlying

kinematic and dynamic models. Furthermore, the problem at sea is less constrained

than motion planning on roads considering the available space to maneuver. In general,

maneuvers at sea are also less dynamic than maneuvers at highways, what relaxes the need

for fast motion planning. Another difference compared to automotive problems is that

communication between the involved ships is usually established. Summarizing, collision

avoidance strategies working at sea are hard to transfer to automotive applications.

Collision avoidance systems in aviation are also well established. Examples can be

found in Tomlin, Mitchell and Ghosh (2001, [114]) and in Richards and How

(2002, [90]). Though, also algorithms used in airborne collision avoidance seem hardly ap-

plicable to automotive problems: besides the fundamentally different dynamics of aircrafts

compared to road vehicles, the third available dimension (elevation) and communication

between aircrafts facilitate the problem drastically. Therefore, robotics, including mani-

pulators, mobile robots, and car-like autonomous vehicles, is probably the field providing

most suitable motion planning algorithms applicable to vehicle guidance. Hence, first an

overview of various approaches to motion planning is given, before potential field methods

are highlighted.

Various Approaches

Mildner (2004, [73]) focuses on wheel based car-like robots and structures the field by

classifying motion planning methods into

•model based,

•rule based,

•geometric, and

•potential field based.

Geometric methods incorporating moving objects and their velocities might more pre-

cisely be referred to as kinematic methods. Besides that, the list could be extended by

probalistic methods and it should be mentioned that a variety of mixed approaches exist

that cannot clearly be assigned to one of the categories above.

1.1 STATE OF THE ART 9

A unique model based method to generate emergency trajectories that avoid collisions

with one oncoming vehicle was put forward by Lachner (1997, [66]): the collision avoi-

dance problem is formulated as differential game based on the modified game of two cars.

The basic idea of the modified game of two cars is to consider the host-vehicle E(Evader)

and an oncoming vehicle P(Pursuer) as parts of one dynamic system. The configuration

of this system is depicted in Figure 1.5.

Figure 1.5: Modified game of two cars

The dynamic states of this system are the lateral distances to the road center xEand

xP, the distance between the two vehicles along the road y, the orientations of the velo-

city vectors φEand φPand the speed of the host-vehicle vE. The speed of the oncoming

car vPis assumed to be constant. The control inputs are the normalized steering-wheel

deflections uEand uPof both vehicles as well as the normalized driving or braking force

ηE. Inequality constraints result from the requirement not to leave the road of width bas

well as from bounds on the control inputs. The characteristic property of this nonlinear

system is that Eonly has control over the inputs uEand ηE, while Pcan only access

uP. The objective of Eis to avoid a collision with P. Thus, a minimum distance dmin

between Eand Phas to be maintained. To synthesize a collision avoidance strategy for

E, maximization of dmin is aspired. It is assumed that Eand Pcannot communicate.

Therefore, Ehas no knowledge about possible strategies of Pand vice versa. However,

to solve the differential game a worst-case scenario is assumed: for the oncoming vehicle

Pan optimal strategy for hypothesized collision intension is presumed. Hence, the task

is to identify the corresponding strategy of the collision avoiding vehicle E. In order to

solve the problem of maximizing the minimal distance dmin between

both vehicles, Lachner (1997, [66]) simplifies the problem by considering the squared

lateral distance d2

final =(xP(tf)−xE(tf))2between both vehicles at the instant of mutual

passing tf. The applied numerical procedure needs a sufficiently accurate estimate of the

start solution because convergence cannot be guaranteed in general. Furthermore, the

numerical procedure is computationally very expensive and cannot be applied for online

motion planning. An exclusion criterion for vehicle guidance is that the generated motion

strategies are only guaranteed to be collision-free if Pbehaves as presumed and strives

for collision!

Tsoularis and Kambhampati (1998, [116]) discuss another straight forward model

based approach. Therein, a discrete time optimization procedure for collision avoidance

is proposed. The nominal path of the robot as well as the trajectories of the obstacles are

assumed to be known. In this approach the robot stays on the nominal path but changes

its velocity in order not to collide with the obstacles.

10 1INTRODUCTION
Velocity obstacles are a different method to generate collision-free trajectories. Herein,
maneuvers are selected in the velocity space in order to avoid collisions based on the
current positions and velocities of robot and obstacles. To guarantee that the planned
maneuver is dynamically feasible, Fiorini and Shiller (1998, [33]) intersect the set of
avoidance velocities with the set of admissible velocities. A trajectory from one position to
another is then generated by searching a tree of feasible avoidance maneuvers computed
at discrete time intervals.
Kindel, Hsu, Latombe and Rock (2000, [60]) describe kinodynamic motion plan-
ning for mobile robots. The concept combines kinematic and dynamic constraints of a
mobile robot with probalistic randomized motion planning.
In automotive research rule based or behavior based approaches are very popular.
These algorithms rely on rules formulated for particular situations. An example could be:
”if an obstacle occurs ahead and braking is not possible, initiate an evasion maneuver”.
An expert system for automated highway driving is discussed by Niehaus and Stengel
(1990, [79]). Other examples can be found in Br¨
udigam (1994, [12]) in combination with
a linear single-track model to generate trajectories for autonomous lane change maneuvers
or in Reichardt (1996, [88]). A fuzzy controller network based on more than 300 rules
that was trained by human drivers in order to drive autonomously in presence of fixed
obstacles is described by Lages (2001, [67]).
In recent research on emergency systems and autonomous vehicles different geometric
or kinematic approaches for evasion maneuvers have been analyzed. Lammen (1993, [68])
considers two separate options for collision avoidance: changing the planned velocity or
changing the parameters of the geometrically planned path. The underlying trajectories
are generated by applying geometric path planning with simple functions such as sinusoids.
An advanced geometric motion planning concept is proposed in Stiller, Simon and
Weisser (2001, [110]). Therein, discrete points satisfying geometric constraints along
the corridor in front of the vehicle are chosen and interpolated by splines. By using the
Bezier-method the course can be adapted locally to sudden changes in the environment.
Also Simon (2003, [104]) proposes geometric path planning based on Bezier-splines. In
doing so, non-valid areas are defined that should not be penetrated by the path in order
to avoid collisions. The method proposed by Ameling (2002, [4]) generates trajectories
consisting of three clothoid segments approximated by third order polynomials. Therefore,
discontinuities in the steering angle when following the planned trajectory are avoided.
Mildner (2004, [73]) extends the approach given by Ameling (2002, [4]) by a velocity
profile along the path. The generation of this velocity profile incorporates kinematic
relations between the curvature and the resulting lateral acceleration as well as the feasible
deceleration of the vehicle. Therein, maneuvers are planned assuming that the driver
is no part of the control loop and therefore has no influence on the driven emergency
maneuver. The vehicle is finally brought to halt in order to hand over control to the
driver. The path planner used by Monemerlo, Thrun, Dahlkamp, Stavens and
Strohband (2006, [75]) in the DARPA (Defense Advanced Research Projects Agency)
”grand challenge” generates possible paths from a smoothed version of a given baseline
trajectory. Different constraints such as the number of obstacles under the path are then
evaluated to select the best path. Figure 1.6 shows the autonomous vehicle ”Stanley”
winning the grand challenge 2006.

1.1 STATE OF THE ART 11

Figure 1.6: Autonomous vehicle Stanley, see for example Monemerlo, Thrun,

Dahlkamp, Stavens and Strohband (2006, [75])

The recently proposed procedure by St¨

ahlin, Schorn and Isermann (2006, [108])

is to a large extend analog to the one proposed by Ameling (2002, [4]); the clothoid

function is replaced by a sigmoid.

In addition to the motion planning approaches presented so far, artificial potential fields

represent a further interesting class of motion planning methods.

Potential Field Methods

Potential field methods for motion planning and collision avoidance, originally introduced

by Krogh (1984, [65]) and Khatib (1986, [57]), are well established and have been applied

many times in robotics. Sample introductory texts covering potential field methods in

robotics are Latombe (1991, [69]), Siegwart and Nourbakhsh (2004, [103]), and

Choset et al. (2005, [18]).

The basic idea is to integrate environmental information of the robot into artificial

potential fields. Therein, the potential function V(r), with rdenoting the position of the

robot, can be composed of attractive and repulsive potentials to encode desirable areas

as well as non-valid areas or obstacles. Then, the artificial potential field causes a virtual

force on the robot. In absence of other forces the robot follows the virtual potential force

F(r)=−∇V(r) in direction of the negated gradient of V(r) towards lower values of the

potential. In this framework non-valid areas and obstacles act as repelling forces. The

robot could be compared to a wanderer in the mountains following the steepest path

downhill, avoiding intermediate peaks representing obstacles.

Therefore, potential field methods provide a straight forward way to calculate collision-

free paths through the environment of the robot based on its position and environmental

sensor data. Compared to rule-based approaches, potential field methods are not limited

to scenarios that are a priori trained. The evolving trajectories of geometric methods

are restricted to the shapes of underlying functions such as sinusoids. In contrary, the

trajectories provided by potential field methods can more flexible adapt to a present

scenario. A further essential advantage when compared to other methods is that there

is no principal limit in the number of obstacles that can be handled by potential field

methods.

However, an inherent drawback is that potential field methods act as a steepest descent

optimization and the robot might get stuck at local equilibria. Critical points are positions

12 1INTRODUCTION
with vanishing gradient ∇V(r)=0. Therein, local maxima, saddle points, and local
minima can be distinguished as indicated in Figure 1.7.
Figure 1.7: Critical points with vanishing gradient ∇V(r)=0
Usually potential functions have isolated critical points. Hence, unless the robot starts
at a local maximum or a saddle point, the potential force Fin direction of the negated
gradient guides the robot towards (local) minima, where the robot can get stuck.
Different strategies evolved to address the problem of local minima within potential
field methods. One approach is to design the governing potential function V(r) in a way
that no local minima can appear. In order to do so, Connolly and Grupen (1993, [19])
proposed so-called harmonic functions to construct the potential field. Harmonic func-
tions have zero curl and satisfy Laplace’s equation !n
i=1
∂2
∂x2
iV(r) = 0 with xibeing the
components of r. Any critical point, ∇V(r)=0, is therefore guaranteed to be a saddle
point. Also Sullivan, Waydo and Campbell (2003, [111]) suggest so-called stream
functions to construct the potential field in a way that all critical points are saddle points
and therefore the robot will always reach its goal.
Chang and Marsden (2003, [16]) present another approach to the guidance problem
from a start to a goal location by means of governing potential fields. Thereby, obstacles
are explicitly excluded from the potential field such that a potential function free of local
minima can be chosen. To avoid collisions with obstacles gyroscopic forces are applied
locally. This approach was extended by Chang, Shadden, Marsden and Olfati-
Saber (2003, [17]) to multiple agent systems with swarming behavior.
Other subsets of potential field methods combine the potential field approach with
methods from artificial intelligence. One idea is to construct the vector field depending
on the present situation by means of heuristics and to apply evolutionary programming
schemes, see for example Kim, Kim and Kwon (2001, [59]).
The concept of virtual bumpers introduced by Hennessey, Shankwitz and Donath
(1995, [48]) is perhaps the first potential field like automotive application. The perimeter
of the vehicle is surrounded by a sensor-based virtual bumper, see Figure 1.8. As the
boundary of this bumper is deflected, a virtual force is generated by the compression of
imaginary springs and dampers. The vehicle controller responds to this virtual force in a
way that the deflection of the bumper is revoked. Schiller, Morellas and Donath,
(1998, [98]) combine this idea with an heuristic rule base to generate motion commands
for the vehicle.

1.1 STATE OF THE ART 13

Figure 1.8: Virtual bumpers and potential field based lane-keeping

The lane-keeping approach of Rossetter (2003, [93]) differs from the concept of vir-

tual bumpers as the vehicle is virtually coupled to the environment, see Figure 1.4 and

Figure 1.8.

Kageyama and Nozaki (1995, [52]) compose a potential field of several hazard fac-

tors such as road edges and obstacles, and conclude that this approach generates control

results comparable to the ones of human drivers. Gerdes, Rossetter and Saur

(2001, [41]) coined the term hazard map to describe this way of encoding the environment

in terms of potential fields.

The natural transfer of environmental sensor data, such as positions and velocities of

tracked objects with respect to the host-vehicle, into a potential field hazard map is one

essential advantage of potential field methods for vehicle guidance; each object is modeled

by a separate repulsive potential field. If further data, for example from a video system,

are available and if the objects can be classified, e.g. as pedestrians, the potentials assigned

to the single objects can be scaled with respect to the hazards associated to the objects.

Figure 1.9 shows a vehicle exposed to a potential field hazard map composed of the borders

of the road and two obstacles.

Figure 1.9: Traffic scenario encoded in potential field hazard map

The potential field methods presented so far are so-called reactive methods, which

means that the robot or vehicle is directly affected by changes in the environment. If

the potential field representing the environment is not carefully constructed as outlined

above, the robot can be guided towards local minima. Furthermore, the robot, pushed

by the virtual force, might react very abruptly to changes. In robotics the dynamics are

often considered as a mass particle, see for example Latombe (1991, [69]), such that the

robot is capable to move in arbitrary directions. The dynamics of a ground vehicle are

more complex. Besides that, instantaneous reactions might lead to inconvenient driving

behavior and affect the stability of the vehicle.

14 1INTRODUCTION

Quinlan and Khatib (1993, [87]) proposed elastic bands as a predictive way of

potential field based motion planning. Elastic bands consist of elastically coupled nodes

that are exposed to the surrounding potential field. In doing so, the virtual potential

forces deform the elastic band instead of directly acting on the robot. Hence, the elastic

band represents collision-free trajectories for the robot. Meanwhile, elastic bands are well

established, see for example Komainda and Hiller (1999, [62]). As the last node of

an elastic band can be located at the goal of the robot, the problem of local minima

is inherently avoided. Furthermore, the predictive way of motion planning smooths the

generated trajectories as sudden reactions to environmental changes are prevented. A

further essential advantage for automotive applications lies in the inherently low curvatures

of the trajectories generated by elastic bands due to the internal potentials that counteract

high offsets between adjacent nodes. First automotive applications were proposed by

Gehrig and Stein (2001, [40]) for platooning and by Hilgert, Hirsch, Bertram

and Hiller (2003, [49]) for autonomous evasion maneuvers. Brandt, Sattel and

Wallaschek (2004, [8]) consider elastic bands for driver assistance. Figure 1.10 shows

a vehicle guided by an elastic band through an hazard map as proposed in this thesis.

Figure 1.10: Vehicle guided by elastic band and hazard map

1.2 Thesis Contributions and Outline

This thesis provides a unified potential field framework for lateral vehicle guidance with

possible applications as in lane-keeping or in collision avoidance. Thereby, motion planning

as well as tracking of the planned trajectories is established by means of potential field

methods. Particular emphasis is put on shared vehicle guidance and therefore on keeping

the driver permanently embedded in the vehicle guidance control loop as demanded by

the h-metaphor. The underlying potential field methods originating in the field of robotics

are adapted and extended for driver assistance.

The motion planning concept incorporates a potential field hazard map that naturally

encodes environmental sensor data such as detected lanes. In addition to hazard maps

proposed so far, also moving objects and their anticipated trajectories are included. In

order to do so, the dynamic states of obstacles are integrated forward in time. Therein,

different behaviors as for example vehicles following their lanes are considered. Therefore,

this new hazard map gives a spatial as well as a temporal representation of the traffic

scenario and is not limited in the number of objects that can be handled, which is a

drawback of many recently proposed geometric motion planning approaches.

1.2 THESIS CONTRIBUTIONS AND OUTLINE 15
Trajectories of low hazard levels are generated by elastic bands that sense the hazard
map like antennae of insects, compare Figure 1.10. In this predictive approach the po-
tential field forces act on the nodes of the elastic band instead of directly affecting the
vehicle as for example in Rossetter (2003, [93]). The last node of an elastic band can
be placed ahead of the vehicle, e.g. on the center of the right lane or on a leading vehicle.
When internal and external potential field forces, acting on the nodes of the elastic
band, are in balance, the elastic band is collision-free. The equilibrium concept proposed
here takes the anticipated trajectories of other traffic participants into account. Hence, the
potential field forces are not only position but also time dependent. This is a substantial
extension compared to previous applications of elastic bands where only snap shots of the
environment were considered and changes in the environment were only incorporated by
replanning. Including prediction of the motion of other traffic participants may lead to
smoother trajectories and is more similar to the motion planning of human drivers, which
supports shared vehicle guidance between driver and assistance system. In general, more
than one collision-free solution exist. However, elastic bands belong to the so-called local
methods and only provide one solution depending on the initial solution, see Quinlan
and Khatib (1993, [87]). Here, the initial solution is determined in cooperation with
the driver. In doing so, the assistance system can for example propose a trajectory that
passes an obstacle on the left; if the driver insists to pass it on the right, the assistance
system adapts and provides corresponding trajectories passing the obstacle on the right
hand side.
The gap between motion planning and guiding the vehicle along the planned trajectories
is closed by combining a potential field feedback controller, which generates a steering
angle by means of a virtual force depending on the tracking error, and a feedforward
controller that calculates a steering angle based on the curvature of the planned trajectory.
Chapter 2 introduces the underlying lateral vehicle dynamics models for motion plan-
ning and vehicle guidance. Therein, also steering characteristics are considered and the
models are compared to experimental data. The concepts of cooperative potential field
motion planning including hazard map and elastic bands and the corresponding numeric
procedures are outlined in Chapter 3. To accelerate the computation, one option is the
reduction of the degrees of freedom of the nodes of the elastic band, which is also shown
within the scope of Chapter 3. The potential field framework is completed by potential
field guidance control, which is discussed in Chapter 4. The guidance controller in combi-
nation with disturbance rejection or feedforward control is shown to be stable in the sense
of Lyapunov. Furthermore, a reachable set for the tracking error depending on the initial
conditions and the parameters of the controller is given. This provides the basis to adapt
the potential field controller to the environment in order to guarantee collision avoidance.
Subsequently, in Chapter 5 the driver assistance system is analyzed in simulator ex-
periments with 16 test subjects. Therein, the interaction with the driver is established
by means of a force feedback steering wheel. In this exploration different configurations
of the assistance torques, reflecting loose rein and tight rein, are inspected. Besides that
the test subjects are distracted by a secondary task in some test drives. Finally, the main
results of this thesis are summarized and an outlook to future research directions is given
in Chapter 6.

Chapter 2

Lateral Vehicle Dynamics

For lane-keeping and collision avoidance systems, lateral vehicle dynamics are predomi-

nant. Therefore, the aim of this chapter is to provide models that cover the dominant

dynamic effects of lateral motion. Since vehicle dynamics has been a well studied field of

research for many years, many very detailed models already exist. However, the models

presented in this chapter are limited in their degree of complexity. One reason for this is

that the generality of vehicle dynamic models tends to decrease with an increasing level of

detail. Thus, results gained from the analysis of a very detailed model are often only ap-

plicable to the particular class of vehicles that is studied, see Karnopp (2004, [56], p. 98).

Besides that, complex vehicle models generally consume more computational effort, which

limits their use for vehicle dynamics control systems.

The models studied in this chapter are planar and neglect effects that are associated

with roll, pitch or heave motion. Therefore, also coupling effects between these degrees of

freedom and the lateral motion are ignored.

First, different sets of reference frames are introduced. Subsequently, the modeling of

the tires, which are the main force transmission elements between the vehicle and the road

is addressed. Steering characteristics are discussed on the basis of the linear single-track

model. After that, different vehicle dynamics models are compared to experimental data.

2.1 Reference Frames and Geometry

For motion planning and vehicle guidance, different inertial frames, such as a road-fixed

frame, are used to describe the motion of the vehicle. Here, an earth-fixed reference

frame !

E: (E∗,Eex,Eey,Eez) is used to represent the inertial frame. Besides that, the

following moving reference frames, as depicted in Figure 2.1, are used:

V : (V∗,Vex,Vey,Vez)"=Vehicle-fixed reference frame

T : (T∗,Tet,Ten,Teb)"=Trajectory-fixed reference frame

FL : (FL∗,FLex,FLey,FLez)"=Front Left wheel-fixed reference frame

FR : (FR∗,FRex,FRey,FRez)"=Front Right wheel-fixed reference frame

RL : (RL∗,RLex,RLey,RLez)"=Rear Left wheel-fixed reference frame

RR : (RR∗,RRex,RRey,RRez)"=Rear Right wheel-fixed reference frame

18 2LATERAL VEHICLE DYNAMICS

Figure 2.1: Reference frames

The vehicle-fixed reference frame !

V: (V∗,Vex,Vey,Vez) is attached to the center of

gravity CG of the vehicle. Therein, the Vexaxis coincides with the longitudinal axis of

the vehicle and points in forward direction. The Veyand the Vezaxes point left and

upwards, respectively. The Vexand Eexaxes of the moving reference frame !

V and the

earth fixed reference frame !

E contain the yaw angle ψ.

The trajectory-fixed reference frame !

T: (T∗,Tet,Ten,Teb) is a so-called Frenet reference

frame, see for example Bronstein and Semendjaev (1997, [11], pp. 229). The axes of

T are also attached to the center of gravity CG but can rotate relative to the vehicle.

Its Tetaxis is always tangential to the trajectory driven by the vehicle, while the Ten

axis is normal to the driven trajectory and points towards the instantaneous center of

curvature ICC. The Tebaxis is called the binormal and forms a right-handed reference

frame with Tetand Ten. The Tetaxis of the trajectory-fixed system !

T and the Vexaxis

of the vehicle-fixed reference frame !

V contain the vehicle side slip angle β.

The wheel-fixed reference frames !"

FL , !"

FR , !"

RL and !"

RR are attached to the center

points of the respective wheels. However, they do not rotate with the wheels about the

eyaxes! Their exaxes point forward in longitudinal direction of the respective wheels.

The eyand ezaxes point to the left and upwards, respectively. The wheel-fixed reference

2.2 TIRE MODELING 19

frames !"

FR and !"

FL attached to the front tires are rotated by the steering angles δLand

δR, respectively, about the ezaxes with respect to the vehicle-fixed reference frame !

The distances of the front and the rear axis to the center of gravity CG are denoted as

aand b, respectively. The track width is called d, as depicted in Figure 2.1. The position

of the vehicle with respect to the inertial frame !

E is described by the position vector of

the origin V∗of the vehicle-fixed reference frame !

ErV∗=ExV∗

Eex+EyV∗

Eey.(2.1)

According to Figure 2.1, the position vectors of the wheel center points in the vehicle-fixed

reference frame !

V become:

VrV∗, F L∗=aVex+d

2Vey,

VrV∗, F R∗=aVex−d

2Vey,

VrV∗, RL∗=−bVex+d

2Vey,

VrV∗, RR∗=−bVex−d

2Vey.

(2.2)

2.2 Tire Modeling

The tire is the main force transmission element between the vehicle and the ground.

Therefore, the characteristics of the tires have significant influence on the dynamic be-

havior of the vehicle. In the next section, some kinematic values describing the physics of

the tire are introduced. Subsequently, a tire model that provides tire forces based on the

tire kinematics is given.

2.2.1 Tire Kinematics

The kinematic values for a wheel q∈{FL,FR,RL,RR}are depicted in Figure 2.2.

Figure 2.2: Tire kinematics

20 2LATERAL VEHICLE DYNAMICS

The velocity Evq∗of the wheel center point q∗with respect to the inertial reference

frame !

E and the longitudinal axis qexof the tire contain the tire slip angle αq. For

low velocities, e.g. for parking maneuvers, the tire slip angle αqalmost vanishes and the

vehicle behaves kinematically nonholonomic with the constraint Evq∗·qey= 0. However,

for cornering maneuvers at high speeds the cornering angle can obtain values up to 15◦,

see for example Kiencke and Nielsen (2005, [58]), and is associated with a lateral force

that guides the vehicle. The lateral slip sq

αof the tire is usually described by means of the

tire side slip angle:

α= tan αq.(2.3)

In the following, degrees of freedom associated to the suspension system will not be

considered. Figure 2.3 shows the free body diagram for the planar, longitudinal wheel

motion.

Figure 2.3: Tire forces and torques

The rotational motion of the wheel is governed by

q˙ωq

y=Mq

d/b −qFCP

xRe.(2.4)

The angular acceleration of the wheel E

q˙ωq

ydepends on the driving or braking torque Mq

d/b

and on the longitudinal tire force qFCP

x, which acts at the contact patch CP. The effective

running radius of the tire Recan be estimated by the distance covered by the slip free

rolling wheel 2πRe.

For rigid wheels the contact patch degenerates to a contact line. For slip free rolling

wheels, ReE

qωq

y=E

qvCP

x, as well as for combined rolling and sliding motion, the tire

forces can be computed by solving the equations of motion of the wheel, see Popp and

Schiehlen (1993, [86], pp. 120). In doing so, forces resulting from sliding friction can for

example be obtained by assuming Coulomb friction.

However, the nonlinear elastic properties of the tire and the complex geometry of

the contact patch complicate the computation of lateral and longitudinal tire forces.

In general, the longitudinal tire forces are described by means of the longitudinal slip.

Though, in practice different definitions for the longitudinal slip exist. Mitschke and

Wallentowitz (2004, [74], p. 19) distinguishes between longitudinal slip for driving

(ReE

qωq

y≥E

qvq∗

x) and for braking (ReE

qωq

y<E

qvq∗

x) wheels:

2.2 TIRE MODELING 21

Driving

x,d =ReE

qωq

y−E

qvq∗

ReE

qωq

(2.5)

Braking

x,b =−

qvq∗

x−ReE

qωq

qvq∗

.(2.6)

In the following, a combination of Equation (2.5) and Equation (2.6),

x=ReE

qωq

y−E

qvq∗

max #Re$$E

qωq

y$$,$$E

qvq∗

x$$%,(2.7)

with sq

x∈[−1,1] is used, compare Rill (1994, [92], p. 55). The values of the longitudinal

slip can now be interpreted as follows:

•sq

x= 1: wheelspin,

•0<s

x<1: driving,

•sq

x= 0: rolling,

•−1<s

x<0: braking,

•sq

x=−1: locking.

2.2.2 Tire Forces: The DugoffModel

One way to incorporate the tire characteristics in vehicle dynamics modeling is to employ

experimental tire data. The drawback of this method is that data for all environmental

conditions are necessary. For this reason, the use of models that describe the physics

of tire road interaction is common and many different tire models for different fields

of application exist and are still subject to numerous research activities. One of the

most frequently used models is Pacejka’s magic tire formula, see Pacejka (2002, [81]).

Pacejka’s model is a curve fitting approach to tire test data. In doing so, many parameters

need to be adapted, which makes the approach less attractive for principal investigations.

Another widespread approach, with less parameters, is the tire model of Burckhardt, see

Kiencke and Nielsen (2005, [58]).

In the following, the Dugoffmodel introduced by Dugoff, Fancher and Segel

(1969, [30]) will be used. The Dugoffmodel is a physical model of the tire road interaction

and has a lesser computational effort when compared with the Pacejka and the Burckhardt

model. The only necessary parameters are the longitudinal stiffness of the tire Cx, the

cornering stiffness Cα, and the adhesion coefficient µ0of the tire and the road surface.

The Dugoffmodel provides a relationship of the kinematic values and the forces acting

at the tires. In doing so, the Dugoffmodel distinguishes explicitly between pure adhesion

and combined adhesion and sliding in the contact patch. Whereas, the assumption of pure

adhesion is only valid for small side slip angles α. In that region the tire deforms linearly.

22 2LATERAL VEHICLE DYNAMICS

Linear Deformation Model

The deformation of a tire qfor small side slip angles αqis illustrated in Figure 2.4.

Figure 2.4: Tire deformation in the linear region according to Mitschke and Wallen-

towitz (2004, [74], p. 35)

The point Asymbolizes a point on the tread of a cornering tire that enters the contact

patch. With the rotation of the tire, Amoves along the centerline of the contact patch

and experiences a deflection with respect to the wheel. The spring elements symbolize

the accompanying lateral force. As the center of the contact area lies behind the wheel

center q∗, the resulting lateral force qFycauses an aligning torque qMz. The offset nyis

called pneumatic trail and accounts in the linear region for approximately one sixth of

the length of the contact patch, nlin

y≈L/6. The resulting equations of the linear model

become:

qFy=−Cq

αtan αq≈−Cq

ααq,(2.8)

qMz=−Cq

αtan αqnlin

y≈−Cq

ααqnlin

y.(2.9)

The linear deformation model implies full adhesion of the contact patch. The longitudi-

nal forces as well as the lateral forces are assumed to be well below the adhesion force,

&qFx2+qF2

y<µ

0qFz, where qFzdenotes the normal load on the tire. In order to con-

sider the lateral forces without taking coupling effects with the longitudinal forces into

account, the longitudinal slip sq

xas well as the cornering angle αqneed to be small.

2.2 TIRE MODELING 23

Combined Adhesion and Sliding

For larger side slip angles αqor longitudinal slip sq

xthe contact patch consists of areas of

adhesion and of areas of sliding as indicated in Figure 2.5.

Figure 2.5: Contact patch for small and for large side slip angles αqaccording to

Mitschke and Wallentowitz (2004, [74], p. 35 and 46)

In the left part the linear tire deformation for small side slip angles αqand complete

adhesion is shown. Again point Aindicates a point entering the contact patch and un-

dergoing a linear deflection from the wheel centerline moving through the contact patch.

For a large side slip angle αqalso sliding zones at the end of the contact patch form as

depicted in the center part of Figure 2.5. A vivid comparison is to consider the tire as a

rotating brush. Bristles that enter the contact patch stick at the ground (adhesion zones)

while bristles that leave the contact patch stroke over the ground (sliding zones). The for-

mation of adhesion and sliding zones in the contact patch is accompanied by a nonlinear

deflection experienced by a point Amoving through the contact patch. These effects are

approximated by the Dugoffmodel as depicted in the right part of Figure 2.5.

The value ¯sqis introduced as an indicator for the transition from linear to nonlinear

tire behavior

¯sq='(Cq

x|sq

x|)2+(Cq

αtan αq)2

µqqFz(1 −|sq

x|)with ¯sq≤0.5→linear and ¯sq>0.5→nonlinear .

(2.10)

24 2LATERAL VEHICLE DYNAMICS

The friction coefficient µqof tire qand the ground becomes

µq=µ0(1−+($$E

qvq∗

x$$&sq

x2+ tan αq2)),(2.11)

where +is the so-called adhesion reduction factor. Dugoff, Fancher and Segel

(1969, [30]) give +=0.011 s/m for adhesion coefficients µ0∈[0.53 ,1.05]. The force

equations of the Dugoffmodel read:

Longitudinal force

qFx=









xsq

1−sq

,¯sq≤0.5

xsq

1−sq

¯sq−0.25

(¯sq)2,¯sq>0.5

(2.12)

Lateral force

qFy=









−Cq

αtan αq≈−Cq

ααq,¯sq≤0.5

−Cq

αtan αq

1−sq

¯s−0.25

(¯sq)2,¯sq>0.5(2.13)

Aligning torque caused by lateral tire forces

qMz=









−Cq

αtan αqnlin

y≈−Cq

ααqnlin

y,¯sq≤0.5

−Cq

αtan αq

1−sq

¯sq−0.25

(¯sq)2nnl

y,¯sq>0.5(2.14)

The tire characteristics resulting from the force equations (2.12) and (2.13) are plotted

in Figure 2.6. Figure 2.6 a) shows the longitudinal, Figure 2.6 b) the lateral tire forces

against longitudinal slip sxand the tire side slip angle α, respectively. The black line in

Figure 2.6 a) indicates a none-cornering tire, α= 0. The black line in Figure 2.6 b) high-

lights the lateral forces for a slip free rolling tire, sx= 0. Note that positive longitudinal

slip sx>0 generates positive longitudinal (driving) forces Fx>0, while positive side slip

angles α>0 produce negative lateral forces Fy<0.

(a) Longitudinal forces (b) Lateral forces

Figure 2.6: Dugofftire characteristics

2.2 TIRE MODELING 25

Focusing on the black line in the left part, it can be noticed that, for small values of

the longitudinal slip sx, the longitudinal forces Fxrise linearly with the longitudinal slip.

Then, the characteristic curve starts to rise on a diminishing scale and finally saturates.

This behavior reflects the transition from pure adhesion to combined adhesion and sliding.

On the right hand side, analogous behavior can be ascertained for the lateral forces.

Noteworthy, that the aligning torques resulting from the pneumatic trail have minor

influence on vehicle dynamics and are usually neclegted, see Mitschke and Wallen-

towitz (2004, [74], p. 556). However, the aligning torque contributes to the steering

torque at the steering wheel experienced by the driver. In general, the pneumatic trail

depends on the distribution of the surface pressure in the contact patch. In the linear

region the pneumatic trail nlin

yaccounts for approximately the sixth part of the length of

the contact patch, 1

6L, as mentioned above. In the nonlinear region the pneumatic trail

depends on the pressure distribution in the range of 1

10 L < nnl

y<1

6L. Approximation

formula can be found in Rill (1994, [92], pp. 66).

Steering Wheel Torque

The steering wheel torque experienced by the driver due to vehicle dynamics effects is

provided by a simple steering model

MSW/V=1

iSVPS .MFL

∗

z+MFR

∗

z−nc#FFL

∗

y+FFR

∗

y%/,(2.15)

see for example Mitschke and Wallentowitz (2004, [74], p. 550). Therein, iS=δSW/δ

describes the relation between steering angle at the steering wheel δSW and the steering

angle at the tires δ, and is referred to as steering ratio. The effect of power steering

on the torque at the steering wheel is incorporated in the steering gain VPS. The term

ncis called caster and refers to the steering geometry; the caster describes the distance

between the intersection points of the vertical wheel axis and the steering axis with the

road. The caster due to the steering geometry is illustrated in Figure 2.7. Therefore, the

driver experiences, besides the aligning torques Mq

z, a further torque at the steering wheel

caused by the lateral tire forces Fq

yand the caster nc.

Figure 2.7: Influence of steering geometry on the steering wheel torque according to

W¨

urtenberger (1997, [125], p. 29)

26 2LATERAL VEHICLE DYNAMICS
2.3 Double-Track Model
The double-track model presented in this chapter is a planar model. Thus, only motions
in the yaw plane (Eex,Eey) will be considered. The geometry of the model is identical to
the one given in Figure 2.1.
2.3.1 Kinematics
The motion of the vehicle with respect to the inertial frame !
"
E will be expressed in the
vehicle-fixed reference frame !
"
V , see Figure 2.1. For brevity the longitudinal and the lateral
speed of the vehicle are defined as Ux:= E
VvCG
xand Uy:= E
VvCG
y, respectively. Therefore,
the velocity of the center of gravity CG with respect to the inertial frame !
"
E reads
E
VvCG =UxVex+UyVeyand E
VωV=EωV
Vez=˙
ψVez.(2.16)
Translational and rotational speeds are combined in the vector of generalized speeds
Vv=0Ux,U
y,˙
ψ1T.(2.17)
The acceleration of the center of gravity CG becomes
E
VaCG =
Ed
dt
E
VvCG =
Vd
dt
E
VvCG +E
VωV×E
VvCG =.˙
Ux−˙
ψUy/Vex+.˙
Uy+˙
ψUx/Vey.(2.18)
The time derivative of the angular velocity in Equation (2.16) yields the angular accele-
ration
E
VαCG =E˙ωV
Vez=¨
ψVez.(2.19)
To track the generalized coordinates Eq=0ExV∗
,EyV∗
,ψ1Tof the vehicle in the inertial
frame !
"
E , the vector of generalized speeds Vvhas to be transformed into !
"
E,
E˙
q=CE,V
Vvwith CE,V =

cos ψ−sin ψ0
sin ψcos ψ0
0 0 1 
,(2.20)
and integrated; Equation (2.20) contains the so-called kinematic differential equations.
To compute the tire forces Fq
xand Fq
y, the side slip angles αqare also of interest. With
Equation (2.16) and the position vectors of the wheel center points q∗with respect to the
vehicle-fixed reference frame !
"
V , as given in Equation (2.2), the velocities of the wheel
center points q∗can be computed according to
Evq∗=EvCG +EωV×rV∗,q∗.(2.21)
Now, the side slip angles αqcan be calculated by comparing the Vexand the Vey
components of Equation (2.21), see also Figure 2.1 and Figure 2.2. This yields for the

2.3 DOUBLE-TRACK MODEL 27

side slip angles at the front tires

αFL = arctan 6Uy+˙

ψa

Ux−d

2˙

ψ7−δLand αFR = arctan 6Uy+˙

ψa

Ux+d

2˙

ψ7−δR,(2.22)

and at the rear tires

αRL = arctan 6Uy−˙

ψb

Ux−d

2˙

ψ7and αRR = arctan 6Uy−˙

ψb

Ux+d

2˙

ψ7.(2.23)

2.3.2 Dynamics

In the following, the Newton-Euler equations of the planar double-track model will be

formulated. In doing so, forces due to inclinations, driving resistances and side wind will

be ignored. Henceforth, the remaining forces acting on the body of the vehicle are the

tire forces. However, since pitch and roll motion are not within the scope of this model,

the tire forces will directly be applied at the wheel center points. Figure 2.8 shows the

free body diagram of the double-track model.

Figure 2.8: Tire forces acting upon the double-track model

The forces qFq∗

xand qFq∗

ywith q∈{FL,FR,RR,RL}can be interpreted as the resulting

suspension forces.

The forces at the front tires, given in the wheel-fixed reference frames !"

FL and !"

FR ,

have to be transformed into the vehicle-fixed reference frame !

V: VFFL

∗=CV,F L

FLFFL

∗

and VFFR

∗=CV,F R

FRFFR

∗with

CV,F L =



cos δL−sin δL0

sin δLcos δL0

0 0 1 

and CV,F R =



cos δR−sin δR0

sin δRcos δR0

0 0 1 

.(2.24)

28 2LATERAL VEHICLE DYNAMICS

The moments of the tire forces acting on the vehicle body with respect to the center of

gravity CG become

VMF L,CG =VrV∗, F L∗×VFFL

∗=VrV∗, F L∗×CV,F L (δL)FLFFL

∗=VMF L,CG

z(δL)Vez

VMF R,CG =VrV∗, F R∗×VFFR

∗=VrV∗, F R∗×CV,F R (δR)FRFFR

∗=VMF R,CG

z(δR)Vez

VMRL,CG =VrV∗, RL∗×VFRL∗=VMRL,CG

zVez

VMRR,CG =VrV∗, RR∗×VFRR∗=VMRR,CG

zVez(2.25)

with the position vectors VrV∗,q

∗of the tires as defined in Equation (2.2). Combining the

mass of the vehicle mand the moment of inertia Izabout the Vezaxis in the mass matrix

M=



m00

0m0

00Iz

,(2.26)

the Newton-Euler equations of the double-track model can be written as

MV˙

v=Q#VFq∗,Vv%=



VFFL

∗

x(δL)+ VFFR

∗

x(δR)+ VFRL∗

x+VFRR∗

VFFL

∗

y(δL)+ VFFR

∗

y(δR)+ VFRL∗

y+VFRR∗

VMF L,CG

z(δL)+ VMF R,CG

z(δR)+ VMRL,CG

z+VMRR,CG



.

(2.27)

The forces and moments on the right hand side of Equation (2.27) depend on the steering

angles δLand δR, which are considered as input variables. Furthermore, they depend, by

means of the Dugofftire model, equations (2.12)-(2.13), on the side slip angles αqand

the longitudinal slip values sq

x. Knowing the generalized speeds Vv, the side slip angles

αqfollow from equations (2.22) and (2.23).

Besides the steering angles δLand δR, the longitudinal tire forces qFq∗

xare also consid-

ered as input variables, i.e. u=0δL,δR,FLFFL

∗

x,FRFFR

∗

x,RLFRL∗

x,RRFRR∗

x1T. Noteworthy

is, that not only the lateral tire forces qFq∗

ycan cause a yaw moment about the Vezaxis.

Different braking forces qFq∗

xat the left and at the right tire of one axle also apply a

yaw moment. As the longitudinal forces qFq∗

xare control input variables this provides a

controllable input to the yaw component of Equation (2.27). Figure 2.9 shows possible

configurations of differential braking.

Figure 2.9: Cornering by differential braking with ∆FF

x=FFR

∗

x−FFL

∗

xand ∆FR

x=FRR∗

x−FRL∗

2.4 SINGLE-TRACK MODEL 29

By setting a driving or braking torque Md/b, Equation (2.4) together with Equa-

tion (2.7) and equations (2.22)-(2.23) provide the corresponding longitudinal slip sq

xand

the side slip angles αq. The longitudinal forces qFq∗

xthen follow from the Dugoffmodel in

Equation (2.12) as illustrated in Figure 2.10.

Figure 2.10: Control input variables

2.4 Single-Track Model

Starting from the double-track model, a further reduction can be achieved by assuming

symmetry about the longitudinal axis, Vex, of the vehicle. In doing so, left and right tires

are lumped together resulting in a single track as illustrated in Figure 2.11.

Figure 2.11: Transition from double- to single-track model

30 2LATERAL VEHICLE DYNAMICS

2.4.1 Kinematics

In particular, combining left and right tires implies not to distinguish between the steering

angles at the left δLand at the right front tire δR, which is reasonable for sufficiently large

radii of curvature, 1/κ>> (a+b), of the driven path. The steering angle of the single-

track model is referred to as

δ:= δL

=δR.(2.28)

Recalling the side slip angles at the front tires according to Equation (2.22)

αFL = arctan 6Uy+˙

ψa

Ux−d

2˙

ψ7−δLand αFR = arctan 6Uy+˙

ψa

Ux+d

2˙

ψ7−δR,

it can be observed that αFL and αFR only differ in the algebraic sign of the term d

2˙

ψin

the denominator. This means that a lateral tire force increasing from this component on

one tire, for example on the left FLFFL

∗

y, is accompanied by a lateral tire force decreasing

from this component on the opponent, for example the right tire FRFFR

∗

y. For the single-

track model same side slip angles at the left and at the right tires are assumed. The yaw

rate contribution d

2˙

ψ, which is small compared to the longitudinal speed, d

2˙

ψ<< Ux, is

therefore neglected in the denominator. With the steering angle δ, see Equation (2.28),

the side slip angles for the single-track model become

αF= arctan 6Uy+˙

ψa

Ux7−δand αR= arctan 6Uy−˙

ψb

Ux7,(2.29)

for the front and for the rear axis, respectively. The kinematic quantities of the single-track

model are visualized in the Figure 2.12.

Figure 2.12: Kinematics of the single-track model

Translational velocity E

VvCG and rotational velocity E

VωV, as defined in Equation (2.16),

translational acceleration E

VaCG and angular acceleration E

VαCG, as given in equations (2.18)

and (2.19), respectively, remain unchanged for the single-track model. Therefore, the

kinematic differential Equations (2.20) also apply for single-track model.

2.4 SINGLE-TRACK MODEL 31
2.4.2 Dynamics
For small values of the longitudinal slip sq
xand equal side slip angles αFand αRat one
axle, see Equation (2.29), the lateral forces at the tires of one axle can be combined into
resulting lateral forces FF
yand FR
yat the front and the rear axle, respectively. Therefore,
the cornering stiffness of the combined tires CF
α=CFL
α+CFR
αand CR
α=CRL
α+CRR
αis
used in the Dugoffmodel, compare Section 2.2.2.
In order to maintain the possibility to apply a yaw moment on the vehicle body by
means of differential braking, the longitudinal tire forces of the double-track model are
combined into resulting and differential (pairs of) forces at one axle
FF
x=FFL
∗
x+FFR
∗
x,
FR
x=FRL∗
x+FRR∗
x,
∆FF
x=FFR
∗
x−FFL
∗
x,
∆FR
x=FRR∗
x−FRL∗
x.
(2.30)
Figure 2.13 shows the free body diagram of the single-track model. Note that the
lateral forces FF
yand FR
yare shown as positive in positive y-direction of the corresponding
reference frames. The side slip angles αFand αRare also depicted as they were positive.
However, when the slip angles are positive, the forces exerted on the tires would actually
be in negative direction. According to Karnopp (2004, [56], p. 102) ”This is a common
problem in describing tire-roadway interactions, which leads either to negative cornering
coefficients in the linearized case, or to a special way of writing the force laws to have
positive coefficients”. In this case the opposing directions of side slip angles and lateral
forces are captured in the lateral force equations of the Dugoffmodel, see Equation (2.13).
Figure 2.13: Dynamics of the single-track model
The moments about the Vezaxis, MF
∆Fx=d
2∆FF
xcos δand MR
∆Fx=d
2∆FR
x, in Fig-
ure 2.13 indicate the influence of differential braking. With the definitions of Equa-
tion (2.30), the Newton-Euler equations of the single-track model expressed in vehicle-
fixed coordinates !
"
V , see equations (2.18) and (2.19), read
mE
VaCG
x=m.˙
Ux−˙
ψUy/=FR
x+FF
xcos δ−FF
ysin δ
mE
VaCG
y=m.˙
Uy+˙
ψUx/=FR
y+FF
xsin δ+FF
ycos δ
IzE˙ωV
z=Iz¨
ψ=aFF
xsin δ+aFF
ycos δ−bFR
y+MF
∆Fx+MR
∆Fx.
(2.31)

32 2LATERAL VEHICLE DYNAMICS

With the vector of generalized speeds Vv, according to Equation (2.17), the mass matrix

M, defined in Equation (2.26), and the gyroscopic term b=[−˙

ψUy˙

ψUx0]Tthe equations

of motion (2.31) can be rearranged and written in matrix notation

MV˙

v+b(Vv)= FR

y



−b



89: ;

uncontrolled term

+FF

y



−sin δ

cos δ

acos δ



89: ;

coupled term

+



x+FF

xcos δ

xsin δ

aFF

xsin δ+MF

∆Fx+MR

∆Fx





89: ;

controlled term

=Qnc #FR

y(Vv)%+Qcoupl #FF

y(Vv),δ%+Qc(u),(2.32)

with u=[δ, FF

x,FR

x,∆FF

x,∆FR

x]Tbeing the vector of control inputs. All uncontrolled

quantities such as the lateral forces are collected in the generalized force term Qnc. The

generalized force term Qcoupl contains coupled controlled and uncontrolled quantities,

while Qconly depends on the vector of control inputs u. The simulation structure of the

nonlinear single-track model is summarized in Figure 2.14.

Figure 2.14: Simulation structure of the nonlinear single-track model

2.5 Linear Single-Track Model

In order to gain some insight by use of linear analysis techniques, the equations of motion of

the single-track model can be linearized for a prescribed longitudinal motion. The most

convenient choice is to assume the longitudinal velocity to be almost constant and the

longitudinal forces to be small. In doing so, the longitudinal degree of freedom is removed

from the model. In the following, two different versions of the linearized equations of

motion are presented. First, the equations of motion are formulated in the vehicle-fixed

reference frame !

V . The remaining two degrees of freedom of the lateral motion are

2.5 LINEAR SINGLE-TRACK MODEL 33

governed by the lateral velocity Uyand the yaw rate ˙

ψ, respectively. Second, a formulation

in the trajectory-fixed Frenet frame !

T is given. The corresponding states of the model

are the side slip angle βand the yaw rate ˙

ψ.

2.5.1 Vehicle-Fixed Formulation

For a prescribed longitudinal motion ˙

Ux≈0, the Newton-Euler equations (2.31) in Vex-

direction can be ignored; the remaining equations of lateral motion are

m.˙

Uy+˙

ψUx/=FR

y+FF

xsin δ+FF

ycos δ

Iz¨

ψ=aFF

xsin δ+aFF

ycos δ−bFR

y+MF

∆Fx+MR

∆Fx.(2.33)

Assuming that for slowly varying or constant longitudinal speed Ux, the longitudinal forces

approximately vanish, FF

x≈FR

x≈0, the equations of motion (2.33) simplify further

m.˙

Uy+˙

ψUx/=FR

y+FF

ycos δ

Iz¨

ψ=aFF

ycos δ−bFR

y+MF

∆Fx+MR

∆Fx.(2.34)

Replacing the lateral tire forces FF

yand FR

yby use of the linear Dugoffmodel, FF

y=−CF

ααF

and FR

y=−CR

ααR, see Equation (2.13), applying the small angle approximations

cos δ≈1,

αF= arctan 6Uy+˙

ψa

Ux7−δ≈Uy+˙

ψa

−δ,

αR= arctan 6Uy−˙

ψb

Ux7≈Uy−˙

ψb

(2.35)

for the steering and the side slip angles, respectively, the equations of motion (2.34) become

m.˙

Uy+˙

ψUx/=−CR

α<Uy−˙

ψb

Ux=−CF

α<Uy+˙

ψa

−δ=

Iz¨

ψ=−aCF

α<Uy+˙

ψa

−δ=+bCR

α<Uy−˙

ψb

Ux=+MF

∆Fx+MR

∆Fx.

(2.36)

In matrix notation with v=0Uy,˙

ψ1Tand u=0δ, MF

∆Fx,M

∆Fx1T, the equations of

motion with respect to !

V read

M˙

v+0G(Ux)+D(Ux)1v=Bu (2.37)

with

M=>m0

0Iz?,G=>0mUx

D=1

Ux>CF

α+CR

αCF

αa−CR

αb

αa−CR

αbC

αa2+CR

αb2?and B=>CF

α00

αa11

(2.38)

34 2LATERAL VEHICLE DYNAMICS

The matrix Gis the gyroscopic matrix. Matrix Dis positive definite and therefore a

damping matrix. The according quadratic form is

vTDv =

α+.Uy+a˙

ψ/2+CR

α.Uy−b˙

ψ/2

>0∀*v*+=0.(2.39)

Rearranging Equation (2.37) yields the state space equations of the linear single-track

model formulated in !

V˙

x=VAVx+VBVu(2.40)

with

VA=





−CR

α+CF

αb−CF

αa

−Ux

bCR

α−aCF

IzUx

−b2CR

α+a2CF

IzUx





and VB=





m00

aCF





,(2.41)

being the system and the control matrix, respectively. The state vector Vx=vand the

vector of control inputs Vu=uare defined as above in Equation (2.37).

2.5.2 Trajectory-Fixed Formulation

Starting again from the Newton-Euler equations (2.31) considering only lateral motion

and assuming that the longitudinal tire forces vanish, FF

x≈FR

x≈0, leads together with

the small angle approximation, cosδ≈1, to

VaCG

y=FR

y+FF

xsin δ+FF

ycos δ

≈FR

y+FF

Iz¨

ψ=aFF

xsin δ+aFF

ycos δ−bFR

y+MF

∆Fx+MR

∆Fx

≈aFF

y−bFR

y+MF

∆Fx+MR

∆Fx.

(2.42)

Now the equations of motion are projected onto the Frenet frame !

T : (T∗,Tet,Ten,Teb)

whose origin T∗is located at the center of gravity of the vehicle CG.!

T is rotated by the

side slip angle βwith respect to !





Tet

Ten

Teb

=



cos βsin β0

−sin βcos β0

0 0 1 



8 9: ;

CT,V





Vex

Vey

Vez

.(2.43)

Therefore, the Tetaxis is always tangential to the path driven by the vehicle, while the

Tenaxis is normal to it. In the Frenet frame !

T , with the abbreviations Ut:= E

TvCG

tand

Un:= E

TvCG

n, the only nonzero component of the velocity of the center of gravity EvCG is

in Tet-direction

EvCG =UtTet.(2.44)

2.5 LINEAR SINGLE-TRACK MODEL 35

With the angular velocity E

TωT=#˙

β+˙

ψ%Tebof !

T with respect to !

E , the acceleration of

the center of gravity CG with respect to !

E can be expressed as

TaCG =

TvCG =

TvCG +E

TωT×E

TvCG

=˙

UtTet+Ut#˙

β+˙

ψ%Ten.(2.45)

Substituting the unit vectors of !

T by the unit vectors of the reference frame !

V , according

to Equation (2.43), Equation (2.45) becomes

VaCG =B˙

Utcos β−Ut.˙

β+˙

ψ/sin βC

8 9: ;

VaCG

Vex+B˙

Utsin β+Ut.˙

β+˙

ψ/cos βC

8 9: ;

VaCG

Vey.(2.46)

Substituting the lateral acceleration in the Newton-Euler equations (2.42) with the ex-

pression from Equation (2.46) and using the linear part of the Dugofftire model, Equa-

tion (2.13), with the small angle approximations for the tire side slip angles as given in

Equation (2.35), the equations of motion become

m.˙

Utsin β+Ut.˙

β+˙

ψ/cos β/=−CF

α<Uy+˙

ψa

−δ=−CR

α<Uy−˙

ψb

Ux=

Iz¨

ψ=−aCF

α<Uy+˙

ψa

−δ=+bCR

α<Uy−˙

ψb

Ux=+MF

∆Fx+MR

∆Fx.

(2.47)

As the velocity component Unin Ten-direction is zero, the velocities Uxand Uyin Equa-

tion (2.47) can be expressed by

Ux=Utcos β−Unsin β=Utcos β

Uy=Utsin β+Uncos β=Utsin β,(2.48)

therefore, the equations of motion for constant or slowly varying speed ˙

Ut≈0 become

mBUt.˙

β+˙

ψ/cos βC=−CF

α<Utsin β+˙

ψa

Utcos β−δ=−CR

α<Utsin β−˙

ψb

Utcos β=

Iz¨

ψ=−aCF

α<Utsin β+˙

ψa

Utcos β−δ=+bCR

α<Utsin β−˙

ψb

Utcos β=+MF

∆Fx+MR

∆Fx.

(2.49)

For small side slip angles, cos β≈1 and sin β≈β, the equation can be solved for ˙

βand

ψ. The linear state space equations of the single-track model formulated in !

T become

T˙

x=TATx+TBTu(2.50)

36 2LATERAL VEHICLE DYNAMICS

with

TA=





−CF

α+CR

αb−CF

αa

mU2

−1

bCR

α−aCF

−a2CF

α+b2CR

IzUt





and TB=





mUt

aCF





,(2.51)

being the system and the control matrix, respectively. The state vector and the vector of

control inputs read

Tx=Bβ,˙

ψCTand Tu=0δ, MF

∆Fx,M

∆Fx1T.(2.52)

2.6 Steering Characteristics

This section provides some terms and definitions based on the analysis of the linear single-

track model. First, the so-called Ackermann steering angle for cornering at low velocities is

introduced. Subsequently, the required amount of steering for driving at higher velocities is

addressed. In this context the terms oversteering,understeering, and selfsteering gradient

are defined for steady state cornering.

2.6.1 Ackermann Steering

For low velocities EvCG∗the cornering angles at the front and at the rear tires αFand

αRalmost vanish. For circular driving on a prescribed radius of curvature 1/κabout the

instantaneous center of curvature ICC with αF=αR= 0, the necessary steering angle

can be determined from the geometry as depicted in Figure 2.15.

Figure 2.15: Geometry of the Ackermann model

2.6 STEERING CHARACTERISTICS 37

The resulting steering angle is called Ackermann angle δA. Noteworthy, unlike most

wheeled robots, see for example Murray and Sastry (1994, [76]) or Siegwart and

Nourbakhsh (2004, [103]), road vehicles are not subjected to nonholonomic constraints

of the kind EvF·Fey= 0 and EvR·Rey= 0 that enforce the cornering angles to vanish.

Road vehicles have rubber tires and the velocity constraints are therefore dissolved by

elastic deformations. However, for low velocities, accompanied by low lateral forces, road

vehicles behave kinematically similar to nonholonomic rigid wheeled robots. Considering

both tracks of a vehicle, the Ackermann geometry would require different steering angles

at the inner and at the outer front wheels. However, for the single-track model the

Ackermann angle δArepresents a mean value

tan δA=a+b

&(1/κ)2−b2

.(2.53)

For small angles, tan δA≈δA, and with b << 1/κthe Ackermann angle becomes

δA=(a+b)κ.(2.54)

The vehicle side slip angle β0for vanishing cornering angles can be expressed by means

of the Ackermann angle, see for example Zomotor (1991, [128], p. 103)

β0=bκ=δA

a+b.(2.55)

This geometric result is consistent with the kinematic and the dynamic equations of the

linear single-track model provided in sections 2.5.1 and 2.5.2. Recalling the small angle

approximation for the tire side slip angles from Equation (2.35) and equating them to

zero gives

αF≈Uy+˙

ψa

−δ!

αR≈Uy−˙

ψb

=0.

(2.56)

Subtracting the second from the first equation in (2.56) leads to

δ=˙

ψ(a+b)

≈(a+b)κ=δA,(2.57)

with ˙

ψ=κUt=κUxcos β≈κUxfor steady state cornering on a circular track and small

side slip angles, cos β≈1.

2.6.2 Self Steering Gradient

Steady state cornering on a given radius of curvature and at fixed (low) velocity requires a

particular steering angle, the Ackermann angle δA, as discussed in Section 2.6.1. However,

if the velocity is gradually increased, the steering angle needs to be adapted. The amount

38 2LATERAL VEHICLE DYNAMICS

of additional steering angle to the Ackermann angle, which is necessary to remain on the

same radius of curvature, depends on the steering characteristics of the vehicle.

For steady state cornering, ˙

β= 0 and ¨

ψ= 0, on a given radius of curvature 1/κthe

steering angle δcan be obtained from the steady state of the equations of motion (2.50)







−CF

α+CR

αb−CF

αa

mU2

−1

bCR

α−aCF

−a2CF

α+b2CR

IzUt





<β

ψ=+





mUt

aCF









∆Fx



=<0

0=.

(2.58)

Eliminating the side slip angle βfrom the set of linear equations (2.58) and substituting the

yaw rate by ˙

ψ=κUtgives for the steering angle δfor pure steering, MF

∆Fx=MR

∆Fx= 0:

δ=(a+b)κ+m(CR

αb−CF

αa)

αCR

α(a+b)U2

tκ.(2.59)

The coefficient of the lateral acceleration E

TaCG

n=U2

tκin Equation (2.59) is called self

steering gradient SG

SG =m(CR

αb−CF

αa)

αCR

α(a+b),(2.60)

see (2004, [1]). With the Ackermann angle, as defined in Equation (2.54), Equation (2.59)

can be rewritten as

δ=δA+SG ·E

TaCG

n.(2.61)

The self steering gradient, SG =∂(δ−δA)/∂E

TaCG

n, describes the change in the steer-

ing angle with respect to the lateral acceleration. In other words, the product of the

self steering gradient SG and the lateral acceleration E

TaCG

n, which is proportional to the

squared speed U2

tof the vehicle for a fixed radius of curvature 1/κ, provides the amount

of additional steering angle to the Ackermann angle δA.

Figure 2.16: Steady state cornering on a large radius of curvature R

2.6 STEERING CHARACTERISTICS 39
The steering characteristics and therefore the self steering gradient SG of a vehicle are
related to the difference between the cornering angles at the front and at the rear tires,
αF−αR. This fact is highlighted in the following. In steady state cornering the inertial
force FCG at the center of gravity CG is balanced by the lateral tire forces FF
yat the front
and FR
yat the rear axle as depicted in Figure 2.16. Notice that the lateral tire forces are
in positive direction but the slip angles are negative.
With the lateral acceleration, E
TaCG
n=U2
tκ, the equilibrium of forces and moments
yields the following approximation, which holds for large radii of curvature 1/κ,
FF
y≈mU2
tκb
a+band FR
y≈mU2
tκa
a+b.(2.62)
Recalling the lateral tire forces in the linear region from the Dugoffmodel, see Equa-
tion (2.13), the lateral forces in Equation (2.62) can be substituted yielding the steady
state cornering angles for a vehicle cornering about the instantaneous center of curvature
(ICC) as depicted in Figure 2.16:
αF=m
CF
α
b
a+bU2
tκand αR=m
CR
α
a
a+bU2
tκ.(2.63)
Therefore, the difference between the cornering angles at the front and the rear tires can
be written as
αF−αR=m(CR
αb−CF
αa)
CF
αCR
α(a+b)U2
tκ.(2.64)
Comparing Equation (2.64) to Equation (2.60) reveals that the product of the self steering
gradient SG and the lateral acceleration E
TaCG
nequals the difference of the cornering angles
at front and rear tires
αF−αR=SG ·E
TaCG
n.(2.65)
Thus, the steering angle becomes, according to Equation (2.61),
δ=δA+SG ·U2
tκ=δA+#αF−αR%.(2.66)
The difference of the cornering angles at the front and at the rear axles equals the
amount of additional steering angle to the Ackermann angle δA. Equation (2.66) is a
linear equation with the slope SG. Figure 2.17 shows the steering angle δdepending on
the lateral acceleration for three different configurations: SG < 0, SG = 0 and SG > 0.
In the linear region SG is a constant, in the nonlinear region SG changes due to the tire
characteristics, see Figure 2.6.

40 2LATERAL VEHICLE DYNAMICS

Figure 2.17: Self steering gradient according to Zomotor (1991, [128], p. 109)

Concluding, the steering characteristics can be interpreted by means of the self steering

gradient and categorized as understeering,neutral steering and oversteering:

•SG > 0(CF

αa−CR

αb<0) understeering

The steering angle needs to be increased with increasing velocity to stay on a con-

stant radius of curvature 1/κ. The vehicle tends to drift straight over the front tires.

The cornering angles at the front tires are higher than at the rear tires (αF−αR>0).

•SG = 0 (CF

αa−CR

αb= 0) neutral steering

The steering angle is independent of the velocity and identical to the Ackermann

angle δ=δA. The cornering angles at the front and at the rear tires are equal

(αF−αR= 0).

•SG < 0(CF

αa−CR

αb>0) oversteering

The steering angle needs to be decreased with increasing velocity to stay on a con-

stant radius of curvature 1/κ. The vehicle tends to break out at the rear wheels. The

cornering angles at the rear tires are higher than at the front tires (αF−αR<0).

2.7 MODEL VALIDATION WITH EXPERIMENTAL DATA 41

2.7 Model Validation with Experimental Data

In this section, experimental test data1are used as benchmark for the nonlinear single-

track model, given in Section 2.4, and for the linear single-track model as presented in

Section 2.5.1. The test drives were performed with the steer-by-wire test vehicle P1 shown

in Figure 2.18.

Figure 2.18: Stanford steer-by-wire test vehicle P1, Laws et al. (2005, [70])

The experimental vehicle P1 features independent left and right front wheel steering

mechanisms as well as independent electric rear-wheel drive. Measurements are taken by

various sensor systems as for example INS and GPS. The most important measurements

that are used as reference to evaluate the vehicle dynamics models are the speed of the

vehicle Uref

x, the left and right steering angles δref

Land δref

R, the yaw rate ˙

ψref, and the lateral

acceleration ˙

Uref

y. Table 2.1 summarizes the vehicle parameters used for the simulations.

Table 2.1: Vehicle parameters

Parameter Symbol Value

Mass m1700 kg

Yaw moment of inertia Iz2500 kgm2

Front axle distance to CG a1.33 m

Rear axle distance to CG b1.17 m

Track width d1.62 m

Cornering stiffness front tires CF

α44000 N/rad

Cornering stiffness rear tires CR

α63000 N/rad

Longitudinal stiffness front tires CF

x69000 N

Longitudinal stiffness rear tires CR

x97000 N

Effective roll radius R00.32 m

Wheel moment of inertia Iwheel 0.9 kgm2

Adhesion coefficient µ00.8. . . 0.9

1The experimental test data are provided by courtesy of the Stanford Dynamics Design Laboratory

headed by Professor J. Christian Gerdes.

42 2LATERAL VEHICLE DYNAMICS

For both models the average value

δref(t)=δref

L(t)+δref

R(t)

2(2.67)

of the measured left and right wheel steering angles is prescribed. Besides the steering

angle ¯

δref (t) the measured speed Uref

x(t) of the vehicle is tracked. In case of the nonlinear

model the prescribed velocity profile is tracked by a PID-loop that controls the driving

and braking torques

Md/b(t)=KP0Uref

x(t)−Ux(t)1+KI

Uref

x(τ)−Ux(τ)dτ+KD

dt 0Uref

x(t)−Ux(t)1.

(2.68)

As the test vehicle P1 is rear wheel driven, the torque Md/b is applied solely to the rear

wheels. Equation (2.68) implies that the torque is generated without time delay, which

means that no powertrain is modeled. However, the focus of the model lies on lateral

vehicle dynamics and the PID-loop only intends to track the velocity profiles of the test

drives.

For the linear model the speed is treated as time varying parameter. Thus, the state

space model, Equation (2.40), with MF

∆Fx=MR

∆Fx= 0, reads

<˙

ψ==





−CF

α+CR

mUref

x(t)

αb−CF

αa

mUref

x(t)−Uref

x(t)

bCR

α−aCF

IzUref

x(t)−a2CF

α+b2CR

IzUref

x(t)





<Uy

ψ=+





aCF





¯

δref(t).(2.69)

The simulation data are compared to the experimental data incorporating the following

criteria proposed by Frik (1994, [39]):

+j,r =

max

i=1...n.$$xij −xref

ij $$/

max

i=1...n$$xref

ij $$(2.70)

¯+j=1/n

i=1 $$xij −xref

ij $$(2.71)

¯+j,r =!n

i=1 $$xij −xref

ij $$

i=1 $$xref

ij $$,(2.72)

with inumbering the data in test drive j. The relative error +j,r describes the maximal

local deviation of the simulation results xij from the measurements xref

ij with respect to

the maximum of the experimental reference in the considered test drive j. The quality of

the global solution is addressed by the mean error ¯+jand the mean relative error ¯+j,r.

2.7 MODEL VALIDATION WITH EXPERIMENTAL DATA 43

In the following, steady state and unsteady state driving maneuvers are analyzed. In

doing so, the criteria in equations (2.70)-(2.72) are evaluated for different test maneuvers

j, for the yaw rate as well as for the lateral acceleration, xij ∈F˙

ψ(ti),˙

Uy(ti)G.

2.7.1 Steady State Cornering Maneuver

In this section experimental data of a steady state cornering maneuver are used to validate

the linear and the nonlinear single-track model. Figure 2.19 shows the steering angle ¯

δref (t)

and the velocity profile Uref

x(t) of the maneuver. The vehicle accelerates from rest to a

constant velocity of approximately 15m/s. The test drive includes two periods of constant

steering angle inputs of 5◦and −5◦, respectively. The steady state phases are highlighted

by gray boxes.

Figure 2.19: Steady state cornering maneuver

Therein, the phases of steady state cornering correspond to circles driven to the left in

case of the positive steering angle and driven to the right for the negative steering angle.

The radii of the circles vary between 30m and 35m.

The data of the steady state cornering maneuver were used to identify the tire para-

meters of the models given in Table 2.1. In order to do so, the ratio of the measured yaw

rate ˙

ψref and the steering angle ¯

δref in steady state cornering was compared to the steady

state yaw rate gain of the linear model, see Mitschke and Wallentowitz (2004,

[74], p. 563),

6˙

ψref

δref 7ss

a+b

Uref

1+(Uref

Uchar )2with U2

char := a+b

SG =CF

αCR

α(a+b)2

m(CR

αb−CF

αa).(2.73)

Inspecting the self steering gradient SG, the ratio of the cornering stiffnesses CF

αand CR

can be adjusted in order to satisfy Equation (2.73). Whereas, the vehicle mass mand the

distances aand bto the center of gravity CG are known from direct measurements.

44 2LATERAL VEHICLE DYNAMICS
The simulation results as well as the measured data for the yaw rate and the lateral
acceleration are given in Figure 2.20.
Figure 2.20: Model comparison for steady state cornering maneuver
Noteworthy, that in the acceleration phase of the test drive, at approximately 5s, the
yaw rate and the lateral acceleration simulated with the nonlinear vehicle model deviate
more from the measurement results than the simulation results of the linear model. This
is due to the fact that the nonlinear tire model combines longitudinal and lateral tire
forces. The longitudinal tire forces are influenced by the driving torque, generated by the
PID-loop, Equation (2.68). Thus, lateral vehicle dynamics are affected when the velocity
of the vehicle changes. Besides that, the accuracy of the nonlinear tire model decreases
at low velocities; for zero velocity the longitudinal slip is not defined.
In the steady state phases, highlighted by gray boxes, it can be observed that the
linear single-track model generates slightly too high absolute values for the yaw rate
as well as for the lateral acceleration. The linearization of the lateral tire forces with
respect to the cornering angles is valid up to approximately one third of the maximum
tire forces, which corresponds to a lateral acceleration of about 4 m/s2, see Mitschke
and Wallentowitz (2004, [74], p. 36-38 and p. 560). In the steady state phases of the
test drive the lateral acceleration lies between 5 and 6 m/s2as presented in the lower part
of Figure 2.20. Therefore, the tires leave the linear region and the linear tire model tends
to generate too high lateral forces, which results in the increased values for yaw rate and
lateral acceleration.
Table 2.2 shows the criteria defined in equations (2.70)-(2.72), evaluated for yaw rate
and lateral acceleration as given in Figure 2.20. The lower relative errors +j,r of the linear
single-track model compared to the relative errors of the nonlinear model document the
deviations in the acceleration phase of the test drive. However, the values of the mean
error ¯+jand of the relative mean error ¯+j,r indicate the higher quality of the global solution
of the nonlinear model, which corresponds to the more accurate results in the phases of
steady state cornering in Figure 2.20.

2.7 MODEL VALIDATION WITH EXPERIMENTAL DATA 45

Table 2.2: Quantitative model comparison for steady state cornering

Linear Nonlinear

Error Single-track model Single-track model

Yaw rate

+j,r 0.22 0.45

¯+j0.04 0.03

¯+j,r 0.13 0.08

Lateral acceleration

+j,r 0.20.57

¯+j0.49 0.45

¯+j,r 0.11 0.09

2.7.2 Unsteady Cornering Maneuver

While in the last section a steady state cornering maneuver was surveyed, this section

deals with a maneuver where the vehicle gradually approaches the limits of adhesion and

finally breaks out. Figure 2.21 shows the steering angle ¯

δref (t) and the velocity profile

Uref

x(t) of the test drive. In the highlighted part of the test drive the speed of the vehicle

Uref

x(t) is almost constant at 17m/s while the steering angle ¯

δref (t) is gradually increased

from 0◦to 11◦.

Figure 2.21: Unsteady maneuver with break out

Subsequently, Figure 2.22 provides the yaw rate and the lateral acceleration of the test

drive and the corresponding simulations with the linear and the nonlinear model.

46 2LATERAL VEHICLE DYNAMICS

Figure 2.22: Model comparison for unsteady maneuver with break out

Focusing the observations on the highlighted phase, it can be noticed that yaw rate

and lateral acceleration initially increase together with the steering angle. Then, the

tires start to saturate and the force characteristic becomes highly nonlinear. Further

increments of the steering angle now cause smaller increments of the yaw rate and the

lateral acceleration. Subsequently, the tires exceed their limits of adhesion and the vehicle

breaks out. The break out is indicated by a step in the yaw rate, which means that the

vehicle starts spinning. In the end, the vehicle is decelerated to rest.

The nonlinear model nicely captures the qualitative behavior of the vehicle cornering

at the limits of adhesion. The remaining differences are partly caused by the velocity

profile of the test drive that is only approximated in the simulations by the PID-loop

in Equation (2.68). However, with rising lateral accelerations effects such as roll and the

associated shifting of weight become more evident and reduce the quality of the simulation

results. These effects can only be captured by more detailed models, that on the other

hand require more simulation time.

Yaw rate and lateral acceleration simulated with the linear model constantly increase

with the steering angle and show no saturation effects. Thus, the break out of the vehicle

is not captured by the linear model. Comparing the experimental data in Figure 2.22

with the simulation results of the linear model illustrates the effect of linearization of the

tire forces in the nonlinear region.

The better solution quality of the nonlinear model becomes evident by comparing the

errors given in Table 2.3. Therein, the high relative errors +j,r for the yaw rate and the

lateral acceleration indicate the high deviations of the linear model from the experimental

data at high lateral accelerations. The better global solution quality of the nonlinear

model is documented by the lower mean and relative mean errors ¯+jand ¯+j,r.

2.7 MODEL VALIDATION WITH EXPERIMENTAL DATA 47

Table 2.3: Quantitative model comparison for unsteady state cornering

Linear Nonlinear

Error Single-track model Single-track model

Yaw rate

+j,r 4.58 0.9

¯+j0.15 0.12

¯+j,r 0.98 0.37

Lateral acceleration

+j,r 3.49 1.52

¯+j0.99 0.67

¯+j,r 1.14 0.54

The simulations above were performed with an adhesion coefficient of µ0=0.8. To

illustrate how sensible the simulation results are to parameter variations the simulation

with the nonlinear model is repeated for µ0=0.9. The resulting yaw rates, presented in

Figure 2.23, show that the adhesion coefficient mainly affects the part of the maneuver

where the vehicle breaks out. The behavior of the vehicle cornering at high lateral accele-

rations but not exceeding the limits of adhesion is still captured. However, obviously the

model does not break out when the adhesion coefficient is increased to µ0=0.9. In this

case, the simulated maximal lateral force is too high.

Figure 2.23: Influence of the adhesion coefficient

Summarizing, the comparison with experimental data confirms the well known state-

ment that the linear model is only reliable up to a lateral acceleration of approximately

0.4g, which is consistent with the literature, e.g. Mitschke and Wallentowitz

(2004, [74]). Beyond 0.4gthe linear model tends to generate too high lateral forces.

For trajectory following this means for example that the linear model tends to follow a

given trajectory at high lateral acceleration better than a real vehicle. The nonlinear

vehicle model includes saturation effects at the tires and therefore generates better results

at high lateral accelerations. However, as indicated by the variation of the adhesion coef-

ficient, µ0, the simulation results at the limits of adhesion are very sensitive to parameter

variations.

Chapter 3

Potential Field based Motion

Planning

Ahazard map and a so-called elastic band are the main elements of the proposed potential

field based motion planning. The hazard potential RVhaz encodes the available sensor in-

formation about the environment of the host-vehicle in terms of potential fields. Therefore,

RVhaz is composed of information about the course of the road, represented by its center-

line Rand its borders ∂Bland ∂Br, information about the obstacles Oj,j=1, . . . , M,

referring to other traffic participants as well as to static objects in the environment, and

information about the host-vehicle V. Besides that the hazard map contains information

about the motion of the host-vehicle and provides a spatial as well as a temporal repre-

sentation of the traffic situation. The elastic band consisting of coupled nodes P0, . . . , PN

scans the hazard map, like a virtual antenna of an insect, for trajectories of low hazard lev-

els. The combination of external hazard potentials and internal potentials that couple the

nodes of the elastic band cause trajectories with inherent low curvatures. The composed

potential field motion planning module is depicted in Figure 3.1, see also Figure 1.9.

Figure 3.1: Motion planning module composed of hazard map and elastic band, based on

sensor information collected at the planning instant t0

50 3POTENTIAL FIELD BASED MOTION PLANNING

In the sense of shared vehicle guidance it is mandatory that the driver has influence on

the motion planning procedure: by setting the accelerometer and braking pedal, which

results in driving and braking torques Md/b, the driver prescribes the velocity profile

Ut(t) of the longitudinal motion of the host-vehicle. In doing so, he may be assisted by

systems such as Adaptive Cruise Control or the Emergency Brake. Besides that the driver

influences the motion planning directly by communicating his intended driving maneuver

via the steering angle δ. The result of the motion planning process is a collision-free

trajectory represented by the position vectors of the nodes of the elastic band.

Similar to driving strategies of human drivers, the motion planning concept introduced

in the following is predictive and incorporates extrapolation procedures to anticipate the

evolution of the traffic scenario. However, to be able to refine the extrapolation of the

traffic scenario and to be able to react to unpredicted events such as new objects within

the detection range, the motion planning is repeated in time intervals of ∆T, where the

planning horizon ∆Tdepends on the update frequency of environmental sensor data. The

principle procedure of handling changes in the environment is sketched in Figure 3.2. The

internal coupling potentials are symbolized by springs connecting the nodes P0, . . . , PNof

the elastic band. In the upper part of the figure the hazard map consists only of potentials

due to the borders of the road, which are also symbolized by means of roundish springs;

the elastic band is in an equilibrium configuration on the right lane. The obstacle on the

center of the right lane is not discovered by the environmental sensors yet. Then, at time

t0, the obstacle is detected and thus contributes to the hazard map. As a consequence of

the changes in the hazard potential, the elastic band is shifted into a new collision-free

equilibrium configuration. The computed trajectory remains valid until new sensor data

are available at time t0+∆T.

Figure 3.2: Elastic band principle

It should be mentioned that the elastic band approach originally introduced by Quin-

lan and Khatib (1993, [87]) for robotic path planning had to be extended significantly

for automotive applications. Examples are the concept of finding an equilibrium configu-

ration of the elastic band in a hazard map, the incorporation of obstacle motion and the

interaction with the driver. However, as the approach presented here also relies on the

elastic coupling of nodes of a virtual antenna exposed to potential fields, the antenna is

also referred to as elastic band.

In this approach, the computation of collision-free trajectories on each planning horizon

includes many different elements that are closely coupled. The hazard map for example

contains extrapolated trajectories of the obstacles, which are evaluated at discrete instants

in time ti. These instants tiin turn are prescribed by the host-vehicle passing the nodes

of the elastic band. As the velocity profile driven by the host-vehicle Ut(t) is determined

by the driver, there is also an interplay with the driver. The shape of the elastic band

also influences the instants tiwhen the host-vehicle passes the nodes P0, . . . , PN. Due to

these mutual dependencies the proposed trajectory is computed by an iterative procedure,

which will be outlined in this chapter.

The introduction of the different motion planning elements is structured as follows:

First, the elements of the hazard map such as the road and the obstacles with the cor-

responding hazard potentials are presented. Then, the formulation of the elastic band

with the internal coupling potentials is given. Subsequently, the interfacing elements be-

tween driver and motion planning are introduced. Next, computational aspects such as

distance computations and the iterative solution of the equilibrium configuration of the

elastic band are addressed. Then, sample simulation results illustrate the application of

the method. Finally, the entire motion planning procedure is summarized as pseudo code.

Figure 3.3: Reference frames and coordinates used in the motion planning process

52 3POTENTIAL FIELD BASED MOTION PLANNING

The reference frames employed in the motion planning process are depicted in Figure 3.3

and are listed in the following :

E : (E∗,Eex,Eey,Eez)"=Earth-fixed reference frame

R : (R∗,Rex,Rey,Rez)"=Road-fixed reference frame fixed at planning time t0

R:(

R∗,¯

Ret,¯

Ren,¯

Rez)"= Auxiliary reference frame shifted tangentially along the

road centerline R

V : (V∗,Vex,Vey,Vez)"=Vehicle-fixed reference frame

Oj:(O∗

j,Ojex,Ojey,Ojez)"=Obstacle-fixed reference frame

As shown in Figure 3.3 the reference frame !

R is oriented in tangent direction of the

road centerline R. It is fixed at the instant of planning t0and remains fixed for the entire

planning horizon ∆T. In the next instant of planning, t0+∆T,!

R is replaced according

to the new position of the host-vehicle. Meanwhile, !

R serves as inertial frame during each

planning horizon. If not stated otherwise, position vectors are denoted with respect to

R∗, e.g. rP=rR∗,P , and *·*denotes the 2-norm in distance computations. The following

abbreviation is used for derivatives with respect to Rx:(·)#=∂

∂Rx(·) or (·)#=d

dRx(·),

respectively.

3.1 Hazard Map

The environmental model represented by the hazard map is composed of potential fields

capturing the course of the road and the motion of obstacles. In the following the un-

derlying models of road geometry and obstacle motion in terms of potential fields are

provided.

3.1.1 Road

In most countries roads are build according to national guidelines. In Germany for exam-

ple, the construction of roads is directed by the ”Richtlinien f¨

ur die Anlage von Straßen

(RAS-L)” (1999, [3]). The knowledge of road construction is used by intelligent environ-

mental sensor systems to identify the course ahead of the vehicle. A broad overview of

vision based automotive environmental detection is given by Dickmanns (2002, [24]).

Road Centerline

According to (1999, [3]), German roads are designed to be continuous in curvature. In

doing so, roads are composed of straight lines, circular arcs and so-called clothoids that

are used as transition curves, where the curvature κof a clothoid varies linearly with its

arc length. Therefore, in each point Pof a clothoid

A2=s·r(3.1)

holds with sbeing the arc length, rbeing the instantaneous radius of curvature, and A

denoting the clothoid parameter; A= 1 characterizes an unit clothoid. From the geometry

3.1 HAZARD MAP 53

illustrated in Figure 3.4 and with Equation (3.1) the following equation arises

ds =r·dτ=A2

sdτ⇒dτ=s

A2ds . (3.2)

Figure 3.4: Clothoid transition element

Integration of Equation (3.2) from arc length s0= 0 to sand assuming that the clothoid

starts in Rex-direction and therefore integrating the gradient angle from τ0= 0 to τyields

τ(s)= s2

2A2.(3.3)

Therefore, differential changes in the coordinates of a point Pon the clothoid projected

on !

R read

dRx= cos (τ(s)) ds = cos (s2

2A2)ds and dRy= sin (τ(s)) ds = sin (s2

2A2)ds . (3.4)

For a clothoid starting in R∗in Rex-direction, τ(0) = 0, Rx(0) = 0, Ry(0) = 0, see

Figure 3.4, tangent orientation and position on the clothoid element in !

R thus become

τ(s)=

κ(¯s)d¯s, (3.5)

Rx(s)=

cos τ(¯s)d¯s=

cos (¯s2

2A2)d¯sand (3.6)

Ry(s)=

sin τ(¯s)d¯s=

sin (¯s2

2A2)d¯s. (3.7)

As the curvature of a clothoid varies linearly with its arc length, κ=κ0+dκ·s, where the

rate of curvature depends on the clothoid parameter, dκ=1/A2, Equation (3.5) yields

the tangent orientation on a clothoid with initial curvature κ0

τ(s)=κ0·s+dκ

2s2.(3.8)

54 3POTENTIAL FIELD BASED MOTION PLANNING
The integrals in equations (3.6) and (3.7) are known as FRESNELs integrals, see for
example Bronstein and Semendjaev (1997, [11], p. 647), and cannot be solved in
closed form. Therefore, Taylor series expansions or numerical integration techniques have
to be applied. However, in practice road curvatures especially on highways are very low.
Table 3.1 gives admissible minimal radii of curvature depending on the vehicles speed
according to (1999, [3]).
Table 3.1: Minimal admissible radii of curvature and arc length of clothoid transition
elements depending on the speed according to (1999, [3], Tab. 5)
Ut[km/h] Minimal
radius [m]
Minimal arc
length [m]
50 80 30
60 120 35
70 180 40
80 250 45
90 340 50
100 450 55
120 720 65
For small initial curvatures κ0and curvature rates dκsmall angle approximations,
cos τ≈1 and sin τ≈τ, can be applied to equations (3.6) and (3.7) and yield by means
of integration together with Equation (3.8)
Rx(s)=s, (3.9)
Ry(s)=1
2κ0·s2+1
6dκ·s3.(3.10)
Summarizing, the course of the road, represented by its centerline Rin the road-fixed
reference frame !
"
R , can be approximated for low initial curvatures κ0and curvature rates
dκby
Ry(Rx)=1
2·κ0·Rx2+1
6·dκ·Rx3.(3.11)
In doing so, straight sections and circular arcs are special cases of Equation (3.11) with
κ0= 0, dκ= 0 and κ0+= 0, dκ= 0, respectively.
Considering the curvature parameters of the road to be a function of the distance along
the road, they are usually estimated recursively by use of Kalman filtering techniques. One
of the first applications for well-structured roads was shown by Dickmanns and Mys-
liwetz (1992, [25]). Meanwhile similar algorithms were also applied to less structured
environments; the polynomial in Equation (3.11) provided for example the basis for the
measuring system used by Creaman and Murray (2006, [22], Eqn. (5)). Therefore,
Equation (3.11) is used to model the estimated course of the road centerline Rin the
hazard map. However, the underlying low curvature assumptions apply predominantly
to highways. For roads with higher curvatures, e.g. in urban environments, the model
should be extended by considering higher order terms of the Taylor series expansions of
equations (3.6) and (3.7).

3.1 HAZARD MAP 55

Borders of the Road

According to the discussion above the road centerline R, estimated by environmental

sensors, is described by the third order polynomial in Equation (3.11) with respect to the

road-fixed reference frame !

R on each planning horizon ∆T. Thus, the position vector to

an arbitrary point ¯

R∗on Rcan be described in cartesian coordinates with respect to !

With an estimation of the road width band the auxiliary reference frame ¯

Rlocated in ¯

R∗

and with normal-tangential coordinates along the course of the road

Ret:= r¯

R∗#

$$r¯

R∗#$$=Rex+Ry¯

R∗#Rey

'1+(Ry¯

R∗#)2,¯

Ren:= −Ry¯

R∗#Rex+Rey

'1+(Ry¯

R∗#)2,(3.12)

the corresponding points at the borders of the road are given by equations (3.11) and

(3.12)

rBq(Rx)=r¯

R∗(Rx)±b

2¯

Ren(Rx) with q∈{l,r}.(3.13)

Road Potential Field

Potential fields are assigned separately to the left and to the right border of the road ∂Bl

and ∂Br, respectively. To evaluate the potential for a node of the elastic band Piwith

position vector ri, see Figure 3.5, the position vectors rBq

ito the corresponding points at

the borders of the road need to be defined

rBq

i:= HrBq∈∂Bq|I

Iri−rBqI

I= min

Rx∈[0,L]I

Iri−rBq(Rx)I

IJ.(3.14)

Solving the minimum problem (3.14), see Equation (3.52), the potential of the borders

of the road experienced by a node Pican be formulated with respect to the road-fixed

reference frame !

RVBq

i=RVBq.ri,rBq

i/.(3.15)

Figure 3.5: Road potential field

56 3POTENTIAL FIELD BASED MOTION PLANNING

Adjusting the Potential Field Minimum

The minimum of the road potential field defines the course of minimal hazard along the

road in absence of obstacles. By scaling the potentials of left and right border ∂Bl,∂Br

separately, adjusting the location of the minimum is a design choice. On a road with two

lanes the minimum could for example be placed on the right lane. However, driving on the

center of the road does not reflect the behavior intended by most of the drivers. In general,

drivers tend to guide the vehicle closer to the road centerline than to the border of the

road. On curved roads the desired lateral position depends on the curvature; Mitschke

and Wallentowitz (2004, [74], pp.653) summarize some empirical studies on human

driving behavior and give a linear relation of the intended lateral offset to the center of the

lane and the curvature κof the road. The curvature κof the road follows from the second

derivative of the describing polynomial in Equation (3.11) with respect to Rx. One choice

to design the potential field of the borders of the road is the logarithmic formulation

RVBq

i=−kBq·ln I

Iri−rBq

I,kBl

kBr=dl

,(3.16)

with dland drbeing the distance of the minimum of the potential to the left and to the

right border ∂Bland ∂Br, respectively. Placing the minimum on the center of the right

lane on a two lane road with width bwould for example lead to kBl

kBr=0.75b

0.25b. In general,

kBland kBrcan be subjected to changes depending on the curvature of the road or due

to strategic decisions as for example lane change maneuvers indicated by the driver.

3.1.2 Obstacles

The anticipated behavior of the obstacles Oj,j=1, . . . , M, is modeled based on sensor

information provided at the instant of planning t0. In general, the following informa-

tion are available after fusion of different environmental sensor systems, see for example

Mildner (2004, [73], p. 22-24):

•position: RrO∗

j(t0),

•velocity: RvO∗

j(t0),

•acceleration: RaO∗

j(t0),

•angular velocity: RωOj(t0),

•orientation: RψOj(t0) and

•geometry: length lOjand width wOjor diameter dOj.

If no information about the acceleration of an obstacle Ojis available, the velocity RvO∗

is assumed to be constant on each planning horizon ∆T.

In robotics growing methods are used to virtually enlarge the dimensions of the robot

or obstacles; examples are given by Sparbert, Kumpel and Hofer (2001, [107]). Here,

a modified growing method is applied to each obstacle yielding a safety area as illustrated

in Figure 3.6. According to the predominant shape of an obstacle it can for example be

3.1 HAZARD MAP 57

approximated by a circular or by a rectangular geometry. The safety area is then created

in such a way that obstacle and host-vehicle do not collide if a reference point on the

planned trajectory of the host-vehicle does not penetrate the safety area. In doing so,

only the distances of points Pi,i=0, . . . , N, on the planned trajectory to the boundary

of the safety area ∂Ojhave to be checked in order to detect collisions. The point ˆ

denotes the point on the border of the safety area ∂Ojwith minimal distance to a point

Pi. In contrary to this modified growing procedure that marks non-valid areas of the traffic

space Quinlan and Khatib (1993, [87]) used a so-called bubble concept, where bubbles

indicated obstacle-free areas.

Figure 3.6: Obstacle geometry and according safety area

The orientation of the obstacle is reflected by the orientation of the obstacle-fixed

reference frame !"

Ojwith respect to the road-fixed reference frame !

R . The rotational

transformation between the two reference frames is given by

0Ojex,Ojey,Ojez1T=COj,R [Rex,Rey,Rez]T(3.17)

with

COj,R =



cos RψOjsin RψOj0

−sin RψOjcos RψOj0

0 0 1 

.(3.18)

For the motion planning, the future configuration of the obstacles rO∗

j(t),RψOj(t) with

t∈0t0,t

f1and j=1, . . . , M is extrapolated based on the initial data given in the listing

above. Assuming that the side slip angle of the obstacle along its future path is small,

βOj≈0, the orientation of the unit vectors Ojex(t) and Ojey(t) with respect to the

predicted path is tangential and normal, respectively. In the extrapolation procedure two

categories of obstacles are distinguished: either an obstacle is assumed to stay in its lane

or it is assumed to depart from it. If the yaw angle RψOj(t0) of an obstacle, depicted in

Figure 3.7, at the instant of planning t0deviates only slightly from the orientation of the

road tangent Rψ¯

R(RxO∗

j),

|RψOj(t0)−kπ−Rψ¯

R(RxO∗

j(t0)) |<+with Rψ¯

R= arctan (Ry¯

R∗(RxO∗

j(t0))

RxO∗

j(t0)),

(3.19)

58 3POTENTIAL FIELD BASED MOTION PLANNING

the obstacle is assumed to stay in its lane, where k= 1 denotes oncoming traffic and

k= 0 indicates obstacles moving in the same direction as the host-vehicle V.

Figure 3.7: Obstacle orientation with respect to the road

In the other case, the motion of the obstacle is extrapolated based on the measured

state at the planning time t0. Both motion pattern of obstacles are principally sketched in

Figure 3.8. Mildner (2004, [73]) gives a range of 10◦. . . 20◦for the threshold +indicating

lane departure intention based on empirical data. More detailed models of driver’s lane

change behavior are provided by Ehmanns (2003, [31]).

Figure 3.8: Obstacle extrapolation in lane and with lane departure

Extrapolation In Lane

The motion rO∗

j(t) of an obstacle staying in its lane is derived now. At the instant of

planning t0the measured obstacle position is given by

rO∗

j(t0)= RxO∗

j(t0)Rex+RyO∗

j(t0)Rey.(3.20)

Assuming small curvatures, RxO∗

j(t0)≈Rx¯

R∗holds, compare to Figure 3.8. With the

road centerline Ras given in Equation (3.11), this results in the position of the obstacle

3.1 HAZARD MAP 59

along its future path

rO∗

j(RxO∗

j)=r¯

R∗(RxO∗

j)+BRyO∗

j(t0)−Ry¯

R∗#RxO∗

j(t0)%C¯

Ren(RxO∗

j),(3.21)

based on the initial obstacle coordinates (RxO∗

j(t0),RyO∗

j(t0)), obtained from the sensor

data. The corresponding yaw angle is given by

RψOj(RxO∗

j)= Rψ¯

R(RxO∗

j)+kπ.(3.22)

Next, the time dependent path extrapolation RxO∗

j(t) has to be determined. The arc

length sO∗

j(RxO∗

j) covered by an obstacle follows from integration of Equation (3.21) with

respect to RxO∗

sO∗

j(RxO∗

j) = sgn.RvO∗

j(t0)·¯

Ret(RxO∗

j)/·

RxO∗

j(t0)

*(rO∗

j)#(ξ)*dξ.(3.23)

The integral in Equation (3.23) can for example be solved by truncating a Taylor series

expansion. Defining the initial conditions at the planning instant,

vO∗

j(t0) :=RvO∗

j(t0)·¯

Ret(RxO∗

j) and aO∗

j(t0) :=RaO∗

j(t0)·¯

Ret(RxO∗

j),(3.24)

and integrating them with respect to time yields the arc length

sO∗

j(t)=1

2aO∗

j(t0)t2+vO∗

j(t0)t. (3.25)

The values of the arc length given in equations (3.23) and (3.25) must be equal

sO∗

j(t)−sO∗

j(RxO∗

j) = 0 ,(3.26)

thus, the relation RxO∗

j(t) is found implicitly. To any time instant t, the corresponding

position of the obstacle RxO∗

jcan be computed by solving the algebraic Equation (3.26).

However, with the small curvature assumption, xO∗

j(t)≈sO∗

j(t)+xO∗

j(t0), the time de-

pending position of an obstacle in its lane can be determined by plugging Equation (3.25)

into Equation (3.21).

Extrapolation With Lane Departure

If the relative angle between the orientation of the obstacle and the road, given in Equa-

tion (3.19), exceeds the threshold +, the obstacle is considered to intend lane departure.

Then, the motion of the obstacle is extrapolated by integrating from the states sensed at

t0with respect to time

rO∗

j(t0),O∗

j(t)=1

RaO∗

j(t0)t2+RvO∗

j(t0)t. (3.27)

60 3POTENTIAL FIELD BASED MOTION PLANNING

The orientation of the obstacle follows according to Figure 3.3 from

cos (RψOj(t)) = Ojet(t)·Rexwith Ojet=

dtrO∗

j(t0),O∗

j(t)(t)

|Rd

dtrO∗

j(t0),O∗

j(t)(t)|.(3.28)

Obstacle Potential Field

The contribution of the obstacles to the hazard map can, analogous to the borders of the

road, be modeled as logarithmically decaying potential

RVO∗

i:= RVO∗

j.ri,rˆ

i/=−kOjln .I

Iri−rˆ

I/.(3.29)

Therein, Pidescribed by its position vector rirepresents a node of the elastic band being

passed by the host-vehicle at ti. As depicted in Figure 3.6 the position vector rˆ

imarks

the point ˆ

Ojon the boundary of the obstacles safety-area ∂Ojhaving the shortest distance

to Pi,

rˆ

i:= rˆ

Oj(ti)=HrQ(ti)∈∂Oj|*ri−rQ(ti)*= min

Q∈∂Oj

*ri−rQ(ti)*J(3.30)

with rQ(ti)=rO∗

j(ti)+rO∗

j,Q(ti) and ti∈[t0,t

f],i=0, . . . , N ,

where rO∗

j,Q(ti) follows from Equation (3.27). Recall that the evaluation of the potential

field of the obstacle at Pirelies on enlarged dimensions of the obstacle including the

dimensions of the host-vehicle, as illustrated in Figure 3.6. This growing method simplifies

the necessary distance computations. For circular obstacles the point ˆ

Ojlies on the

connecting line of O∗

jand Pi, see Figure 3.6. For rectangular obstacles a case differentiation

given in Section 3.4.1 is applied.

3.1.3 Hazard Map Composition

Finally, the hazard map is composed of the hazard potentials given by the borders Bland

Brof the road and by the Mobstacles Oj. Hence, the hazard potential experienced at a

node Piof the elastic band with position vector rireads

RVhaz

i:= RVhaz #ri,rBl

i,rBr

i,rO1

i,...,rOM

=RVBl.ri,rBl

i/+RVBr#ri,rBr

i%+

j=1

RVOj.ri,rˆ

i/.(3.31)

It is important to note, that in the composition of the hazard map the motion of obstacles

and host-vehicle is spatially and temporally coupled by the nodes of the elastic band

sensing the hazard potential. This coupling is illustrated in Figure 3.9.

3.1 HAZARD MAP 61

Figure 3.9: Spatial and temporal composition of the hazard map

The nodes Pi−1,P

i,and Pi+1 of the elastic band are depicted in a deformed configu-

ration. To each of these nodes the corresponding instants of time ti−1,t

i,and ti+1 are

assigned at which the host-vehicle passes the corresponding node: Pi−1↔ti−1,P

i↔ti,

and Pi+1 ↔ti+1. The instants of passing the nodes of the elastic band depend on the ve-

locity profile of the host-vehicle intended by the driver. The computation of these instants

is addressed in Section 3.3.2 by Equation (3.51).

At the same time, the extrapolated motion of obstacle Ojis discretized according to

the instants when the host-vehicle passes the nodes of the elastic band: rˆ

Oj(ti−1),rˆ

Oj(ti),

and rˆ

Oj(ti+1). Then, the repulsive continuously differentiable potential field of obstacle

Oj,j=1, . . . , M, is evaluated at the position riof the node Piand therein depends on

the position of the obstacle rˆ

Oj(t),

VOj

i.ri,rˆ

Oj(t)/∀t∈[t0,t

f],i=0, . . . , N . (3.32)

The idea of incorporating moving obstacles in the motion planning approach is now that

the potential field of the obstacle VOj

ishould only act on those nodes of the elastic band,

which correspond to the same instant of time

VOj

i.ri,rˆ

Oj(ti)/.(3.33)

This particular coupling of obstacle motion and the trajectory of the host-vehicle by

repulsive potential fields is visualized by means of springs connecting the nodes of the

elastic band with the according positions of the obstacle, Pi−1↔ˆ

Qj(ti−1), Pi↔ˆ

Qj(ti),

Pi+1 ↔ˆ

Qj(ti+1), in Figure 3.9.

62 3POTENTIAL FIELD BASED MOTION PLANNING

3.2 Elastic Band

The predictive search for trajectories of low hazard levels within the hazard map is per-

formed by an elastic band as shown in Figure 3.10.

Figure 3.10: Elastic band Kand consecutive elastic band K+1, see also Section 3.2.2

To obtain solutions of inherent low curvature the nodes Piwith i=0, . . . , N of the

elastic band are coupled by internal potentials, symbolized by springs. Assuming these

springs to be linear, each node is immersed in a quadratic internal potential field,

RVint

i(ri−1,ri,ri+1)=1

2ki−1(*ri−ri−1*−li−1)2+1

2ki(*ri+1 −ri*−li)2,

with i=1, . . . , N −1.(3.34)

Therein, kiand liare the stiffness parameter and the unstretched length of spring i,

respectively. For simplicity, kiand liare chosen to be constant for the entire elastic band.

As shown in Figure 3.10 the elastic band is adapted on each planning horizon ∆Twhen

new environmental information is available, K→K+ 1. Furthermore, it is a suitable

option in automotive applications to consider only the lateral displacement of the nodes

of the elastic band, which accelerates the computation, see Section 3.4.3. In this case

the distances ∆xbetween the nodes Pi,i=0, . . . , N, along the road are prescribed, see

Figure 3.10.

3.2.1 Equilibrium Configuration

To derive the potential field forces acting on each node Piof the elastic band, the direc-

tional derivatives

R∇ri:= (∂

∂Rxi

,∂

∂Ryi)T

,(3.35)

are defined. Then, the hazard and the internal potential field forces acting on Pifollow

from

Fhaz

i:= −R∇ri#RVhaz

i%and Fint

i:= −R∇ri#RVint

i%,(3.36)

3.2 ELASTIC BAND 63
with RVhaz
iand RVint
ibeing the hazard and the internal potential fields defined according
to Equation (3.31) and Equation (3.34), respectively. The hazard forces Fhaz
irepresent the
environmental information contained in the hazard map and are evaluated as described
in Section 3.1.3 by discretizing the estimated motion of the obstacles depending on the
discretization of the elastic band. The internal forces Fint
icouple the displacements of
the nodes Pito enforce trajectories with inherently low curvatures. Additionally, also
constraint forces Fc
ican be introduced, allowing only lateral displacements. In the equi-
librium configuration the sum of internal, hazard, and (if applied) constraint forces on
all nodes of the elastic band vanishes. The corresponding free body diagram is shown in
Figure 3.11.
Figure 3.11: Free body diagram of node Pi
The trajectory is obtained by solving the equilibrium condition1
Fint
i#ri−1,ri,ri+1%+Fhaz
i#ri,rBl
i,rBr
i,rˆ
O1
i,...,rˆ
OM
i%
+.Fc
i#ri,rBl
i,rBr
i,rˆ
O1
i,...,rˆ
OM
i%/=0(3.37)
for the position vectors riwith i=0, . . . , N of the elastic band as proposed by Brandt,
Sattel and Wallaschek (2004, [8] and 2006, [10]). Since Equation (3.37) is nonlinear,
the position vectors are obtained numerically, which is discussed in Section 3.4. The
presented elastic band approach belongs to the so-called local methods. Hence, if more
than one solution exist, the trajectory depends on the initial solution provided to the
numerical algorithm. In other words: if the initial solution passes an obstacle Ojto the
right, the final trajectory will also pass Ojon the right hand side. In general, a maximum
of 2Mpossible local solutions exists for Mobstacles. Thus, the computation of all possible
solutions especially in presence of many obstacles is too time consuming. Here, the initial
solution is generated in cooperation with the driver and is addressed in Section 3.4.2.
Compared to the work of Quinlan (1993, [87]), who shifted only some nodes of the
elastic band affected by an obstacle, there are different advantages of searching the equilib-
rium configuration in the hazard map: if the environment changes only slightly, compared
to the last measurement, this will result in an equilibrium of the elastic band, which is
close to the last solution. A further crucial advantage is that the equilibrium configuration
very naturally provides collision-free trajectories, even for large numbers of obstacles Oj,
j=1, . . . , M; geometric methods for example are often restricted to just one obstacle,
1As explained later, the equilibrium condition is not evaluated for fixed nodes and only unconstraint
directions of movement are considered. The constraint forces in Equation (3.37) are only introduced here
for the sake of completeness.

64 3POTENTIAL FIELD BASED MOTION PLANNING

which is not suitable for automotive applications. Formulating the hazard potentials of the

borders of the road and of the obstacles according to Equation (3.16) and Equation (3.29),

respectively, the following proposition holds:

Proposition 3.1. If a collision-free trajectory locally exists and if the elastic band is in

an equilibrium configuration, Fint

i+Fhaz

i=0, then no node Pi,i=0, . . . , N, of the elastic

band lies within the safety area of an obstacle Oj,j=1, . . . , M, while the node is being

passed by the host-vehicle Vat time ti. Therefore, each equilibrium configuration of the

elastic band in presence of Mmoving obstacles is collision-free.

Proof. Taking the directional derivatives R∇ri, see (3.35), of the potential fields of the

borders of the road RVBq

iand of the obstacles RVO∗

i,j=1, . . . , M, according to Equa-

tion (3.16) and Equation (3.29), respectively, yields potential field forces FBqand FOj

being proportional to the reciprocal distance of a node Pi,i=0, . . . , N, to the closest

point on the borders ∂Bqor on the boundaries ∂Ojat the instant tiwhen the host-vehicle

Vpasses Pi,

FBq∼I

Iri−rBq

−1and FOj∼I

Iri−rˆ

−1,(3.38)

Therefore, the repulsive forces acting on a node Pithat approaches the borders of the

road ∂Bqor the boundary of a safety area ∂Ojof an obstacle grow infinitely,

FBq

!ri−rBq

→0

−−−−−−−−−−−−−−→ ∞and FOj

ri−rˆ

→0

−−−−−−−−−−−−−−→ ∞.(3.39)

To be in equilibrium, Fint

i+Fhaz

i=0, the potential field forces due to the borders of the

road FBqand due to obstacles FOjhave to be balanced. As the potential field forces, acting

on a node Pi, are, according to Equation (3.39), infinitly high if ri∈∂Bq∨ri∈∂Oj, they

can only be balanced if Piis located on at least two boundaries ri∈∂Oj∧ri∈∂O¯

jwith

j+=¯

j. This in turn means that locally no collision-free solution exists, see Figure 3.12.

Otherwise, no node Piof the elastic band can be located on the boundaries ∂Bqor ∂Ojif

the elastic band is in an equilibrium configuration.

Figure 3.12: Left: Pion the boundary of two obstacles Ojand O¯

j, Right: Pion the

boundary of one obstacle Oj

3.3 COOPERATIVE MOTION PLANNING 65

3.2.2 Node Placement

In order to prescribe the boundary slope of an elastic band or to place a single node Pi,

e.g. the first or the last, at particular positions in the traffic space, position constraints,

ri·Rex=const., ri·Rey=const., need to be satisfied. This is guaranteed by introducing

constraint forces Fc

iacting on the fixed nodes Pi, see Equation (3.37). The placement

of the first node P0of the elastic band is of particular interest. In general, the vehicle,

commonly guided by a guidance controller, see Chapter 4, and the driver, will not perfectly

track the planned trajectory. Therefore, at each instant of replanning t0the vehicle will

show a lateral deviation ∆yas well as an angular deviation ∆ψto the previously planned

trajectory as depicted in Figure 3.10. Based on this tracking error, the guidance controller

and the driver set their control inputs such as the steering angle δ. Placing the first node

of the elastic band P0at the current position of the vehicle would instantaneously cancel

the tracking error. Thus, the vehicle guidance controller would be exposed to unintended

discontinuities. To avoid this problem in the replanning process from an elastic band K

at t0to an elastic band K+ 1 at t0+∆T, the first node of the new elastic band P(K+1)

0is

chosen depending on the last elastic band and on the current position of the vehicle RrV∗

as illustrated in Figure 3.10,

r(K+1)

0=r(K)

iwith Rxi= min{Rxi∈P$$Rxi≥RxV∗

i},i=0, . . . , N . (3.40)

The first node of the previous elastic band, that lies in front of the vehicle, indicated by

a higher Rex-coordinate than the one of V∗is selected as the first node of the new elastic

band.

3.2.3 Constraining Longitudinal Displacements

For lane-keeping, lane-changing, and evasion maneuvers the longitudinal displacements of

the nodes are not essential. Beyond that, the efficiency of the numerical algorithm to find

the equilibrium of the elastic band, see Equation (3.37), can be increased by considering

only lateral displacements. The longitudinal position of a node Pithen only depends on

the discretization parameter ∆xof the elastic band, see Figure 3.10,

ri·Rex=const. =i·∆x , i =0, . . . , N . (3.41)

Multiplying the force equilibrium of Equation (3.37) by Reyand ignoring fixed nodes such

as P0, the constraint forces Fc

iare eliminated, resulting in

Fint

i,y +Fhaz

i,y =0,i=1, . . . , N . (3.42)

3.3 Cooperative Motion Planning

This section describes the shaping of the elastic band in cooperation with the driver. For

lateral guidance, the driver’s steering intention defines the search direction for a collision-

free trajectory in the hazard map, whereas the maneuver strategy influences the position

of the last node of the elastic band. The longitudinal vehicle guidance remains fully

66 3POTENTIAL FIELD BASED MOTION PLANNING

with the driver. However, the extrapolation of the vehicle’s states based on the driver’s

longitudinal driving intent also influences the motion planning for lateral vehicle guidance.

3.3.1 Driver’s Steering Intention

As aforementioned, the elastic band method is a local method. Thus, equilibrium configu-

rations elsewhere than in the search direction, prescribed by the initial solution, are not

necessarily found. In order to reflect the drivers command, the search direction is coupled

on the steering angle δ(t0) and on the orientation of the vehicle RψOj(t0), as depicted in

Figure 3.13. For this reason, the first nnodes, n=0, . . . , N, of the initial solution (not of

the elastic band!) are fixed according to the anticipated motion of the vehicle for a fixed

steering angle δ(t0).

Figure 3.13: Defining the search direction by means of the Ackermann model

If for example the driver intends to pass an obstacle on the left hand side, then the

motion planning should, if possible, generate a trajectory that passes the obstacle on the

left. As only the rough direction of search is given by the initial solution, it is not necessary

to extrapolate the motion of the vehicle very precisely. Therefore, the positions of the first

nnodes of the initial solution lie on a circle determined by use of the Ackermann model,

see Section 2.6.1. This is illustrated in Figure 3.13. With the steering angle δ0fixed at

the instance of planning t0and the distance from front to rear axle, a+b, the radius of

curvature according to the Ackermann model becomes

RA=a+b

tan δ0

.(3.43)

With the yaw angle RψOj(t0) the coordinates of the instantaneous center of curvature

ICC expressed in the road-fixed reference frame !

R read

(RxICC

RyICC )=(−RAsin RψOj(t0)

RAcos RψOj(t0))+.rV* .(3.44)

3.3 COOPERATIVE MOTION PLANNING 67
Depending on the algebraic sign of δ(t0) and RψOj(t0) different cases have to be distin-
guished. In Figure 3.13, the situation is depicted for δ(t0)>0 and RψOj(t0)>0. For a
prescribed Rex-position of a node Pi, as it is illustrated in Figure 3.10, the Rey-component
on the circular arc follows from
Ryi=Ktan RψOj(t0)·Rxi,δ=0
RyICC −sign (δ)·&R2
A−(RxICC −Rxi)2,δ+=0 (3.45)
with Rxi=i·∆x.The procedure to determine the positions of the remaining nodes
Pi,i=n+1, . . . , N, of the initial solution is independent of the driver and is given in
Section 3.4.2.
3.3.2 Vehicle State Extrapolation
The equilibrium of the elastic band, Equation (3.37) or Equation (3.42), and therefore the
configuration level of the planned trajectory depends on the planned velocity profile of the
host-vehicle V. The velocity profile in turn depends on the driver who controls throttle
and braking pedal. For the motion planning procedure the velocity profile is gained by
extrapolation assuming constant longitudinal acceleration,
RaV∗:= RaV∗(t0)·Vex=const. (3.46)
with RaV∗being positive for accelerating and being negative for braking. The extrapolation
starts at t0= 0 with s#t0%= 0. Therefore, the speed and the covered path length at node
Pibecome
Ut#ti%≈Ux#ti%:= RaV∗ti+RvV∗#t0%·Vexand (3.47)
si:= s#ti%=
RaV∗
2t2
i+RvV∗#t0%·Vexti.(3.48)
The positions of the obstacles rO∗
iare evaluated at time tiwhen the host-vehicle passes
node Pi, see Figure 3.9. The host-vehicle’s speed at the instant of planning RvV∗#t0%and
the acceleration RaV∗(t0) set by the driver are known. Since the equilibrium configuration
of the elastic band is determined iteratively, see Section 3.4.3, the position vectors of the
previous iteration step r(k)
iare also known. Hence, the covered path length si, traveled by
the host-vehicle until passing node Piat time ti, is prescribed and can be estimated by
s(k+1)
i=
i
E
j=1
*r(k)
j−r(k)
j−1*(3.49)
with kdenoting the iteration step. Solving Equation (3.47) for tiand inserting the result
into Equation (3.48) yields with Equation (3.49) the instant of passing node Pi

68 3POTENTIAL FIELD BASED MOTION PLANNING

ti=









Ut(t0),RaV∗=0

−Ut(t0)

RaV∗−L.Ut(t0)

RaV∗/2+2

RaV∗(si−s1),RaV∗<0

−Ut(t0)

RaV∗+L.Ut(t0)

RaV∗/2+2

RaV∗(si−s1),RaV∗>0

.(3.50)

Note that, from an algorithmic point of view, the computation is accelerated by computing

the instants tiin ascending order, which results in

ti=ti−1+









si,i−1

Ut(ti−1),RaV∗=0

−Ut(ti−1)

RaV∗−L.Ut(ti−1)

RaV∗/2+2

RaV∗si,i−1,RaV∗<0

−Ut(ti−1)

RaV∗+L.Ut(ti−1)

RaV∗/2+2

RaV∗si,i−1,RaV∗>0

(3.51)

with si,i−1=si−si−1and Ut(ti−1)=RaV∗·(ti−1−t0)+Ut(t0), see also Sattel and

Brandt (2006, [96]).

3.3.3 Driver’s Maneuver Strategy

Principally, each node Piof the elastic band can be placed in a particular position in the

traffic space. Decisions on the node placement have to be taken by a higher planning

instance. However, the last node PNis of particular interest. It can for example be

placed on the center of the right lane or on the center of a neighboring lane if the driver

commands to change lanes by setting the indicator. Formally, assigning a position to the

last node reads rN:= rdes, where rNis then excluded from the equilibrium search, see

Equation (3.37). If vehicle following is intended, the last node PNof the elastic band can

be placed on the leading vehicle. The path driven by the leading vehicle could serve as

initial solution to the elastic band in that case.

3.4 Algorithm and Computation

In the last sections the general procedure of path planning based on elastic bands was out-

lined. Now, some algorithmic aspects of distance computations, finding an initial solution

and zero finding in order to solve for the equilibrium configuration of the elastic band, see

Equation (3.37), are addressed. Subsequently, the numerical procedure is summarized as

pseudo-code.

3.4.1 Distance Computation

The external forces, Equation (3.36), are derived from the hazard map, Equation (3.31).

To evaluate the external forces at node Pi, the corresponding position vectors to the

borders of the road rBr

i,rBl

iand to the obstacles rˆ

ihave to be determined.

3.4 ALGORITHM AND COMPUTATION 69

Distance to Borders of the Road

A necessary condition for the minimum distance, defined in Equation (3.14), is

d*ri−rBq(Rx)*2

dRx=0 ⇒BRxi−RxC+BRyi−RyBq(Rx)CRyBq#(Rx) = 0 (3.52)

with rBqbeing the position vector of points on the border Bqof the road, see Equa-

tion (3.13). Solving Equation (3.52) results in Rx=Rxˆ

Bqwith q∈{l, r}.

Distance to Obstacles

As the distance computation to circular obstacles is straight forward, the remaining prob-

lem lies in the distance computation between a point Piof the elastic band and the

boundary ∂Ojof a rectangular obstacle Oj, as shown in Figure 3.6. Therefore, the posi-

tion vector Rrˆ

i, see Equation (3.30), of the point ˆ

Ojon the border ∂Oj, having minimal

distance to Pi, has to be determined. This computation is facilitated by using the obstacle-

fixed reference frame !"

Oj. In order to do so, the position vector of Piis transformed

OjrO∗

j,Pi=COj,R #Rri−RrO∗

j%with COj,R =



cos RψOjsin RψOj0

−sin RψOjcos RψOj0

0 0 1 

.(3.53)

As illustrated in Figure 3.14, the surrounding space of a rectangular obstacle can be

divided into the four quadrants of the obstacle-fixed reference frame !"

Oj.

Figure 3.14: Computation of distance to obstacle Oj

In each quadrant again four cases have to be distinguished for the distance computa-

tions. In the following, these cases are discussed for the first quadrant; the determination

of ˆ

Ojin the remaining quadrants works analogously only differing in the algebraic signs.

70 3POTENTIAL FIELD BASED MOTION PLANNING

Case I:

Pilies besides Ojwith .0≤OjrO∗

j,Pi·Ojex≤LOj

2/∧.OjrO∗

j,Pi·Ojey≥WOj

2/, then

OjrO∗

j,ˆ

Oj(ti)=OjxiOjex+WOj

2sign(Ojyi)Ojey.

Case II:

Pilies closest to a corner of Ojwith .OjrO∗

j,Pi·Ojex≥LOj

2/∧.OjrO∗

j,Pi·Ojey≥WOj

2/,

then

OjrO∗

j,ˆ

Oj(ti)=LOj

2sign(Ojxi)Ojex+WOj

2sign(Ojyi)Ojey.

Case III:

Pilies in front of Ojwith .OjrO∗

j,Pi·Ojex≥LOj

2/∧.OjrO∗

j,Pi·Ojey≤WOj

2/, then

OjrO∗

j,ˆ

Oj(ti)=LOj

2sign(Rxi)Ojex+RyiOjey.

Case IV:

Pilies inside Ojif .0<OjrO∗

j,Pi·Rex<LOj

2/∧.OjrO∗

j,Pi·Rey<WOj

2/, then

no distance is computed.

Finally, the position vector OjrO∗

j,ˆ

Oj(ti) is transformed into the road-fixed reference

frame !

Rrˆ

Oj(ti)= RrO∗

j(ti)+CR,Oj

OjrO∗

j,ˆ

Oj(ti) with CR,Oj=(COj,R)T.(3.54)

3.4.2 Initial Solution

In general more than one trajectory of low hazard levels exist. Hence, the numerically

determined equilibrium depends on the initial solution, which in turn may be chosen

according to the application. For vehicle following it is suitable to consider the trajectory

driven by the leading vehicle as initial solution. However, for individually guiding the

host-vehicle, the initial solution has to be generated automatically at each replanning

instant t0. The procedure described in the following incorporates the hazard map as well

as the lateral displacements to the preceding nodes of the elastic band. To find the initial

solution, the road is discretized in Rex- and in Rey-direction. The equidistant discretization

∆x:= L

N,∆y:= b

J,(3.55)

recalling Lbeing the length of the elastic band, Nthe number of nodes, bthe width of

the road, and Jthe number of possible lateral positions. The grid of possible positions

for the nodes in the initial solution is visualized in Figure 3.15.

3.4 ALGORITHM AND COMPUTATION 71

Figure 3.15: Grid structure of initial solution: * Nodes from Ackermann model, •nodes

from initial solution

Be reminded that the first nnodes of the initial solution P0, . . . , Pnare prescribed

according to the Ackermann model, as explained in Section 3.3.1. The lateral positions

of these nodes are not restricted to the grid points. In order to define an initial solution,

the position vectors at the Kth replanning instant,

r(K)

i=i∆xRex+.j∗

i∆y+Ry¯

R∗(i∆x)−b

2/Rey,i=n+1, . . . , N , (3.56)

for the remaining nodes Pn+1, . . . , PNhave to be determined using the road definition

given in Equation (3.11). Therefore, the row index, j∗

i∈{0, . . . , J}, describing the lateral

position of node Pion the grid, has to be found. In order to do so, the hazard potential

Vhaz, including the borders of the road and the obstacles, is evaluated at the grid points

Vhaz =





V0,n V0,n+1 · · · V0,N

V1,n V1,n+1 V1,N

.....

VJ,n VJ,n+1 · · · VJ,N





.(3.57)

Therein, the instants ti, when the host-vehicle passes a node Pi, are estimated by Equa-

tion (3.51) with si,i−1=∆x. Each column iof the array Vhaz in Equation (3.57) refers to

the hazard potential experienced by node Pion all possible lateral positions on the grid.

Each row of the array Vhaz in Equation (3.57) corresponds to one lateral position on the

grid. Next, Vhaz is normalized according to

Vhaz

j,i =

Vhaz

j,i −min

j∈{0,...,J}#Vhaz

j,i %

max

j∈{0,...,J}#Vhaz

j,i %−min

j∈{0,...,J}#Vhaz

j,i %with S

Vhaz

j,i ∈[0,1] ,(3.58)

72 3POTENTIAL FIELD BASED MOTION PLANNING

with Vhaz

j,i=const. being an (J+ 1) ×1 array containing the values of the hazard potential

evaluated for all possible positions of node Pi.

In order to keep the curvature low and to avoid high lateral displacements between two

adjacent nodes, the initial solution is not solely generated by the hazard potential. The

positions of the nodes of the initial solution are determined in ascending order from node

Pn+1 to node PN. Therein, the array of all possible lateral displacements from a node

Pi−1, that is already fixed, to the following node Pireads

R∆yi:= Ryi−Ryi−1·01,...,11T.(3.59)

Next, the array of all possible lateral displacements from a node Pi−1to the following

node, Equation (3.59), is normalized

R∆Syj,i =

R∆yj,i −min

j∈{0,...,J}(R∆yj,i)

max

j∈{0,...,J}(R∆yj,i)−min

j∈{0,...,J}(R∆yj,i)with R∆Syj,i ∈[0,1] ,(3.60)

Finally, the initial solution is generated by minimizing the weighted sum of hazard poten-

tial S

Vhaz

j,i and lateral displacement R∆Syj,i to the preceding node for all remaining nodes of

the elastic band Pn+1, . . . , PN. The index of the lateral position of node Pibecomes

j∗

i:= Fj∈[0, . . . , J]$$$B(1 −γ)·S

Vhaz

j,i +γ·R∆Syj,iC

= min

j∈{0,...,J}B(1 −γ)·S

Vhaz

j,i +γ·R∆Syj,iCJ.(3.61)

Together with Equation (3.56) the initial solution for the elastic band is given. The

parameter γcan be used to adjust the weight of lateral displacement to the preceding

node and the influence of the hazard map. Noteworthy, positions on the grid that would

collide with an obstacle are not selected for the initial solution, r(0)

i/∈Oj.

3.4.3 Equilibrium Solution

In order to find the equilibrium configuration of the elastic band, the equilibrium condition

Equation (3.37) or Equation (3.42) has to be solved for the position vectors riof the nodes

Piwith i=1, . . . , N. Due to the geometry as well as to the hazard potential this is a

nonlinear algebraic problem with the 2Ncomponents of the position vectors ribeing the

unknowns. For the numerical solution the problem can be stated in general form as

F(z)=0(3.62)

with the array of position vectors z=[rT

1,...,rT

N]Tand the array of resultant forces at

each node F=[FsumT

1,...,FsumT

N]T, whereas Fsum

i=Fint

i+Fhaz

i+Fc

iis composed of

internal, hazard, and constraint forces acting on node Pi. However, if the position of a

node Piis fixed, its position vector is a priori known. Therefore, the zero finding problem

in Equation (3.62) can be reduced by the number of fixed nodes. The start node P0for

3.4 ALGORITHM AND COMPUTATION 73

instance is generally fixed, thus its position vector r0does not appear in Equation (3.62);

analogously all nodes that should remain at a fixed position riare removed.

The zero finding problem in Equation (3.62) is solved by employing a Newton-Raphson

method, wherein the Newton-Raphson iteration can be interpreted as a linear approxima-

tion

F(z)≃˜

F(z) := F(z(k))+J(z(k))(z−z(k)),(3.63)

obtained by truncating the Taylor series expansion of Fat z(k)after the linear term and

then solving the resulting linear equation ˜

F(z)=0, calling the solution z(k+1), see for

example Ortega and Rheinboldt (1970, [80], p. 181-185). The matrix Jdenotes the

Jacobian of F. Performing an iteration step leads to

z(k+1) =z(k)−J−1(z(k))F(z(k)),k=0,1,2, ... . (3.64)

Starting from an initial vector z(0), sufficiently close to the solution, the iteration converges

quadratically to a root z∗. As the hazard forces of the left and of the right border of

the road, ∂Bland ∂Br, become infinite when a node Piapproaches the borders, see

Equation (3.39), it is ensured that, if the road is not blocked by obstacles Oj, equilibrium

solutions only exist on the road. The iteration is terminated if a sufficient accuracy +is

achieved,

+≥max .∆z(k)

i/:= max #0−J−1(z(k))F(z(k))1i%.(3.65)

To avoid that the Newton-Raphson algorithm shifts nodes Piof the elastic band,

ri/∈Oj∧ri∈]∂Bl,∂Bl[ , into non-valid areas such as into obstacles or outside the borders

of the road, the step size of each node is controlled independently. Thus, the step size

is limited by the distance to the closest object, which is either an obstacle Ojor one of

the borders of the road ∂Blor ∂Br. This does not take high computational effort as the

necessary distance computations have already been performed in the evaluation of the

external potentials at the nodes of the elastic band. Besides that, a fixed maximal step

size ∆max is applied to all nodes in order to avoid large displacements in one iteration

step.

The internal forces and the hazard forces Fint

iand Fhaz

i, respectively, as given in Equa-

tion (3.36), contribute in a different manner to the Jacobian matrix J. Splitting the hazard

forces into their components due to the road and due to obstacles, the Jacobian matrix

reads

J=Jint +Jroad +Jobs =∂Fint

∂z+∂Froad

∂z+∂Fobs

∂z.(3.66)

As the internal forces Fint

idepend on the position of a node rias well as on the positions

of the preceding and the subsequent node ri−1and ri+1, the contribution of the internal

forces to the Jacobian matrix has tridiagonal form. The forces due to the potential of the

borders of the road depend only on the position of the considered node ri. Therefore, the

74 3POTENTIAL FIELD BASED MOTION PLANNING

borders of the road contribute only on the main diagonal to the Jacobian matrix,

Jint =







Jint

1,1Jint

1,20

Jint

2,1Jint

2,2Jint

2,3

Jint

3,2Jint

3,3Jint

3,4

.........

int

N,N−1Jint

N,N







,Jroad =





Jroad

1,10

Jroad

2,2

0...

Jroad

N,N





.(3.67)

The hazard forces caused by moving obstacles depend on the extrapolated trajectories

of both, the obstacles and the host-vehicle, see Section 3.1.3. Therefore, the positions

of all preceding nodes, already passed by the host-vehicle in the extrapolation, influence

the external hazard forces due to moving obstacles on a node Pi. This coupling is due

to the fact that the course of the elastic band to a node Piinfluences the time ti, when

the host-vehicle passes that node. In the computation of the hazard forces, the estimated

trajectories of the obstacles are evaluated at the instants tiand therefore depend on the

position vectors of the preceding nodes r0,...,ri−1. Hence, the contribution of the hazard

forces due to moving obstacles in the Jacobian matrix has a triangular structure,

Jobs =







Jobs

1,1

Jobs

2,1Jobs

2,20

Jobs

3,2Jobs

3,3

.......

Jobs

N,1· · · Jobs

N,N−1Jobs

N,N







,(3.68)

whereas a non-moving obstacle Ojresults, like the borders of the road ∂Bland ∂Br, in

a diagonal contribution to the Jacobian matrix. To demonstrate the influence of moving

obstacles Ojon the Jacobian J, the example of a pedestrian crossing the road at constant

speed is considered.

Example: A pedestrian PE may be modeled by a circular safety area of diameter Dj

walking at constant speed vPE∗(t0)measured at t0. Extrapolating the trajectory of the

pedestrian according to Equation (3.27) for constant speed and evaluating the position

vectors of the pedestrian rPE(t)at the instants ti, when the host-vehicle Vpasses the nodes

Piof the elastic band with constant speed Ut, estimated according to Equation (3.49) and

Equation (3.50) and plugging the result into equations (3.29) and (3.33), respectively,

gives

VPE

i(r0,...,ri)=

−kPE ln 6I

Iri−rPE(t0)−vPE∗(t0)6!i

k=1 *rk−rk−1*

−t07I

I−Dj

27.(3.69)

3.4 ALGORITHM AND COMPUTATION 75
It can be seen that the hazard potential of the pedestrian VPE
ievaluated at a node Piof
the elastic band also depends on the preceding nodes P0, . . . , Pi−1. Therefore, also the haz-
ard forces due to the pedestrian FPE show the same dependencies, so the structure of the
Jacobian matrix Jis analog to Equation (3.68). If the pedestrian does not move,
vPE∗=0, it becomes obvious from Equation (3.69) that the dependence on the preced-
ing nodes r0,...,rivanishes and the hazard potential of the pedestrian VPE would only
contribute to the main diagonal of the Jacobian J.
In order to accelerate the computation in each iteration step, the off-diagonal elements
in Equation (3.68) caused by moving obstacles are neglected
˜
Jobs =










Jobs
1,1
Jobs
2,20
Jobs
3,3
0...
Jobs
N,N










.(3.70)
However, the motion of the obstacles is still considered in each iteration step by means of
the hazard forces of the obstacles Fobs in Equation (3.64). In general, the approximation
of the Jacobian, Jobs ≈˜
Jobs, may result in more iteration steps, but, as the number of
iterations is relatively low, this drawback is outbalanced by the reduction of computations
in each iteration step.
A further substantial reduction of computational effort is achieved by restricting the
node’s degrees of freedom to the lateral direction Ry. Thus, only the equilibrium conditions
in the direction of free displacements have to be considered. These are obtained by pro-
jecting the equilibrium equations onto the free directions, as described by Equation (3.41)
and Equation (3.42). What remains is to solve these equations with the Newton-Raphson
procedure (3.64) at each node for Ryi, as Rxi=i∆xis prescribed by the discretization of
the elastic band. In finding the equilibrium for each node in Rx- and in Ry-direction, the
submatrices of the Jacobians, (3.67) and (3.68), have the structure
Ji,j =6∂Fx
∂xj
∂Fx
∂yj
∂Fy
∂xj
∂Fy
∂yj7and thus: dim #Jint
i,j %= dim #Jroad
i,j %= dim #Jobs
i,j %=2×2.
(3.71)
For Nnodes Pi, the Jacobian J, Equation (3.66), therefore has dimension 2N×2N. If only
the lateral positions of the nodes Piare considered, the submatrices in equations (3.67) and
(3.68) are reduced to scalars of the type ∂Fyi
∂yj. Thus, considering only lateral displacements
reduces the number of components in the Jacobian Jby factor 4.
Note that for the transition from one planning update Kto the next motion planning
step K+ 1 the configuration of the elastic band has to be stored in the assistance system
with respect to the earth-fixed reference frame !
"
E or with respect to the last road-fixed
reference frame !"
R(K).

76 3POTENTIAL FIELD BASED MOTION PLANNING

3.5 Simulations

The potential field motion planning procedure introduced in this chapter is now demon-

strated for some traffic scenarios.

3.5.1 Scenario I: Entering Traffic

In this simulation example the functioning of the proposed motion planning method is

highlighted. One of the main advantages of the presented approach is, that it works in

a predictive rather than in a reactive manner. In order to illustrate the effect of the

extrapolation of obstacle motion, as proposed, the same scenario is simulated twice: in

the first simulation run the extrapolation is turned offand the obstacle is treated as a

static obstacle in each update of the elastic band; in the second simulation run the motion

of the obstacle is extrapolated based on its previous behavior. However, to have a ”fair”

comparison, the obstacle changes its behavior during the simulation and therefore does

not travel in a way that can perfectly be predicted. The scenario set up is depicted in

Figure 3.16. The scenario description is completed by the data given in Table 3.2, while the

parametrization of the hazard map and the elastic band motion planning is summarized

in Table 3.3.

Figure 3.16: Scenario I: A vehicle entering the highway

In this scenario two vehicles are involved: the host-vehicle traveling along a highway at

a speed of 108 km/h and a second vehicle, considered to be the obstacle, that turns into the

3.5 SIMULATIONS 77

highway from a side road. Entering the highway the obstacle has a speed of 18 km/h. Due

to the difference in speed, the obstacle is fast approached by the host-vehicle. However,

the obstacle accelerates at a rate of 4 m/s2. The host-vehicle detects the obstacle after

1.4 s. The position of the obstacle at t=1.4 s is sketched in Figure 3.16. At t= 2 s the

obstacle has completed the entering maneuver and travels down the highway.

Table 3.2: Scenario data

Symbol Value Description

b7 m road width

blanes 3.5 m lane width

κ00m

−1initial road curvature

˙κ0m

−1s−1road curvature rate

lO14 m obstacle length

wO12 m obstacle width

For the entire scenario the speed of the host-vehicle remains constant at 108 km/h. For

this reason the motion planning procedure initiates a passing maneuver. However, due to

the high acceleration of the obstacle, the obstacle reaches the speed of the host-vehicle

before the passing maneuver is completed. The host-vehicle returns to the right lane and

follows the obstacle. The driven paths are depicted in Figure 3.17 with and without ex-

trapolation of the obstacle motion. It becomes evident that the driven trajectory without

extrapolation is more wavy than the one with. This is due to the fact that in each re-

planning the new trajectory differs significantly from the previously planned, because the

environment has changed (the obstacle occurs in a different position). Using the extra-

polation, the motion of the obstacle was anticipated and therefore the difference between

two consecutively planned trajectories is smaller.

Figure 3.17: Motion planning with and without extrapolation of obstacle motion

78 3POTENTIAL FIELD BASED MOTION PLANNING

Table 3.3: Hazard map and elastic band motion planning parameter

Symbol Value Description

l1.5 m node distance

l01.35 m unstreched spring length

kint 30000 N/m spring constant

kBl,kBr750, 250 scaling factor of the potentials of the borders

of the road

kO1000 scaling factor of obstacle potential

lP100 m planning distance

∆T0.1 s planning update interval

∆max 1.5 m maximal displacement in each iteration step

+0.05 m tolerance

3.5.2 Scenario II: Crossing Animal

In the last scenario it was seen, that it can be beneficial to incorporate extrapolation

into the predictive motion planning in order to keep variations between two consecutively

planned trajectories as small as possible. In the second scenario an even more dramatic

example is given: the host-vehicle without extrapolation collides with an animal crossing

the road, while the host-vehicle with extrapolation avoids the collision. The scenario is

depicted in Figure 3.18.

Figure 3.18: Scenario II: An animal crossing the road

The host-vehicle drives at a constant speed of 60 km/h on a road modeled according

to Table 3.2. At a distance of 50m ahead of the host-vehicle an animal, approaching the

road at a speed of 5 km/h from the right shoulder, is detected. After 1.4 s the animal

becomes aware of the host-vehicle and accelerates. After 3 s the animal suddenly stops in

panic and remains at the left lane of the road. The parameters of the motion planning

algorithm are chosen identical to the last scenario, according to Table 3.3. The safety area

of the animal is modeled by a circle with a diameter of 0.5 m.

3.5 SIMULATIONS 79

In Figure 3.19 the simulated paths with and without extrapolation are shown. In the

left column the scenario with extrapolation is given: at t=0 s the host-vehicle has not yet

detected the animal and intends to follow the right lane. In the next depicted instant

at t=1.5 s the animal was incorporated in the motion planning. It can be observed, that

the motion of the animal towards the left lane was already anticipated by the motion

planning procedure, as the host-vehicle turns to the right. Finally, the host-vehicle passes

the animal on the right lane.

In the right column the simulation is repeated without extrapolation. Initially the

animal is considered to be a static obstacle on the right lane. Therefore, the motion

planning algorithm intends to pass it on the left. However, as the animal moves towards

the left border of the road, also the planned trajectory of the host-vehicle is shifted to

the left with each adaptation of the trajectory. Finally, the host-vehicle collides with the

animal at approximately t=3.1 s.

Figure 3.19: Comparison of algorithm with and without extrapolation

80 3POTENTIAL FIELD BASED MOTION PLANNING

3.5.3 Scenario III: Passing Maneuver with Oncoming Traffic

The two preceding scenarios included traffic traveling in the same direction as the host-

vehicle and an animal crossing the road. Now, a passing scenario is given including a static

obstacle as well as oncoming traffic. The scenario is depicted in Figure 3.20. The vehicle

in front of the host-vehicle suddenly brakes and stops 40 m ahead of the host-vehicle. At

this instant the host-vehicle has a speed of 108 km/h; thus, braking is hardly feasible. The

host-vehicle evades the static vehicle and remains at constant speed for the rest of the

scenario. It has to return to the right lane before it collides with the oncoming vehicle.

Figure 3.21 shows the trajectories simulated with and without extrapolation. In con-

trary to the last scenarios no significant difference in the result of the two motion planning

algorithms can be noted. Both algorithms intend to pass the resting vehicle and return

to the right lane, to the minimum of the hazard potentials of the borders of the road.

Curvature and curvature rate of the road are κ=0.003 1/m and ˙κ= 10−61/ms, respec-

tively. For static obstacles there is no difference between the algorithm with and without

extrapolation.

Figure 3.20: Scenario III: Passing maneuver with oncoming traffic

3.5 SIMULATIONS 81

Figure 3.21: Passing maneuver simulated with and without extrapolation

3.5.4 Pseudo Code of Motion Planning Algorithm

Finally, the potential field motion planning algorithm is summarized as pseudo-code.

82 3POTENTIAL FIELD BASED MOTION PLANNING
Initialize: K=0
planning instant: t0
1. Get sensor data:
(a) road: #b, (Vxi,Vyi)∈R%⇒r¯
R∗,Eqn. (3.11)
(b) obstacles: .rO∗
j(t0),RvO∗
j(t0),RaO∗
j(t0),RωOj(t0),l
Oj,w
Oj/,see Fig. 3.1
(c) driver/vehicle: #δ(t0),RvV∗(t0),RaV∗(t0)%,see Fig. 3.1
2. Get previous path:r(K−1)
0, . . . , r(K−1)
N
3. Generate hazard map:RVhaz Eqn. (3.31)
3.1 Generate potential of the road: RVBqEqn. (3.16) with Eqn. (3.11)-(3.15) and (a)
3.2 Generate potential of the obstacles: !M
j=1 RVOjin Eqn. (3.29)
3.2.1 Identify type of obstacle motion by Eqn. (3.19) : IN LANE / LANE DEPARTURE
3.2.2 Extrapolate obstacle motion: rO∗
j(t) with t∈[t0,t
f]
If IN LANE: rOj(t),RψOj(t)Eqn. (3.20)-(3.26) with (b)
If LANE DEPARTURE: rOj(t),RψOj(t) Eqn. (3.27)-(3.30) with (b)
4. Generate initial solution:
4.1 Set: ˜
r(K)
0Eqn. (3.40) ,˜
r(K)
1, . . . , ˜
r(K)
nEqn. (3.43)-(3.45) and (c) , see Fig. 3.13, 3.15
4.2 Compute distances: (rBl
j,i,rBr
j,i ) Eqn. (3.52) , and .rˆ
O1
j,i , . . . , rˆ
OM
j,i /Eqn. (3.54)
for i=n+1, . . . , N and j=0, . . . , J, see Fig. 3.15
4.3 Evaluate hazard potential: RVhaz
j,i =RVhaz #rj,i,rBl
j,i,rBr
j,i,rO1
j,i , . . . , rOM
j,i %Eqn. (3.57)
4.4 Set: ˜
r(K)
n+1, . . . , ˜
r(K)
NEqn. (3.56) and Eqn. (3.58)-(3.61)
5. Compute equilibrium solution:
Set k=0 and max .∆z(0)
i/>#, set z(0) =>.˜
r(K)
1/T
, . . . , .˜
r(K)
N/T?T
according to 4.
while: max .∆z(k)
i/>#
compute distances:
borders of the road: zB(k):= B#rBl
1%T,#rBr
1%T,#rBl
2%T, . . . , #rBr
N)TCTEqn. (3.52)
obstacles: tiEqn. (3.51) ⇒rˆ
O1
i, . . . , rˆ
OM
ifor i=1, . . . , N Eqn. (3.53), (3.54)
zO(k):= B#rˆ
O1
1%T, . . . , #rˆ
O1
N)T,#rˆ
O2
1%T, . . . , #rˆ
OM
N%TCT
compute J(k)=Jint.z(k)/+Jroad.z(k);zB(k)/+˜
Jobs.z(k);zO(k)/Eqn. (3.67), (3.68)
compute F(k)=F.z(k);zB(k),zO(k)/Eqn. (3.36)
solve: J(k)∆z(k+1) =−F(k)⇒∆z(k+1)
set maximal step size: ∆z(k+1)
i,max = min .∆zi,max,*ri−rBl
1*, . . . , *ri−rˆ
OM
1*/
check step size: if ∆z(k+1)
i≥∆z(k+1)
i,max ⇒∆z(k+1)
i=∆z(k+1)
i,max
z(k+1) =z(k)+∆z(k+1)
k=k+1
end
next replanning: at t0=t0+∆T , K =K+1

Chapter 4

Potential Field based Vehicle

Guidance Control

In the last chapters, vehicle dynamics models and potential field based motion planning

methods were introduced. In the next steps guidance laws for trajectory tracking have

to be formulated and the guidance strategy has to be coordinated with the driver. In

the sense of a consistent framework for motion planning and vehicle guidance, the control

problem is addressed by potential field methods. However, it has to be emphasized that the

longitudinal guidance of the vehicle remains with the driver. Consequently, the velocity

profile of the planned trajectory is not affected by the control laws given in this chapter;

the vehicle is guided with respect to the path P, which represents the configuration level

of the elastic band. In order to do so, the guidance potential field is formulated in terms

of the lateral path tracking error. Therein, the reference position is that position on

the path Pbeing closest to the current position of the host-vehicle. Subsequently, a

generalized virtual guidance force is derived from the guidance potential. By means of

the vehicle dynamics models given in Chapter 2, the generalized virtual guidance force is

finally mapped onto the control inputs of the vehicle.

Principally, the curvature of the path Pacts as a disturbance that has to be regulated by

the guidance controller. However, as the planned path Pand its curvature κGare known,

the disturbances can be compensated by feedforward control. Therefore, the potential

field controller only has to regulate initial disturbances, for example due to side wind or

bank. For the potential field controller, in combination with feedforward control, stability

in the sense of Lyapunov can be shown. Furthermore, a Lyapunov function is employed to

provide bounds for the maximal tracking error depending on the initial disturbance. For

a known initial configuration, the Lyapunov function can be used to guarantee collision

avoidance by guaranteeing a deviation from Pbeing smaller than the distance to the

closest obstacle.

4.1 General Concept of Potential Field Guidance

The main idea of the potential field controller consists in deriving a generalized virtual

guidance force from the potential of the path tracking error G∆qlat (t). Therein, the

84 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

generalized virtual guidance force

VQguid

lat (t)=f(G∆qlat (t)) (4.1)

is composed of a lateral force and a moment in the yaw plane, respectively. In Figure 4.1

an example of a potential field, symbolized by a spring regulating the lateral deviation

from the path, is shown.

Figure 4.1: Design example of a guidance potential field, see Section 4.6.5

With the controller matrix Kguid the potential in the lateral deviation G∆qlat (t) is

chosen to be quadratic

GVguid(G∆qlat) := 1

2G∆qT

lat Kguid

G∆qlat .(4.2)

The generalized virtual guidance force with respect to the guidance frame !

G is then

computed by the taking gradient

GQguid

lat := −G∇

∆qGVguid(G∆qlat)=−Kguid

G∆qlat .(4.3)

Therein, the origin G∗of the guidance frame !

G is identical to that point on Phaving the

smallest distance to the actual position of the vehicle, represented by V∗, see Section 4.3.

To map the generalized guidance force on the control inputs it has to be transformed into

the vehicle-fixed reference frame !

VQguid

lat =>VFguid

VMguid

z?=CV,G

GQguid

lat with CV,G =



cos G∆ψ(t) sin G∆ψ(t)0

−sin G∆ψ(t) cos G∆ψ(t)0

0 0 1 

.

(4.4)

4.2 MAPPING A VIRTUAL GUIDANCE FORCE ON CONTROL INPUTS 85

4.2 Mapping a Virtual Guidance Force on Control

Inputs

The aim of this section is to provide a map of the generalized virtual guidance force

VQguid

lat on the control inputs uof the vehicle. In doing so, maps on the control inputs of

the nonlinear as well as on the control inputs of the linear single-track model are given.

4.2.1 Nonlinear Mapping

To determine the corresponding control inputs of the vehicle model for a given generalized

virtual guidance force VQguid

lat , the equations of motion of the nonlinear single-track model

in Equation (2.32) with differential braking are recalled

MV˙

v+b(Vv)=Qnc #FR

y(Vv)%+Qcoupl #FF

y(Vv),δ%+Qc(u).(4.5)

The term Qcoupl #FF

y(Vv),δ%depends on the lateral tire forces at the front tires FF

yas

well as on the steering angle δ. The lateral tire forces are not directly accessible, while the

steering angle is one of the control inputs. Therefore, the term Qcoupl is approximately

separated into two parts, one depending on the lateral forces, the other depending on the

steering angle. In doing so, the lateral tire forces in

Qcoupl #FF

y(Vv),δ%=FF

y[−sin δ,cos δ, a cos δ]T(4.6)

are substituted by the lateral tire forces, FF

y=−CF

ααF, generated by the Dugoffmodel

in the linear region, see Equation (2.13), compare also Brandt, Sattel and Wal-

laschek (2005, [9]). Together with the small angle approximation, αF≈.Uy+˙

ψa

Ux/−δ,

for the front cornering angle, defined in Equation (2.29), Equation (4.6) becomes

Qcoupl ≈−CF

α6Uy+˙

ψa

Ux7

8 9: ;

:=¯αF





−sin δ

cos δ

acos δ

+CF

αδ



−sin δ

cos δ

acos δ

.(4.7)

Applying the small angle approximations, sinδ≈δ, cos δ≈1, and ¯αFδ≈0, to the first

addend in Equation (4.7) the following separation arises,

Qcoupl ≈−CF

α¯αF



a



89: ;

:=Qcoupl

+CF

αδ



−sin δ

cos δ

acos δ

.(4.8)

Now, a procedure analog to the one proposed by Rossetter (2003, [93], pp. 16) can be

applied: the zero term, Qcoupl

nc −Qcoupl

nc , is added to the right hand side of the equations

of motion (4.5),

MV˙

v+b(Vv)=Qnc +Qcoupl

8 9: ;

:= ¯

Qnc

−Qcoupl

nc +Qcoupl +Qc

8 9: ;

:= ¯

.(4.9)

86 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

The right hand side of the equations of motion is now approximately decoupled; the

generalized force term ¯

Qnc only depends on the states of the vehicle, while ¯

Qcabsorbed

the controllable part of the couple term Qcoupl and depends on the control inputs u.

Figure 4.2: Equivalence of generalized guidance force and control inputs: steering angle

and differential braking

Finally, equivalence of the generalized virtual guidance force VQguid

lat and the controllable

part on the right hand side of the equations of motion ¯

Qc(u) is required

VQguid

lat ≡¯

Qc#uguid%.(4.10)

The required equivalence of the generalized lateral guidance force VQguid

lat containing the

lateral force VFguid

yand the yaw moment VMguid

zand the control inputs, being the steering

angle δand the yaw moments due to differential braking at the front and at the rear axle

∆Fxand MR

∆Fx, respectively, is visualized in Figure 4.2. Notice that longitudinal forces

at the tires other than the differential braking forces are not considered as only lateral

vehicle guidance is intended.

To sum up, the required equivalence in Equation (4.10) gives two equations while three

control inputs for lateral vehicle dynamics are available. Hence, an additional condition is

necessary to solve for the control inputs. A natural choice is to prescribe the distribution

of differential braking at front and rear axle. In doing so, a static characteristic curve can

be defined or the distribution can be chosen to be dependent on the dynamic handling

characteristics of the vehicle.

4.2.2 Linear Mapping

Recalling the equations of motion (2.37)

M˙

v+0G(Ux)+D(Ux)1v=Bu (4.11)

of the linear single-track model, it is obvious that the control inputs uare already de-

coupled from the dynamic states of the vehicle. Thus, the required equivalence of the

generalized virtual guidance force VQguid

lat and the right hand side of Equation (4.11),

VQguid

lat =>VFguid

VMguid

z?≡Bu

guid =>CF

αδ

aCF

αδ+MF

∆Fx+MR

∆Fx?(4.12)

4.3 PATH TRACKING ERROR 87

gives again two equations for the three control inputs δ,MF

∆Fx, and MR

∆Fx. Again, pre-

scribing a distribution, MF

∆Fx=f#MR

∆Fx%, of the differential braking gives the additional

condition to solve Equation (4.12) for the control inputs.

The structure of the potential field guidance control is depicted in Figure 4.3. The

path P, representing the configuration level of the trajectory planned by the driver and

the potential field motion planning, see Chapter 3, is compared to the actual states of the

vehicle Rqand R˙

q. Recall that the path Pis only recalculated in intervals of ∆T, where

Klabels the version of P; the pseudo-code of the motion planning is given in Section 3.4.

The comparison of the actual vehicle states and the planned path P, in Figure 4.3 denoted

as !

- , yields the path tracking error and is addressed in the next section. Based on the

path tracking error G∆qlat the guidance potential field and therefore the virtual guidance

force is computed. The virtual guidance force GQguid

lat is then mapped on the control inputs

uguid of the vehicle.

Figure 4.3: Structure of potential field guidance control

4.3 Path Tracking Error

The motion planning provides the path Pfor the next planning horizon t∈[t0,t

f]1with

respect to !

rR∗,G∗=RxRex+RyG∗(Rx)Reywith tangent orientation RψG= arctan (RyG#(Rx)) .

(4.13)

Therein, the notation (·)#=d

dRx(·) is used. For any instant t∈[t0,t

f] position and

orientation of the vehicle in !

R may be given by

rR∗,V ∗

(t)=

RxV∗

(t)Rex+RyV∗

(t)Reyand RψV(t).(4.14)

The origin G∗of the guidance frame !

G is identical to that point on Phaving the smallest

distance to the actual position of the vehicle, represented by V∗; this situation is depicted

in Figure 4.4.

1Note that tfdenotes the point in time until that the trajectory is planned, while ∆Trefers to the

time after that the motion planning is updated.

88 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

Figure 4.4: Geometry of vehicle guidance problem

Therefore, the desired position and orientation of the vehicle on Pare given by

rR∗,G∗(t)= RxG∗(t)Rex+RyG∗#RxG∗(t)%Reyand RψG(t) = arctan #RyG∗##RxG∗(t)%%,

(4.15)

where the coordinate RxG∗(t) satisfies the minimum condition

rG∗,V ∗=HrG,V ∗|I

IrR∗,V ∗−rR∗,GI

I= min

Rx∈[0,L]I

IrR∗,V ∗−rR∗,G (Rx)I

IJ.(4.16)

Defining the set of mutually perpendicular unit vectors of !

Get:= rR∗,G

∗#

|rR∗,G

∗#|=Rex+RyG∗#

Rey

'1+(

RyG∗#)2and Gen:= −RyG∗#

Rex+Rey

'1+(

RyG∗#)2,(4.17)

being tangential and normal to P

$$rR∗,V ∗−rR∗,G∗$$·Get#RxG∗%=0,(4.18)

must hold and results in

0RxV∗−RxG∗1+0RyV∗−RyG∗(RxG∗)1RyG∗#(RxG∗) = 0 .(4.19)

Therefore, the arrays containing the actual state and the desired state of the vehicle with

respect to !

R can be defined as

Rq(t) := 0RxV∗

(t),RyV∗

(t),RψV(t)1Tand Rqdes (t) := 0RxG∗

(t),RyG∗

(t),RψG(t)1T.

(4.20)

4.4 GUIDANCE KINEMATICS 89

The tracking error becomes

R∆q(t)= Rq(t)−Rqdes (t).(4.21)

To formulate potential field control laws in terms of the guidance frame !

G the tracking

error has to be transformed according to

G∆q(t)=CG,R

R∆q(t) with CG,R =



cos RψG(t) sin RψG(t)0

−sin RψG(t) cos RψG(t)0

0 0 1 

.(4.22)

As the longitudinal guidance of the vehicle remains with the driver, only the lateral

tracking error, see Figure 4.4, is necessary for guidance control and is defined as

G∆qlat (t) := [ G∆q(t)](y, ψ)=>G∆y(t)

G∆ψ(t)?=

>−sin RψG(t)(RxV*(t)−RxG*(t)) + cos RψG(t)(RyV*(t)−RyG*(t))

RψV(t)−RψG(t)?.

(4.23)

Notice that the first equation

G∆x(t) = cos RψG(t)(RxV*(t)−RxG*(t)) + sin RψG(t)(RyV*(t)−RyG*(t)) (4.24)

is identically fulfilled since R∆xG*(t) = 0 is demanded and Equation (4.19) holds.

4.4 Guidance Kinematics

In the last section the path tracking error G∆qlat of the vehicle with respect to the planned

path P, represented by the guidance frame !

G , was derived. Based on G∆qlat a guid-

ance potential GVguid was defined in order to generate a generalized virtual guidance force

VQguid

lat , which is mapped onto the control inputs uof the vehicle model. However, so far

lateral guidance control was only addressed at the configuration level. In order to study

stability properties of the proposed potential field control scheme, the perturbation equa-

tions of motion have to be considered. The discussion of stability properties is prepared

by deriving the underlying guidance kinematics.

90 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

In order to derive the lateral guidance state variables, G∆˙yand G∆˙

ψ, describing the

deviation from the desired velocitiy of G∗and from the desired yaw rate ˙

ψdes of !

G , the

velocity of the vehicle, represented by V∗, with respect to !

GvV∗=

dtrG∗,V ∗=

dtrR∗,V ∗−

dtrR∗,G∗(4.25)

dtrR∗,V ∗+GωR×rR∗,V ∗−>Rd

dtrR∗,G∗+GωR×rR∗,G∗?(4.26)

dtrR∗,V ∗−

dtrR∗,G∗+GωR×#rR∗,V ∗−rR∗,G∗%(4.27)

=RvV∗−RvG∗+GωR×rG∗,V ∗(4.28)

is considered.

Figure 4.5: Lateral guidance kinematics: host-vehicle and guidance frame !

G rotating

about their instantaneous centers of curvature ICC and ICCG, respectively

The velocity of the guidance frame !

G can be expressed by means of its rotation about

its instantaneous center of curvature ICCG, see Figure 4.5. With

GvG∗=RωG×rICCG,G∗(4.29)

and RωG=−GωR, Equation (4.28) becomes

GvV∗=RvV∗−RωG×#rICCG,G∗−rG∗,V ∗%.(4.30)

4.4 GUIDANCE KINEMATICS 91
Next, Equation (4.30) is projected on the unit vectors of !
"
G . With the transformation
given in Equation (4.4), RvV∗becomes
R
GvV∗=(Uxcos G∆ψ−Uysin G∆ψ)Get+(Uxsin G∆ψ+Uycos G∆ψ)Gen.(4.31)
Substituting the position vectors in Equation (4.30) by rICCG,G∗=−1
κGGenand by
rG∗,V ∗=G∆yGen, see Figure 4.5, and the angular velocity by RωG=˙
ψdes Gez, yields
G
GvV∗=>Uxcos G∆ψ−Uysin G∆ψ−˙
ψdes (1
κG
−G∆y)?
8 9: ;
:= G∆˙x
Get
+BUxsin G∆ψ+Uycos G∆ψC
89: ;
:= G∆˙y
Gen.
(4.32)
With the constraint G∆˙x= 0, the desired yaw rate ˙
ψdes follows from the Get-component
in Equation (4.32)
˙
ψdes =Uxcos G∆ψ−Uysin G∆ψ
1/κG−G∆y=κGUxcos G∆ψ−κGUysin G∆ψ
1−κGG
∆y.(4.33)
Assuming only small angular deviations, cos G∆ψ≈1 and sin G∆ψ≈G∆ψ, from the
planned path P, and ignoring the higher order terms, κG·G∆y≈0 and κG·G∆ψ≈0,
the guidance state variables can be written in matrix form
>G∆˙y
G∆˙
ψ?=>Uy
˙
ψ?+>G∆ψ
−κG?Ux(4.34)
with G∆˙
ψ=˙
ψ−˙
ψdes. By taking the time derivative of Equation (4.34), the deviation
from the planned path Pon the acceleration level becomes
>G∆¨y
G∆¨
ψ?=>˙
Uy
¨
ψ?+>˙
ψUx−κGU2
x+G∆ψ˙
Ux
−˙κGUx−κG˙
Ux?.(4.35)
Notice that in the first equation of (4.35), G∆˙
ψwas replaced by the second equation
of (4.34), G∆˙
ψ=G˙
ψ−κGUx, and that the term Ux˙
ψ, see for example Equation (2.18), was
taken into account in the derivative. With G∆˙
qlat =BG∆˙y, G∆˙
ψCT,G∆¨
qlat =BG∆¨y, G∆¨
ψCT,
v=BUy,˙
ψCT, and ˙
v=B˙
Uy,¨
ψCT, the guidance states in Equation (4.34) and the time
derivatives of the guidance states in Equation (4.35) read
G∆˙
qlat =v+>G∆ψ
−κG?Ux,(4.36)
G∆¨
qlat =˙
v+>˙
ψUx−κGU2
x+G∆ψ˙
Ux
−˙κGUx−κG˙
Ux?.(4.37)

92 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

4.5 Guidance Dynamics

Guidance dynamics describe the motion of the vehicle with respect to the planned path P.

In order to do so, the deviation from the path on the acceleration level, Equation (4.37),

is multiplied by the mass matrix M

MG∆¨

qlat =M˙

v+M>˙

ψUx−κGU2

x+G∆ψ˙

−˙κGUx−κG˙

Ux?.(4.38)

Recalling the equations of motion of the linear single-track model from Equation (2.37)

M˙

v=−0G(Ux)+D(Ux)1v+Bu,(4.39)

M˙

vin Equation (4.38) can be replaced by Equation (4.39), leading to

MG∆¨

qlat =−0G(Ux)+D(Ux)1v+Bu+M>˙

ψUx−κGU2

x+G∆ψ˙

−˙κGUx−κG˙

Ux?.(4.40)

Next, inserting v=G∆˙

qlat −0G∆ψ,−κG1TUxfrom Equation (4.36) yields

MG∆¨

qlat =−0G(Ux)+D(Ux)1G∆˙

qlat −0G∆ψ,−κG1TUx+Bu

+M>˙

ψUx−κGU2

x+G∆ψ˙

−˙κGUx−κG˙

Ux?.(4.41)

Rearranging Equation (4.41) in terms of G∆¨

qlat,G∆˙

qlat,G∆qlat and the disturbance vector

z=[κG,˙κG]T, gives the equations of motion with respect to the planned path P

MG∆¨

qlat +D(Ux(t)) G∆˙

qlat +KG∆qlat =Bu +E(Ux(t)) z(4.42)

with the damping matrix D, see Equation (2.39), and the stiffness matrix K, being

D=





α+CF

−CR

αb−CF

αa

−CR

αb−CF

αa

αb2+CF

αa2





and K=<0−.CR

α+m˙

Ux+CF

α/

0CR

αb−CF

αa=(4.43)

and the control matrix Band the disturbance matrix E, being

B=>CF

α0

αa1?and E=>CR

αb−mUx2−CF

αa0

αa2−CR

αb2+˙

UxIz−IzUx?.(4.44)

Notice that a known distribution of differential braking at front and rear axle is assumed,

see Equation (4.12), and therefore Bis quadratic. The control inputs are u=[δMF/R

∆Fx]T.

4.6 STABILITY ANALYSIS AND CONTROLLER DESIGN 93

4.6 Stability Analysis and Controller Design

The general idea of a potential field controller guiding a vehicle along a planned path

Pbased on the lateral and angular deviation was lined out. Subsequently, guidance

kinematics and guidance dynamics were formulated in order to provide the equations of

motion of the linear single-track model with respect to P. Now, stability issues of the

system are addressed and in this connection a design example of a potential field controller

is discussed. The stability analysis is based on Lyapunov’s direct method, which is briefly

recalled first.

4.6.1 Lyapunov’s Direct Method

Lyapunov’s direct method is related to the fact that the energy of a dissipative mechanical

system with no external force input decreases along any trajectory. The basic idea is to

show that a scalar energy-like function Lexists, which is positive definite and decreases

in a monotone way with time. As Ldepends on the states of the system, the existence

of a function Lindicates that the states will saddle down to an equilibrium x. The pro-

cedure of showing stability in the sense of Lyapunov for linear as well as for nonlinear

systems employs the following results, see Vincent and Grantham (1997, [119], p. 217):

Lyapunov’s Stability Theorem: The equilibrium solution at x=0to a dynamical

system ˙

x=f(x)is stable if a C1function L(x)can be found satisfying the following

conditions in some open region X containing the origin as an interior point:

(i)L(0) = 0,

(ii)L(x)>0for all nonzero x∈Xand

(iii)˙

L(x)≤0for all x∈X.

The equilibrium solution x=0is globally asymptotically stable if a function Lcan be

found such that (i)-(iii) hold, and

(iv)˙

L(x)<0for all nonzero x∈X

and in addition

(v)L(x)→∞ as *x*→∞,regardless of direction.

Condition (v) is referred to as ”growth” condition by Vincent and Grantham (1997,

[119], p. 217) or as radial unboundness by Slotine and Li (1991, [105], p. 65).

A main advantage of Lyapunov’s direct method in stability analysis is that it does not

require to integrate the equations of motion. Besides that the procedure can be applied

to higher dimensional nonlinear systems. However, the main challenge lies in finding a

Lyapunov function for the considered system; some techniques for constructing Lyapunov

functions are discussed by Vincent and Grantham (1997, [119], pp. 221). A natural

choice used in Section 4.6.2 is to consider the sum of kinetic and potential energy of the

system as candidate Lyapunov function. Then, it has to be shown that the candidate

function is a Lyapunov function and therefore satisfies Lyapunov’s stability theorem. In

the following Lyapunov’s direct method is applied to the linear equations of motion of the

94 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL
single-track model with respect to the planned path P. Lyapunov candidate functions for
linear systems are often formulated as quadratic forms
L=xTPx.(4.45)
In that context the definiteness of matrices such as Pbecomes important in order to
check if the candidate function is a Lyapunov function. Here, Sylvester’s theorem is help-
ful Vincent and Grantham (1997, [119], p. 213):
Sylvester’s Theorem: A symmetric matrix, P=PT, and its corresponding quadratic
form in Equation (4.45) are positive definite if and only if the determinants of the principle
minors are all positive:
|p11|>0,$$$$p11 p12
p21 p22 $$$$>0, . . . , $$$$$$$
p11 · · · p1n
.
.
.....
.
.
pn1· · · pnn
$$$$$$$
.(4.46)
Noteworthy, the assumption that Pis symmetric is not restrictive as Equation (4.45)
can be written equivalently in terms of a symmetric matrix. Considering the symmetric
matrix Psym =1
20P+PT1=PT
sym and on the other hand the skew symmetric matrix
Pskew =1
20P−PT1=−PT
skew, Equation (4.45) can be rewritten as
L=xT[Psym +Pskew]x.(4.47)
Since xTPskew x≡0, it follows
L=xTPx=xTPsym x.(4.48)
4.6.2 Stability Analysis
An important prerequisite to apply Lyapunov’s direct method is to ensure that no energy is
transferred into the system by external disturbances. Otherwise the existence of Lyapunov
functions cannot be guaranteed. Hence, the control input in Equation (4.42) is split into
two parts, u=uguid +udr. The first part uguid is the map of the generalized virtual
guidance force VQguid
lat , see Equation (4.4), regulating the path tracking error G∆qlat (t),
which is given in Equation (4.23). The second part udr is then used to reject external
disturbances z=[κG,˙κG]T. For linear time-variant systems as the equations of motion
with respect to the planned path P, recall Equation (4.42)
MG∆¨
qlat +D(Ux(t))G∆˙
qlat +KG∆qlat =Bu +E(Ux(t)) z,(4.49)
the disturbance rejection, see F¨
ollinger (1994, [36], p. 518), is given by
Budr +E#Ux(t)%z=0⇒udr =−B−1E#Ux(t)%z.(4.50)

4.6 STABILITY ANALYSIS AND CONTROLLER DESIGN 95

The resulting control structure is depicted in Figure 4.6.

Figure 4.6: Control loop with potential field control and disturbance rejection

Thus, disturbances due to curvature κGand curvature rate ˙κGare eliminated from the

system. The potential field guidance controller only has to compensate for the remaining

disturbances, as for example due to side wind or bank. Recalling the generalized virtual

guidance force and the linear map on the control inputs

GQguid

lat =−Kguid

G∆qlat with Kguid =kguid >kguid

11 kguid

kguid

21 kguid

22 ?and Buguid ≡GQguid

lat

(4.51)

from Equation (4.3) and Equation (4.12), the equations of motion of the closed-loop-

system become

MG∆¨

qlat +D#Ux(t)%∆˙

qlat +0K+Kguid1∆qlat =0with Ux(t)>0.(4.52)

Requiring the longitudinal speed of the vehicle Ux(t), controlled by the driver, to vary

slowly and thus neglecting terms incorporating the longitudinal acceleration ˙

Uxin the

equations of motion (4.52), the stiffness and the disturbance matrix become

K=>0−(CR

α+CF

α)

0CR

αb−CF

αa?and E=>CR

αb−mU

2−CF

αa0

−CR

αb2+CF

αa2−IzUx?(4.53)

and the following propositions hold.

96 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL
Proposition 4.1. For any generalized virtual guidance force GQguid
lat =−Kguid G∆qlat with
the components of Kguid satisfying the following conditions
(i)stiffness scaling factor kguid >0,
(ii)lateral stiffness kguid
11 >0,
(iii)lateral-heading coupling stiffness kguid
12 ∈Rchosen arbitrarily ,
(iv)lateral-heading coupling stiffness kguid
21 =kguid
12 −CF
α+CR
α
kguid and
(v)heading stiffness
kguid
22 >1
kguid B#CF
αa−CR
αb%+1
kguidkguid
11 #CF
α+CR
α−kguidkguid
12 %2C,
and with
T(G∆˙
qlat) := 1
2G∆˙
qT
latMG∆˙
qlat (4.54)
being the kinetic energy of the lateral motion and with
V(G∆qlat) := 1
2G∆qT
lat0K+Kguid1G∆qlat (4.55)
being a potential energy-like function, the function
L(G∆˙
qlat,G∆qlat)=T(G∆˙
qlat)+V(G∆qlat) (4.56)
is a Lyapunov-Function to the linear-time-variant dynamical system in Equation (4.52)
with the equilibrium point G∆qlat0 = [0,0]T,G∆˙
qlat0 = [0,0]T∀t≥t0.
Proof. To be a Lyapunov-Function the following conditions must hold, see Section 4.6.1,
see also Sattel and Brandt (2007, [95]):
1.) Lis positive definite:
The kinetic energy T, given in Equation (4.54), is positive definite. Thus, the compo-
nents of the controller stiffness matrix Kguid have to be chosen in order to guarantee that
the potential energy-like function V, see Equation (4.55), and therefore the closed loop
stiffness matrix #K+Kguid%is positive definite. In order to do so, it is assumed that the
closed loop stiffness matrix is symmetric, #K+Kguid%=#K+Kguid%T, which is equi-
valent to condition (iv) in Proposition 4.1. Then, Sylvester’s theorem (4.46) guarantees
#K+Kguid%to be positive definite for kguid >0, kguid
11 >0, and det #K+Kguid%>0,
which is satisfied by conditions (i), (ii), and (v) in Proposition 4.1.
2.) ˙
Lis negative definite:
The time derivative of Lgives ˙
L=G∆˙
qT
latMG∆¨
qlat +G∆˙
qT
lat(K+Kguid)G∆qlat .Sub-
stituting MG∆¨
qlat by Equation (4.52) yields ˙
L=−G∆˙
qT
latD(Ux)G∆˙
qlat. Since D(Ux) is
positive definite, see Equation (2.39), it follows that ˙
L<0 holds.

4.6 STABILITY ANALYSIS AND CONTROLLER DESIGN 97
Proposition 4.2. The equilibrium point G∆qlat0 = [0,0]T,G∆˙
qlat0 = [0,0]T∀t≥t0of
Equation (4.52) is Lyapunov-stable.
Proof. With Proposition 4.1 a Lyapunov-Function L, see Equation (4.56), is given for the
linear-time-variant system in Equation (4.52). Then, it follows from Lyapunov’s stability
theorem, see Section 4.6.1, that the equilibrium point is stable.
It was shown for the linear-time-variant single-track model that the general potential
field controller design given in Section 4.1 yields closed loop dynamics that are stable
in the sense of Lyapunov if the guidance stiffness matrix Kguid satisfies the conditions
given in Proposition 4.1. These conditions can be used for controller design. An example
implementation is given later in Section 4.6.5.
4.6.3 Steady State Tracking Error
In the last section stability in the sense of Lyapunov with respect to the planned path P
was shown for the linear single-track model guided by a general potential field controller in
combination with a disturbance compensation. Now, the remaining steady state tracking
error of this control concept is addressed. The equations of motion of the closed loop
system were
MG∆¨
qlat +D#Ux(t)%G∆˙
qlat +0K+Kguid1G∆qlat =Budr +E(Ux(t)) z.(4.57)
In steady state the time derivatives of the lateral tracking error vanish, G∆˙
qlat =0
and G∆¨
qlat =0. Hence, solving the equation of motion of the closed loop system in
Equation (4.57) for the lateral steady state tracking error gives
G∆qlat =0K+Kguid1−10Budr +E(Ux(t)) z1.(4.58)
As the disturbance compensation was defined as
udr =−B−1E#Ux(t)%z,(4.59)
see Equation (4.50), the steady state tracking error with respect to Pis identical zero
G∆qlat =0.(4.60)
Summarizing, the potential field control concept has no steady state tracking error
if the disturbances due to curvature κGand curvature rate ˙κGof the planned path are
rejected.
4.6.4 A Bound on the Tracking Error - Collision Avoidance
By use of Lyapunov’s direct method it was shown that the closed loop system under
guidance of the potential field controller is stable at all speeds Uxand has no steady state
tracking error if disturbances due to the planned trajectory Pare compensated. However,
the Lyapunov function L, given in Equation (4.56), is extremely powerful as it can be

98 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

used to bound the tracking error depending on the initial configuration. This fact was

already noticed by Rossetter (2003, [93], p. 71) in context with lane-keeping. According

to Lyapunov’s stability theorem the energy-like Lyapunov function

L(G∆˙

qlat,G∆qlat)=T(G∆˙

qlat)+V(G∆qlat) (4.61)

cannot increase over time as ˙

L≤0 holds by definition. Thus, the maximal possible

deviations from the planned path P, representing the configuration level of the elastic

band, can occur when the entire initial energy is transferred into the potential energy-like

term

V≤L(G∆˙

qlat(t),G∆qlat(t)) ≤L(G∆˙

qlat(t0),G∆qlat(t0)) .(4.62)

Recall that in each adaptation step from elastic band Kto elastic band K+ 1 the

trajectory does, in general, not start from the current position of the vehicle in order to

avoid discontinuities in the control inputs, see Figure 4.7 and Equation (3.40).

Figure 4.7: Initial deviation from elastic band

To determine the maximal possible lateral deviation G∆ymax from the elastic band with

respect to the given initial configuration

V=1

2G∆qT

lat(t)0K+Kguid1G∆qlat(t)=L(G∆˙

qlat(t0),G∆qlat(t0)) (4.63)

has to be satisfied. In two dimensions, G∆qlat =[

G∆y, G∆ψ]T, constant contours for the

positive definite quadratic form Vare ellipses as sketched in Figure 4.8.

Figure 4.8: Qualitative presentation of the bound for the lateral deviation G∆qlat

4.6 STABILITY ANALYSIS AND CONTROLLER DESIGN 99

With the total initial energy being transferred into the potential energy-like term V,

the necessary condition for finding the maximal possible lateral deviation G∆ymax is

∂V

∂G∆ψ=˜

k12 G∆y+˜

k22 G∆ψ=0,(4.64)

where S

K:= K+Kguid denotes the closed loop stiffness matrix being composed of the

stiffness matrix of the equations of motion Kand the stiffness matrix of the potential

field controller Kguid. Thus, the maximal possible lateral deviation G∆ymax occurs for

G∆ψ=−˜

k12

k22

G∆y. (4.65)

Substituting Equation (4.65) into the potential V, see Equation (4.63), yields

G∆y2

max =2L(G∆˙

qlat(t0),G∆qlat(t0))

k11 −˜

k12/˜

k22

.(4.66)

For a given closed loop stiffness matrix S

K, Equation (4.66) gives the maximal possible

lateral deviation G∆ymax to the planned path Pdepending on the initial configuration.

However, the result can also be used to ensure collision avoidance! With Gdmin describing

the closest position of an obstacle with respect to the elastic band P, see Figure 4.8,

k11 −˜

k12

k22

=2L(G∆˙

qlat(t0),G∆qlat(t0))

Gd2

min

(4.67)

gives a condition for the components of S

Kguaranteeing collision avoidance between the

host-vehicle following the elastic band and the closest obstacle. This result is in particular

useful as it does not include longitudinal dynamics. The kinetic energy of the longitudinal

motion, at least for highway scenarios, is much higher than the kinetic energy of the lateral

motion. Therefore, including longitudinal dynamics in the Lyapunov function would lead

to more conservative results. However, it has to be emphasized that the preceding analysis

only covers linear vehicle dynamics. Furthermore, it was assumed that disturbances are

fully rejected.

4.6.5 Sample Controller Design

From the general potential field control concept introduced in the last sections, different

implementations can be derived by choosing the parameters of Kguid according to Propo-

sition 4.1. One possible implementation is illustrated in Figure 4.9. The spring with

stiffness klat symbolizes the potential GVguid(G∆qlat)=1

2G∆qT

lat Kguid

G∆qlat in the lateral

deviation, evaluated at the look ahead point LA. In order to scale the lateral guidance

force GFguid

ywith respect to the guidance moment GMguid

z, the guidance force is applied at

the point LG with lLG =λ·lLA. This particular controller was proposed by Rossetter

(2003, [93]).

100 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

Figure 4.9: Sample potential field controller

The generalized guidance force reads

GQguid

lat ≡>GFguid

GMguid

z?=−>klat klat lLA

λklat lLA λklat l2

LA ?

8 9: ;

Kguid

>G∆y

G∆ψ?(4.68)

=−>klat#G∆y+lLA G∆ψ%

λklat#G∆y+lLA G∆ψ%lLA ?,

where the expression #G∆y+lLA G∆ψ%corresponds, for small angles sin G∆ψ≈G∆ψ, to

the length of the spring depicted in Figure 4.9. The guidance moment GMguid

zis scaled

by shifting the point of attack LG of the lateral guidance force.

Applying the conditions in Proposition 4.1 or the underlying requirements kguid

11 >0,

det #K+Kguid%>0, and #K+Kguid%=#K+Kguid%Tdirectly, it can be shown, that the

equilibrium point is stable for

klat >0 (4.69)

lLG >CF

αa−CR

αb

α+CF

=: lNSP (4.70)

lLA =lLG +CF

α+CR

klat

.(4.71)

4.6 STABILITY ANALYSIS AND CONTROLLER DESIGN 101
The point lNSP =CF
αa−CR
αb
CR
α+CF
αin Equation (4.70) has physical significance in being the neutral
steer point NSP. The neutral steer point is that location on the centerline of a vehicle
where an external force produces no steady state yaw velocity. This concept is often used
to discuss sidewind sensitivity of a vehicle, see Zomotor (1991, [128], p. 131), and has
a natural interpretation when considering virtual forces and stability. For stability, the
virtual force must be applied in front of the neutral steer point of the vehicle. This ensures
that the vehicle will rotate in the same direction as the virtual control force points.
The equivalence condition in Equation (4.12) to map the generalized guidance force
GQguid
lat , see Equation (4.68), on the control inputs reads
VQguid
lat =CV,G
GQguid
lat =−>klat#G∆y+lLA G∆ψ%cos G∆ψ
λklat#G∆y+lLA G∆ψ%lLA cos G∆ψ?(4.72)
≡>CF
αδ
aCF
αδ+MF
∆Fx+MR
∆Fx?.
According to the h-metaphor, Flemisch et. al. (2003, [35]), the intended maneuver
should be coordinated with the driver by use of haptic control elements that the driver
can access and interpret, in this particular case the steering wheel. However, the map of
the potential field guidance law in Equation (4.72) includes steering as well as differential
braking, which can hardly be coordinated with the driver. For this reason MF
∆Fx=MR
∆Fx=0
is chosen. Another beneficial side effect of avoiding differential braking on the guidance
level is that conflicts with systems on the stability level, see Donges (1982, [28]), such
as ESP that use differential braking cannot occur. Consequently, the guidance control
concept can be combined in a very modular way with systems that act on the stability
level. The map on the control inputs, Equation (4.72), results in
VQguid
lat =−>klat#G∆y+lLA G∆ψ%cos G∆ψ
λlLAklat#G∆y+lLA G∆ψ%cos G∆ψ?≡>CF
αδ
aCF
αδ?.(4.73)
To fulfill both conditions in Equation (4.73), it is required that the lateral guidance force
is applied on the front axle, λlLA =lLG =a. Summarizing, the parameters of this sample
controller according to equations (4.69)-(4.71) and Equation (4.73) become
klat >0,l
LG =a > lNSP and lLA =lLG +CF
α+CR
α
klat
.(4.74)
The controller stiffness klat can for example be chosen according to Equation (4.67).
4.6.6 Feedforward Control
In Section 4.6.2 disturbances due to the curvature κGand due to the curvature rate ˙κGof
the planned path Pwere completely rejected by Equation (4.50). However, some remarks
should be made:

102 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

•As already stated for the potential field guidance controller, the intended maneuver

should be coordinated with the driver in accordance to the h-metaphor. Therefore,

also the disturbance compensation should only use those control inputs that can

be accessed by the driver, in this particular case the steering wheel. However, the

disturbance rejection given in Equation (4.50) uses steering as well as differential

braking.

•The access of differential braking by the disturbance rejection in Equation (4.50)

as part of the guidance control scheme can lead to confusion with systems on the

stability level such as ESP that also access differential braking.

•It cannot be guaranteed that the yaw moment implied by the steering angle δand

by differential braking MF/R

∆Fxhave the same orientation.

•The potential field controller acts on the configuration level and does not require a

very costly interpolation of the nodes of the elastic band (for a fairly high number of

nodes even a linear interpolation is sufficient). However, the discussed disturbance

rejection requires smoothness for curvature and curvature rate. To be continuous in

the curvature rate ˙κG, a spline interpolation would require interpolation polynomials

of order 7 or higher is necessary. Besides the high computational effort, higher order

polynomials tend to oscillate between the nodes of the elastic band.

To address these issues, a feedforward control only incorporating the curvature κG, but

not the curvature rate ˙κG, of the planned path Pcan be applied. Furthermore, using

only the steering angle δas control input, the driver can directly interpret the intended

maneuver of the feedforward controller. By recalling the Ackermann steering angle δA

from Equation (2.54) and the self steering gradient SG, refer to Equation (2.60), a possible

feedforward control reads

δff=δA+SG ·R

GaCG

n=(a+b)·κG+m(CR

αb−CF

αa)

αCR

α(a+b)·U2

xκG.(4.75)

The feedforward steering angle only depends on the curvature κGof the planned path P.

For low speeds Uxthe Ackermann angle dominates, while for understeering vehicles the

part incorporating the self steering gradient SG increases with the speed of the vehicle.

The control scheme including potential field guidance and feedforward control is visualized

in Figure 4.10.

Figure 4.10: Control loop including potential field guidance and feedforward control

4.7 SIMULATIONS 103

4.7 Simulations

The potential field guidance controller described in Section 4.6.5 in combination with dis-

turbance rejection, see Equation (4.50), and with feedforward control, see Equation (4.75),

is now illustrated for a double lane change scenario. The course given by splines of seventh

order is followed at a longitudinal speed of Ux= 30 m/s. The intended course and the

corresponding curvature κGand curvature rate ˙κGare shown in Figure 4.11.

Figure 4.11: Double lane change maneuver with curvature κGand curvature rate ˙κG

The parameters of the nonlinear single-track model with differential braking used in

the simulation are summarized in Table 4.1.

Table 4.1: Vehicle parameters

Parameter Symbol Value

Mass m1700 kg

Yaw moment of inertia Iz2500 kgm2

Front axle distance to CG a1m

Rear axle distance to CG b1.25 m

Track width d1.54 m

Cornering stiffness front tires CF

α63000 N/rad

Cornering stiffness rear tires CR

α63000 N/rad

Longitudinal stiffness front tires CF

x160000 N

Longitudinal stiffness rear tires CR

x160000 N

Effective roll radius R00.3m

Wheel moment of inertia Iwheel 0.9 kgm2

Adhesion coefficient µ00.87

104 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

The potential field guidance controller is parametrized by the values given in Table 4.2,

while the parameters of the feedforward controller as well as of the disturbance rejection

follow from the vehicle model in Table 4.1.

Table 4.2: Controller parameters

Parameter Symbol Value

Look ahead distance lLA 35 m

Attack point of lateral guidance force lLG (= a) 1 m

Controller stiffness klat 4500 N/m

Figure 4.12 shows the intended and the driven course for the vehicle guided by the

potential field controller in combination with the feedforward law in Equation (4.75)

based on Ackermann steering and the self steering gradient. The resulting steering angle,

δ=δguid +δff, as well as the control inputs from the potential field controller δguid and

from the feedforward controller δffare depicted in the second part of the figure. In steady

state, at constant curvatures κG, compare to Figure 4.11, the vehicle is guided by the

feedforward controller. In transient phases from one constant curvature κGto another the

resulting steering angle is dominated by the potential field controller regulating deviations

from the intended course. The lateral deviation G∆yas well as the angular deviation are

fairly low with maximal values of about 0.2 m and 0.5◦, respectively.

Figure 4.12: Double lane change at 30 m/s with potential field control δguid in combination

with feedforward control δffaccording to Equation (4.75)

The lateral acceleration Vayand the side slip angle βexhibited by the vehicle are given

in Figure 4.13. The low level of the lateral acceleration, compare to Section 2.7, indicates

that the vehicle dynamics are in the linear range.

4.7 SIMULATIONS 105

Figure 4.13: Lateral acceleration and side slip angle

Figure 4.14 shows the same maneuver driven by the potential field controller in com-

bination with the disturbance rejection according to Equation (4.50). The disturbance

rejection accesses the steering angle as well as differential braking. For simplicity, a fixed

distribution applying differential braking solely at the rear axle is assumed. In contrast

to the combination of potential field control and feedforward control, the steering angle of

the potential field controller δguid and the disturbance rejection δdr counteract each other.

Besides that, also the differential braking ∆FR

xand the steering angle provided by the

disturbance rejection δdr cause yaw moments in opposite directions.

Figure 4.14: Double lane change at 30 m/s with potential field control δguid in combination

with disturbance rejection δdr

The lateral and the angular deviation are in the same range as for the potential field

controller in combination with feedforward control. But the amount of the control inputs

is higher. As the resulting steering angle was limited to about 0.5◦for the guidance

concept consisting of potential field control and feedforward, the resulting steering angle

of potential field guidance in combination with the disturbance rejection is about twice as

much. Additionally, high differential braking forces of about 2 kN have to be generated.

With increasing speed these effects become even more severe.

106 4POTENTIAL FIELD BASED VEHICLE GUIDANCE CONTROL

4.8 Comment on Human Vehicle Guidance

In general, human drivers handle the task of vehicle guidance on two layers: anticipating

(open-loop) and compensating (closed-loop control), see for example Mitschke and

Wallentowitz (2004, [74], p. 649). On the anticipation layer the curvature of the road

is estimated by means of visual information and the according steering angle is chosen. On

the compensation layer lateral deviations from the track are regulated. On roads with low

curvature, good vision conditions and in absence of obstacles the anticipating control is

dominant. Compared to the control scheme proposed in the last sections, the feedforward

controller acts on the anticipation layer, while the potential field controller acts on the

compensation layer. However, one difference between the potential field control concept

with feedforward compared to human drivers is that drivers usually choose a reference

point in some look-ahead distance. This concept could be adopted for guidance control

by evaluating the path tracking error in a look-ahead distance. Then, the driven path

would additionally be smoothed by the controller. However, large deviations at the look-

ahead point, caused by high curvatures of the planned path, can cause local deviations

of the vehicle from the planned path in the vicinity of obstacles. This problem becomes

particular severe if obstacles are within the look-ahead distance, e.g. in passing maneuvers.

Then, the guidance controller could cause collisions as illustrated in Figure 4.15.

Figure 4.15: Over-steering of the planned path by a guidance controller with look-ahead

The situation depicted in Figure 4.15 could occur if the elastic band guides the host-

vehicle around an obstacle and demands, after passing it, to return to the previously

occupied lane: at time t1, the controller predicts a deviation to the left at the look-ahead

point, which is caused by the curvature κGof the elastic band. Therefore, the controller

demands to steer the vehicle to the right, as indicated at time t2. However, due to the

curvature κGof the planned path the deviation at the look-ahead point becomes larger.

This results in further corrections by the controller. The controller oversteers the planned

path due to the look-ahead, which finally leads to a collision at time t4.

The reference point for the control scheme proposed in the last section was that point

on the planned path Pbeing closest to the actual position of the vehicle, see Section 4.3.

For situations as the one sketched in Figure 4.15 this strategy is similar to drivers reducing

their look-ahead distance for maneuvers with high intended curvatures. However, in lane-

keeping situations, in the absence of obstacles, the proposed control scheme differs from

the strategies usually followed by drivers. To resemble the behavior of human drivers, the

proposed control scheme could be applied in presence of obstacles, while in lane-keeping

situations without obstacles the look-ahead point could be shifted forward. This would

be analog to drivers balancing between anticipating and compensating control strategies.

Chapter 5

Shared Vehicle Guidance between

Driver and Assistance System

Potential field motion planning and vehicle guidance control provide the basis for driver

assistance on the guidance level. In order to guide a vehicle cooperatively with the driver,

the guiding strategies of driver and assistance system have to be coordinated via a Human

Machine Interface (HMI). In the following, the vehicle guidance control loop including

driver, assistance system, human machine interface, and a first driving simulator explo-

ration of the potential field based assistance system are presented.

5.1 Vehicle Guidance Control Loop

Driver and assistance system should work cooperatively. Therein, the driver should always

be able to override the automation as for example recommended by Griffiths and

Gillespie (2004, [44]). In particular, the driver should permanently be kept in the control

loop as demanded by the h-metaphor of Flemisch et al. (2003, [35]). Focusing on

acceptance, the approach of continuous haptic interaction is further supported by studies

showing that systems without feedback to the driver, automatically engaging in vehicle

guidance, are not accepted as reported by Wolf, Z¨

ollner and Bubb (2005, [124]).

Figure 5.1 shows how potential field motion planning, lateral guidance control, and

driver are embedded in the overall control structure. Therein, the driver remains com-

pletely responsible for the longitudinal vehicle dynamics, whereas driver and assistance

system cooperate in the lateral vehicle guidance. Throttle and brake pedal forces FP/D

of the driver are the inputs to the powertrain. The resulting driving and braking torques

Md/bare control inputs for the longitudinal guidance, which is solely controlled by the

driver. The assistance system displays guidance information to the driver via an additional

steering wheel torque MSW/A. Hence, the resulting steering angle δSW, set cooperatively

by the driver and the assistance system, corresponds to the equilibrium of torques at the

steering wheel. As depicted in Figure 5.1, these torques consist of torques due to vehicle

dynamics MSW/V, see for example Equation (2.15), the assistance torque MSW/A, and the

torque exerted by the driver MSW/D. The motion planning module adapts to the driver’s

commands based on the vehicle’s states and acceleration [Rq,Vv,Va] as well as on the

steering angle δ, as explained in Chapter 3. New trajectories are generated after planning

108 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

intervals of ∆T<100 ms. By this means environmental changes can also be incorporated

into the motion planning.

Figure 5.1: Control loop of potential field driver assistance for shared vehicle guidance

The guidance controller regulates position and heading deviation between the desired

vehicle configuration, given by Rqdes, and the actual vehicle configuration Rq. Even with-

out additional driver commands, the potential field controller would be able to guide the

vehicle along the predicted trajectory by means of the proposed steering angle δguid. How-

ever, in order to keep the driver in the loop, δguid is not directly applied to the wheels. In

fact, the proposed steering angle δguid, multiplied by the steering ratio iS, is compared to

the steering angle δSW, which is currently set at the steering wheel. In order to close the

control loop, the difference ∆δSW between the proposed and the set steering angle is used

for haptic rendering. In doing so, the feedback characteristic acts as an impedance, pro-

viding the assistance torque MSW/Adisplayed to the driver. Griffiths and Gillespie

(2004, [44]) define that impedance by using proportional control for a small handwheel

that could be turned by one hand. In that case, the force feedback could be considered as

a virtual torsion spring between the proposed and the actual steering angle. Noteworthy

is, that simulator studies with alternative approaches were also conducted, see for example

Penka (2001, [84]) or Pohl and Ekmark (2003, [85]). In these contributions the feed-

back to the driver directly depends on the lateral deviation of the vehicle to the desired

trajectory. However, approaches without guidance controller seem hardly applicable to

evasion maneuvers. For this reason, here too an approach based on a virtual spring is

chosen. Though, the stiffness of the virtual torsional spring is adapted as explained in

Section 5.2.3. Most notably, the resulting assistance torque MSW/Acan always be over-

come by the driver.

5.2 DRIVING SIMULATOR EXPLORATION 109

5.2 Driving Simulator Exploration

The proposed control structure for shared vehicle guidance was studied in a driving sim-

ulator exploration. First, the haptic human machine interface used is presented, followed

by the driving simulator setup and first lane-keeping and collision avoidance experiments.

5.2.1 Haptic Human Machine Interface

In the simulator exploration, the haptic human machine interface was realized by a force

feedback steering wheel. To determine the requirements on the steering wheel, a brief

introduction on human haptics is given first. Therein, the origin of the term haptic,

which is broadly used to describe the mechanical interaction of human beings with their

environment, goes back to the Greek word hapteshai, meaning ”to touch”. Furthermore,

Gillespie (2005, [42]) categorizes haptic interfaces by the terms tactile and kinesthetic

as depicted in Figure 5.2. The adjective tactile describes perceptions primarily served

by sensors located in the skin, so-called cutaneous mechanoceptors, while the term kines-

thetic refers to perceptions served by sensors in muscles and joints, called muscle spindles

and Golgi organs, respectively. In general, manipulation involves tactile as well as kines-

thetic perceptions. Nevertheless, haptic interfaces might be classified by their primary

function. Thus, ignoring tactile perceptions such as sensing the surface, a force feedback

steering wheel could be characterized as kinesthetic interface. Furthermore, a force feed-

back steering wheel is grounded, which refers to the fact that it is rigidly attached to its

environment.

Figure 5.2: Taxonomy of haptic terms, Gillespie (2005, [42])

Besides the high demand on the visual channel, already mentioned in Section 1.1.3, the

processing speed of haptic stimuli is another benefit of haptic human machine interaction.

The haptic channel is the only one using the spinal cord for low level control, while acoustic

and visual information is interpreted by the brain. For this reason, haptic stimuli can be

processed within about 50 ms. The processing of visual or acoustic information takes

about 200 ms and thus four times longer, see for example Penka (2001, [84], p.4).

However, in order to define the requirements on the force feedback steering wheel,

further information about human haptic performance is necessary. The amplitude of

positional detection thresholds for example depends on the rate of change of positional

clues. Therein, very slow motions generally provide subliminal stimuli that are not per-

ceived consciously. Though, the resolution of positional clues rises with the frequency.

According to Penka (2001, [84], p. 6) amplitudes of 10−1mm can be perceived at 10 Hz,

while a frequency of 200 Hz allows a resolution up to 10−4mm.

110 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

Here, the force feedback steering wheel displays assistance torques generated by a DC-

motor. Therefore, a torque resolution beyond the human detection thresholds is desirable

in order to study arbitrary force feedback characteristics. Penka (2001, [84], p. 7) cites a

study, which gives torque detection thresholds in the range of 0.0036 Nm to 0.0054 Nm at

a base torque of 0.5 Nm and a range of 0.0072 Nm to 0.009 Nm at a base torque of 1 Nm.

These quantities were gained for active side sticks. For steering wheels Buschardt

(2003, [15]) reports a ”just noticeable difference” in the range of 0.4 Nm to 0.5 Nm and

torques of more than 0.8 Nm that are reliably detected by the driver. Further information

on detection thresholds can for example be found in Dosher, Lee and Hannaford

(2001, [29]).

Besides the ability to display small haptic effects in the range of the human detection

threshold, it should be possible to display haptic information up to a hard stop. Also the

ability to display high torque amplitudes is desired. For this reason the high performance

”Active Steering Wheel System” developed by Stirling Dynamics (2005, [115]) was

chosen for the simulator experiments. In a range up to 8 Nm, a DC-motor is precisely

controlled by a feedback controller based on a torque sensor. Beyond the range of 8 Nm

the DC-motor generates torques up to 38 Nm in open loop mode. Characteristic technical

data of the ”Active Steering Wheel System” are collected in Table 5.1.

Table 5.1: Technical data of the ”Active Steering Wheel System”, (2005, [115])

Performance

Max. Torque 38Nm at 12DC and 25◦C

Torque Response 38Nm at 300◦/s

20Nm at 600◦/s

Maximum Velocity 1000◦/s

Input Signals

Open Loop Torque Demand Range ±35Nm

Accuracy -

Closed Loop Torque Demand Range ±8Nm

Accuracy ±0.001Nm

Output Signals

Column Angle Range ±1600◦

Accuracy ±0.1◦

Column Rate Range ±5rps

Accuracy ±0.001rps

Column Torque Range ±8Nm

Accuracy ±0.001Nm

5.2 DRIVING SIMULATOR EXPLORATION 111

5.2.2 Driving Simulator Setup

The hardware and software architecture of the fixed-based driving simulator is summa-

rized in Figure 5.3. The visualization is established by three Epson EMP-9100R

'projec-

tors, which are rigidly mounted at the ceiling and project the traffic scenario on three

2.05 m×2.62 m screens. The test subjects control the driving speed by means of Logitech

MOMOR

'throttle and brake pedals, while for lateral guidance a force feedback steering

wheel is used as described in the last section. The pedals are directly connected to the

server by the Universal Serial Bus (USB), while the force feedback steering wheel signals

are converted from the Controller Area Network (CAN) to USB first. The simulation

is distributed on three dual processor PCs at 3.2 GHz clock rate and 2 GB main memory

each. One computer acts as server, the two clients are connected to the server via ethernet

using the User Data Protocol (UDP).

Figure 5.3: Simulator hardware and software architecture

The software architecture is based on the Straightforward Modular Prototyping Library

(SMPL++)1, which is a C++ library that can be used conveniently for rapid prototyping

of driver assistance systems. Due to the modular design new modules can very easily be

integrated into the framework. Therein, the different modules communicate via a Shared

Memory as depicted in Figure 5.3. The Shared Memory provides a common data basis

with read and write access for the different modules.

1SMPL++ was provided by courtesy of the group headed by Dr. F. Flemisch at the German Aerospace

Center (DLR)

112 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

In order to realize an equal processing load, the modules can be distributed on the

available computers. Furthermore, the execution of the modules can be ranked by a

priority flag. Figure 5.3 shows a possible distribution of the single modules on the three

machines. The computationally intensive Motion Planning and one of the three Vision

modules share for example an entire machine as indicated by the shading in Figure 5.3.

For the exploration the modules depicted in Figure 5.3 were included. Therein, Motion

Planning based on methods described in Chapter 3, Guidance Control in a version with

look-ahead, see Section 4.8, and Vehicle Dynamics play the same role as in Figure 5.1.

The modules Steering Wheel and Throttle/Brake control the interfaces to the driver.

While the Throttle/Brake module only reads the driver’s inputs, the Steering Wheel

module also contains the force feedback characteristic and computes the assistance torque

MSW/Adisplayed to the driver. The Scenario module contains the course of the road as

well as information about possible obstacles. Further on, the Virtual Sensor emulates

a sensor concept providing information about the course of the road and the obstacles.

This information is used by the Motion Planning and Guidance Control modules. The

three channel visualization is generated by the three Vision modules. The Experiment

Control allows the instructor to control the exploration, e.g. different assistant torque

characteristics or scenarios can be chosen. Simulation data are saved for analysis purposes

by the Storage module.

Figure 5.4 a) shows a photo of the fixed-based driving simulator. In the foreground a

test subject in front of the force feedback steering wheel, in the background the experiment

control station with the instructor can be seen. Figure 5.4 b) shows the scenery presented

to the test subjects.

(a) Test subject and experiment control (b) Scenery displayed to the test subjects

Figure 5.4: Driving simulator

5.2 DRIVING SIMULATOR EXPLORATION 113

5.2.3 Lane-Keeping and Collision Avoidance Experiments

In the following, the exploration of the potential field based driver assistance system in

the fixed-based driving simulator is described. In particular, the haptic communication

between the test subjects and the assistance system was studied. In previous simulator

studies by Penka (2001, [84]), Buld and Kr¨

uger (2002, [13]), and Pohl and Ekmark

(2003, [85]) the assistance torque depended directly on the lateral deviation from the lane.

One outcome of these studies is that soft assistance torques in a limited range of tolerable

deviation lead to the best results. However, the results vary significantly from subject to

subject.

Noteworthy, Bender and Landau (2006, [6]) observed in experiments that many

drivers do not react at all in critical situations. Hence, they propose that the assistance

system should overrule the driver in those situations. However, according to the previous

discussions, see Chapter 1, the driver should remain in the vehicle guidance control loop

and always be able to override maneuvers proposed by the assistance system. Neverthe-

less, within the scope of the h-metaphor the degree of assistance should be adapted to

the situation. In critical (tight rein) situations the driver should indicate by means of

sufficiently high steering torques that he is aware of the situation.

Here, the assistance torque MSW/Ais generated based on the difference ∆δSW between

the steering angle proposed by the assistance system and the actual set steering angle

at the steering wheel. A similar design for lane-keeping was studied by Steele and

Gillespie (2002, [109]) and Griffiths and Gillespie (2004, [44] and 2005, [45]). This

design can be considered as a virtual torsional spring providing a torque MSW/Adepending

on the steering angle deviation ∆δSW.

Force Feedback Characteristic

In the exploration, a combination of a continuous base torque depending on the deviation

∆δSW from the proposed steering angle (virtual spring) and a vibration, initiated after a

certain deviation threshold was studied. The intention of the base torque is to commu-

nicate the proposed steering strategy to the driver, while the vibration aims at ensuring

the drivers attention when a maneuver is performed that does not match with the tra-

jectories planned by the assistance system. The base torque has a linear characteristic

without a threshold but with saturation. The vibration can be shaped in different ways:

saw tooth and rectangular signal shapes can for example provide additional directional

information. However, in pretests it turned out that these signal shapes are often misin-

terpreted as disturbances such as side wind. Hence, the directional information is rather

counterproductive. Therefore, in the experiment sinusoidal vibrations were chosen, which

are often similar to the mental model that drivers create for chattermarks. In doing so,

two different types of sinusoidal vibration were used. In absence of obstacles a sinusoidal

(lane-keeping) vibration with an amplitude of 2 Nm at a frequency of 40 Hz was generated

beyond a threshold of $$∆δSW$$≥15◦. If the virtual environmental sensors detected obsta-

cles, a more aggressive (collision avoidance) vibration was applied in order to ensure the

driver’s attention: beyond a threshold of $$∆δSW$$≥15◦a vibration with an amplitude of

5 Nm at a frequency of 20 Hz was chosen. This design led to a limited range of deviations

114 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

of ∆δSW where the test subjects only experienced the continuous base torque as assistance

torque.

For evaluation purposes of the proposed force feedback design and to gather information

for further improvements of the force feedback characteristic, two different variants were

studied. In order to be able to interpret different results in the exploration, the number

of different parameters between both variants was limited. For this reason, only the

continuous base torques were altered for soft Assistance (sA) and for hard Assistance

(hA). For soft assistance a slope of 1 Nm/rad and for hard assistance a slope of 8 Nm/rad

was used. In both variants the base torque saturated at ±5 Nm. The parameters of the

assistance torque settings are summarized in Table 5.2; Figure 5.5 shows the resulting

combinations of soft and hard assistance torques MSW/Awith lane-keeping and collision

avoidance vibration plotted against the steering angle deviation ∆δSW and against the

time t. Besides that, also test drives where noAssistance torque (nA) was applied were

performed.

(a) Hard assistance (hA) with lane-keeping

vibration

(b) Soft assistance (sA) with lane-keeping

vibration

dance vibration

(d) Soft assistance (sA) with collision avoi-

dance vibration

Figure 5.5: Soft and hard assistance torque combined with lane-keeping and collision

avoidance vibration

5.2 DRIVING SIMULATOR EXPLORATION 115

Table 5.2: Parameters of assistance torques

Base torque Vibration

Assistance Slope Saturation Situation Amplitude Frequency

hard (hA): 8 Nm/rad ±5 Nm

soft (sA) : 1 Nm/rad ±5 Nm

hard (hA): 8 Nm/rad ±5 Nm

soft (sA) : 1 Nm/rad ±5 Nm

lane-keeping: 2 Nm 40 Hz

collision avoidance: 5 Nm 20 Hz

Test Group

The experiment was conducted with a heterogeneous group of sixteen test subjects as

depicted in Table 5.3. The test group was split half into female and male test subjects

as well as into experienced and unexperienced drivers, half of the test subjects were older

than 35.

Table 5.3: Test subjects

Subject-No. Age Gender Driving experience in km

1 26 male 260,000

2 40 male 550,000

3 27 male 150,000

4 39 male 750,000

5 27 female 60,000

6 23 female 50,000

7 36 female 2,000

8 31 female 75,000

9 56 female 216,000

10 28 female 35,000

11 26 female 2,000

12 42 male 1,200,000

13 23 male 25,000

14 49 male 500,000

15 64 male 1,500,000

16 56 female 750,000

Driving and Secondary Task

In this exploration the main focus lay on the lateral vehicle guidance. For this reason, the

test subjects were asked to drive at a constant speed of 100 km/h. In order to avoid that

the subjects get used to the virtual test track, two different tracks with a different order

of left and right turns with radii between 800 m and 1000 m were used. Each of the test

tracks depicted in Figure 5.6 had a length of approximately 6 km and therefore a driving

time of 4 minutes.

116 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

(a) Test track 1 (t1) (b) Test track 2 (t2)

Figure 5.6: Virtual test tracks used in exploration

In each test drive four 2 m x 2 m static obstacles suddenly appeared in front of the

virtual test vehicle, see for example Figure 5.4 b). The lateral position of the obstacles as

well as the distances in front of the vehicle varied. The obstacles appeared in distances

between 80 m and 104 m ahead of the vehicle such that evasion maneuvers at a speed of

100 km/h are always feasible.

To analyze the assistance system under different conditions, the test subjects were asked

to handle a secondary task in some of the test drives. In doing so, situations where drivers

are inattentive or distracted could be simulated and the effect of the assistance system in

these situations could be studied. The secondary task was presented on a display besides

the steering wheel. Figure 5.7 shows a test subject in two different experimental configu-

rations. In Figure 5.7 a) the subject is solely driving in the simulator and noSecondary

task (nS) has to be handled. In Figure 5.7 b) the subject is driving with Secondary task

(wS). The experimental setup was configured in a way that the test subjects could not

focus on the driving and on the secondary task at the same time.

(a) Subject solely driving (b) Subject driving and handling

secondary task

Figure 5.7: Driving with and without secondary task

In the secondary task, the test subjects got an array of ”T”s as shown in Figure 5.8.

Within this array one different letter (E, F, H, I, L) appeared at a random position.

The test subjects had to identify this letter and to press the corresponding key on the

keyboard. For each configuration the test subjects had 10 s time to identify the different

letter before the screen cleared and proceeded to the next configuration.

5.2 DRIVING SIMULATOR EXPLORATION 117

Figure 5.8: Secondary task

Experimental Design

The combination of the different force feedback characteristics (nA, sA, hA), with or

without secondary task (wS, nS), with a particular test track (t1, t2) generates the test

settings as collected in Table 5.4. Noteworthy, the parameters of motion planning and

vehicle guidance control are kept constant in all test settings.

Table 5.4: Test settings

Test setting Label

no assistance (nA) / no secondary task (nS) / test track 1 (t1) nA/nS/t1

no assistance (nA) / with secondary task (wS) / test track 1 (t1) nA/wS/t1

hard assistance (hA) / no secondary task (nS) / test track 1 (t1) hA/nS/t1

hard assistance (hA) / with secondary task (wS) / test track 2 (t2) hA/wS/t2

soft assistance (sA) / no secondary task (nS) / test track 2 (t2) sA/nS/t2

soft assistance (sA) / with secondary task (wS) / test track 2 (t2) sA/wS/t2

Before the experiment started, the test subjects had the opportunity to get used to

the driving simulator on a training test track, which differs from track 1 and track 2

used in the exploration. This training period was divided into three sessions: first the

subjects drove without assistance and secondary task in order to get familiar with the

simulator environment. Second, they practiced the secondary task and finally, they drove

and handled the secondary task in parallel. Noteworthy, in the training phase no obstacles

appeared. There was no particular time limit for the training. The training phase ended,

when the subjects felt confident in the driving as well as in the secondary task. Before

the main exploration started the performance of the test subjects in the secondary task

without driving was measured as baseline.

118 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

In order to avoid influences of the order in which the different test settings were pre-

sented to the subjects, the different test conditions were combined into four different

sequences as collected in Table 5.5. Thereby, test settings with equal assistant variants

(nA, hA, sA) are kept consecutively in each sequence. The test subjects were randomly

assigned to one of the four sequences. Drive I and drive II were in all cases without

assistance torques.

Table 5.5: Test sequences

Sequence 1 Sequence 2 Sequence 3 Sequence 4

Drive I: nA/nS/t1 nA/nS/t1 nA/nS/t1 nA/nS/t1

Drive II: nA/wS/t1 nA/wS/t1 nA/wS/t1 nA/wS/t1

Drive III: sA/nS/t2 hA/nS/t1 hA/wS/t2 sA/wS/t2

Drive IV: sA/wS/t2 hA/wS/t2 hA/nS/t1 sA/nS/t2

Drive V: hA/nS/t1 sA/nS/t2 sA/wS/t2 hA/wS/t2

Drive VI: hA/wS/t2 sA/wS/t2 sA/nS/t2 hA/nS/t1

Before the test drives began, the subjects were reminded that the primary task consists

in driving at a speed of 100 km/h and in remaining at the right lane. However, potential

obstacles had to be evaded. Besides that, the subjects should perform in the secondary

task as good as possible.

The simulation data of each test drive were stored for subsequent analysis. Besides

these objective data, the subjects had to fill out a detailed questionnaire after each test

drive. In this questionnaire topics ranging from acceptance of driver assistance systems in

general down to acceptance of the presented system in particular situations such as lane-

keeping or collision avoidance were covered. Besides that, the test subjects were asked for

suggestions how to improve the presented assistance system. The answers are collected in

the appendix.

5.2.4 Results and Discussion

One of the most significant results of the exploration is the mean lateral deviation from

the planned trajectories. Figure 5.9 depicts mean values and standard deviations for

all sixteen subjects. Hereby, the first two bars give the results for drive I and drive II

without assistance, with and without secondary task, respectively. The third and forth

bar indicate the mean lateral deviation with hard assistance (hA), again with and without

secondary task (wS/nS). Finally, the same quantities are given for soft assistance (sA). It

can be seen that the lateral deviations from the planned path, e.g. the center of the lane

if no obstacles are present, grow if the subjects are distracted by a secondary task. This

observation is independent whether the test drives are conducted with assistance or not

and what kind of assistance is used. Most interesting, it can be noted that the assistance

torques in general contribute in a way that the deviations can be reduced. Here, the hard

assistance performed better in minimizing the deviations.

5.2 DRIVING SIMULATOR EXPLORATION 119

Figure 5.9: Mean lateral deviation from the planned path for different degrees of assistance

In Figure 5.10 the mean lateral deviation of all test subjects is dissolved for pure

lane-keeping situations and for collision avoidance situations. Here, collision avoidance

situations refer to situations where obstacles were detected including the evasion maneuver

and the return to the right lane. In this case the above observations still hold, but it

becomes obvious that the intentions of test subjects and potential field guidance system

were less coordinated for collision avoidance than for lane-keeping. This result aligns with

observations during the exploration. In general, the test subjects intended to change the

lane earlier than the assistance system when they were aware of an obstacle ahead. On

the other hand, the assistance system proposed to return to the right lane more quickly

than intended by the test subjects. Here it might be interesting to adapt the parameters

of the vehicle guidance controller to the situation as indicated in Section 4.8.

Figure 5.10: Lateral deviation from planned path depending on the degree of assistance,

dissolved for lane-keeping and collision avoidance

However, another interesting result is the number of collisions with obstacles in the

different test settings given in Table 5.6. In each test drive four obstacles appeared. As

the test settings in Table 5.4 were presented to each of the sixteen subjects, 64 collisions

were in principal possible in each test setting. The number of actual collisions shows that

120 5SHARED VEHICLE GUIDANCE BETWEEN DRIVER AND ASSISTANCE SYSTEM

the test subjects in general managed to avoid collisions if they were not distracted by

the secondary task. The only exception was one test subject who collided twice with soft

assistance. With secondary task and without assistance 19 collisions were detected. This

number was dramatically reduced in the test settings with the assistance system. Here,

the hard assistance configuration with 4 collisions performed best.

Table 5.6: Total number of collisions of all test subjects

with secondary task (wS) no secondary task (nS)

no assistance (nA) 19/64 0/64

soft assistance (sA) 6/64 2/64

hard assistance (hA) 4/64 0/64

These objective data promise that shared vehicle guidance with an assistance system

based on the proposed potential field framework can contribute to a better driving perfor-

mance. In particular, in situations where drivers are distracted and not aware of potential

hazards, the system might help to avoid accidents. However, the discussed objective

quantities contain no information how far the assistance system is accepted by the test

subjects. These aspects are covered by the questionnaire that the test subjects answered

after each test drive. For the complete set of questions and the collected answers of the

test subjects refer to the appendix.

In general, the acceptance of the system for lane-keeping as well as for collision avoi-

dance was satisfying. The cooperation with the assistance system was more harmonic in

lane-keeping situations, which was also reflected in the test subject’s comments. In order

to get an information basis how to adapt the level of assistance, the test subjects were

asked for their opinion concerning the strength of the assistance torque. As Figure 5.11

shows, the test subjects rated the hard assistance as too hard, and the soft assistance

as too soft. This tendency is more pronounced in collision avoidance, where lane-change

maneuvers are demanded, than in lane-keeping situations. Most of the test subjects felt

patronized by the hard assistance variant up to comments like ”I was driven”. On the

other hand, the test subjects formulated problems in interpreting the intention of the

assistance system if the assistance torque was too low.

(a) Lane-keeping (b) Collision avoidance

Figure 5.11: Mean ratings of the assistance torque by the test subjects

5.2 DRIVING SIMULATOR EXPLORATION 121

To estimate the driving demand, the questionnaire as well as the performance of the test

subjects in the secondary task can be analyzed, see for example Buschardt (2003, [15],

pp. 240). However, most of the test subjects showed a higher performance in the secondary

task with rising duration of the experiment, which might indicate learning effects. There-

fore, only the subjective rating of the influence of the assistance system on the driving

demand is considered, see Figure 5.12. The test subjects in general agreed that the driving

demand was reduced by the assistance system. This effect was more pronounced for hard

assistance.

Figure 5.12: Influence of the assistance system on the driving demand

Summarizing, the results of the exploration are promising in terms of reducing the

number of collisions and providing a good lane-keeping performance as well as in the re-

duction of driving demand experienced by the test subjects. However, in further studies

the force feedback characteristics should be adapted. In doing so, the hard and the soft

variant, studied in this exploration, might serve as upper and lower bound. In between

the characteristics might individually be customized. Also a combination of the haptic in-

teraction with visual or acoustic warnings, particular in critical situations, see for example

Fricke, de Flippis and Th¨

uring (2006, [38]), should be considered. Another point is

to tune the parameters of the motion planning and the guidance controller such that the

proposed maneuvers are closer to the driver’s intention. Besides that, another important

step is the experimental validation of the proposed system in real traffic situations.

Chapter 6

Conclusion

6.1 Summary

Modern driver assistance systems gradually engage in vehicle guidance. First examples

on the market are lane departure warning and lane-keeping systems. Nevertheless, more

advanced systems including the capability of evasion maneuvers are still subject to in-

tensive research. One key issue that has to be addressed is the motion planning that

enables advanced driver assistance systems to function at the guidance level. However,

most research projects in vehicle motion planning only consider autonomous driving or

emergency scenarios where the driver is overruled completely by the safety system.

In this thesis, methods for cooperative vehicle guidance of driver and assistance system

were contributed. In particular, a unified framework of motion planning and tracking

control was provided. This unified framework for shared vehicle guidance is established

by the potential field methods given in Chapter 3 and 4. In doing so, the concept of

searching for equilibrium configurations of an elastic band in a hazard map, which en-

codes the environmental information, was introduced and shown to provide collision-free

solutions. Therein, the elastic band, originally introduced in robotics, acts like a virtual

antenna of an insect sensing trajectories of low hazards in the environment. Necessary

numeric procedures were outlined to compute the equilibrium configurations. For auto-

motive applications the motions of other traffic participants are anticipated by means of

extrapolation methods. Hence, the proposed potential field motion planning is predictive

spatially as well as temporally. This in turn is not only beneficial to the motion planning

itself, but it also resembles the predictive motion planning of human drivers and facilitates

the coordination between driver and assistance system.

Information about the planned trajectory, in particular the curvature, which can be

considered as a disturbance, can be incorporated into the tracking algorithms. In order

to do so, the potential field guidance controller, combined with disturbance rejection or

feedforward control, was shown to provide closed-loop dynamics that are stable in the

sense of Lyapunov. Besides that, the given Lyapunov function provides bounds for the

tracking error depending on the initial conditions and on the parameters of the controller.

Therefore, the maximal deviation from the planned trajectory can be predicted and the

gains of the controller can be chosen to guarantee collision-free vehicle guidance.

124 6CONCLUSION

The steering angle, proposed by the guidance system, is communicated to the driver

via a force feedback steering wheel. In parallel, the driver’s steering intension is used

to shape the planned trajectories. The interactive guidance concept was experimentally

tested in a fixed-based driving simulator with a test group of sixteen subjects. Therein,

different configurations of the assistance torques, reflecting loose rein and tight rein of

the h-metaphor, were analyzed. Distracting the subjects with a secondary task, it turned

out that the performance of the test subjects in tracking the planned trajectories in test

drives using the guidance system was higher than in test drives without. Besides that,

the number of collisions with obstacles decreased in test drives with the driver assistance

system.

Summarizing this thesis provided a unified potential field framework for shared vehicle

guidance of driver and assistance system. The underlying potential field methods cover

motion planning as well as trajectory tracking. The concept can be applied to complex

traffic scenarios and is not, contrary to other methods, limited in the number of obsta-

cles that can be handled. Lane-keeping and collision avoidance assistance are sample

applications of this guidance concept.

6.2 Future Work

This thesis establishes a basis from which different directions for further theoretical and

experimental research can be taken. Some examples are: The elastic band, sensing the

hazard map for trajectories of low hazard levels, substantially contributes to the predic-

tive character of the motion planning concept. Therein, internal potentials distribute

local deformations over the entire elastic band in order to avoid locally high curvatures.

Here, a further interesting approach could be the extension of the elastic band approach

by introducing additional bending stiffnesses. Of particular interest is also the further

incorporation of vehicle dynamics in the motion planning procedure. The stiffness para-

meters of the elastic band could for example be adapted according to the drivability of

a trajectory. Nevertheless, it should be reminded that evaluating drivability in the end

relies on the adhesion factor between tires and ground, which is hard to determine.

The potential field controller can be parameterized in order to keep the tracking error

smaller than the distances to passed obstacles and therefore avoid collisions. An interesting

point for further research is the automatic adaptation of the guidance controller with

respect to the hazard map, which demands further stability analysis.

However, substantial next steps are also the incorporation of longitudinal vehicle guid-

ance into the potential field concept and test drives to verify the results gained in the

driving simulator. Further research is also demanded in the field of haptic interaction

between driver and assistance system.

Appendix:

Exploration Questionnaire

In the following, a translated extract of the questionnaire used in the exploration, see

Chapter 5, is presented; the exploration was originally conducted in German. Instruc-

tions given to the test subjects and demographic data are omitted here for brevity. After

each test drive, the subjects were asked to express their degree of accordance to the state-

ments given in each question by marking a small box. In the following, the boxes are

replaced by numbers that indicate how many of the test subjects chose a particular de-

gree of accordance. First, that part of the questionnaire is presented, which was used after

the initial test drives without assistance system. In Drive I the subjects could completely

focus on the driving task, while in Drive II they were additionally stressed by a secondary

task.

Extract of the questionnaire

Drive I

The vehicle always behaved as expected.

”I abs. disagree.” 0 0 0 2 7 7 ”I abs. agree.”

It was easy to avoid obstacles.

”I abs. disagree.” 0 0 1 0 5 10 ”I abs. agree.”

It was easy to keep the lane.

”I abs. disagree.” 0 1 0 1 9 5 ”I abs. agree.”

Drive II

The vehicle always behaved as expected.

”I abs. disagree.” 0 2 3 2 3 6 ”I abs. agree.”

It was easy to avoid obstacles.

”I abs. disagree.” 1 0 1 9 3 2 ”I abs. agree.”

It was easy to keep the lane.

”I abs. disagree.” 2 3 2 6 3 0 ”I abs. agree.”

126 APPENDIX

How intensively were you stressed by the secondary task?

not at all 0 1 1 2 6 6 very intensively

The following questions were posed after each test drive with driver assistance (Drives

III-IV). The tuple of four numbers denotes the total number of votes given by the test

subjects in the corresponding test drive; the answers are arranged in the following order:

1. hard assistance / no secondary task (hA/nS)

2. hard assistance / with secondary task (hA/wS)

3. soft assistance / no secondary task (sA/nS)

4. soft assistance / with secondary task (sA/wS).

Drives III-VI

Please answer the following questions with respect to collision avoidance assistance.

In presence of obstacles the driver assistance system perceptibly engaged in vehicle guidance.

yes 15|14|10|8 no 1|1|6|7

How should the assistance system engage in the steering?

earlier 0|1|0|12|2|3|3 10|12|7|10 3|0|3|10|0|1|0later

softer 1|2|0|07|7|0|07|6|6|60|0|4|70|0|4|2harder

How should warnings, by means of vibration at the steering wheel, be realized?

earlier 0|1|1|13|3|4|5 12|11|8|80|0|1|10|0|0|0later

softer 0|3|0|07|3|2|17|9|8|81|0|2|30|0|1|3harder

How should it be possible to overrule the assistance system?

easier 0|0|0|08|10|5|27|5|5|10 0|0|4|30|0|0|0harder

Please answer the following questions with respect to lane-keeping assistance.

In absence of obstacles the driver assistance system perceptibly engaged in vehicle guidance.

yes 11|13|10|13 no 2|0|5|2

How should the assistance system engage in the steering?

earlier 0|0|1|10|2|2|0 12|14|10|13 3|0|2|21|0|0|0later

softer 1|0|0|05|6|1|0 10|10|7|11 0|0|5|30|0|2|2harder

How should warnings, by means of vibration at the steering wheel, be realized?

earlier 0|0|1|11|1|1|1 12|14|12|13 3|1|1|10|0|0|0later

softer 0|0|0|04|4|1|0 10|11|8|10 2|1|5|40|0|1|2harder

How should it be possible to overrule the assistance system?

easier 1|1|0|08|6|3|37|9|8|11 0|0|4|20|0|0|0harder

APPENDIX 127

Please rate how far you agree to the following statements.

The vehicle always behaved as expected.

”I abs. disagree.” 0|0|0|01|0|2|01|5|2|54|5|1|35|1|6|45|5|5|4”I abs. agree.”

It was easy to avoid obstacles.

”I abs. disagree.” 0|0|0|00|2|0|10|4|0|33|2|5|36|4|5|87|4|6|1”I abs. agree.”

It was easy to keep the lane.

”I abs. disagree.” 0|0|0|00|0|0|10|1|0|71|6|1|17|4|9|68|5|6|1”I abs. agree.”

I always knew what the assistance system intended to do or me to do.

”I abs. disagree.” 0|0|1|23|3|4|61|5|3|24|1|4|45|4|2|23|3|2|0”I abs. agree.”

I felt overruled by the assistance system.

”I abs. disagree.” 4|2|7|85|3|3|42|5|2|22|4|4|21|2|0|02|0|0|0”I abs. agree.”

I always recognized when the assistance system engaged in vehicle guidance.

”I abs. disagree.” 0|0|0|00|0|3|10|0|0|20|0|2|16|8|7|10 10|8|4|2”I abs. agree.”

The system assisted me in lane-keeping.

”I abs. disagree.” 0|0|2|10|0|1|01|0|2|22|2|2|36|9|7|67|5|2|4”I abs. agree.”

The system assisted me in collision avoidance.

”I abs. disagree.” 0|0|2|00|1|2|21|1|1|32|1|0|37|7|7|56|6|4|3”I abs. agree.”

I always had trust in the assistance system.

”I abs. disagree.” 0|0|1|11|3|4|35|2|2|55|4|4|32|4|3|33|3|2|1”I abs. agree.”

I was annoyed by the assistance system.

”I abs. disagree.” 7|4|6|62|4|4|63|6|1|33|2|3|11|0|2|00|0|0|0”I abs. agree.”

The assistance system took at least one wrong decision.

”I abs. disagree.” 7|5|7|74|1|5|42|4|2|02|1|0|31|1|1|20|4|1|0”I abs. agree.”

In total, the assistance system relieved the driving task to me.

”I abs. disagree.” 0|0|1|11|0|1|03|3|2|24|3|4|73|4|7|55|6|1|1”I abs. agree.”

The assistance system additionally stressed me.

”I abs. disagree.” 6|5|5|46|5|4|51|3|5|33|2|2|30|1|0|00|0|0|1”I abs. agree.”

In addition, the assistance system should give visual warnings.

”I abs. disagree.” 5|5|5|65|3|5|32|1|0|00|2|2|22|3|4|42|2|0|1”I abs. agree.”

In addition, the assistance system should give acoustic warnings.

”I abs. disagree.” 3|4|3|36|4|4|40|2|1|04|1|3|50|3|2|03|2|3|4”I abs. agree.”

How intensively were you stressed by the secondary task?

not at all -|0|-|0-|1|-|2-|0|-|0-|4|-|2-|8|-|7-|3|-|5very intensively

128 APPENDIX

Please answer the following questions about your opinion about lane-keeping and

collision avoidance assistance in general.

I think lane-keeping and collision avoidance are, in general, valuable.

”I abs. disagree.” 0|0|0|00|0|0|01|0|0|01|1|4|17|9|5|97|6|7|6”I abs. agree.”

I think lane-keeping and collision avoidance, in general, might increase traffic safety.

”I abs. disagree.” 0|0|0|00|0|0|01|0|0|01|1|3|26|10|5|68|5|8|8”I abs. agree.”

I think lane-keeping and collision avoidance might, in general, increase driving comfort.

”I abs. disagree.” 0|1|0|00|0|1|02|0|0|12|1|5|44|9|4|48|5|6|7”I abs. agree.”

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