On Generic Road Vehicle Motion Modelling and Control Johan Andreasson Ph.D. Thesis

On Generic Road Vehicle Motion Modelling and Control Johan Andreasson Ph.D. Thesis
On Generic Road Vehicle Motion
Modelling and Control
Johan Andreasson
Ph.D. Thesis
TRITA-AVE 2006:85
ISSN 1651-7660
ISBN 91-7178-527-2
ISBN 978-91-7178-527-5
Postal address
Visiting address
Telephone
Internet
Royal Institute of Technology
Teknikringen 8
+46 8 790 6000
www.ave.kth.se
Vehicle Dynamics
Stockholm
Telefax
SE-100 44 Stockholm
+46 0 790 9290
AT X
Typeset in L
E
c 2007, Johan Andreasson
°
Abstract
With the increased amount of on-board electric power driven by the ongoing hybridization, new ways to realize vehicles are likely to occur.
This thesis outlines a future
direction of vehicle motion control based on the assumptions that: 1) future vehicle
development will face an increased amount of available actuators for vehicle propulsion and control that will open up for an increased variety of possible congurations, 2)
the onboard computational power will continue to increase and allow higher demands
on active safety and drivability that will require a tighter interaction between sensors
and actuators, 3) the trend towards more individualized vehicles on common platforms
with shorter time-to-market require design approaches that allow engineering knowledge to be transferred conveniently from one generation to the next.
A methodology to facilitate the selection of vehicle congurations and the design
of the corresponding vehicle motion controllers is presented. This includes a method to
classify and map congurations and control strategies onto their possible inuence on
the vehicle's motion. Further, a structured way of implementing and managing vehicle
and subsystem models that are easy to recongure and reuse is suggested and realised
in the developed VehicleDynamics Library. In addition, generic ways to evaluate vehicle congurations, especially the use of the adhesion potential to identify safety margin
and expected limit behaviour are presented.
Special attention is given to how the characteristics of a vehicle conguration can
be expressed so that it can be used in vehicle motion control design.
A controller
structure that enables a generic approach to this is introduced and within this structure,
two methods for control allocation are proposed, via tyre forces and directly. The rst
method uses a developed mapping of available actuators as constraints onto the achievable tyre forces and inverse tyre models to calculate the actuator inputs. The second
method allocates the actuator inputs directly for an adapted problem that is linearized
around the current operating point. It is shown that the methods are applicable to a
variety of different vehicle congurations without redesign. Therefore, the same controller can manage a variety of vehicle congurations and there is no need to recognize
and treat each different situation separately.
Finally, a road map on how to continue this research towards a possible industry
implementation is given. Also suggestions on more detailed improvements for modelling and vehicle motion control are provided.
I
Acknowledgements
The research work presented has been carried out at the division of Vehicle Dynamics
at the Royal Institute of Technology, KTH, in Stockholm. I am pleased to acknowledge
the nancial support from the Swedish taxpayers through the Gröna Bilen National
Research Programme within the FCHEV framework.
There are a number of persons to whom I would express my gratitude. First of all
to my supervisors Annika Stensson Trigell and Lars Drugge for the faith you put in me,
it has been a long journey and I'm glad I joined in. Bengt Jacobson, you also deserve
a lot of credit for initiating me to the area and giving great feedback on my licentiate
thesis.
Leo Laine, I had a great time working with you the rst years and I'm glad you
ended up being more than a colleague to me. Jonas Fredriksson, it was great having
you involved. Tilman Bünte and Christian Knobel, I hope we get a chance to work
more together in the future.
Rajiv Gupta at General Motors and Saab Automobile, Johan Wedlin at Volvo Cars
(now at Volvo AB), Anders Bodin at BAE Systems Hägglunds and Lars Carlhammar at
Volvo Technology for valuable comments and feed-back during the steering committee
meetings and to the staff at Ford Motor Company and General Motors for interesting
and lively discussions. Göran Johansson and Sture Eriksson also deserve credit for
keeping up good spirits within the FCHEV framework.
I am grateful to Martin Otter for being a continuous source of inspiration and
knowledge, without my time at DLR my mind would be much poorer. The Dynasim
crew, Sven Erik Mattsson for taking time to explain the fuzzy stuff that happens within
Dymola, Dag Brück for introducing me to version handling, Hans Olsson for being
such a Modelica magician and to Hilding Elmqvist whos belief in my work made the
tight cooperation possible.
Past and present colleagues at KTH, I've had a great time at and after work thanks
to many of you. Feel free to take credit later on as well.
Modelon fellows, Hubertus Tummescheit and Jonas Eborn deserve a fair share
of the attention for the inspiring atmosphere and Magnus Gäfvert also for great fun
during the often too long nights of development work.
I'm looking forward to the
continuation.
Mats Jonasson, seeing a Ph.D. project from a supervisors perspective has brought
me much insight, as has the outcome of our work. Kanehira Maruo, you deserve credit
III
for your enthusiastic support and for continuously providing me with the latest news
on alternative fuels and hybrid technologies.
Megan Bingham, without your enthusiasm, patience and spare energy, this thesis
would neither be half as good nor half as nished by now.
The guy in Egypt who tricked me into buying way too much of that peculiar tea, I
don't really know what is in it but it sure has boosted my performance.
Finally to my family for the love and care they gave and to all of you both expected and unexpected friends that supported me through this last turbulent year, that
except for thesis work included intense development of the VehicleDynamics Library
and leading the house community through a takeover and a complete renovation from
which my apartment has suffered long enough now. The spare room on couches and
mattresses was also highly appreciated, I hope I will be able to pay you back soon.
THANK YOU!
Stockholm, January 2007
Johan Andreasson
Contents
1 Introduction
1.1 Scenario outline . . . . . . .
1.2 Upcoming technologies . . .
1.3 Thesis outline . . . . . . . . .
1.4 Origin of the presented work
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2 Principles of tyre force generation
2.1 Motivation . . . . . . . . . . . . .
2.2 Tyre load carrying and suspension
2.3 Pure slip eects . . . . . . . . . .
2.4 Combined slip . . . . . . . . . . . .
2.5 Load sensitivity . . . . . . . . . . .
2.6 Inclination eects . . . . . . . . .
2.7 Transient behaviour . . . . . . . .
2.8 Additional eects . . . . . . . . . .
2.9 Tyre models . . . . . . . . . . . . .
3 Principles of vehicle motion control
3.1 Motivation . . . . . . . . . . . .
3.2 Means of actuation . . . . . . .
3.3 Sensoring . . . . . . . . . . . . .
3.4 Control structure . . . . . . . . .
3.5 Application . . . . . . . . . . . .
4 The
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VehicleDynamics Library
Background . . . . . . . . . . . . . . .
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Implementation issues . . . . . . . . .
Recommended extensions of Modelica
Example vehicles . . . . . . . . . . . .
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VI
5 Methods for vehicle motion analysis
5.1 Background . . . . . . . . . . . . . . . . . .
5.2 Eects on lateral acceleration . . . . . . . .
5.3 Eects on yaw moment . . . . . . . . . . .
5.4 Phase-plane . . . . . . . . . . . . . . . . . .
5.5 Open loop stability tests . . . . . . . . . . .
5.6 Combined lateral-longitudinal performance
Contents
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6 Methods for generic vehicle motion control design
6.1 Outline . . . . . . . . . . . . . . . . . . . . . . .
6.2 Denition of vehicle motion . . . . . . . . . . . .
6.3 Vehicle motion to global forces . . . . . . . . . .
6.4 Direct allocation of wheel inputs . . . . . . . . .
6.5 Allocation of forces . . . . . . . . . . . . . . . .
6.6 Actuator commands . . . . . . . . . . . . . . . .
6.7 Application of force allocation . . . . . . . . . .
6.8 Application of direct allocation . . . . . . . . . .
6.9 Remarks on the presented methods . . . . . . .
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7 Scientic contribution
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8 Concluding discussion
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9 References
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Appendices
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A Nomenclature
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B Glossary
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C Modelica in brief
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Chapter 1
Introduction
This chapter outlines the scenario for future vehicle development that forms the
background for this thesis work. The opportunities and restrictions of this future are also discussed in addition to presenting a generic approach to vehicle
motion modelling and control as an important way to formalize and reuse vehicle dynamics knowledge. A strategy for approaching changes in vehicle motion
control is also proposed.
1.1
Scenario outline
Ever since the introduction of Anti-lock Braking Systems (ABS), computerized
control of vehicle motion has not only advanced dramatically, it has evolved
and expanded to become an integral part of automotive design. Drivability and
driving safety, among other factors, are transforming from what used to be a
straightforward matter of static tuning into a complex interaction of systems
from multiple physical domains. So far, there is no reason to believe that this
trend is going to diminish and the effects of this transformation are already
tangible. In fact, recent studies show that stability control systems have already
reduced the occurrence of skidding accidents by 25% [1]. Therefore, keeping
up with the pace of future developments in vehicle motion control is absolutely
crucial for the survival of any automotive manufacturer.
At least three aspects require imperative consideration as a result of this
evolution. First, new or renaissance technologies such as electric propulsion
will open up new possibilities in vehicle design, creating a seemingly endless
number of decisions to be made when choosing a vehicle conguration. In fact,
the only remaining assumption for future vehicles and their components is the
1
2
Chapter 1
Introduction
existence of tyres. Second, these new possibilities in combination with the increased onboard computational power will allow for higher demands on active
safety and drivability which in turn will require a tighter interaction between
sensors and actuators. Third, the trend towards more individualized vehicles on
common platforms with shorter time-to-market require design approaches that
allow engineering knowledge to be transferred conveniently from one generation to the next.
Some of these aspects can already be seen in production vehicles, prototypes and published research as will be described more thoroughly in Section 1.2. The problem remains, however, in the sheer quantity and pervasive
inuence of choice. In a highly competitive market, the ability to make those
choices on development and conguration in an effective, efcient and timely
manner will ultimately dene either success or failure.
The work presented in this thesis suggests a comprehensive strategy for
approaching the proliferation of choice in the new world of computerized control in order to maintain a competitive development pace. This is based on a
generic approach to modelling and control of road vehicle motion as outlined
in Section 1.3.
1.2
Possibilities and challenges
of upcoming technologies
For a long period of time, ground vehicles have basically been carts with the
horses replaced by an engine and transmission. Even in the majority of today's
vehicles, despite the increasing amount of control, vehicle design is still limited
by the use of a hydro-mechanical coupling between the power source and the
wheels. With the introduction of electric propulsion, this coupling is loosened,
making way for many more possibilities in vehicle design.
An example of this evolution is illustrated in Figure 1.1, [2]. The Autonomy
concept shows how the use of fuel cells together with in-wheel motors could
be used to design a exible, skateboard like, chassis upon which a variety of
bodies could be attached.
Another solution aiming in that direction is the Autonomous Corner Module
(ACM) presented in [3]. The idea is to replace the lower A-arm of a suspension with two linear actuators, allowing them to control steering and camber
in a redundant way. Along with active suspension and in-wheel motors, these
modules are able to individually control each wheel's steering, camber, suspension and spin. By packing these as autonomous units they could be easily be
exchanged if, for example, more performance were needed as illustrated in Fig-
Section 1.2
Upcoming technologies
3
Figure 1.1: The skateboard-like concept Autonomy [2] illustrates how all functionality
could be gathered into a platform allowing for exchangeable bodies.
4
Chapter 1
Introduction
Figure 1.2: The Autonomous Corner Module concept [3] suggests merging all actuators
involved in the vehicle motion control into interchangeable units.
ure 1.2. Recently, Siemens has also presented a similar concept called eCorner,
Figure 1.3 [4].
More hands-on interest in wheel motors and their possibilities can also be
found in ongoing vehicle dynamics research. At Tokyo University a research
vehicle [5] is used to explore the benets of wheel motors in comparison to
a traditional driveline when designing vehicle motion control. The layout of
this vehicle is to a great extent similar to the Autonomy concept in Figure 1.1.
Additionally, Michelin has also presented research vehicles with wheel motors
as shown in Figure 1.3 and in fact, the idea is far from new. At the end of the
19th century, working for Lohner, Ferdinand Porsche mounted electric wheel
motors on their vehicles, Figure 1.4. The main problem was, and still remains
the electric energy storage. To avoid the 1800kg batteries required at that time,
the Lohner was designed as a series Hybrid Electric Vehicle (HEV) using an
Internal Combustion Engine (ICE) and a generator to supply the electric power.
So, the HEVs that are gaining in popularity today are far from a new invention, even if the incentives of their creation have changed today. Since the
primary target today is a reduction in emissions and fuel consumption, series
ICE congurations as applied in the Lohner are less likely to be successful.
Therefore, the introduction of wheel motor concepts depends on the development of Fuel Cells (FC), batteries and/or hydrogen storage units. HEVs with
Section 1.2
5
Upcoming technologies
Figure 1.3: Implementations of the corner module concept that comprise an in-wheel
motor (1), steering (2), suspensioning (3) and brake (4). Siemens VDO eCorner [4]
(left) and Michelin HY-LIGHT (right).
partial electrical propulsion are already on the market.
The main benet of
these concepts, when compared to traditional vehicles, is that the electric machines can be used to handle load variations. This in turn means that the ICE
can be dimensioned after mean power instead of max power and that it can
work at optimal loads. Another advantage is that electric power can be regenerated and stored in the battery while braking. Both these aspects can help to
reduce fuel consumption and emissions.
From a vehicle motion control point-of-view there are two other interesting
advantages. Instead of one ICE propelling some or all wheels via a complex
mechanical-hydraulic driveline, electric motors can be used to distribute the
drive and brake torques. An extreme case of this is the use of wheel motors
which would make a traditional drive train obsolete. This was a main incentive
for the Lohner concept.
The second, indirect advantage of electric propulsion is the additional onboard electric power that increases the potential for replacing mechanical and
hydraulic actuators by electrical ones.
This will aid the introduction of by-
wire techniques in ground vehicles which in turn allows for more computerized
control.
It will also allow for easier algorithmic partitioning and tighter in-
tegration of actuators to achieve better vehicle performance. In a publication
from the mid-80's [7], the process of adopting to these benets is divided into
three stages: The rst stage covered the development of stand-alone components where electronics replaced mechanics such as electronic ignition in the
mid 60's. The second stage covered special purpose independent systems such
as engine control and was believed to last until the late 80's. The last stage,
6
Chapter 1
Introduction
Figure 1.4: Lohner equipped with front wheel hub motors from the late 19th century [6].
predicted to start in the 90's and lasting into 2000, would be categorized by
engineering solutions that take advantage of the additional benets that come
with the tighter integration of available actuators. It was stated that "designers
will escape from the mechanical function replacement and add-on approaches
that have categorized stage 1-2".
In [8] the concept of Integrated Vehicle Control was introduced, posing the
question: What would the car be like if the microprocessor had been invented
before the automobile? Since then, numerous works on the subject of combining steering, traction, brakes and suspension have been published under names
like Active Chassis Control, Integrated Vehicle Control, Integrated Chassis
Management, Vehicle Dynamics Management and Global Chassis Control, etc.
A conclusion from this, also supported in Chapters 2 and 3, is that in order to
reach break-troughs in drivability and driving safety, an integration of several
actuation principles has to be performed. A wider perspective of this theme is
often referred to as driver assistance systems and spans from a range from planning assistance to collision mitigation and post-crash activities. This has been
a popular theme for overview papers during the last years (e.g. [9, 10]), while
other approaches discuss selected vehicle congurations or control approaches
(e.g. [11, 12]).
It is notable that the process of adapting to the benets of replacing mechanical with computerized functionality can be, and in many cases is, driven
independently of electric propulsion. For example, synergetic control of steer-
Section 1.3
7
Thesis outline
ing [13] and suspension [14], aiming for improved safety and achieving a better
compromise between handling and comfort, has lately been introduced more
broadly on the market. Naturally, the ability to synchronize different functionalities and identifying and eliminating conicting effects is a competitive factor
that has gained corresponding interest. So essentially, the new possibilities to
improve vehicle performance that comes with the increased amount of computerized control can just as well be a competitive disadvantage if not approached
properly.
The general difculty remains in the huge amount of possible conguration
which in turn require a systematic and generic approach. It can also be assumed
that for HEVs, subsystems that traditionally carry brand specic functionality
such as ICEs and gearboxes will be less important for vehicle characteristics.
One solution to this is to nd new subsystems for carrying brand specications.
Such work is seen, for example, when it comes to FCs, where the manufacturer
that can rst bring these into production vehicles for a reasonable price will
probably gain in market shares.
This is however a very costly process that
often requires many manufacturers to cooperate and thus, a manufacturer must
not only be the rst to derive the new technologies but also to be the best to
implement it in its vehicles.
At the same time, it is obvious that there is a lot of knowledge to be found in
conventional vehicles, both concerning requirements on drivability and driving
safety and solutions for how to fulll these. This knowledge has to be adapted
to new technologies. The better a manufacturer manages this, the greater their
advantage will be. Along with the increasing amount of outsourcing and modularization, as in the ACM case, competing brands will share both more hardware and software. It can be expected though that brand characteristics have to
be specied on a functional level, which in turn will require new methods for
dening and evaluating performance. Thus, a methodology to efciently select
suitable congurations and design the corresponding vehicle motion controllers
that can also carry brand specic functionality is a vital tool for being successful in a competitive market.
1.3
Thesis outline
The assumed future scenario outlined above brings new possibilities to improve
vehicle performance. It is believed that if an automotive manufacturer does not
explore and turn these possibilities into advantages soon enough, that they will
lag behind and someone else will prot.
Consider the schematic illustration to the left in Figure 1.5 consisting of a
8
Chapter 1
Introduction
Figure 1.5: Current approach to vehicle motion control design, left, and a desired ultimate process, right. f indicates the vehicle, g the vehicle control and h the evaluation
criteria divided into excitation h1 and response comparison h2 .
vehicle, f , the vehicle control, g, and the evaluation criteria, h, simplied to
blocks. The current approach to vehicle motion control is to design a controller
for a given vehicle conguration to fulll requirements that have been dened
based on the expected performance, i.e. g( f , h( f )). All of this is based on more
general demands such as load carrying abilities, fuel consumption, looks and so
on. The result is a process where the feedback from evaluations are used to tune
the parameterizations of f and g while structural changes require a redenition
of the parts in this process. With the preceding motivation in mind, the aim
of this thesis is to lay a foundation for a methodology for selecting f , g and h
based on the demands as illustrated to the right in Figure 1.5.
This work is based on two main strategies: The rst is to make each of
the blocks f , g and h as generic as possible to minimize additional costs when
setting up a new conguration. In Figure 1.6, this is illustrated by a complete
re-implementation (top), a reusable partitioning of f , g and h (middle), and
the merging f and g to functional units F that also allows software and hardware functions to be exchanged. The second strategy is to provide a range of
levels-of-detail that allow a successive selection process so that the amount of
congurations can be decreased as the implementation effort increases.
To increase the accessibility of this work it has been partitioned into chapters that can be read separately. Chapter 2 contains an overview of tyre behaviour relevant for vehicle motion control. This is included since the majority
of actuators that affect the vehicle motion do so via the tyres and it is thus necessary to understand the effects of potential actuators on the tyre force generation.
It is also believed that the inclusion of this chapter will increase accessibility
for readers with a background outside the traditional vehicle dynamics domain.
Section 1.3
Thesis outline
9
Figure 1.6: By identifying and partitioning commonalities between different congurations these can be reused to minimize effort and errors when setting up new congurations.
10
Chapter 1
Introduction
Chapter 3 reviews the current status of vehicle motion control and suggests
a classication method to estimate the ability of different congurations. This
aims both to increase understanding of the underlying phenomena that can be
used for vehicle motion control. It also gives the means for the rst step in the
selection process with the lowest level-of-detail.
Chapter 4 presents the selected approach to generic vehicle design, i.e. the
partitioning of f above.
The chapter focuses on implementation issues es-
pecially those relating to reusability and various levels-of-detail.
Chapter 5
presents the limitations with the currently applied evaluation methods, h above,
and suggests variants that are adopted to the increased amount of actuators.
In Chapter 6, different approaches to the design of generic vehicle motion controllers are proposed. This corresponds to dening a reusable g above.
Chapters 7 and 8 nally highlight the contribution, summarize the work, propose related research topics and outlines additional work required for the methodology to be a part in daily development work.
1.4
Origin of the presented work
This thesis is based on both previously unpublished material and a set of earlier
publications, listed below. In each following chapter or section, references to
these publications are given to indicate the origin of the contribution.
Appended papers
A
J. Andreasson and L. Laine, Driving Dynamics for Hybrid Electric Vehicles Considering Handling and Control Architecture. Vehicle System
Dynamics. Volume 41, pages 497-506, 2004.
In this paper, a generic vehicle motion control architecture is suggested,
corresponding to g above. The information ow from the driver's intentions to vehicle motion is especially considered. The idea is introduced
and veried, that the driver's intentions are transformed into a global
force equivalent that is distributed to each wheel.
B
J. Andreasson, L. Laine and J. Fredriksson, Evaluation of a Generic Vehicle Motion Control Architecture. In Proceedings of World Automotive
Congress FISITA, Barcelona, Spain, 2004.
This paper introduces the force restriction concept as an abstraction in
Section 1.4
Origin of the presented work
11
order to apply constraints to the force allocation problem from [A], corresponding to the actual vehicle conguration, i.e. a mapping from f to
g. This is evaluated for three different cases.
C
L. Laine, J. Andreasson and J. Fredriksson, Reusable Functional Partitioning of Tractive Force Actuators Applied on a Parallel Hybrid Electric
Vehicle. In Proceedings of 7th International Symposium on Advanced
Vehicle Control, AVEC'04, HAN University, Netherlands, 2004.
Here, the methodology to abstract a conguration to a constrained optimization problem from [B] is applied for an energy management task.
It is also illustrated how this task can cooperate with the vehicle motion
controller according to [L].
D
J. Andreasson and T. Bünte, Global Chassis Control Based on Inverse
Vehicle Dynamics Models. To be published in Vehicle System Dynamics, Volume 44, 2006.
This publication approaches vehicle motion control as an inverse for
feed-forward control with an additional feed-back part. The inverse is
separated into a dynamic part dening the resulting forces on the body
as a function of the desired motion. These forces are then allocated directly to the wheel actuators instead of via the tyre forces as in [A,B,F,N].
E
J. Andreasson, C. Knobel and T. Bünte, On Road Vehicle Motion Control
- Striving towards synergy. In Proceedings of 8th International Symposium on Advanced Vehicle Control, AVEC'06, Taipei, Taiwan, 2006.
In this work, different approaches to road vehicle motion control involving several subsystems are surveyed and categorized. To give additional
perspective, aeronautics and robotics areas are also considered.
F
M. Jonasson and J. Andreasson, Exploiting Autonomous Corner Modules to Resolve Force Constraints in the Tyre Contact-Patch. Submitted
for publication.
This article uses a variant of the force allocation method proposed in [B]
to exploit the abilities of the ACM concept. Some attention is also give to
the additional freedom that allow new types of manoeuvre trajectories to
be dened.
12
Chapter 1
Introduction
Additional publications
G
J. Andreasson, A. Möller and M. Otter, Modelling of a Race Car with
Modelica's MultiBody library. In Proceedings of 1st International Modelica Workshop, Lund, Sweden, 2000.
H
J. Andreasson and J. Jarlmark, Modularised Tyre Modelling in Modelica.
In Proceedings of 2nd International Modelica Conference, Oberpfaffenhofen, Germany, 2002.
I
J. Andreasson, The VehicleDynamics Library. In Proceedings of 3rd International Modelica Conference, Linköping, Sweden, 2003.
J
L. Laine and J. Andreasson, Modelling of Generic Hybrid Electric Vehicles. In Proceedings of 3rd International Modelica Conference, Linköping,
Sweden, 2003.
K
M. Beckman and J. Andreasson, Wheel Model Library in Modelica for
Use in Vehicle Dynamics Studies. In Proceedings of 3rd International
Modelica Conference, Linköping, Sweden, 2003.
L
L. Laine and J. Andreasson, Generic Control Architecture applied to a
Hybrid Electric Sports Utility Vehicle. In Proceedings of 20th Electric
Vehicle Symposium, EVS'20, Long Beach, California, 2003.
M
H. Elmqvist, S.E. Mattsson, H. Olsson, J. Andreasson, M. Otter, C.
Schweiger and D. Brück, Realtime Simulation of Detailed Vehicle and
Powertrain Dynamics. In Proceedings of the SAE World Congress 2004,
Paper no 2004-01-0768, Detroit, Michigan, 2004.
N
J. Fredriksson, J. Andreasson and L. Laine, Wheel Force Distribution for
Improved Handling in a Hybrid Electric Vehicle using Nonlinear Control.
In Proceedings of 43rd IEEE Conference on Decision and Control, CDC,
Bahamas, 2004.
O
J. Andreasson, Hybrid Electric Vehicles, Aspects on Driving Dynamics and Control Architecture. Licentiate thesis, KTH Vehicle Dynamics,
ISBN 91-7283-759-4, 2004.
P
T. Bünte and J. Andreasson, Integrierte Fahrwerkregelung mit minimierter
Kraftschlussausnutzung auf der Basis dynamischer Inversion.
ceedings of Autoreg 2006, VDI Berichte no 1931, 2006.
In Pro-
Section 1.4
Q
Origin of the presented work
13
J. Andreasson and M. Gäfvert, The VehicleDynamics Library - Implementation and Application. In Proceedings of 5th International Modelica
Conference, Vienna, Austria, 2006.
R
M. Gäfvert, J. Svedenius and J. Andreasson, Implementation and Application of a Semi-Empirical Tire-Model in Multi-Body Simulation of
Vehicle Handling.
In Proceedings of 8th International Symposium on
Advanced Vehicle Control, AVEC'06, Taipei, Taiwan 2006.
S
L. Laine and J. Andreasson, Control Allocation based Electronic Stability Control System for a Conventional Ground Vehicle. To be submitted.
14
Chapter 1
Introduction
Chapter 2
Principles of
tyre force generation
This chapter explains how forces are generated in the tyre-road contact, partitioned after how they could be used by different actuators. This is a base for
the approach used in the subsequent chapters. Focus is on a qualitative rather
than quantitative description of the mechanics.
2.1
Motivation
For literally all road vehicles, tyres are the sole contact between the vehicle and
the road and the main source for force generation. Thus, tyres are relied upon
to carry the vehicle's load with as little resistance as possible while at the same
time generating as much longitudinal and lateral forces as possible whenever
needed. Inevitably, these are conicting aims which require different solutions
depending the application. A truck tyre, for example, has lower rolling resistance but poorer grip than a car tyre, since fuel economy is considered more
important for trucks than for cars.
Despite looking much like a simple rubber doughnut attached to the rim,
tyre construction is in fact vastly more complex in order to meet these demands,
Figure 2.1. The complexities of tyre behaviour are a direct result of tyre construction and vary greatly with each tyre's type and running condition. This
has been thoroughly studied for decades and in-depth explanations are found in
e.g. [15, 16, 17, 18]. This chapter presents an overview of the phenomena that
create the resulting tyre forces to give a base for understanding what effects
15
16
Chapter 2
Principles of tyre force generation
Figure 2.1: Despite its outside looks, the tyre is a complex component.
there are to consider when dealing with vehicle motion control. The partition
of the chapter is based on possible wheel actuator inputs, where the focus is on
the qualitative rather than quantitative behaviour of tyres.
2.2
Tyre load carrying and suspension
Even though how a tyre carries and suspends the vehicle load might seem to be
a pure comfort issue, the way the load is distributed through the tyre will in fact
have a great impact on its abilities to generate contact forces.
A main assumption for a tyre on a road is that all deformation due to the
contact takes place in the tyre and that it carries the load as a spring. However
since the tyre's vertical stiffness normally is about a factor 10 stiffer (passenger
car) than the suspension, one often refers to the mass suspended only by the
tyre as being unsprung. A common misconception is that the air in the tyre
carries the load. Instead, the suspension of the vehicle mass is managed by the
carcass, consisting of the belt, radial cords and beads. The plies in the radial
cord work like the spokes of a bicycle rim; a pretension must be exerted by
pressurized air inside the tyre to carry the load.
It is notable that the steel belts are applied at an angle and this angle has a
substantial effect on tyre behaviour when loaded. In the contact patch, the pretension decreases which causes the radial cords to deform giving a deection of
the tyre sides. Due to this angle, the tyre belt has peripheral elasticity and the
reduction of pretension will cause the belt length to decrease as illustrated in
Figure 2.2, left. Without this elasticity, the tyre belt would have to fold in order
to t the patch, causing both discomfort and an unpredictable pressure distri-
Section 2.2
Tyre load carrying and suspension
17
Figure 2.2: Tyre deformation: The leftmost picture illustrates how the tyre belt compresses in the peripheral direction to t along the patch instead of the original arc
length. The second picture illustrates the symmetrical pretension caused by the pressurized air when the tyre is unloaded. When the tyre is loaded, third picture, the preload
is redistributed so that the rim is hanging in the bead [20, 17]. The fourth picture shows
the deformation caused by a drive torque and the corresponding pressure distribution.
bution.
Instead this compression gives uneven longitudinal horizontal force
distribution within the contact patch.
The compression of the belt will also affect the efcient rolling radius,
Re , dened as the travelled distance per revolution for a free rolling wheel.
1
An approximation
is achieved by studying the circumference of the loaded
tyre [17, 19], giving
Re
=
vx
ω
≈
2R0 + R
3
(2.1)
where R0 is the undeformed radius and R is the loaded radius or distance from
the wheel centre to the ground.
For the continued discussion, the pressure distribution within the contact
patch is of substantial interest. Assuming the tyre to be an ideal membrane
would give a constant pressure distribution over the whole patch, equal the
compressed air pressure in the tyre. In reality, the tyre's bending stiffness introduces a pressure distribution, depending on surface conditions, wheel angles,
load, drive torque and more, Figure 2.2. Closely connected to this is also the
1
Due to centrifugal effects, rolling radius is also speed dependent which is of signicant
importance at high speed driving.
18
Chapter 2
Principles of tyre force generation
size and location of the contact patch itself. The bending stiffness is also a
major reason it is important to have the right air pressure since it will impose
2
unwanted pressure distributions . This is also one of the reason the tyre is load
sensitive as will be explained later on.
Along with bending stiffness, there is also damping that causes an asymmetric force distribution for a rolling tyre that gives a redistribution of the pressure distribution, i.e. rolling resistance. For a driven tyre, this effect is exaggerated whereas a braked tyre has a reduced or even opposite asymmetry. Since the
damping properties of a tyre directly affect the rolling resistance it is difcult
to achieve high vertical damping without increasing fuel consumption.
To understand the importance of pressure distribution, for the generation of
longitudinal and lateral tyre forces, the brush model is of substantial help. It
will be explained briey in the next section and is then applied analogously in
the preceding ones.
2.3
Pure slip eects
The pure slip characteristics is dened for either pure longitudinal or pure lateral slip and the so called brush model gives a rst outline of the understanding
of tyre characteristics. Consider Figure 2.3 where the tyre is considered to be
covered with bristles like on a brush. For a rolling tyre, a bristle is undeformed
when coming into contact with the road. Then as it travels trough the contact
patch, the inner end will follow the tyre belt while the outer end strives to stick
in contact with the point where it rst met road. If the tyre is sliding sideways
at an angle
α
this will cause a lateral deformation of the bristle. Assuming
the bristle is elastic, this will go on as long as the bristle force is less than the
maximum friction force, dividing the contact patch into stick and slip regions.
Since the friction is load dependent, the pressure distribution is now of signicant interest. The generated side force corresponds to the sum of the bristle
forces as illustrated by the shaded areas in Figure 2.3 where some different
cases are shown to indicate this. It is notable that unless the slip angle is high
or the friction is low, the center of the resulting force is located behind the middle of the contact patch, causing an aligning moment that tends to turn the tyre
towards the sliding direction, i.e. it is stabilizing. It is also notable that since
the tyre forces builds up towards the rear, a lot of tyre force potential indicated
by the pressure distribution cannot be used.
In reality, the behaviour is more complex and to make a qualitative reasonable tyre model, the belt deformation has to be considered as well which
2
A comprehensive illustration of the tyre belt effects can be found via [21].
Section 2.4
Combined slip
19
Figure 2.3: Principle of the brush model. When free rolling, bristles follows the tyre
rotation (rst from left). A tyre at a slip angle forces the outer end of the bristles to
follow the road causing them to stretch (second) until the force exceeds the available
friction. This causes the bristle to slide which in turn delimits the generated force.
The third and fourth pictures show simplied illustrations of this principle with the
limitation due to pressure distribution indicated on the latter.
tend to make analytical models complex [17, 22, 16]. For handling studies,
empirical tyre side slip characteristics, either tabular or curve-tted are still
dominating. A typical example of lateral pure slip steady state characteristics,
f y (α), is shown as a solid line in Figure 2.4, left.
Just as for generating lateral forces by steering the tyre, the generation of
longitudinal forces by braking or driving can be understood by considering
bristle deformations.
2.4
Combined slip
The interdependency between longitudinal and lateral characteristics brings a
further aspect to the nonlinearity of a tyre. Again, relating to the bristles on a
brush model, the maximum force that can be transferred to the contact friction
is resulting from both lateral and longitudinal forces which leads to a circular maximum force transfer. The whole tyre has similar characteristics but a
few signicant differences can be seen as illustrated in Figure 2.4. First of all,
the circle is rather an ellipse since the maximum force tends to be different in
lateral and longitudinal directions due to anisotropic in the tyre construction.
Therefore, this dependency between lateral and longitudinal forces is often referred to as the friction ellipse which is used not only for single tyres but also
to describe the potential of a whole vehicle as will be describe further in Chapter 5. As a second effect, the ellipse is rather an egg since there is no symmetry
around zero longitudinal force. One reason for this is the change in contact
20
Chapter 2
Principles of tyre force generation
Figure 2.4: Illustrations of combined force characteristics. Lateral ( f y ) and longitudinal
( f x ) force dependency on slip angle (α) and longitudinal slip (κ), left. A friction ellipse
with iso-slip lines, right.
force distribution that depends on the tension in the tyre belt; a brake torque
will move the pressure distribution rearwards where the slip is higher, allowing
more side force to be generated. Accordingly, driving would force the distribution forward with the opposite effect.
2.5
Load sensitivity
The tyre's load sensitivity is one of the most important effects when it comes to
tuning as described in Chapter 3. Tests show that there is a nonlinear relation
between tyre load and lateral force for a given slip angle, Figure 2.5. Any book
on vehicle dynamics would point out this effect but an explanation of why this
occurs is harder to nd which is probably because some effects work in favour
of increased horizontal force and some against. For a tyre with linear vertical
stiffness, doubling the deformation would also double the load but should not
double the contact area which would indicate an increased pressure which due
to the anisotropic behaviour of rubber could explain why the horizontal force
is not doubled. Additionally, since the load dependent friction primarily affects
the maximum deformation allowed before sliding, the increased load only affect the parts of the contact patch that otherwise would be sliding which also
would limit the force increase.
However, measurements [17] show only a modest increase of contact pressure and as the contact length increases, the bristle deformation due to slip
should instead increase quadratically. This suggests that at least the side stiffness should more than double which would be a step in the wrong direction.
A third aspect that is in line with the observed behaviour is the uneven distribution of longitudinal forces in the contact patch, mentioned in Section 2.2.
Since the amplitude of this distribution increases with increased load and has
Section 2.6
21
Inclination eects
Figure 2.5: Resulting tyre load sensitivity. Doubling the load will not give twice as
much lateral force and thus, the load distribution will affect the lateral force distribution.
to be carried by the bristles which limits the obtainable lateral force [17]. An
other factor that affects that affects the behaviour on large load variations are
the effects on pressure distribution, e.g. between wall and air pressure stiffness.
2.6
Inclination eects
Inclining the wheel has at least three kinds of impact on the tyre force generation but before discussing these, it is worth clarifying the difference between
γ,
inclination,
and camber,
3
work, inclination
ε,
which sometime tend to be mixed up. In this
is referred to as the angle between the road normal and the
vertical axis of the wheel and camber is the angle between the vertical axes of
the chassis and the wheel. On a at road without steer angles applied, the difference is the roll angle,
ψ.
Naturally, when discussing tyre road contact, focus
is on inclination while when tuning suspension, camber is used.
First, although inclination means tilting the wheel, it causes lateral force
according to a principle similar to steering but slightly more difcult to grasp.
Consider once again a rotating wheel but this time tilted at an angle
γ,
caus-
ing the inner end of the bristle to follow an elliptic curve if seen from above,
Figure 2.6, left. Still, just as for steering, the outer end strives to stick in contact with the road. This time, the width of the shaded area increases towards
the center and then reduces again, giving a shape more similar to the pressure
4
distribution . As the available friction decreases, this effect will be more im3
Inclination is sometimes called tyre camber.
4
The same result is achieved if one considers turn slip. In fact, inclining the tyre will result
in a component of the wheel spin in the turn direction.
22
Chapter 2
Principles of tyre force generation
Figure 2.6: When applying the brush model approach to the tyre, the bristles will deform in an elliptic pattern that will generate a side force distribution similar to the load
distribution (A,B). Depending on whether the tyre belt is deformed with maintained
road contact (C) or not (D), the inclination can also be use to improve performance by
compensating for the side force with an inclination angle (E).
portant, which also suggests that using camber instead of steering should be
advantageous on low friction surfaces. Note also that the center of the lateral
force is located at the tyre patch center, giving little or no aligning moment.
Second, consider a tyre exposed to lateral force. Depending on its shape,
the tyre will deform differently. For some tyres, it is advantageous to compensate for the deformation with an inclination angle to get a better patch, Figure 2.6, right.
Third, tilting the tyre towards the inner side will cause the outer side to have
a larger rolling radius that the inner side, imposing an aligning moment and a
yaw velocity.
For empirical and semi-empirical models and measurements, these effects
occur lumped as inclination (or tyre camber) force characteristics as exempli5
ed in Figure 2.7 . Notable here is that there is an inclination stiffness just
as for the steering. An effect that is heavily used in racing is the increase of
maximum lateral force when an inclination angle is applied. However, the gain
tends to decrease with the increase of the slip angle which often is referred to
as roll-off.
2.7
Transient behaviour
The need to consider transient effects varies greatly with the application and
most semi-empirical tyre models include some kind of rst order dynamics
5
Consider e.g. [21] for sample measurement results.
Section 2.8
23
Additional eects
Figure 2.7: Typical slip-force characteristics for different inclination angles (left) and
the inclination dependency at a constant
α for different vertical loads [23] (right).
that typically depend on speed and load. This is often sufcient for studying
responses due to slip angle or wheel speeds that occur in handling manoeuvres [17]. Typical rules of thumbs are 1/3 of a revolution for longitudinal force
and twice for the lateral force where the latter varies more, depending on tyre
dimension and side stiffness.
An important exception is the high frequency of an ABS controller that
normally requires more detailed dynamics and the general advice is to think
twice before considering or applying fast dynamics in vehicle models designed
for handling since they would also require modications of other subsystems.
2.8
Additional eects
In addition to the effects presented above, tyre force generation is also dependent on temperature, pressure, adhesion, etc. Temperature dependencies are of
signicant interests in racing applications but are seldom considered in published tyre models. Adhesion potential, or friction, is often handled with the
similarity method, stating that the shape of the tyre force characteristics remain
similar independent of surface conditions.
With the presented effects in mind, it is also important consider that they
might change as tyres develop. If the so called tweel concept, Figure 2.8, would
enter the market, effects involving the side wall would certainly change.
2.9
Tyre models
As seen from the discussion above, tyre behaviour is quite complex and it is
essential that model based analysis is performed with representations that prop-
24
Chapter 2
Principles of tyre force generation
Figure 2.8: The tweel concept uses spokes to replace the side wall and the pressurized
air.
erly describe the situations studied. Tyre models have evolved to meet increasing demands as the possibilities with simulation in general have grown. Brush
approaches were explored and it turned out that detailed models were needed to
grasp the characteristics even from pure slip conditions. As a result, empirical
models where assigned, based on various types of curve ts. A famous example is the so called Magic Formula that was developed as a joint effort by Volvo
AB and Delft University with the objective of describing the tyre characteristics with a limited set of parameters [24]. The original model used curve-ts
for the pure slip characteristics and an analytical formulation for the combined
slip conditions and Michelin later suggested a curve t based description also
for the latter characteristics. With the introduction of belt dynamics, this model
eventually developed to contain more than one hundred parameters that are to
be adopted to measurements.
As a reaction to this evolution, other models based on the original thought
from [24] have been developed.
In a series of publications, e.g. [25, 19], a
model now called TMeasy is described where a few steady state pure slip measurement points is used to inter- and extrapolate a combined transient characteristics. Another approach is introduced in [26] where the extrapolation is
made given arbitrary functions describing pure slip characteristics. The idea is
that only parameters that are easily obtained should be included in the model.
These models have been implemented in the environment described in Chapter 4. Other approaches are found in e.g. [27].
As the computational performance has grown, the analytically dened models have also gained in interest. Unlike the models above which are based on
a single contact point, models like F-Tire [28] and RMOD-K/CDTire [29, 30]
discretise the contact patch. This is relevant especially for high frequency anal-
Section 2.9
Tyre models
25
ysis and when travelling over surfaces where the unevenness is in magnitudes
of the contact patch or less. An alternative method is introduced by the SWIFT
model that lters the road surface so that single contact point calculations can
be applied [16].
As a result of this variety, the choices of tyre model is far from obvious. A
reoccurring dilemma is the lack of reliable tyre data and a very relevant question
in the tyre model selection process is whether or not it is possible to acquire
parameterizations. In general, tyre models are capable of handling combined
slip, normal tyre load variations, rst order dynamics and modest inclination
changes on at or nearly at roads. When dealing with large deformations,
high loads, inclination angles over a few degrees, varying tyre temperatures,
standing still, soft grounds or uneven roads, more dedicated models are often
required.
26
Chapter 2
Principles of tyre force generation
Chapter 3
Principles of
vehicle motion control
This chapter explains the fundamental principles for vehicle motion control
and outlines the means for future development. Considering also the controller
structure, a method to classify different vehicle motion control concepts is suggested.
This chapter is in parts based on Paper E.
3.1
Motivation
Controlling the vehicle motion and achieving good handling performance and
safety is essentially about nding a suitable set of tyres for the task and adapting the chassis to have the tyres work in their optimal range. In racing, this
would be straightforward but when designing passenger cars, the same settings
have to withstand variety of different tasks and road conditions. Additionally
one has to anticipate that the consumer would use any tyres that have approximately the same dimensions. For commercial vehicles, the load conditions
make things even more tricky. As a means to improve performance and safety,
computerized control has great potential to overcome compromises required
by static tuning. In general, the aim with Vehicle Motion Control (VMC) is to
improve handling, safety during acceleration/deceleration and ride comfort as
described in Figure 3.1, [31]. Most of these systems are based on the direct or
indirect control of the mechanism affecting tyre forces to generate the desired
27
28
Chapter 3
Principles of vehicle motion control
Figure 3.1: Outline of vehicle motion control, [31].
vehicle motion as discussed in the previous chapter. However, the sheer variety,
especially in naming of proposed vehicle motion control systems creates a need
for a common classication system to avoid confusion. This is classically illustrated by the Electronic Stability Program (ESP) introduced in the Mercedes
S-series in 1995 which is reproduced and marketed under more than 10 different names by various brands. A common classication system would clarify
the functional aspects and design techniques in a market where the possible
congurations are seemingly endless.
This chapter focuses on a systematic way to categorize the functional aspects of various approaches to road vehicle motion control. The classication
is divided into two parts where the rst considers the following means of actuation:
individuality Is one actuator applied to one wheel, an axle or the whole vehicle?
tyre input Does the actuator affect the wheel spin, wheel angles, wheel load
or something else?
degree of freedom Which of the vehicle's degrees(s) of freedom are affected.
longitudinal, lateral, vertical, roll, pitch and/or yaw motion?
inuence on tyre force Is the effect on the tyre force direct or indirect?
Section 3.2
Means of actuation
29
energy demand Is the system active or passive?
feedback Does the system act in closed or open loop?
adaptability Can the system adapt or is it static to changes that are not state
feedback?
activity Is the system continuously active or active only through intervention?
These are explained in more detail in Section 3.2. The second part of the classication considers the control structure partitioning, covering unication, coexistence and merging as explained in more detail in Section 3.4. By this classication, a tool is achieved that makes it is easier to understand and estimate
what performance can be expected from a given VMC. This is essential for understanding more complex, integrated approaches, especially in cases of overactuation. This is explored in detail in Paper E addressing not only road vehicle
motion control but also useful references in robotics and aeronautics. These examples will also provide the foundation for the work presented in Chapter 6.
3.2
Means of actuation
As explained earlier, one part of the classication considers which degrees of
freedom are affected. From a stability point-of-view, yaw motion is often the
primary target since the basic idea is to prevent excessive over- and understeering. Even so, it is important to keep track of how the other degrees of
freedom are affected. For example while braking it would be unacceptable to
prolong the stopping distance while improving stability.
Before digging too deep into the actual actuator congurations it is rst
relevant to consider which congurations actually make sense since they will
reveal other aspects that also need to be classied. There are various approaches
to this (e.g. [32, 33, 31]) and one popular way to present ndings is based on
the resulting friction ellipse. Figure 3.2 illustrates one example showing four
different congurations and in what range they are effective.
It can be seen from the gure that roll moment is efcient also on the lateral
motion, revealing a reason to be systematic and distinguish between whether
the effect is indirect or direct as is illustrated for some typical means to affect
lateral tyre force in Figure 3.3. Consider the steering example in Figure 3.2,
it can be used directly to generate side force and control lateral acceleration,
change the vehicle slip angle and control the yaw motion. As a comparison,
the roll moment is used directly to control the vehicle's roll motion and thus
the dynamic load transfer. The distribution of roll moment between front and
30
Chapter 3
Principles of vehicle motion control
Figure 3.2: Effective range of different vehicle motion control concepts, illustrated on a
resulting friction circle where Gx and Gy indicates longitudinal and lateral acceleration,
respectively. From left to right: Steering, left/right drive torque proportioning, roll
moment and front/rear drive torque proportioning, respectively [32].
Figure 3.3: Direct (left) and indirect (middle and right) means to effect the lateral tyre
force.
rear indirectly controls yaw motion by affecting the side force through redistribution of vertical loads. This indirect effect can however only be used to affect
the cornering characteristics of the axle and thus cannot be used to affect the
vehicle's slip angle to the same extent as steering.
With the increasing numbers of possible actuators it is also relevant to have
a more systematic separation between 'axle' and 'wheel'. The term 'four wheel
steering' (4WS) is widely used for systems where the steering angles on each
axle's wheels are coupled.
To separate this from systems with wheels that
are independently steerable, it is suggested to refer to coupled systems as axle
steering. This separation principle is also highly relevant for driveline congurations as four/two wheel torque distribution and two axle torque distribution
have different effective ranges, gure 3.2.
From Figure 3.2 the importance in distinguishing what the driver controls
Section 3.2
Means of actuation
31
Figure 3.4: A comparison of active and passive systems where for a damper, x would
be the (compression) velocity and y the (compression) force. The shaded areas indicate
the maximum operating range for a closed loop system and the curve shows open loop
characteristics.
and what is left for the VMC can also be seen. Again, steering is a classical
example: It is concluded in [32] that 4WS is effective at lower acceleration
levels. This is however not necessarily true in the general case but in the case
referred to here, the analysis is done on a design that can only affect the rear
axle steering angle while the front axle is directly coupled to the driver. If also
the front axle steering could be inuenced by the VMC, the effective range
would have been wider. So, being precise about what authority the driver has
helps avoid confusion. Full authority as is the case in [32] means that the VMC
has to adapt to the driver input while sharing authority allows the VMC to
overrule the driver to a limited extent. No authority requires that the meaning
of the driver input is changed as for the throttle-by-wire case where the throttle
pedal position can be re-interpreted as a desired acceleration or velocity.
In the context of vehicle control, a clear denition of 'active' and 'passive'
is also highly relevant due to the confusion it tends to cause when 'active' in
the sense of adding energy to the system is mixed up with 'active' in the sense
of being able to affect/inuence the system. Here it is suggested to use the
term 'active' for the rst case and elaborate more on how the system is inuenced, by using the control engineering terms of open loop, adaptive open loop,
closed loop, and adaptive closed loop control. These aspects are summarized
for general circumstances in Figures 3.4 and 3.5. In the rst gure, an open
loop system occurs as a curve along which it can operate while a closed loop
system is indicated by its maximum operating range.
To illustrate how the suggested control engineering terms in the second gure are applied, consider rear axle steering which gained interest in the late
80s. Honda claims they were rst on market with the Prelude in 1987, adopting a non-linear but speed insensitive steering ratio between the front and rear
32
Chapter 3
Principles of vehicle motion control
Figure 3.5: Summarization of control engineering to separate open loop from closed
loop and static from adaptive, respectively.
axles. Small angles gave equally directed steering and larger angles gave op1
posite steering (static open loop) . Taking it one step further, the rear steering
gain can depend on vehicle speed which has been used for e.g. the Mazda 626
and BMW 850i [33]. This allowed high and low speed characteristics to be
separated (adaptive open loop). Systems that use measurements or estimates
of the vehicle state further improve the abilities and can be used to control for
example yaw and side slip (static closed loop). Adding ability for the VMC
to adjust to changes in for example vehicle velocity or driver behaviour leads
to the forth category (adaptive closed loop) and depending on which of these
approaches is selected, different performances can be achieved.
The active-passive classication now directly relates to the energy con2
sumption and is highly relevant for suspensions since having an active system
carrying the vehicle load can be very power-demanding. For such systems, it
might even be relevant to separate between slow (less power) and fast (more
power) active systems [34]. A nal remark related to this concerns when the
control is active. For steering, the most common approach is to have a continuous control while yaw control through brakes mostly works as an intervention
during a limited time at critical situations. The latter approach is necessary to
not ruin fuel consumption and wear but may also lead to abrupt changes of the
characteristics as illustrated in Figure 3.6. This behaviour has both advantages
and drawbacks since on one hand, the changes occur suddenly and uncomfortably but on the other hand, they also give a clear indication to the driver that
the limits have been passed.
1
In this sense it is same as 'tuning' mechanical parameters.
2
Suspension types that are passive and adaptive open loop or (adaptive) closed loop are often
called semi-active.
Section 3.2
33
Means of actuation
TC
VDM
VSC
ABS
Figure 3.6: Intervention control through several systems may lead to abrupt changes
of the characteristics (left) while continuous control gives a smoother behaviour [35].
The rest of this section is used to describe how different actuator congurations can be classied by how they are used to affect the vehicle's motion.
Accordingly, the sequel is divided into wheel spin, wheel angle, wheel load and
other possibilities for a VMC to act.
Wheel spin
Wheel spin control is achieved by either braking or driving the wheel and this
section illustrates how means to affect the wheel speed affect the vehicle motion.
The most fundamental control of vehicle brakes is the brake force distribution. The load transfer while braking is dependent on the deceleration. In
other words, as the total brake force increases, distribution to the rear brakes
should decrease in order to maintain rear cornering stiffness, Cα,34 , and thereby
stability. Traditionally, this is achieved by hydraulic devices but since the introduction of anti-lock braking systems (ABS), this can also be done electronically,
allowing for features like adaptive tuning.
ABS was the rst step towards an electronically based safety system for
two reasons. First, the system assured steerability when excessive brake force
was applied by preventing the wheels from locking, allowing novice and panic
drivers to steer and brake at the same time. This is a great safety improvement
by itself and also a good example of how a brake that is intended for longitudinal control of the vehicle can indirectly affect the lateral control, based on
the friction ellipse characteristics as described in the previous chapter. Second,
by introducing wheel speed sensors and a valve to reduce the brake pressure
individually over each brake, the foundation for more advanced safety systems
was laid. To be even more useful, the brake system had to be able to apply
34
Chapter 3
Principles of vehicle motion control
more brake force and not just reduce it. This was rst introduced to allow the
brakes to be activated in order to prevent wheel spin and thereby reduce the risk
of spin out accidents.
Except for being able to apply an accelerating torque, the effect of the driveline on the vehicle's yaw stability is basically the same as the brake system.
The distribution of drive forces between front and rear can be used to balance
a car if the car is all wheel driven. On a traditional car this is mainly done
using various combinations of clutches and differential gears whereas a future
car with electric motors at each wheel could apply the different drive forces
directly. Direct yaw control through (mainly rear) differentials that can distribute drive torque together with brake intervention has been a popular subject
in Japan. This also can be seen in performance production vehicles such as the
Subaru Impreza, Mitsubishi Evolution and Honda Legend.
Being able to control the difference in wheel speed between left and right
can be of substantial interest not just when accelerating. A differential that is
able to redistribute torques between the left and right wheels, or even lock the
axles together, will introduce a constraint between the wheels that affects vehicle characteristics. Forcing both sides to spin with the same velocity causes
the outer wheel to generate brake forces and the inner to generate drive forces
which tends to straighten the vehicle. By feeding the differential with information of longitudinal acceleration and/or throttle and brake, tip-in/tip-out stability can be improved and is commonly used to balance race cars for turn
negotiation.
So, both the driveline and the brake systems can be used to affect yaw
stability of the vehicle in two ways; directly by applying different longitudinal
forces on the left and right sides and indirectly because of the friction ellipse
characteristics of the tyres.
Both these effects are used in ESP and similar
systems; oversteering triggers braking of the inner (direct effect) rear (indirect
effect) wheel while oversteering triggers the outer (direct effect) front (indirect
effect) wheel.
Wheel angles
Changing wheel angles primarily means affecting the generated lateral tyre
force.
As seen in Chapter 2, this can be done either by a change in slip or
3
inclination angles where the rst is by far the most common approach . As
mentioned in the previous section, it is relevant to make the distinction between
3
One of the rare examples of steering by pure camber control for cars was seen on a research
prototype from Fiat in the early 90's.
Section 3.2
Means of actuation
35
wheel steering and axle steering where the latter implies that both wheels are
steered together.
For axle steered vehicles, static toe is of signicant importance since it allows the load transfer to induce a side force even at straight driving. This can
be used both to increase stability (toe-in) or to improve response time (toeout). The main drawbacks with static toe are increased rolling resistance and
wear. To maintain the desired effect this can be compensated for by setting
the (slower) kinematic properties of the suspension so that the roll introduces
a change of steering angle, referred to as roll steer or steer gain. Additional
effects are achieved by designing the elastic properties so that tyre forces induce steer angle changes. An often mentioned example of this is the so called
Weissach axle used in Porsche's 928 suspension but in fact, these properties are
present in most suspensions.
It is therefore clear that there are many ways to achieve passive steering
control. Camber is handled in a similar way except for two signicant differences. First, on a at surface, the inclination
ε depends on both camber γ and
roll as described in Section 2.6. This means that to achieve inclination gain
one rst has to compensate for roll. Second, since the tyre forces act in the
road plane, only positive camber elasticity can be achieved on the outer wheel,
which is normally not desired.
Additionally, the same wheel travel is used
both to handle roll and bounce which constrains the possible camber and toe
changes.
Compliance properties in general are also very coupled so that a good steering compliance gives poor camber characteristics and sensitivity to drive and
brake forces. The properties are also affected by comfort constraints and uneven roads which make compromises even more complex. By test driving a
couple of different brands, and sometimes even models, it gets quite obvious
that tuning is not enough, the driver feedback is an additional constraint to be
considered. With adaptive and/or closed loop systems, there is a potential take
advantage of adjustable toe and camber, therefore reducing, or even eliminating
the disadvantages mentioned above.
Wheel loads
As seen in Section 2.2, tyre load has a nonlinear scaling effect on tyre characteristics which is in many cases an efcient way to tune the handling behaviour
by the use of stabilizers. Consider Figure 2.5 where the resultant side force for
an axle with equal and unequal load distribution is shown. When negotiating
a curve, the more load transfer an axle has to carry, the less efcient the force
generation will be. For standard cars with the same tyre dimensions in the front
36
Chapter 3
Principles of vehicle motion control
and rear, this is often the most practical way to balance a vehicle.
While the relative front and rear roll stiffness affects the balance, the total
roll stiffness together with roll axle location and dampers will affect the dynamic load transfer. Additionally, the roll and pitch motion will give changes
in inclination and steer angles. Actually, even without load sensitivity and any
wheel angle changes, the load transfer will still introduce a yaw moment of the
magnitude Mz
= mghax ay as will be discussed in Chapter 5.
Therefore, although traditionally suspension control has primarily been considered means to improve ride, its possibilities to indirectly improve tyre-road
contact and the ever-present compromise between good ride (low vertical accelerations) and good handling (small load variations) make it highly relevant
for approaches to VMC design that combines different types of actuators as
seen in e.g. [36].
Other possibilities
The approaches discussed above all relate to conventional tyre behaviour but
there are of course other ways to affect vehicle motion. One way is to apply
aerodynamic devices to generate downforce to increase tyre load and thereby
indirectly improve the tyre force generation. Direct aerodynamic control of
vehicle motion was adopted in the early days of automotive engineering when
ns were applied for high speed stabilization such as for the Tatra 77.
Another approach to improve the tyre force generation is to modify the
contact between the vehicle and the ground. The Mercedes F400 Carving is
equipped with hybrids of car and motorcycle tyres. They apply camber change
not only to achieve the conventional advantages described in Section 2.6 but
also to be able to shift from standard surface adhesion with the car part of the
tyre to high surface adhesion with the motorcycle part of the tyre. This allows
for performance improvement while keeping the rolling resistance low during
normal driving, [10].
3.3
Sensoring
Any closed loop or adaptive system must rely on sensors or observers to get
the information needed. This is a huge topic in itself and the estimation of road
surface adhesion and the vehicle's lateral velocity have been subject to much
effort throughout the recent years. This is not further treated in this work but
it is wise to keep this problem in mind when considering VMC. In particular
its robustness to imperfect sensoring is important. Does the implementation
Section 3.4
Control structure
37
require a precise value of a state/condition that is hard/expensive to measure?
And if so, how precise?
3.4
Control structure
Along with the increasing availability of actuators, the need to distribute effort
wisely has gained signicant interest during the last decade. Distribution as a
concept is far from new. Steering linkages, differential gears and anti roll bars
have performed these tasks mechanically for decades. What has changed is
that now several actuation concepts can affect the same degree of freedom of
the vehicle's motion. This is both an advantage and a problem [37] that has
been studied since the late 80's, leading to a variety of different proposals. The
following classications deal with the partitioning of the control tasks and how
the different sub-controllers cooperate.
Several systems trying to control the same actuator(s) require that the actuator input signal(s) are unied. In arbitration, only one system at a time is
allowed to act which makes the unication easier but at the same time limits
performance. A natural extension of this is to coordinate the signal by allowing
several systems to be active at the same time. This implies a greater coordinator knowledge in the sense that it must somehow understand the effects of the
combinations. In [12, 38], means to provide this knowledge in terms of neural
nets and fuzzy logic are discussed.
Contemporary systems are often organized to handle one type of actuator
on multiple or all wheels, requiring them to have internal distributions. Most
common among these are the brake system and driveline. Since the generation
of a tyre's lateral, longitudinal and vertical forces are highly interdependent,
several such coexisting systems that act on different sets of actuators can interfere with each other. Figure 3.7 illustrates the differences between parallel,
hierarchical and cooperative coexistence.
A typical example of this is yaw
control by rear axle steering and traction/brake force distribution which gained
interest in Japan during the 90's. This conguration also is referred to in the
sequel.
Parallel coexisting concepts which rely on the development of the individual systems that ensure against critical interferences. Ideally, this requires
no additional cost in form of computational power or modication of existing subsystems when including new ones. It may however suffer from poor
performance since the benets from the combination effects cannot be drawn.
When the applied systems can be separated in terms of control objective and
frequency range, this method can still yield satisfactory results. This is sug-
38
Chapter 3
Principles of vehicle motion control
Figure 3.7: Coexistence approaches: Parallel with no information transfer, hierarchical
with a unidirectional information ow and cooperative with an information exchange.
gested in [39] where effort is spent on a parallel coexistence, based on the
conclusion that traction and brake intervention is suitable for yaw rate control,
while active rear steering controls body side-slip. Another way to implement
parallel coexistence is to have each subsystem consider the actions from the
other as disturbances as suggested in e.g. [40].
Hierarchical coexistence is a step towards more awareness with a unidirectional information ow where one system acts independently and the others
adapt. With the conguration mentioned above, [41, 42] have the steering acting independently while the resulting yaw moment from driveline and brakes
adapts to it. A similar approach is also seen in [43]. Cooperative coexistence
takes this one step further as it is based on mutual information exchange as
suggested by e.g. [44, 34, 45].
As the information exchange between the subsystems in a cooperative coexistence increases, the next step is to merge the functionality as illustrated in
Figure 3.8. Partial merging has been a popular way to deal with the combination above, where the rear steering angle and resulting yaw moment from the
driveline and brakes are merged [46, 47]. The distribution of tractive and brake
torques is then done independently of the steering. This has a signicant advantage in that the partitioning can be done so that the number of inputs and
outputs are the same, allowing conventional multi-variable control approaches
to be applied.
If the merging results in a subsystem that comes to distributing signals to
several subsystems that are independent of each other, it may be called allocation and can occur in several steps. Allocation often occurs in merged or
partially merged structures. A typical application is to control rear axle steer
angle and resulting yaw moment from driveline and brakes, where the wheel
torques in turn are allocated, [46, 47]. A similar approach is used in [48] to
Section 3.5
Application
39
Figure 3.8: Merging approaches as an alternative to coexistence, Figure 3.7. Partial
merging means that some of the original functionality (A,B,C) is merged and some is
not.
combine brake force and load distribution control.
Actually, the road vehicle has been over-actuated since individual wheel
brakes started to be used to control the vehicles yaw motion [49]. Due to the
simplicity of the controllers working principle, a rule based allocation was feasible, but with an increasing number of available actuators affecting the vehicle
motion, systematic allocation becomes more and more important. Aeronautic
and robotic controllers have had to handle the challenge of over-actuation for
decades and the approaches in these elds are, in many cases also relevant also
for ground vehicles as is shown in Chapter 6 and a more detailed overview is
given in Paper E.
3.5
Application
So far, this chapter has explained the principles relevant to vehicle motion control and motivated why they are important to distinguish between.
Applied
together, this leads to a tool that can be used to better understand what could
be expected from a particular VMC and what could not as will be illustrated in
the sequel.
The proposed classication criteria are set up as a form in Table 3.1 where
each box corresponds to some benets and limitations. This can be used as a
check-list when reviewing publications on vehicle motion control to help distil
the information and nd benets and drawbacks that are not always mentioned.
For example, typically a VMC that implements two parallel coexisting controllers should require a motivation for the separation or an explanation of how
interference is avoided. Another example of a red ag is a VMC that uses indirect effects to achieve zero side slip. These and many other issues are made
40
Chapter 3
Principles of vehicle motion control
Continuity
Continuous
Intervention
Individuality
Wheel
Axle (f/r)
Vehicle
Wheel input
Spin
Angle
Load
Degrees of freedom
Longitudinal
Lateral
Vertical
Roll
Pitch
Yaw
Inuence on tyre force
Direct
Indirect
Energy demand
Active
Slow
Passive
Feedback
Open loop
Closed loop
Adaptability
Static
Adaptive
Recongurable
Driver authority
Full
Shared
None
Unication
Arbitration
Coordination
Coexistence
Parallel
Hierarchic
Cooperative
Merging
Separated
Partially merged
Merged
Sensoring
Required inputs
Robustness
Table 3.1: Application of the vehicle motion control classication as an evaluation
form.
more obvious simply by lling in the form.
Chapter 4
The VehicleDynamics Library
This chapter presents the VehicleDynamics Library that is developed in order
to have a platform suitable for developing and evaluating generic vehicle dynamics models.
This Chapter is in part based on Paper Q.
4.1
Background
Vehicle dynamics modelling has traditionally been considered purely an issue
of mechanical simulation and analysis and as such, there are a variety of tools
on the market today designed for this type of modelling, such as ADAMS [50],
SIMPACK [51], veDYNA [52] and CarSim [53].
However, with the future
scenario of multiple actuators and increasingly computerized controllers, as
described in Chapter 1, there now exist additional factors which affect the selection of modelling tools.
To begin with, the vehicle has to be considered as a multidisciplinary system where parts and whole subsystems should easily be exchangeable. Additionally, it should be possible to describe different parts with various levels of
detail such as, for example, connecting a detailed actuator model to a simple
chassis model or vice versa. Finally, since there will be a multitude of ways to
implement control, any given structure must not be limited to or even based on
a layout for conventional vehicles.
1
Modelica
1
has the ability to meet these demands.
It is a publicly avail-
Comprehensive overviews of the language are given in for example [54] and [55]. See also
Appendix C for a brief introduction.
41
42
Chapter 4
The VehicleDynamics Library
able, object-oriented language for the modelling of large, complex and heterogeneous physical systems, maintained and improved by the non-prot Modelica Association [56]. The potential of Modelica, together with the tool Dymola [57], as a mean to efciently describe vehicle models is veried in Paper G. The conclusion is also supported in [58, 59, 60] and Modelica has also
been successfully used for ight applications which faces many similar challenges [61].
Within this future scenario there exists a practically innite potential for
different vehicle motion controllers yet the time and resources required to explore this potential creates a huge limiting factor. As a direct answer to this
difculty, a library for vehicle dynamics studies was developed, maintained
and made publicly available by the author during the years of 2002-2004 using
Modelica as a base. This was done on the initiative and support from Hilding
Elmqvist (Dynasim AB) and Martin Otter (DLR Research Center Oberpfaffenhofen). This VehicleDynamics Library provides means for extensive modelling
and simulation with a scope reaching beyond a purely mechanical analysis.
Due to the great interest and demand the library has now been replaced by a
version developed and maintained by MODELON AB [62] under a commercial
license. The current version of the VehicleDynamics Library is substantially
extended and improved, based on user experiences and feedback from the now
deprecated earlier version.
Along with the Modelica language specication [63], the Modelica Association also provides the Modelica Standard Library (MSL) that contains interfaces and basic components for mechanics, electricity, signal ow, thermodynamics, and more. The VehicleDynamics Library, using this as a platform,
moves beyond the basics to provide a thorough, detailed collection of models,
classes and denitions specically designed for simulating and analyzing vehicle motion control systems. Section 4.2 explores the basic design ideas, which
are also relevant to general library design. Section 4.3 addresses some of the
vital enhancements necessary to build, simulate and analyze vehicle dynamics
models using Modelica. A more detailed explanation of the implementation
can be found in [64].
4.2
Basic ideas
The VehicleDynamics Library contains several hundreds of models and other
classes where the denition of a typical full vehicle model uses tens of thousands of equations. Additionally, models often share a majority of the properties with other similar ones, for example different versions of a particular sus-
Section 4.2
Basic ideas
43
Figure 4.1: An efcient model partitioning that ensures reusability requires that both
functional (horizontal) and controller-plant (vertical) aspects are considered.
pension. To be able to handle this with a minimum effort of maintenance and
code repetition, a suitable structure is needed for both the models and the library
itself. For models with more than a handful of parameters, having descriptive
names for them all is not feasible and to maintain consistency a suitable naming
convention is needed. Additionally, a suitable range of model delities and a
means to perform analysis efciently are crucial for the usability of the library.
This section is partitioned accordingly to explain how these issues are dealt
with.
Structuring considerations
The partitioning of models into sub-models and the packaging of these has
turned out to be a key point when designing a model library that is efcient to
use. The experiences with the development of this library and is predecessors,
can be summarized as done in Figure 4.1 where the grey area illustrates the
system that should be modelled.
The rst mistake was to only consider one aspect at a time. A purely functional partitioning led to an architecture where each sub-function could easily
be exchanged by another sub-function of the same type. But, the models often
suffered from bad performance and the reason was in many cases that there
were no natural distinctions between the functions. This is illustrated by the
plant for sub-functions A and B in the gure. A typical example is the longitudinal and lateral force generation in a tyre. The idea of functional partitioning
requires either to accept the decreased performance or to merge sub-functions
A and B. A compromise is to extend the information exchange or other con-
44
Chapter 4
The VehicleDynamics Library
nections between the sub-functions to compensate for the sub-optimal cut. All
variants work badly, the rst for obvious reasons, the second because the merging tends to continue in the next steps which lumps everything back to one
model, and the third because it tends to cause an immense overhead since every
ad-hoc addition to one sub-function has to propagate through all others of the
same type. In [65] these thoughts are supported from a more general software
perspective.
On the other hand, considering only the vertical controller-plant partitioning in the gure tends to give lumped, specialized models that are hard to reuse.
The method that gave best results was to use both approaches as a supporting
grid and dene the models inside-out. This lead to the adoption of the following
guidelines in the development of the VehicleDynamics Library:
1. Models should be partitioned into sub-models that corresponds with the
(user-imagined) partitioning of the real component.
2. Sub-models should be designed as any other model and not be limited to
a specic application.
3. Models that perform the same task should share common connectors and
parameters.
4. The package structure and the model structure should correspond.
5. Sub-packages that occur in several packages should have a consistent
appearance.
To illustrate how these items are reected in the library, a standard car consists
of chassis, brakes, engine, transmission, driveline and driver interface.
The
chassis then consists of a front and rear suspension, a body and four wheels
etc. which form a logical hierarchy (1). A shaft or a leaf spring model is not
limited to be only used in vehicle applications (2). All suspensions share the
same connector set to mount chassis and wheels and those that have a steering
wheel share the same steering connector (3).
packages are found inside the
Chassis
The
Suspensions
and
package which is inside the
Wheels
Vehicles
package (4). The last item (5) requires a more thorough explanation given in
the sequel.
Since Modelica is an object oriented language, it also include inheritance
which is an effective means for avoiding code duplication and ensuring compatibility. Here it is essentially used for interface and template denitions as
sketched in Figure 4.2. The notation is an adaptation of UML [66] as described
in Appendix C.
Section 4.2
Basic ideas
Figure 4.2: The template usage applied to a conventional car model.
45
46
Chapter 4
An interface is a
The VehicleDynamics Library
partial model that consists only of common connectors
and parameters. Typically there is a base model that is common for all models
in the package and a standard model that is adapted to the common components. For example, a standard rack steering mechanism contains a steering
wheel connector since it is the common approach but the base does not since it
is not necessarily used in a by-wire solution.
2
A template
is also partial and consists of interface instances of the sub-
models as placeholders with connections made. By extending a template model
and lling the placeholders with actual component models, new models can be
implemented in an efcient way that is easy to maintain. By the design of the
template, the appropriate constraints can be applied so that only components
that make sense are selectable.
Interface and template models are always stored in the
Interfaces and Tem-
plates sub-packages to make them easy to nd. Additional standard sub-packages
are Experiments, Components, Internal and Variants packages. These are described in more detail in Appendix C.
As for the sub-packages, there are also component types that are found
throughout the library.
None
indicates null implementations that are mainly
Basic and Ideal indicate
Typical represents a typical variant.
used in templates as a way to exclude components.
basic and ideal implementations and
Naming convention
As a model gets more detailed, the number of parameters to be set increases
rapidly and it is in many cases not feasible to give each parameter a descriptive name. Therefore a consistent naming convention that gives understandable
short names are necessary. This is also important when trying to nd and plot
variables. The naming convention adapted here is based on assigning a property
to one or several parts
[property][index][part](..)(_[wheel number])(_[unit])
where a property is given a standard letter such as
f for force or r for posi0 for
tion. The index is used to indicate conditions for the property such as
construction conguration and
start for start value. The part is noted by an
W for wheel, S for spring, L for link and C for
initial capital letter, for example
chassis.
Some properties, such as construction hard points or elasticities might be
dened in the intersection of several parts which are then added in alphabetical
order. If several parts of the same type are present, they are identied with a
2
Some would probably call this an 'architecture' or a 'wizard'.
Section 4.2
rW_1
f0S_kN
r0CL1_2
i22L1_3
47
Basic ideas
Parameter naming
Location of the front left wheel.
Spring pretension force in [kN].
Construction location of the connection joint between chassis
and link 1 at front right wheel.
Inertia element i22 of link 1 at rear left wheel.
Table 4.1: Applications of the parameter nomenclature.
Figure 4.3: The three levels of detail that are used. Only planar dynamics (left), additional roll, pitch, bounce and wheel spin dynamics (middle) and full representation
(right).
number. For example, in a double-wishbone, the links are numbered
L1, L2,
etc., starting from the front lower link. When necessary, a number at the end,
separated by an underscore denes which wheel is referred to, numbered from
front and left towards right and rear. Finally, if the variable or parameter does
not display an SI unit, this is specied at the end. Some examples of parameter
names are given in Table 4.1 The complete specication is given in [64].
Levels of detail
As was already mentioned, a full vehicle model involves a large number of both
equations and parameters and the chassis model is a major contributor. So from
a vehicle motion control perspective, it makes sense to also have simpler chassis models to work with. Here three different levels of detail are adapted, see
Figure 4.3. The rst is a planar model with three degrees of freedom, static load
48
Chapter 4
The VehicleDynamics Library
distribution and no wheel spin dynamics, making it possible to start the development at low computational cost and allowing quick iterations and batch runs.
In the second level, degrees of freedom for wheel spin, roll, pitch and bounce
gives more accurate response to lateral and longitudinal acceleration changes
and allow for more detailed actuator models. Thanks to the wheel's lack of vertical dynamics, the delity can still be kept low. Both of these types of models
currently run only on at roads since the vertical dynamics of the tyres are not
considered. The third level, uses a full representation of the vehicles degrees
of freedom and the suspension can be dened with linkages or as tabular constraints and is thus available to handle interaction with an uneven road. The
two latter levels of detail share the same interface and it is thus possible to add
the same driveline and brake models, minimizing modelling effort.
Analysis
A vital part of any system design is a thorough understanding of the involved
subsystems; an analysis of a full vehicle is often useless if the behaviour of
involved subsystems is not sufciently understood. To isolate component behaviour and to avoid unnecessary computational cost, test rig experiments are
great aids. In the library, experiments are gathered in the relevant sub-packages
so that suspension examples are found under
Suspensions.Examples
and so
forth.
To make the test rig experiments easier to work with they are designed, parameterized and animated as real test rigs where applicable. Various controllers
can be attached to a test rig to govern the experiment and facilitate the test procedure. Figure 4.4 shows a tyre test rig that can be used to study quasi-static
and dynamic properties under combinations of sweeps and constant levels of
slip angle, slip ratio, camber, load, and velocity.
Vehicle analysis can be performed on a road using either a driver or a robot
to control the vehicle which allow simulations to mimic real test procedures in
order to generate, for example, cornering characteristics diagram as described
in Chapter 5. There are also ideal models of for example drivelines that allow
precise wheel spin velocities or torques to be applied when performing the road
tests. The second option is to, just as for a component, constrain the vehicle in
different test rigs to generate various kinds of measures that requires extensive
laboratory equipment to be performed in reality.
In Figure 4.5 such an experiment is exemplied using a shaker rig. The
model allows the level under each wheel to be adjusted individually. Depending on the chosen input, this can be used to analyze chassis out-of-plane characteristics such as roll and pitch dynamics, spring and damper settings and roll
Section 4.3
Implementation issues
49
Figure 4.4: Tyre test rig.
moment distributions. Additionally, it can for example be used to analyze driveline motion and study joint angles, shaft lengths, and other variables during
various suspension-travel scenarios to verify or tune the geometric design. Examples of handling analyses that can be performed with the library are shown
in the next chapter.
4.3
Implementation issues
While the previous section dealt with the basic ideas of modelling and library
structuring, this section deals with some specic implementation issues that are
of particular relevance for this application. This especially includes enhancements to the existing mechanics models, the tyre-road interaction and the signal
exchange for applications involving controllers.
Mechanics
The Modelica Standard Library (MSL) contain models for 1D translational
(MSL.Translational) and rotational (MSL.Rotational) mechanics and a multibody package for 3D mechanics modelling (MSL.MultiBody) which all to-
50
Chapter 4
The VehicleDynamics Library
Figure 4.5: Vehicle mounted in a test rig.
gether form a great foundation for mechanics modelling. For vehicle dynamics
applications, there are aspects that have instigated a signicant enhancement
of the standard contents that are discussed in this section, relating mainly to
the description of rotation in space, the selection of states in a model and the
inclusion of nonlinear effects.
The
Rotational package is based on a 1D description consisting of rotation
angle around an axis and the torque around the same axis. This description has
already been successfully used for some automotive driveline simulations[67].
There have also been some developments in describing the three-dimensional
effects of components in the rotational library such as coriolis forces [68] but
no satisfactory approach was available for slightly more complex components
such as shafts with different joint geometries.
One possibility would be to use
MSL.MultiBody but there are some signif-
icant drawbacks. The orientation is described either as a sequence of three an3
gles or with quarternions as described in [70]. The rst has the drawback that
there are singular positions of the sequence of angles used to describe the states
of the orientation. The quarternions method instead yields four state candidates
requiring dynamic state selection. A common drawback with both representations is that the number of revolutions is not accounted for so to represent
axial, or close to axial revolution, the multi-body representation is ineffective.
3
A comprehensive overview of object-oriented multi-body dynamics modelling is given
in [69].
Section 4.3
51
Implementation issues
Additionally, due to the representation, building shafts with
MultiBody
joints
often introduces mechanical loops slowing down performance.
Rotational3D was introduced with a composite interface denition consisting of a 1D MSL.Rotational ange resolved in a
MSL.MultiBody frame so that the rotation is around the x-axis and relative to
To overcome this, a package
the y-axis. This allowed the minor rotations of the axle mounts to be separated
for the axle spin. The selected representation allows a consistent interface to
both
MSL.Rotational and MSL.MultiBody.
Unlike many other tools, Dymola uses symbolic manipulation to resolve
kinematic constraints which essentially means that for a linkage model without
included elasticities, the number of states corresponds to the number of degrees
of freedom and needs to be chosen with care.
This is best illustrated with
an example; consider the double wishbone linkage in Figure 4.6. The leftmost
version has only ideal joints and the ve links used constrain ve of the original
six degrees of freedom of the upright motion and there are thus two states that
needs to be selected. If Dymola is able to select the states on its own, there are at
least four combinations that could come up but only two of these turn out to be
reasonable. Having the state in the suspended body is not reasonable for at least
three reasons. Firstly there is the numerical accuracy, if the vehicle is moved
a considerable distance from its reference position, the resolution of the wheel
travel would be low since it means subtracting two large numbers from each
other. Second, since the motion of the upright would be dened relative to the
world frame, meaning that if the vehicle would roll enough, the representation
would be numerically tough and eventually singular. Third, which is common
with having the state in the strut, there are multiple solutions for some state
values which may lead to erroneous congurations as illustrated in Figure 4.7.
For some suspension linkages, multiple solutions could also occur when the
state selection is in the joints. Experience has shown, however, that this is far
easier to deal with by simply supplying reasonable start values.
As seen from the discussion above, the ability to explicitly chose states
is of signicant importance and it is not enough to be able to select all or no
states in a joint as illustrated by the elasto-kinematic linkage in Figure 4.6. The
kinematic linkage needs two states to specify the one degree of freedom which
can be set in either of the joint pairs. Using bushings instead of inner joints
gives ten additional degrees of freedom, i.e. eleven in total. However, since
each of the pair of bushings have twelve potential states, 22 out of 24 have to
be selected.
MSL.MultiBody has been designed with ease of use in mind so the available models have the state selection lumped together for all degrees of freedom.
52
Chapter 4
The VehicleDynamics Library
Figure 4.6: State selection for a kinematic (left) and an elasto-kinematic (right) double
wishbone suspension linkage.
Figure 4.7: If s would be chosen as state, it is difcult to separate between wanted (left)
and unwanted (right) congurations.
unique.
Choosing
φ
instead makes the congurations
Section 4.3
53
Implementation issues
In the above case, this would only allow the user to specify 12 of the 22 states,
leaving the rest for Dymola to gure out, which works in some cases but not
always. To overcome this issue, a set of joint models were developed with advanced users in mind, where each joint state candidate can be set independently
of the others. Additionally, a body model without potential states was also implemented which enforces the possible states to have relative joint motion. Also
the quaternion representation was disabled since it require additional code and
events for the dynamic state selection.
The third extension considers the implementation of various nonlinearities.
MSL.Translational, MSL.Rotational and MSL.MultiBody have been complemented with components such as springs and dampers with tabular characteristics and hysteresis.
Tyre-road interaction
Tyres are critical components in determining the response of a road vehicle to
driver inputs. Tyres are also infamously difcult to model and calibrate with
experimental data, mainly because of the complexity of tyre behaviour as discussed in Chapter 2.
To reduce modelling effort and still achieve comparable results the tyre
model structure was designed so that it complies with the idea of reusable models throughout the delity range specied in Section 4.2. This was achieved by
separating the wheel model tasks and partitioning them into four sub-models;
visualization, hub dynamics, contact calculation and tyre forces.
The rst is purely a programming issue while the second is straightforward implementation using available mechanical components. The tyre force
model is a huge research area in itself and within this work, available models
from [16, 19] and Paper R have been used where the focus has been on single contact point models aimed at handling studies. The contact calculation is
closely related to the road denition and these issues are further discussed in
the sequel.
Any application including wheels requires some kind of road or ground
interaction. Here, the road is dened as an arbitrary surface
¡G
G G G
ry
rx
,
Together with the surface normal
¢
0
7→
¡0
G 0 G 0 G
ry
rz
rx
,
,
¢
(4.1)
G
ez , this denes the relation between a point
resolved in the world frame, 0, and the ground frame G.
This is a relation
between vectors of length 3 and 2 respectively and to compensate for this, the
offset along the road normal is added giving the nonlinear relation between an
54
Chapter 4
arbitrary point r̄
The VehicleDynamics Library
∗ and the corresponding point on the road surface,
0
r̄
∗
= 0 r̄G (G r̄G ) + z0 eˆG
(G r̄G ).
z
(4.2)
This equation is solved by Dymola and it is worth noticing that there are road
and tyre combinations that could lead to multiple solutions. These problems
have not yet occurred in practice though. The ground also contains information
0
G
about surface x-direction ex and surface friction µ.
To be able to conveniently couple reference trajectories to a ground for a
complete specication of a manoeuvre, a path P is dened relative the surface
as
¡P P ¢
rx
7→
¡G
P G P
ry
rx ,
¢
(4.3)
that in combination with the surface information gives the path information
resolved in the world frame,
¡P P ¢
rx
7→
¡0
P 0 P 0 P
rx , ry , rz
¢
(4.4)
Additionally, there is information about the reference speed along the path.
This information can be used either by driver models or directly as a reference
trajectory for a VMC as described in Chapter 6.
The tyre-road interaction is implemented using dynamic name look-up through
inner-outer constructs. This means that a model, in this case the ground
outer within the tyre model so that the ground information
can be accessed. For each model with an outer prex there should be an
inner model with the same interface higher up in the model hierarchy. Since
the
can be dened as
functions are used for the information access, there can be an arbitrary and beforehand unspecied amount of calls to the ground model and as a result, there
can be an arbitrary number of wheels on the vehicle combination(s) simulated.
It also allows tyre models with detailed, discretised models of the tyre-road
contact patch to be used. Also, only the
inner model has to be exchanged to
change ground denition.
The actual realization of the functions then decides what the road will look
like.
This can be done in many different ways.
The chosen approach is to
dene a table based function and to complement it with an interface for more
user friendly inputs such as curvature, banking, inclination and road camber,
etc.
This is implemented as the
RoadBuilder
package that contains utilities
to create tables for the 3D-road model through numerical integration, either
specialized for e.g. a double lane change or more generic. Figure 4.8 illustrates
a typical input for the latter: Curvature is the inverse of the radius of the center
line (1). Road slope at the center line is the altitude increment along the road
Section 4.3
55
Implementation issues
Figure 4.8: Example of possible
RoadBuilder user inputs.
Note that the inputs in the
example are exaggerated to make them clearer.
heading direction (2). Banking is the altitude increment perpendicular to the
road center line (3). Road camber is the downwards-outwards increment from
the center line which is present on roads to allow efcient water drainage (4).
Left (5a) and right (5b) road edges are dened so that positive values are offset
to the left of the center line. Note that the right edge always should have a
greater value than the left edge. The start position (6) and heading angle (7)
are the position and angle of the center line at zero distance, respectively. The
track position (11) is dened as an offset from the road's center line that a driver
model may follow by varying the steering input. If the offset is 0, the driver
will follow the road center line and accordingly there is a velocity reference for
the longitudinal control. The user can also specify the maximum allowed error
due to linear approximation of a curve segment (8) and a maximum allowed
distance between two consecutive grid points, occurring only for long straights
(9). The resolution (10) denes the minimum distance between two grid points
that
RoadBuilder considers as separate when merging the inputs.
there are cones that can be positioned along the road (12).
Additionally
56
Chapter 4
4.4
The VehicleDynamics Library
Remaining issues and recommended extensions
of the Modelica language and related tools
A great advantage of Modelica is that it is not a static denition but keeps
evolving with the user's needs. In the following section, some major issues are
emphasized that have to be resolved from the perspective of the development
of the VehicleDynamics library.
4
One current limitation is the lack of so called member functions . It should
be possible to assign functions to a model that allow other models to exchange
information with it. The current approach only allows other models to get information. Consider the tyre-road interaction described above. Currently only
rigid ground models are allowed since the tyre cannot affect the ground, only
vice versa. To be able to handle deformable grounds with the current approach,
member functions seem to be a necessity. Another nice aspect is that it would
give ability to e.g. animate tyre traces.
The ability to dene constraints is a key in most mechanics modelling activities. Constraints occur, for example, in joints and gears as x2
= f (x1 ) where
xi in the latter case could be the rotation angle of the gear box anges. For many
models, these can be described analytically, which makes it possible for tools
like Dymola to solve them either as x2
= f (x1 ) or x1 =
f
−1 (x
2
) depending on
the problem. To make tasks like index reduction easer for the tools, Modelica
allows the time derivatives of the functions to be specied independently. This
means that x2 can be specied separately as d f dt (x1 , x1 ) which is extremely
valuable when f is dened as a table. For more advanced applications, this
is however not always enough and also the inverse of f needs to be explicitly
dened. Take for example a tabular representation of a mechanism that has
a nonlinear coupling T between a translation s and a rotation
φ.
To handle
the various applications, there currently have to be two models, one dening
φ = T (s)
and one for s
= T −1 (φ).
It is notable that there has to be a nomen-
clature to dene what inverse is given which might be tricky for functions with
multiple inputs and/or outputs.
Related to this is also the ability to initialize when there are multiple solutions. Currently, this works ne as long as states can be selected properly as
described in Section 4.3. An example of when there are not enough states to
be dened is a four link mechanism as illustrated in Figure 4.9. With the state
dened by the indicated angle, there are still two possible congurations and to
ensure that the correct one is selected, there has to be a means of excluding unwanted solutions. This possibility would also be useful in other domains such
4
See e.g. documentation for C++ or another object-oriented programming language.
Section 4.4
Recommended extensions of Modelica
57
Figure 4.9: The four bar linkage has two possible congurations for a given angle.
as thermodynamics and might eventually also be necessary for the applied road
description.
Any complex model involving some kind of control sooner or later runs into
the need to structure information exchange and it is important that denitions
used in the actual products can be reused.
More specically, the following
desires were identied:
1. It should be possible to predene a set of available signal names on the
bus.
2. The set should be replaceable.
3. It should be possible to use multiple sets at the same time.
4. It should be possible to have at sets and sets that have a hierarchy that
differs from the model hierarchy.
5. Efcient diagnosis for fault detection of connections should be available.
6. The formulation should allow GUI support from the tool.
7. It should be possible to model the imperfects of the actual bus such as
band width limitations and delays.
8. No extra connections should have to be drawn to have components exchange signals with the bus.
Currently in Modelica and Dymola, there is limited support to model this. The
expandable connector
allow signals to be added through
connect
statements and is suitable when gathering signals into cables. However, once
a sensor or actuator is connected, the signal name cannot be changed which
means that in order to connect multiple instances of a component, the bus must
mimic the hierarchical structure of the model. Furthermore, if a component that
needs to communicate with the bus is to be introduced deep down in a hierarchy,
such as a sensor in a suspension, one rst has to make sure that each hierarchical level contains a dedicated connector. With this representation, there is also
58
Chapter 4
The VehicleDynamics Library
no natural transmitter-receiver relationship which requires ad-hoc solutions for
diagnosis and when modelling limitations. Another approach used in this work
is available as the
SignalBuses package5 which is based on the inner-outer
semantics. This denition makes it possible to handle many of the desires above
but since the name lookup of
inner-outer has to be hard-coded in the model
denition, it is not possible to have multiple sets of buses without multiple bus
model denitions.
An important remark concerns model and library design, especially for
the Modelica Standard Library (MSL). Probably for historical and the following backwards compatibility reasons,
MSL is a mix of various nomenclatures
VDL work,
and approaches to modelling and structuring. As a result, for the
nomenclatures and structuring guidelines had to be dened from scratch and a
large set of
MSL
had to be re-implemented to be useful for
VDL.
For a long
time already, Dymola has also been the leading tool on the market and there is
a lot of code within
MSL that does not follow the Modelica specication.
To
really open up for new tools, this has to be corrected.
The last comment is especially relevant for the work presented in Chapter 6
and concerns the ability to automatically generate a mathematical description
suitable for the allocation. Simulation tools like Dymola derive Jacobians to
improve simulation performance, analytically when possible and numerically
otherwise. If these were accessible when dening the models, a lot of work in
setting up the vehicle motion controller could be done automatically which in
turn would be a great means of evaluating different vehicle congurations. This
is also coupled to the denition of derivatives as explained above. Currently
only time derivatives can be specied which makes it inefcient and tedious
to access partial derivatives. If these could be specied and accessed directly,
advanced modelling would be much more convenient.
4.5
Example vehicles
To illustrate the usage in some more detail, some examples are given. Consider
rst Figure 4.10 that shows the layout of a typical experiment. It consists of a
atmosphere (1), a world reference (2) and a road model (3) that dene the environment. Additionally there is the driver (4) and the vehicle (5) that interacts
through a set of connections. The dashboard connector (6) supplies the driver
with information about speed, engine speed (when applicable) and can also be
used to have the driver change modes of e.g. adjustable dampers. The steering
wheel (8), the pedals (9) and the gear shift (10) are used by the driver to control
5
Available from http://www.modelon.se.
Section 4.5
Example vehicles
59
Figure 4.10: Standard sedan (5) in experiment with instruction following driver (4)
on a at surface (3) with constant atmospheric conditions (1). The world object (2)
species reference frame and gravity. Driver-vehicle interaction is handled through the
connections (6-10).
the vehicle and to get feedback. Connection (8) represents the motion of the
vehicle and is used for orientation and sensing of vehicle states.
The layout of the vehicle is shown in Figure 4.11 consisting of dashboard
(1), engine (2), transmission (3), brakes (4), chassis (5) and driveline (6). The
same layout is also represented in Figure 4.2.
Below, three examples are given, showing the implementation of the ACM
concept, interaction with a controller in Simulink and a six wheel all terrain
vehicle, respectively.
ACM concept
Figure 4.12 shows the implementation of the ACM concept described in [3] and
used for the simulations illustrated in Figures 5.6.2 and 5.8. Unlike the standard
car in Figure 4.11, this vehicle has a completely different layout with four corner modules (2) that replaces both suspensions, brakes and powertrain. Instead,
60
Chapter 4
The VehicleDynamics Library
Figure 4.11: Layout of a standard vehicle with dashboard (1), engine (2), transmission
(3), brakes (4), chassis (5) and driveline (6).
each corner module contain its own motor, brake, suspension and control.
Interaction with Simulink
Models dened in Modelica can also be incorporated in Simulink [71] for convenient controller design. This example illustrates the incorporation of a chassis
model in an existing Simulink structure [72] done in Paper S which revealed
some signicant advantages with the adopted approach.
First, the Simulink
structure was implemented with wheel spin velocities as inputs to the chassis, rather than the wheel torque which is more common. Since the denition
in
VDL is equation based, adopting the chassis to this input was seamless, see
Figure 4.13. Second, the controller required information about current and limit
tyre forces which easily could be extracted thanks to the open code. Third, deriving a set of models with different levels of detail in
VDL only took fractions
of time compared to the hand coded S-function in the original structure.
All terrain vehicle
The all terrain vehicle in Figure 4.14 is another example of how a non-conventional
vehicle conguration can be set up with
VDL. In this case, the vehicle is parti-
tioned into chassis (3) and powertrain (2) and the VMC (1) has direct inputs for
the reference trajectory (4-6), i.e. the conguration is set-up without driver and
driver interface. The powertrain consists of six in-wheel motors. The chassis
Section 4.5
Example vehicles
61
Figure 4.12: Layout of the ACM model implementation with wheel (1), corner module
(2), driver interface (3), vehicle motion controller (4), signal bus (5), car body (6) and
motion denition (7). The corner module in turn contains suspensioning actuators (8),
friction brake (9), suspension linkage (10), corner module controller (11) and wheel
motor (12). The summary record (13) contain variables of common interest.
62
Figure 4.13: The Adapted
(right).
Chapter 4
VDL
The VehicleDynamics Library
model in Dymola (left) and exported to Simulink
Section 4.5
Example vehicles
63
is equipped with three suspensions (8) where the front and rear can be steered
and each wheel (9) also has a sensor for the load which is transmitted to the bus
(12).
Figure 4.15 shows two congurations negotiating a slope entry. The left
vehicle has xed suspension pretension while the right uses the pretension to
balance the load distribution. As a result, the latter vehicle manage to maintain
a better contact between tyres and ground as seen in the gure.
64
Chapter 4
The VehicleDynamics Library
Figure 4.14: Layout of the all terrain vehicle model with VMC (1), driveline (2), chassis (3), reference trajectory input (4-6) and signal bus (7). The chassis in turn consists
of three suspensions (8), six wheels (9), a body (10), denition of motion (11), signal
transmitters (12) and summary variables (13).
Section 4.5
Example vehicles
65
Figure 4.15: Two all terrain vehicles climbing a hill. Without (left) and with (right)
wheel load distribution. The yellow (lighter) tyre colour indicates loss of tyre-ground
contact.
66
Chapter 4
The VehicleDynamics Library
Chapter 5
Methods for
vehicle motion analysis
This chapter presents available methods for analyzing vehicle motion and related characteristics that are relevant when comparing the behaviour of different vehicle congurations. It also proposes a means for adaptation to the
increased degrees of freedom.
5.1
Background
As is the case with most engineering analysis methods, ground vehicle motion
analysis was initiated by real problems, in this case starting during the 30's
with investigations of the shimmy phenomenon [73]. This came as a natural
effect of increasing speeds and as a result, the fundamental theory, even of
today's analysis methods, was founded in the rst half of the last century. More
comprehensive information can be found in [16, 74, 75, 76, 77], the latter also
discussing the relation with rail vehicle research.
The activity caused by wheel shimmy was induced both by vehicle steering
instability and oscillations in landing gear.
Based on experimental work on
how lateral forces depend on side slip [78] a break through was reached with
the theoretical work in [79]. The model today known as the bicycle or onetrack model was formulated and analyses presented on, among other topics,
the sensitivity to forward speed for some congurations.
Since then, this model has been rened in addition to the means and methods for its analysis. Below different suggested methods for a analyzing vehicle
67
68
Chapter 5
Methods for vehicle motion analysis
behaviour are discussed. Many of the presented methods share a common focus on yaw stability and response to steering input. In later years, as handling
has come to involve brake force distribution more and more, some care has
been given to the combined lateral and longitudinal characteristics. As the use
of control increases, and thereby the number of possible inputs, the ability to
present it on a paper sheet diminishes. To deal with this, traditional analysis
based on pure steering input has to be extended and other approaches have to
be chosen as will be seen in the latter part of the chapter.
5.2
Eects of steering on lateral acceleration
One of the most intuitive measures of vehicle steering characteristics is the
understeer-gradient, Kus , dened as how much more steering angle,
δ,
has to
be used to reach a change in lateral acceleration, ay , compared to the angle
required to negotiate the turn at low speed,
Kus
δA
=
δA .
∂δ ∂δA
−
∂ay ∂ay
(5.1)
is called the Ackermann angle and is derived from a pure geometrical ap-
proach. Thus a neutral steered vehicle needs no change of steering angle as
the speed through the curve increases while an under-steered vehicle (Ku s
will require an increasing angle.
The over-steered vehicle (Ku s
< 0)
> 0)
on the
other hand will require a decreasing angle that for the linear case eventually
will reach zero resulting in instability.
These effects can be illustrated in a handling diagram, Figure 5.1. It is
based on axle characteristics normalized with vertical load and small angle approximations giving
f y,12
f z,12
=
f y,34
f z,34
=
ay
(5.2)
g
α34 − α12 = δ12 − δ34 − Lρ = δ12 − δ34 − δA
(5.3)
This allows the vehicle characteristics from the speed and curvature dependent
ay which makes it easy to spot changes in Kus . Another representation of the
dependency between ay and
δ
is referred to as a cornering characteristics di-
agram, illustrated in Figure 5.2. Unlike the handling diagram, the cornering
characteristics diagram is derived at a constant radius turn which is negotiated
at different speeds. This means that there is a constant
δA
as is indicated in the
gure.
The presented graphs and measurements all relate to steady state conditions
and thus give limited information about the dynamics of a change. To handle
Section 5.2
Eects on lateral acceleration
69
Figure 5.1: Normalized axle characteristics (left) and the resulting handling diagram
(right). Note that the characteristics are adopted to illustrate both under and oversteer
and do not corresponds to real data.
Figure 5.2: Cornering characteristics diagrams for the axle characteristics used in Figure 5.1.
70
Chapter 5
Methods for vehicle motion analysis
Figure 5.3: A moment method diagram showing yaw moment for the straight line case
(ρ
= 0) using axle characteristics from Figure 5.1.
Iso-lines for steering and side slip
are dotted and dashed, respectively.
this, other methods have been suggested, based on the resulting yaw moment
for different driving situations as will be discussed in the next section.
5.3
Eects of vehicle state on yaw moment
To better indicate the dynamics of a vehicle, the aligning moment for different
driving conditions can be studied as suggested in [80]. The test is conducted
by forcing the vehicle along a track with xed yaw velocity for a set of steering angles and body side slip angles, but leaving roll, pitch and bounce DOF
unconstrained. The aligning moment and either lateral acceleration or lateral
force are recorded for plotting, resulting in the grid shape in Figure 5.3, which
1
is referred to as a moment method diagram . Since, the resulting yaw moment
would generate a yaw acceleration if it would be relaxed it can be used to indicate yaw stability and to facilitate the comparison with the handling diagram,
similar data has been used. As seen in the gure, the aligning moment changes
direction for high lateral force which corresponds to the change from under to
over steer in the handling and cornering characteristics diagrams.
To grasp the effect of accelerating or braking while cornering, several of
these sheets have to be produced. A variant of this method referred to as the
1
See e.g. [76] for a comprehensive explanation
Section 5.4
71
Phase-plane
Figure 5.4: The
β-method
illustrates how increased side slip may lead to decreased
aligning moment which gives an indication of the stability range. Axle characteristics
are from Figure 5.1.
β-method,
was presented in [81] and is popular in vehicle dynamics control
applications. It got the name from the choice of having the side slip angle
β
on the x-axis and plotting the aligning moment for different steering angles,
Figure 5.4. The coupling to moment method diagrams can be seen through
the substitution of the iso-β lines to iso-y which makes the coupling between
side slip and aligning moment more obvious; if the aligning moment does not
increase as the side slip increases, there is no self alignment of the vehicle.
By studying the characteristics of the
β
plot for several different longitudinal
acceleration levels, a strategy to control wheel torque distribution to directly
affect the yaw moment was designed.
5.4
Phase-plane
As seen from the methods above, using forces and torques gives a hint of the
dynamics. For a system with two states, a phase-plane is a practical way to give
a more complete illustration of the dynamics, but the additional information
comes at the price of a specied steering angle as seen in Figure 5.5.
Phase planes can also be used to show traces of a manoeuvre as exemplied
β vs β is seen for the two vehicles performing
in Figure 5.6.1, where traces of
the evasive lane change in Figure 5.7. For a front axle steered vehicle, this
72
Chapter 5
Methods for vehicle motion analysis
Figure 5.5: Phase plane representation of the vehicle characteristics used in Figure 5.1.
The square indicates the stable equilibrium at normal steady state cornering and the
circle corresponds to the drifting equilibrium.
information is valuable for grasping the vehicle's lateral behaviour and it is
seen that
β
can be used to trace instability which corresponds well with the
insight from Section 5.3.
For vehicles with additional rear axle steering, high slip angles or slip angle
changes are not necessarily a sign of instability. Consider the same evasive
lane change performed with two other strategies as illustrated in Figure 5.8. A
of high magnitudes also for stable
pure lateral motion without yaw gives (β, β)
manoeuvres. At the same time, the
β = 0 strategy would give little information
until the limit is reached.
Instead, by studying the combination of sideslip and yaw, this additional
degree of freedom can be accounted for as shown in Figure 5.6.3-4. Here, a
motion along the vertical axis corresponds to a purely lateral translation ψ
=0
while the horizontal axis means that the vehicle has no change in slip angle.
, means that the vehicle starts spinning which also can be
= −β
Motion along ψ
can be calculated from lateral
seen in Figure 5.7.3. A further advantage is that β
acceleration and yaw rate sensors.
Again, the additional degree of freedom
comes at a cost, in this case it is hard to spot drift in the slide slip angle. A
phase plane using
β34
that also allows not only β but also front β12 and rear
β and ψ
axle slip to be traced , is illustrated in Figure 5.6.5-6, where iso lines for
front and rear axle slip have been indicated. These are based on linearisation of
and v sin(β34 ) = v sin(β) − bψ
giving the inclinations
v sin(β12 ) = v sin(β) + f ψ
Section 5.4
73
Phase-plane
Figure 5.6: Phase plane diagrams with traces of
, (ψ,
β)
(β, β)
and
(β, ψ)
for the ma-
noeuvres in Figures 5.7 (left column) and 5.8 (right column). The dotted (blue) lines
correspond to the light (blue) vehicle and the lled (red) lines correspond to the dark
(red) vehicle. The lines forming the grid in diagrams 5 and 6 indicate constant
β34 .
β12 and
74
Chapter 5
Methods for vehicle motion analysis
Figure 5.7: Simulation result of an evasive lane change manoeuvre with two vehicles,
one with indirect yaw control through wheel load distribution (red/darker) and one
without (blue/lighter).
Section 5.5
Open loop stability tests
75
Figure 5.8: Simulation result of an evasive lane change manoeuvre with two vehicles, one with zero side slip control (blue/lighter) and one with zero yaw control
(red/darker).
−v/ f
5.5
and v/b, respectively.
Open loop stability tests
A seen in the previous section, a phase plane trace is a good way to visualize
the dynamics of the driver-vehicle behaviour. Another way to detect instability
is to excite the vehicle by a predened input, adapted especially to the steering.
Typical examples are the J-turn which basically is the response to step or quick
ramp input.
An ongoing work [82] is to nd an input that is severe enough to generate
an instable response of the vehicle if not properly tuned. An established way
to trigger both yaw instability and roll-overs of a vehicle is to excite the roll
dynamics and get it in counter-phase to the steering input. This is mimicked
by applying a sine period with either a dwell time or increasing amplitude and
the response is then evaluated with special respect to phase delay since this is
crucial for driver induced oscillations.
76
Chapter 5
Methods for vehicle motion analysis
Figure 5.9: Resulting limits described as Mz (ax , ay ), a vehicle without any possibilities
to distribute the forces (left) and an idealized vehicle that can perform arbitrary force
distribution (right).
5.6
Combined lateral-longitudinal performance
When combining the lateral and longitudinal behaviours, the representation of
the input characteristics gets trickier.
As seen in Section 5.3 the
β-method
considers the longitudinal acceleration through applying several plots and this
could also be done for any moment method. As the number of inputs increases,
an alternative is to concentrate on the resulting limits of a conguration rather
than each individual contribution, for example the span of achievable ax , ay and
Mz as illustrated in Figure 5.9. A projection of these limits onto the ax -ay -plane
gives the so called g-g diagram while a cut through the ay -Mz plane gives the
outer bounds of the moment diagram from Section 5.3.
For a vehicle without any control and only driver inputs for steering, accelerator and brake, the shape is an extrusion through several moment method
diagram for different longitudinal accelerations. Without possibilities to distribute the forces, the ability to control the yaw moment will be limited and the
resulting limits will be edgy, Figure 5.9, left.
Looking closer to the plot also reveals that for high lateral accelerations,
the yaw moment can not be zero.
This means, for example, that to reach
steady state braking while cornering, less acceleration can be achieved than
what would be suggested just by looking at the maximum possible horizontal
force.
In this sense a perfectly balanced vehicle would have maximum ay giving
no Mz but will also be more sensible to disturbances and give a less predicable
behaviour when the limit is reached. Another limitation is given when ax ay
6= 0
Section 5.6
77
Combined lateral-longitudinal performance
∗
as shown in [81, 83]; there is a contribution to the yaw moment, Mz , from the
load transfer that causes imbalance through the lateral forces,
∗
Mz
= Fy
l12
l
µ
1+
h
l12
¶
ax
l34 + Fy
l34
l
µ
1−
h
l34
¶
ax
l12
= mhax ay .
(5.4)
For braking in a curve, it is seen that the load transfer causes a destabilizing
moment which may lead to severe oversteering.
As the ability to control the distribution of the forces between the wheels
increases, and thereby the yaw moment, the shape will be closer to a sphere
as seen to the right. This implies the possibility to achieve higher acceleration
levels if the control can be made to compensate for disturbance sensibility and
load transfer effects.
The proposed presentation has the advantage of being compact and yet giving a good overview of what performance can be expected. Still, to understand
how the performance can be achieved, input-output characteristics are needed.
The above descriptions all assume that the main input to affect the handling
is through the steering wheel.
For integrated control of several actuators, a
more biased classication is needed. But again, the information that can be
presented in graphical form is heavily constrained by the two dimensions of a
piece of paper.
Restricting the presentation to one wheel limits the possible degrees of freedom considerably. Here this is illustrated for the ability to generate yaw moment through steering vs. driving/braking.
Consider Figure 5.10, where the
yaw moment contribution from one wheel is described as a function of lateral
and longitudinal slip. Studying the slopes for each wheel already gives some
vital information about in which situations driving or steering is the better and
also, at what wheel this is most important. For low lateral accelerations, steering is much more efcient than drive and brake torque distribution but closer
to the lateral saturation limit, it gains in importance. It can also be seen that
the levers given by the wheel position relative to the centre of gravity are vital;
braking the inner rear wheel is the most efcient way to increase yaw moment
by braking. To instead decrease the yaw moment, the outre front wheel should
be used. This effect is also used in present yaw stability systems.
When adding more degrees of freedom to each wheel such as vertical load
or camber, the visualization gets tricky but the mathematical denition is straightforward:
Ku
=
∂Mz
∂u
(5.5)
where u is the set of available actuator degrees of freedom. This allows for
a scan of the potential of a conguration for a given state which can also be
78
Chapter 5
Methods for vehicle motion analysis
Figure 5.10: Contribution from each tyre to the total yaw moment.
Section 5.6
Combined lateral-longitudinal performance
79
Figure 5.11: Adhesion potential for the conventional front-steered vehicle, solid (red)
line, and the ACM vehicle dotted (blue) line, from Paper F.
used for vehicle motion control design as will be explained further in the next
chapter.
The efciency of a conguration can be evaluated by the friction potential
usage for each wheel
µ
ηi =
f
f max
¶
(5.6)
i
which reveals the safety margin. In Figure 5.11, this is illustrated for an evasive lane change with a conventional vehicle and an ACM conguration. Even
though the manoeuvre is the same, the ACM manages to keep η lower and more
evenly distributed between the wheels, suggesting an additional margin which
in turn allows for higher performance. The higher
η at the rear wheels is also
notable in this case which suggests that the robustness against inaccuracies is
low since overestimates of the adhesion potential would cause the rear wheels
to loose grip rst.
80
Chapter 5
Methods for vehicle motion analysis
Chapter 6
Methods for generic
vehicle motion control design
This chapter presents two methods for designing generic vehicle motion controllers. The developed methods can also be used to evaluate different congurations since a generic vehicle motion controller has the capacity to handle
multiple vehicle congurations.
6.1
Outline
As seen from the earlier chapters, vehicle motion control is a challenging eld
where the many coupled and non-linear tyre effects result in both pitfalls and
potential for improvement. Work focusing on a given combination of actuators
(see e.g. [43, 48, 84]) require redesign and re-calibration for each and every
conguration evaluated. As a result of this time-consuming process, the assessment is limited to a few combinations, often as an extension of existing
systems like ESP, with a limited number of additional actuators.
This chapter discusses methods where the tyre characteristics are synthesized into a generic description of vehicle behaviour based on the presentations
in earlier chapters. The aim is to provide generic functions g for the design
methodology outlined in the introduction, that are generic in the sense that they
are invariant for different actuator congurations and even for diverse types
of vehicles. Another aim is to dene a qualied vehicle motion control structure that can be implemented in real vehicles. The description is modularized
according to Figure 6.1.
81
82
Chapter 6
Methods for generic vehicle motion control design
Figure 6.1: The generic vehicle motion control principle where each number refer to
section and chapter numbers.
The leftmost box represents the generation and denition of a reference
trajectory and is further discussed in Section 6.2. Omitting the feed-back part,
the rest of the VMC can be considered as an inverse of the vehicle model, i.e.
giving u as a function of yre f with some complicating properties:
The non-minimum phase that is present in more detailed vehicle models is
often related to the kinematic and elasto-kinematic properties of the suspensions, such as the caster trail. One way to treat this is by means of approximate
inverses as suggested in e.g. [85]. The second concern is the non input-output
uniqueness. On one hand, this can mean that there are too few actuators to be
able to control all of the available degrees of freedom, which means that the
demands have to be relaxed. On the other hand, it can also mean that there
are more actuators than degrees of freedom, leading to manifold solutions if no
requests other than the actual motion are added.
Common for the methods suggested here is to approach these issues by
the separation of the inverse problem into two parts: The rst is to dene the
desired resulting forces and moments on the vehicle body from the desired and
actual motion. Two different approaches to this are described in Section 6.3.
The second part deals with the allocation of actuator effort to fulll these forces
and moments, either through direct allocation of wheel inputs, Section 6.4, or
through rst dening the resulting forces at each wheel and then using another
inverse model to dene the wheel inputs, Section 6.5.
Depending on vehicle conguration, the wheel input then has to be transformed into proper actuator reference signals. An example on how this can be
done is given in Section 6.6.
Section 6.2
83
Denition of vehicle motion
Figure 6.2: Denition of vehicle motion.
6.2
Denition of vehicle motion
For both feed-forward and -backward control, it is important to have a suitable
trajectory denition. This section presents some different denitions with their
drawbacks and advantages, illustrated in Figure 6.2.
Consider rst the denition mostly used for motion state equations, where
and their derivatives, reeach degree of freedom is represented by vx , vy , ψ
spectively. This representation has the advantage that it is well dened for all
motions but gives no information about the trajectory when standing still. This
means that it is less suitable to use as a reference trajectory at low speeds.
An alternative representation is to use the speed v along the trajectory that
is dened by curvature,
ρ, and vehicle side slip β.
With this representation,
β is
decoupled from the trajectory making it easier to relax this degree of freedom.
A drawback is that the mapping of the forces acting on the vehicle requires a
nonlinear transformation. This can be dealt with by resolving the whole motion
control along the trajectory instead of in the vehicle frame. Again, this leads to
difculties at low speeds.
A third alternative has the same quantities as the rst case but is resolved in
the trajectory frame instead of in the vehicle frame. A further possibility is to
have v,
represent the vehicle motion which resembles the second case.
β and ψ
In general, the rst case is suitable for the feedback trajectory and the second
is appropriate for dening reference trajectories while the third variant works
as a compromise between the two rst. To reach a consistent representation
throughout the speed range, it is recommended to only apply feed-forward of
the reference trajectory for lower speeds.
The actual generation of reference trajectory is a huge task in itself since
it includes the interpretation and interaction with human drivers which is out
of the scope of this thesis. To avoid having to incorporate human behaviour,
84
Chapter 6
Methods for generic vehicle motion control design
Figure 6.3: Possible trajectories for an evasive lane change.
a set of various trajectory denitions is set up to span possible real scenarios,
illustrated for an evasive lane change in Figure 6.3. These are zero side slip
β = 0,
pure lateral motion ψ
=0
and responses from conventional vehicles,
either from measurements or models. The advantage with these representations
are that they span both positive and negative slip angles for any given curvature.
For the incorporation of a human driver interpreter, some way to give feedback is necessary. The VMC can assist in this by giving information about how
close the vehicle is to its limits, preferably in terms of a limit trajectory but at
least as resulting global force as described in Figure 5.9.
6.3
Vehicle motion to global forces
As discussed above, transforming the desired motion into required global forces
can be considered as a part of a total vehicle inverse but the step itself can be
carried out both with and without an inverse model.
To illustrate this, two
different cases will be discussed in more detail.
Consider rst the implementation as a PI-controller. The allocation part of
the VMC will have to take the limitations of the vehicle into account which
Section 6.3
Vehicle motion to global forces
85
Figure 6.4: PI-control strategy with anti-windup to generate desired global forces Fre f
from a reference trajectory yre f using the actual trajectory y and generated global forces
F.
Figure 6.5: IDOB strategy with Q-loop and an inverse model of the body dynamics
completed anti-windup to generate desired global forces Fre f .
would lead to integrator windup when the control signals are limited or saturated. The adopted strategy to avoid this is referred to as tracking and is further
presented in e.g. [86]. The idea is to make sure that the integral is kept at a
proper value when the allocator reach the limits, so that the controller is ready
to resume action as the control error changes. The PI-controller structure is
depicted in Figure 6.4 and is used in Papers B and N.
Figure 6.5 shows the second case based on the Inverse Disturbance Observer (IDOB) design [85, 87]. It consists of a feed-forward part, which is an
approximate inverse of the vehicle, and a so-called Q-loop that provides high
gain feedback. This control structure has been shown to be efcient for generic
nonlinear MIMO tracking tasks [87] and is extended here with anti-windup.
In the feed-forward part, the reference motion, yre f , determines the motion
for the inverse model together with the feedback. For the given vehicle mass
and inertia, the vector F of total torques and forces to be generated by the
0
chassis, uniquely depends on y and is computed by an inverted lean model.
86
Chapter 6
Methods for generic vehicle motion control design
Measured or observed vehicle states, x, can be used to directly set states of this
inverse model instead of internal integration. This corresponds to the technique
of dynamic inversion which is established, for instance, in the nonlinear control
of aircraft [88].
A transfer function diagonal matrix Q is used in a subordinate loop yielding
high gain for the feedback control, where Q j are unity gain low pass lters. This
results in y tracking yre f . The bandwidth of the Q j -lters are the main tuning
parameters for performance and stability of the feedback control [87]. This
approach is used, for example, in Papers D and P.
6.4
Direct allocation of wheel inputs
This approach is adopted in Papers D and P.
As explained earlier, there are different approaches to allocating the actuator
activities and this section considers the direct allocation of global forces, F, to
wheel inputs, u. In general, vehicle motion control may cover intentional effects on roll, pitch and bounce dynamics. It is also possible to add actuators
for improved downforce by means of aerodynamic control surfaces and camber control for instance. For the sake of clarity, the example in this section is
limited to cover individual steering, traction/braking and load control of a four
wheeled vehicle:
u
where
δi
= [τ1 , δ1 , fz1 , τ2 , δ2 , fz2 , ..., τ4 , δ4 , fz4 ]T ,
denotes steering angles,
τi
(6.1)
drive/brake torques and f zi tyre loads. For
certain congurations such as conventional front steering, some driver inputs d
may directly act on the vehicle.
Depending on the conguration, the desired motion can be fullled to a
different extent.
Further criteria for allocation of horizontal tyre forces via
actuator commands may be incorporated, resulting in an optimization problem
of multiple criteria. In the continuing discussion, the optimization algorithm is
assumed to be executed at a xed sample interval
∆t .
The main task is to achieve the desired total torques and forces acting on
the chassis, F, yielding an allocation problem
F
= Bf(u, x, p)
(6.2)
where the matrix B depends on the wheel locations, f(u, x, p) is the vector of
horizontal forces at each wheel, depending on the actuator commands u the
vehicle states x and the parameter vector p. For simplicity, p and x are omitted
in the notation.
Section 6.4
87
Direct allocation of wheel inputs
Additional costs
Equation 6.2 only requires the motion to be fullled, which for many congurations yields an under-determined problem that therefore has manifold solutions,
as mentioned earlier. Additional criteria are added to benet from these degrees
of freedom as explained below, giving, in most cases, an over-determined prob1
lem which can be solved for u in the least-squares sense .
To avoid manifold solutions, an additional vector valued criterion is em2
ployed to keep the utilization of the tyre grip potential at every wheel low and
preferably equal, giving
0
min |η
u
|2 with η0 (u) = η(u) − ζηmin · 1.
(6.3)
At the wheel i, the degree of utilization of the tyre grip potential is denoted
ηi
ηi
with 0
≤ ηi ≤ 1, η = [η1 , η2 , . . . , ηn ]T
= [1, 1, . . . , 1]T .
denoted ηmin . It
and 1
preceding the actual optimization sample is
The lowest
is used as a
reference value for the actual optimization sample together with a heuristically
adjustable decay parameter
towards lower
η-values.
ζ with 0 ≤ ζ < 1. The latter directs the optimization
The main motivation for the auxiliary criterion is pri-
marily to keep the safety margin high. Additional benets of this are lower tyre
wear and reduced fuel consumption. An incorporation of energy management
aspects for HEVs is done accordingly in [72].
Linear allocation
The resulting optimization problem is nonlinear with linear constraints but for
the further discussion, it is assumed that abrupt changes of actuator positions
are unwanted. Thus, the optimization problem can be linearized while imposing adequate constraints on actuator rates, yielding smooth actuator commands
which evolve along an (at least locally) optimal solution. This approach has
already been successfully applied to the real time control of industrial robots as
described in Paper E.
Thus, u is directly allocated by a linearisation of equations (6.2,6.3) about
∗
the current operating point u , giving
·
f(u)
η0 (u)
¸
·
≈
∗)
f(u
¸
η(u∗ )
∂
+
∂u
·
1
These are also applicable for the force allocation case.
2
This approach is similar to for example [89].
f(u)
η(u)
¸
˜
u
u=u∗
(6.4)
88
Chapter 6
˜
using u
Methods for generic vehicle motion control design
= u − u∗ resulting in an afne formulation3 .
The complete allocation problem covering all criteria and constraints is to
˜
nd u
= u˜ o pt
such that
T
T
˜ − b) W W (Au
˜ − b)
min (Au
(6.5)
˜
u
with a suitable diagonal weighting matrix W and subject to the constraints
˜ −d
Cu
˜ −h
Gu
Here,
·
A
·
and
b
=
=
B
0
0
I
Ft ot
ζηmin · 1n×1
¸
=0
(6.6)
≥ 0.
(6.7)
∂
∂u
¸
·
·
−
f(u)
η(u)
B
0
0
I
¸
,
(6.8)
u=u∗
¸·
f(u
∗)
η(u∗ )
¸
.
(6.9)
Additional criteria like comfort relevant issues could be included in the same
manner by adding rows in A, b, and W.
Equation 6.6 with d
= 0 opens up the possibility of convenient adaptation
to the actual vehicle conguration, exemplied here by a front rack steering
mechanism. The representing constraint couples the steering angles of the front
right and rear wheel, described as f rack (δ1 , δ2 )
= 0.
This equation is linearized
about the current operating point, contributing a row
Ci
= [0, ∂ frack /∂δ1 , 0, 0, ∂ frack /∂δ2 , 0, 0, 0, 0, ...]
(6.10)
to C. Other constraints are added correspondingly by supplying their relevant
partial derivatives, giving a new row for each constraint. Additionally, for all
congurations in the planar example, three constraints for tyre loads are present
representing the force and torque balances for roll, pitch and bounce.
By means of Equation 6.7 with G
= [-I, I]T
both actuator position and rate
limitations can be considered with
h
Here,
= [− min(umax − u∗ , ∆t · u max ), max(umin − u∗ , ∆t · u min )]T .
∆t
is the sample interval of the optimizer and
(u, u )min,max
(6.11)
are present
actuator limits and actuator rate limits, respectively.
3
The afne form allows for the approach from [90] to be applied which is explored in detail
in [72].
Section 6.4
89
Direct allocation of wheel inputs
As seen above, the allocation requires that the Jacobians of f(u) and
η(u)
can be calculated. To assure this, a rather simple tyre model based on Magic
Formula characteristics, non-linear load dependency and coupling between lateral and longitudinal force was used in Papers B and N. Tyre and wheel dynamics are excluded and the contact forces are given as analytic functions of
τi , δi
and f zi , making it possible to automatically generate the required partial
derivatives.
Although the linearisation performed here is motivated by the desired smooth
behaviour, it also imposes problems nding good solutions when the search direction is close to zero due to a sign change. To illustrate this, two examples are
given. First consider a wheel on a vehicle in a turn with no traction or braking
applied. According to the tyre ellipse, both braking and driving will reduce the
side force but since
∂ fy /∂ fx
is zero or at least very small in this region, it is
difcult for the optimization to gure out that a change in f x could help lower
fy .
Second, consider driving with the same or similar vertical load on both
wheels on an axle. Now the difference in
∂ fy /∂ fz
will be small, again making
it difcult for the optimizer to use e.g. an adaptive roll bar. Means to overcome
these difculties are further discussed in below.
Nonlinear allocation
As seen from the previous discussion, there are some pitfalls when linearizing
the tyre characteristics and this section suggests approaches to overcome these.
The main reason for the linearisation is to simplify the optimization problem
for the allocation but still, there are means to perform non-linear optimization.
Consider rst the simplest approach, gridding up the relevant space of combinations of u and evaluating each grid-point. Depending on the density of the
grid, this can be quite costly since the number of evaluations, N, are
n
N
= ∏ ki
(6.12)
i=1
where ki is the number of grid points for each ui . The advantage to this is that
the computational costs are foreseeable and that the search is robust against
non-convexities. A further use of this method is for off-line allocation which
could also be used to dene rule based approaches.
A second alternative is to complement the linear optimization with evaluations in those directions where
∂F/∂u
is small. This could also be achieved
by doing numerical linearisations with large enough step-sizes or through also
evaluating the hessian,
∂2 F/∂u2 .
90
Chapter 6
Methods for generic vehicle motion control design
Figure 6.6: The resulting limits for longitudinal, lateral and vertical force can be approximated by a semi-ellipsoid.
Finally, there are also approaches that allow non-linear optimizations to be
applied directly, either with constraints or by supplying articial constraints
through additions to the cost function.
6.5
Allocation of forces
This approach is adopted in Papers B, F and N.
As seen from Section 6.4, direct allocation of the wheel inputs yields a nonlinear optimization problem with linear constraints. By allocating the tyre forces,
f, the task can be transformed to a linear allocation problem
= Bf
(6.13)
g(f)
=0
(6.14)
h(f)
≥0
(6.15)
F
but with nonlinear constraints
that depend on the conguration and the road conditions. Consider rst the
constraints imposed by the physical limits of the tyre-road contact, which can
be approximated with a semi-ellipsoid as illustrated in Figure 6.6.
For any vehicle, some additional constraints that are dened by the conguration will be present. Even for a chassis with many different actuators like
the ACM concept, there will be constraints on the vertical load since it must
sum up to the vehicle weight and handle roll and pitch. Additionally, there are
limits on how much a wheel can be turned as well as available power that will
limit the usage of lateral and longitudinal forces.
Section 6.5
91
Allocation of forces
Figure 6.7: Force allocation constraints: Physical limitations of the tyre-road contact
solely, with steering and camber constraints, with traction limitation and combined,
from left to right. For simplicity, the inuence of vertical load is omitted in the illustration.
For the representation above to be useful, additional conguration depen4
dent constraints must be available . To deal with this, the restrictor concept is
introduced to dene how the tyre friction ellipse is further restricted by actuator
limitations, Figure 6.7. The restrictors are active on either one or several Wheel
Units (WU) as dened below.
Limitations that only affect the force generation for one WU are dened as
intra-restrictors. Examples of these and how they affect the boundary problem
are listed in Table 6.1. A common intra-restrictor is no steering that compresses
the ellipsoid to a shell, indicating a possibility to change the lateral force by
changing either the longitudinal or vertical force. In the longitudinal direction,
a difference can be made between wheel units that are able to drive and brake
min
and those wheel units that can only brake. The rst case generates limits f x,i
fx
<
max
f x,i
while the latter further restricts
min
f x,i
<
fx
<
< 0.
Other limitations occur due to coupling effects between wheel units, called
inter-restrictors as exemplied in Table 6.2. This could be a mechanism that
couples the steering angle of two wheels which is the standard way to steer
a vehicle today. The resulting coupling of these forces are not obvious but a
rst estimate is that the ratio between achieved and maximum lateral force for
both wheel units should be equal. This yields an additional equality constraint
equation. If the steering rack is directly controlled by the driver, Rack Steer,
the effect on the limitations is identical to a No Steer intra-restrictor. A modied version of this approach with linear lateral force constraints is tried out in
Papers B and N. An example of how the restrictor concept is applied a conguration to derive the resulting optimization problem is shown in Figure 6.8.
4
In Paper B this is also connected to a suggested vehicle architecture but here, the concept is
kept brief.
92
Chapter 6
Methods for generic vehicle motion control design
Intra-restrictors
No Steer: Wheel Unit (WUi )
with suspension that does not
allow the wheel to be steered.
Drive and Brake: WUi that
is able to apply both driving
and braking torque, typically
equipped with a wheel motor.
Brake: WUi that is able to apply only braking torque, typically equipped with a disc
brake.
Special case of Drive
and Brake.
Table 6.1: Description of some intra-restrictors and how they affect the force distribution problem. The equations should be considered as estimates of the effects of the
restrictors and are dened as simply as possible.
Section 6.5
93
Allocation of forces
Inter-restrictors
Differential:
Distributes the
equivalent force from input to
WUi and WU j equally.
Actuated Differential:
Dis-
tributes the force from input k
to WUi and WU j with a maximum force difference
∆f.
Rack Steer: Mechanism that
couples the steering angles of
WUi and WU j . The angle can
not be controlled by the VMC.
Actuated Rack Steer: Mechanism that couples the steering
angles of WUi and WU j . The
angle can be controlled by the
VMC.
Table 6.2: Description of some inter-restrictors and how they affect the force distribution problem. The equations should be considered as estimates of the effects of the
restrictors and are dened as simple as possible.
94
Chapter 6
Methods for generic vehicle motion control design
Figure 6.8: Application of the restrictor concept on a conventional vehicle, Paper B.
By keeping the allocation at a force level it is easier to nd the resulting limits for the vehicle as needed by the driver interpreter. Also, the partitioning of
the actuator generation into more steps makes it easier to isolate subtasks. For
example, a vehicle with both individual steering and camber control requires
only a resulting limit for the lateral force. The contribution from steering and
camber can be handled internally at each wheel. It is also easy to transform the
forces to the trajectory frame, making it easier to separate
β from ρ as described
in Section 6.3.
On the other hand, this technique makes the representation of some restrictors complex. To exemplify, a non-steerable wheel results in a shell like,
non-convex shape which is hard for any optimizer to handle. Now consider
the additional complexity induced by a rack steer mechanism where the shell
shapes on each side are interdependent. Even for very simple mathematical
representations, this is difcult to solve. In Paper F, this method is applied to
the ACM concept with the constraints linearized to planes. This approximation
is however only possible as long as the resulting limitations are convex.
Section 6.5
95
Allocation of forces
Inverse of wheel forces
A common way to express the tyre force generation is as a function of load,
camber, longitudinal slip and slip angle
( fx , fy ) = f ( fz , γ, κ, α) ,
(6.16)
and from this an inverse
(κ, α, γ) =
f
−1
( fx , fy , fz )
(6.17)
is to be dened. First it is important to notice that f z is always an input which
qualies it as any other parameter that cannot be affected. So from two inputs,
three outputs are to be calculated. An approach to deal with this is to dene an
optimal camber angle
γ∗ so that
γ∗ =
f γ ( f x , f y , f z , α).
(6.18)
There are two ways to implement this, either the calculations are done in sequence so that (κ, α) is calculated rst and then the optimal camber angle. This
will not yield an exact result which in many cases however, is not a problem.
The second option is to substitute Equation 6.18 into the tyre force model to
reduce this degree of freedom, giving
(κ, α) =
In [23] the assumption is made that
f
−1
( fx , fy , fz )
γ∗ =
(6.19)
f γ ( f y , f z ) which makes both of the
above approaches coincide.
The actual denition of the inverse functions can be done either analytically,
table based, or through the application of non-linear solvers to the original tyre
model. This rst approach inevitably results in rather simple representations,
typically as an inverted polynomial in Paper A or as trigonometric functions in
Paper F. The resulting inverse has a few basic parameters which makes it easier
to tune. However it has limited possibilities for grasping generic tyre behaviour.
Using tabular information allows for more arbitrary characteristics but suffer
from more difcult tuning if the original tyre model is not available.
If the
original tyre model is used, more accurate inverse behaviour can be achieved at
the cost of needing an external check to make sure that the input to the solver
really is achievable.
96
Chapter 6
Methods for generic vehicle motion control design
Figure 6.9: The linkage used in the ACM concept. The two actuators that replace the
lower A-arm can be used to both control steering and camber.
6.6
Actuator commands
Once the desired wheel properties u are calculated, this has to be transformed
∗
into proper actuator commands, u . Again, an inverse problem has to be solved,
which looks quite different depending on the conguration used. General approaches to generate these automatically from detailed models are discussed in
e.g. [85, 61, 91]. When kept on a kinematic position level, this is straightforward as illustrated below for the ACM linkage, Figure 6.9. The following set
of equations fully species the position of the linkage hard points from either
s1, s2, phi or the wheel inputs, steering delta, camber
gamma and the suspension travel, z.
the actuator inputs
e = normalize(r0CL3 - r0CL4);
T1 = rotation({0,0,1},delta)*
rotation({1,0,0},-gamma)*
rotation({0,1,0}, -chi);
T2 = rotation(e, -phi);
rL1U =
rL2U =
rL3L4U
rL1L2U
rL3L4U + T1*(r0L1U-r0L3L4U);
rL3L4U + T1*(r0L2U-r0L3L4U);
= r0CL3 + T2*(r0L3L4U-r0CL3);
= rL3L4U + T1*(r0L1L2U-r0L3L4U);
Section 6.7
97
Application of force allocation
rL1L2U[1] - rL3L4U[1] = 0;
rL1L2U[3] = z;
s1 = length(rL1U - rCL1) - length(r0L1U - r0CL1);
s2 = length(rL2U - rCL2) - length(r0L2U - r0CL2);
This approach does not take any actuator limits into account which is both a
drawback and an advantage. For example, checking the length limits on the
actuators has to be done separately which is an advantage since the actuator
limits do not further limit the solution space. Therefore this approach is more
tolerant to over-requests.
6.7
Application of the force allocation method
As an illustration of the force allocation method, two different chassis congurations are tested.
The rst allows individual steering and traction/braking
and could typically be realized as the ACM concept. The second is a conventional rear wheel driven vehicle with rack steer and individual brakes that can
be controlled by the vehicle motion controller.
The vehicles are modelled according to the rst level of detail presented
in Section 4.2 with three degrees of freedom for longitudinal, lateral and yaw
motion with static load distribution. The tyre model is a simple Magic Formula
implementation, [24], where the scale factor D is dependent on the nominal tyre
force, f xi,yi , road surface condition, µi , and vertical load, f zi , such that D = f xi,yi
nom
at µi = 1 and f zi = f zi .
nom
nom
The data used in the simulations is dened in Table 6.3 along with actuator
limitations. The actuators are assumed to be ideal. The weight matrix, W, is
chosen such that errors in Fy and Mz are weighted higher than Fx , i.e. the lateral
stability has higher priority than short stopping distance. The same constant
settings of the control parameters are used for all congurations.
To illustrate the ability of the different congurations, the results of entering
on split-µ while braking is shown in Figures 6.10 and 6.11. As seen from Figure 6.10, none of the conguration manage to maintain the required retardation
as the vehicle enter the split-µ surface at t
= 4s.
Still, the rst conguration
manage considerably better than the second one and the reason is the better
means to keep the resulting yaw moment low. In Figure 6.11 this is clearly
seen where the right side lateral forces are close to zero for the second conguration while the rst allocates around 2kN with opposite sign. This allow the
right wheels to brake harder than the left ones while still maintaining a low resulting yaw torque which also is seen in the achieved resulting retardation force
98
Chapter 6
Methods for generic vehicle motion control design
Vehicle data
Total mass, m
1600 kg
Yaw inertia, I
1800 kgm
2
Height of COG over ground, h
0.5 m
Vehicle dimensions, (w,l12 ,l3 4)
(0.85,1.1,1.5) m
Nominal tyre force, f x
nom
4000 N
Nominal tyre force, f y
nom
3500 N
nom
Nominal tyre load, f z
4000 N
Magic Formula parameters, (B, C , E )
min
Mechanical brake, ( f xi
Electric machine, (
Differential, f
max
min
f xi
, fximax )
, fximax )
(4,1.6,-10)
(-5000,0) N
(-1500,1500) N
6000 N
KP , diag(kP1 , kP2 , kP3 )
(10119,14311,15179)
KI , diag(kI1 , kI2 , kI3 )
(4000,8000,9000)
lt , diag(lt 1 , lt 2 , lt 3 )
(0.01,0.01,0.01)
Table 6.3: Data used for the force allocation example.
in Figure 6.10.
6.8
Application of the direct allocation method
As an illustration of the direct allocation method, a mid-sized four-wheeled
passenger car is set up with different actuator combinations to investigate their
effect on the performance according to Section 6.4. For the sake of clarity, this
presentation is limited to cover the most common actuators and their effects on
the horizontal motion. For a vehicle with n wheels, u is dened as
u
= [τ1 , δ1 , fz1 , τ2 , δ2 , fz2 , ..., τn , δn , fzn ]T ,
corresponding to Table 6.4, where
δi
denotes steering angles,
(6.20)
τi
drive/brake
torques and f zi tyre loads. For certain congurations (e.g. conventional front
steering), some driver inputs d may directly act on the vehicle. The actuators
are sorted by type and a pair is chosen from each group for the evaluation,
(A,B), (1,3) and (0,I), giving eight combinations from front axle steering and
velocity control (A,1,0) to front and rear axle steering, individual drive and
brake and variable tyre load distribution (B,3,I).
For all congurations, the same structure and tuning parameters are used,
except for the necessary straight-forward adaptation of the matrix C representing the conguration-immanent constraints in Equation 6.6. Note also that a
Section 6.8
20
0
15
Fx [kN]
vx [m/s]
99
Application of direct allocation
10
−5
5
−10
0
0
2
4
6
0
2
4
6
4
6
4
6
time [s]
0.1
1
0.05
0
Fy [kN]
dψdt [rad/s]
time [s]
0
−1
−0.05
−2
−0.1
0
2
4
6
0
2
time [s]
0.1
1.5
0.05
1
Mz [kNm]
β [rad]
time [s]
0
−0.05
−0.1
0.5
0
0
2
4
time [s]
6
−0.5
0
2
time [s]
Figure 6.10: Comparison of performance of two different congurations while braking
on split-µ. The grey thick line is reference trajectory while the solid, the dashed are the
results of the rst and second conguration, respectively. Plotted variables are from
top left towards the right and bottom: longitudinal speed, global longitudinal force,
yaw rate, global lateral force, slip angle and global yaw torque. Results are from Paper
B.
100
Chapter 6
Methods for generic vehicle motion control design
front right wheel
2
1
1
0
0
fx fy [kN]
fx fy [kN]
front left wheel
2
−1
−2
−1
−2
−3
−3
−4
−4
−5
0
2
4
−5
6
0
2
time [s]
rear left wheel
2
1
1
0
fx fy [kN]
0
fx fy [kN]
6
rear right wheel
2
−1
−2
−1
−2
−3
−3
−4
−4
−5
4
time [s]
0
2
4
time [s]
6
−5
0
2
4
6
time [s]
Figure 6.11: Distribution of the global forces in Figure 6.10. Solid and dashed lines
represents longitudinal and lateral forces while grey thick lines and narrow black represents congurations 1 and 2, respectively. Results are from Paper B.
Section 6.8
101
Application of direct allocation
Actuator congurations
conguration
A:
B:
C:
1:
2a:
2b:
3:
I:
0:
actuators
description
δ1 = δ2
δ1 = δ2 , δ3 = δ4
δ1 , δ2 , δ3 , δ4
∑ τi
τ1 < 0, τ2 < 0, τ3 , τ4
τ1 , τ2 , τ3 < 0, τ4 < 0
τ1 , τ2 , τ3 , τ4
f ( f z1 , f z2 , f z3 , f z4 ) = 0
−
Front axle steering
Front and rear axle steering
Individual wheel steering
Vehicle speed
Individual brake and rear wheel drive
Individual brake and front wheel drive
Individual drive and brake
Variable tyre load distribution
No actuator/control
Table 6.4: A subset of conceivable actuator congurations for a four wheeled car.
Note that for a general case, there is no need to limit the setup, either for four-wheeled
vehicles or for these actuators. Indices refer to wheels (1: front left, 2: front right, 3:
rear left, 4: rear right).
precise reference motion is given such that all congurations may directly be
compared without considering any driver (human) related property.
For the given trajectory, the reference side slip angle
βre f
is calculated by
a perfectly inverted single track model [85] whose parameters match both the
controlled vehicle and the inverse model.
To not give the rear axle steered
congurations extra advantage, the inverse single track model has only front
axle steering. All reference values in the simulation plots are taken from this
reference model.
The selected manoeuvre is cornering and braking on a split-µ surface. After
straight driving at 22m/s for 0.5s, the vehicle enters a curve to the left with
radius R
= 80m, giving alat ≈ 6m/s2 , and at t = 5s, braking with along =-5m/s2
is applied while continuing the cornering. While the single track model uses
µ
= 1, the µ-values of the vehicle and the inverse model used are set µ1,3 = 1.3
= 0.8 for the outer wheels. Used data is given in
for the inner wheels and µ2,4
Table 6.5.
The performance of the different congurations is shown in Figure 6.12.
η of tyre grip potential are displayed where
η is represented by ηmean = |η|2 and ηdi f f = |η − ηmean 1|2 .
Vehicle motion and the utilization
As expected, the biggest difference in the lateral velocity, vy , occurs between vehicle congurations with and without rear axle steering, B and A congurations respectively. The A congurations do not manage to keep the preferred vy even during pure cornering (no braking) due to the split µ conditions,
but it can be seen that especially the variable tyre load distribution can improve
102
Chapter 6
Methods for generic vehicle motion control design
25
0.1
0
20
15
10
5
vy [m/s]
vx [m/s]
−0.1
reference
(A,1,0)
(A,1,I)
(A,3,0)
(A,3,I)
(B,1,0)
(B,1,I)
(B,3,0)
(B,3,I)
−0.2
−0.3
−0.4
−0.5
0
2
4
time [s]
6
8
2
4
time [s]
6
8
2
4
time [s]
6
8
2
4
time [s]
6
8
0.3
6
0.25
5
a [m/s2]
0.2
4
y
dψ/dt [rad/s]
7
0.15
3
0.1
2
0.05
1
0
2
4
time [s]
6
0
8
0.9
0.25
0.8
0.2
0.7
ηdiff [1]
ηmean [1]
0.6
0.5
0.4
0.15
0.1
0.3
0.2
0.05
0.1
0
2
4
time [s]
6
8
0
Figure 6.12: Comparison of performance of some congurations specied in Table 6.4
using direct allocation. Plotted variables are from top left towards the right and bottom:
longitudinal speed, lateral speed, yaw rate, lateral acceleration, mean value of grip
usage and mean value of grip usage variation. The windows show magnications of
interesting regions. From Paper D.
Section 6.9
103
Remarks on the presented methods
Vehicle data
Total mass, m
1296 kg
Yaw inertia, I
1750 kgm
2
Height of COG over ground, h
0.5 m
Vehicle length, (w,l12 ,l34 )
(0.85,1.25,1.32) m
Nominal tyre force, f x
nom
4000 N
Nominal tyre force, f y
nom
4000 N
nom
Nominal tyre load, f z
4000 N
Magic Formula parameters, (B, C , E )
(30,1.3,-1.5)
IDOB diag(tψ
, tβ , tax )
(0.05,0.03,0.5)
Table 6.5: Data used for the direct allocation example.
performance at this stage.
Although a small difference can be spotted for vy between the different B
congurations, the greatest difference here is instead the tyre potential usage,
ηmean .
While all A congurations reach a peak value of 0.7-0.73, (B,1,0) and
(B,1,I) peaks at about 0.82. Adding additional torque distribution reduces ηmean
to about 0.74 which is in parity with the A congurations.
It is also notable that all A congurations have a dip in ay and vy at t
= 5s
because of the additional yaw moment when the braking starts. The B congurations avoid this by adding an extra steering angle at the rear wheels while
the A congurations need to have a higher slip angle than
βre f
to be able to
generate enough side force at the rear axle.
Conguration (A,1,I) also shows seemingly odd behaviour at t
ηmean and ηdi f f
= 7s where
suddenly increases. This is in fact due to the circumstance that
the rear inner wheel is just about to lose contact with the ground, which limits
the load distribution.
6.9
Remarks on the presented methods
In this chapter, two main approaches to allocation are presented, direct allocation and allocation via resulting wheel forces. These are tried out in Papers D
and P and Papers B, F and N, respectively. Their drawbacks and advantages
are highly related to how a conguration imposes constraints on the allocation
problem.
When the possibility to control the lateral force of the wheel is limited
to very narrow non-convex regions, for example no steering or camber, it is
harder to solve through force allocation than through direct allocation. Direct
104
Chapter 6
Methods for generic vehicle motion control design
Figure 6.13: By relaxing the cost for non-low
β,
the braking performance on split-µ
can be enhanced considerably for a vehicle with front axle steering. The thicker (grey)
line shows the reference, the solid (red) line shows the constrained
(blue) line shows the relaxed
β.
β
and the dotted
From Paper P.
allocation on the other hand is more difcult for a vehicle with many degrees
of freedom.
Consider, for example, the ACM case where both camber and
steering angles can be adjusted.
In the direct allocation case, both of these
angles have to be accounted for in the optimization, yielding a large problem.
If the allocation is done on a force level, both these effects occur primarily in
the lateral force limitation and once a desired wheel load and longitudinal force
is given, there is a suitable combination of camber and steer angles that can be
calculated without involving the allocation.
It is also seen that the VMC is able to nd suitable solutions for different
manoeuvres without re-tuning which suggests that these methods would make
situation recognition obsolete. At the same time, it is seen that performance
can be considerably enhanced by shifting the weights, e.g. W in Equation 6.5.
Therefore, rather than eliminating situation recognition as a problem, it has to
be redened as a shift of priorities. A typical example is shown in Figure 6.13
where braking on a split-µ surface is performed. For congurations with lim-
β low reduces the possibility
β may shorten the stopping distance consid-
ited abilities to counter steer, keeping the side slip
to brake efciently while relaxing
erably.
In general, indirect effects are the hardest to grasp for the optimizer. One
reason for this is that it is easy to omit essential characteristics when dening
the constraints and the cost functions. Consider for example the use of wheel
load distribution to control the yaw stability of a vehicle. A simple but dedicated method may easily include the effects of roll dynamics which are harder
to grasp in an optimization problem where the roll motion is relaxed. On the
other hand, for a real vehicle with the ability to control the roll motion, it is
Section 6.9
Remarks on the presented methods
105
reasonable to assume that the trajectory denition also would include demands
on this. Once this is grasped, it facilitates the control of vehicles with more
than four wheels since the degrees of freedom for a vehicle where the body can
be assumed to be rigid is n − 3 where n is the number of wheels. A typical
example where this is important is the load distribution for the six-wheeled allterrain vehicle illustrated in Figure 4.15 since it can both improve mobility and
high speed stability.
Related to this is how different frequency domains can be handled within
the proposed methods. Since the handling dynamics is relatively slow, fully
including the ride in the control allocation would impose that whole system
would have to work much faster which in turn would be costly. Instead it is
suggested to approach this problem by separating the frequency domains so that
the output from the VMC in these cases are characteristics rather than reference
signals. The following examples illustrates some aspects that are believed to be
suitable extensions for further research.
Consider rst the incorporation of actuated suspensions that can be used
both for leveling and ride improvement. The outcome of the VMC would in
this case rst of all be desired ride height and load distribution. Additionally,
which is tightly coupled the driving situation, the VMC could dene target
values for stiffness and damping or supply weights for small vertical motions
under normal driving or low load variations in critical situations.
Characteristics can also be used as criteria in the control allocation such
as the improvement of high speed stability for the six wheeler. By introducing a cost for deviation from the target value for the understeer-gradient in the
cost function, the VMC might also be able to work as a tuner to for example
minimize its own workload.
106
Chapter 6
Methods for generic vehicle motion control design
Chapter 7
Scientic contribution
The over-all scientic contribution of this work is the outline of a methodology
to facilitate the selection of vehicle congurations and the design of the corresponding vehicle motion controllers. This is a huge research area and the main
contributions of this work are:
•
A method to classify and map congurations and control strategies onto
their possible inuence on the vehicle motion (Chapter 3).
•
A structured way of implementing and manage vehicle and subsystem
models that are easy to recongure and reuse (Chapter 4). This contribution is also applicable to other domains.
•
Generic ways to evaluate vehicle congurations, especially the use of
the adhesion potential
η to identify safety margin and expected limit be-
haviour as described in Section 5.6.
•
A structure that enables a generic approach to vehicle motion control
(Section 6.1), and two methods for control allocation; direct allocation of
actuator commands (Section 6.4) and force allocation where the congurations are mapped onto the ability to generate tyre forces (Section 6.5).
•
A road map on how to reach the over-all goal presented in Figure 1.5.
This is outlined in Chapter 8 with suggestions on more detailed improvements for modelling (Section 4.4) and vehicle motion control (Section 6.9).
107
108
Chapter 7
Scientic contribution
Chapter 8
Concluding discussion
This thesis outlines a future direction of vehicle motion control based on the
increased use of actuators and computational power to improve drivability and
driving safety. The works referenced in the introduction demonstrate a transformation of these among other factors, from what used to be a matter of tuning
parameters into a complex interaction of systems from multiple physical domains. So far, there is no reason to believe that this trend is going to diminish
and the effects of this transformation are already tangible.
A vital part of the ongoing change is the seemingly endless conguration
possibilities that come with an increased set of available actuators. Especially
when introducing at least partial electric propulsion, the additional onboard
power will enable an increased use of electromechanical actuators. It can also
be seen that current ad-hoc approaches will eventually be unable to keep up
with the pace of development as new technologies are introduced. If automotive
manufacturers do not explore and turn these possibilities into advantages soon
enough, it is believed that they will lag behind and someone else will prot.
A methodology is proposed that addresses this issue based on two main
strategies: The rst is identifying and partitioning congurations to maximize
reusability and minimize the additional cost when setting up a new conguration. The partitioning of vehicle congurations and control strategies have been
proposed in particular. The second strategy is to provide a range of levels-ofdetail that allow a successive selection process so that the amount of congurations can be decreased as the implementation effort increases. The proposed
levels-of-detail span from an evaluation form to detailed simulation models.
It is further shown how different actuator congurations can be abstracted
to mathematical formulations that allow generic vehicle motion control by optimization to be applied. These formulations are based on the expected inuence
109
110
Chapter 8
Concluding discussion
of an actuator on the vehicle dynamics, typically through the tyres. By this
approach, the same controller can manage a variety of congurations and there
is no need to recognize and treat each different situation separately.
At the
same time, it is found that the behaviour strongly depends on the cost function
specication. So when interpreting the driver commands, dening a desired
trajectory is not enough, also the priorities during limit conditions has to be
made clear.
To fully benet from the methodology in product research and development
it is recommended to continue the work by conducting a pilot project in close
cooperation with an automotive manufacturer. This could be done as sketched
below:
Selecting conguration
1. Identify a relevant product and specify the requirements. Ideally these
requirements should be deduced from brand specic characteristics and
driver preferences.
2. Identify possible congurations with respect to packing, available propulsion technology etc.
3. Evaluate these congurations according to the methodology presented in
Chapter 3 with the form from Table 3.1.
4. Set up the conguration as planar and study properties such as the resulting limits diagram, Figure 5.9.
5. Select a controller, either force or direct allocation as is described Chapter 6.
6. Focus on the steady-state criteria rst since they best correspond to the
low level-of-detail. Study properties such as friction potential usage
η.
Compare with a conventional implementation from the manufacturer.
7. Increase the level-of-detail to also include roll, pitch, and vertical body
motion but keep vertical wheel motion excluded. Verify the behaviour
from the previous step. Pay more attention to transient criteria, actuator
rates and power consumption. Compare with a conventional implementation from the manufacturer. If the congurations contain actuators to
affect high frequent motion such as active closed-loop suspension components, verify that performance with e.g. a quarter car model.
111
8. Implement the remaining conguration(s) as a full 3D model that also
includes wheel motion and verify the behaviour from the previous step.
Compare with a conventional implementation from the manufacturer.
9. Improve the 3D model with more detailed actuator and sensor models
and verify the behaviour.
10. Build a prototype.
Dening the on-board controller
For this purpose, the controller can either be kept generic which enables
the handling of, for example, hardware failures. It can also be compiled
into a controller dedicated for the particular conguration by evaluating
xed variables to reduce the optimization problem. If a more traditional
dedicated version is chosen, it is recommended to rst use the form from
Table 3.1 to verify that the selected controller structure is suitable. The
general purpose controller can be used as a formal specication so that
for any given combination of states, the dedicated controller can be compared and veried against the generic one.
Of course, there will be questions to resolve along the road. Figure 8.1
illustrates the some key issues and how they relate to the different possible applications of the presented generic vehicle motion control approach. For the
process of selecting congurations, this will most certainly start already when
dening the requirements. The demands of today have evolved in synergy with
the vehicle, and additionally, the drivers have adapted to how current vehicles
function which makes this issue a research area on its own. Additionally, the
driver-vehicle interaction needs proper attention since aspects such as steering
feel will need redenition. It would be very interesting to investigate whether
the feedback can be built on the resulting limits proposed in Section 5.6. One
way to approach the problem could be to study what traces in the vehicle's motion space
(ax , ay , ψ̈)
are acceptable and if it is possible predict how a driver
would traverse this space. Such an approach could work as a frame for connecting this research with for example driver preferences and Human-Machine
Interfaces (HMI) and Model Predictive Control (MPC). In relation to this question, the weighting of the cost function for the allocation and how it should
depend on situation and driver behaviour and mode also must be given attention.
Another concern is sensoring which may be a problem if the conguration
is to be implemented with the generic vehicle motion controller on-board as
112
Chapter 8
Concluding discussion
Figure 8.1: Applications of the presented work with identied additional requirements
to be addressed in future research work. Direct on-board usage (left), as a mean to
select conguration (middle) and as a specication for conguration-dedicated controllers (right).
113
a general purpose controller. The road friction coefcient is used as a parameter in the model-based approaches which might be less robust and thereby
require measurements or estimations of higher accuracy than today. Therefore,
the sensor fusion for improved state estimation and dynamic inversion (model
adaptation) will continue to be a relevant research topic. For congurations that
allow the side slip angle,
β, to be controlled, improved accuracy and reliability
of this estimation might also be required.
As the additional degrees of freedom to control the wheels and tyres increase, tyre models have to be adapted to cope with the new operating ranges.
Typical examples of this are increased camber angles and variable tyre pressures. Purely empirical models are hard to tune and the present purely analytical models are very detailed at high computational cost. Two other approaches
to consider further are semi-empirical models and models with approximately
or qualitatively correct behaviour throughout the considered range. An implementation could be based on the model presented in Paper R with extensions
to belt deformations as indicated in e.g. Figure 2.7. Another possibility would
be to make limited extensions to the brush model with multiple but yet limited
sets of bristles.
In addition to the points outlined in this work, there are several other aspects to consider before a potential generic vehicle motion controller can reach
production.
In recent decades the rst generations of vehicle motion con-
trollers have evolved, and the suppliers of these systems have grown strong.
As more and more of the vehicle characteristics are software dened, the roles
of OEMs and suppliers have to be claried in order for the integration to reach
a wider market. Despite ongoing standardization work [92], publications show
that suppliers seek to retain a signicant part of the top level vehicle control
(e.g. [93, 94, 84]) while OEMs aim at reclaiming the controllers that also affect
brand-specic characteristics down to a smart actuator level [95]. In [34], this
is identied as increasing total cost, and to avoid this, a compromise is suggested based on cooperative control where the stability functionality is kept by
the suppliers.
A crucial issue of vehicle motion control design is the complexity of the total system which vastly depends on its architecture. A straighter hierarchy may
allow for reduced complexity but ultimately, an integrated approach requires
high transparency. This might be a signicant limitation unless the design is
done within the same company, either an OEM or a supplier. In a longer term
perspective, partitioning has to evolve to maximize performance for cost, otherwise it will open up the market for new operators instead.
114
Chapter 8
Concluding discussion
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Appendix A
Nomenclature
Typeface
Italics are used when introducing a term that is explained in the Glossary.
Typewriter is used for expressions in Modelica and Modelica keywords.
Sans serif is used for to denote Modelica packages and components outside a programming scope, e.g. MSL.MultiBody.
Scalar entities are written slant ed, e.g. x.
Physical vector entities are over-lined, e.g. v̄.
Lumped entities such as sets of states, reference values or estimates are
denoted with bold lower case letters, e.g. u.
Matricides are denoted with bold CAPITAL letters, e.g. A.
Schemes
A vector quantity x of B resolved relative A are dened as
A
B
x̄C where C
is an index.
A transformation y from A to B is denoted
123
B
yA , i.e.
B
x
=B yA A x.
124
Appendix A
Figure A.1: Vehicle nomenclature.
Nomenclature
125
Indices
The following indices are used
, , , , ..
1 2 3 4
Indication of wheel starting from front left towards right and
rear. Composites such as
12 , 34 , 24
mean front axle, rear axle and right
hand side, respectively.
, ,
x y z
Components of a vector pointing forwards, leftwards and upwards,
respectively.
lim
Limits
re f
Reference
min,max
Minimum and maximum values
126
Appendix A
Greek
Nomenclature
F resulting force
α tyre slip angle
F global forces
β vehicle side slip angle
g equality constraint
γ inclination angle
h height
δ steering angle
h inequality constraint
η adhesion usage
f
i inertia element, gear ratio
f max
ε camber angle
m mass
κ tyre spin slip
M resulting moment
µ tyre-road adhesion coefcient
K gradient
ω angular velocity
l length
φ angular rotation
n direction
ϕ roll
r position
ψ yaw
s linear displacement
ρ curvature
rcm location of center of mass
θ pitch
t torque
ζ angular acceleration
u input
v velocity
Latin
w width
a acceleration
x states
B geometry matrix
y output
c stiffness
Cα cornering stiffness
d damping
f force
f tyre forces
∂ fy
∂α |α=0
Appendix B
Glossary
ABS Anti-BlockierSysteme, translated brush model Model describing the tyre
into English as Anti-lock Brak-
patch as bristles on a brush, see
ing System.
Section 2.3.
Ackermann angle The steering angle
camber Angle between wheel normal
required to negotiate a turn at low
and chassis z axis, dened as pos-
speed, see Section 5.2.
itive if the wheel is leaning outwardsupwards, c.f.. inclination.
active Used for components that are
able to add energy to the system,
cardinality Number that denes how
many other components are at-
see Chapter 3.
tached, used in Modelica to readaptive Used for a system that can
produce structural information.
change or tune its working princlass Specication that can be instan-
ciple to adapt to changes.
tiated into an object.
allocation Distribution of signals from
several actuators that affect the
closed-loop System that uses state feedback, see Chapter 3.
same degree of freedom, see Section 3.2.
coexistence Type of control structure
classication, see Chapter 3.
arbitration Unication that only allows one subsystem to act at a
time, see Chapter 3.
component In a modeling context, component is used for an instance of
axle Term use to specify that both wheels
on an axle or suspension are effected by the same actuator, see
a model. C.f.. class and object.
connector Modelica keyword for a model
Section 3.2.
with variables that are transmitted between connected models
127
128
Appendix B
to allow interactions between the
components to be specied.
Glossary
FC Fuel Cell.
y-by-wire Se x-by-wire.
cooperative Type of coexistence where
there is cooperation and bidirec-
friction ellipse The combined lateral
and longitudinal limits of a wheel,
tional information exchange be-
see Sections 2.4 and 5.6. For a
tween subsystems.
whole vehicle, this is often recoordination Unication that allow sev-
ferred to as a g-g diagram.
eral systems to act at a time, see
Chapter 3.
ganging matrix A matrix that couples
two degrees of freedom, often to
daisy-chaining When several actuators
mimic a mechanical constraint.
are available for the same degree
of freedom and one is run to saturation before the next is engaged.
g-g diagram Diagram showing lateral
vs longitudinal acceleration. It
can be used to trace vehicle per-
degrees of freedom The set of independent displacements that fully
specify the motion of conguration.
DOF Degree(s) Of Freedom.
formance especially in racing applications since one goal is to always have a high acceleration magnitude. It can also be used to describe the resulting vehicle limits.
direct Term to specify that the effect
on the tyre force is direct, CR. in- global forces The resulting forces and
moments on a vehicle.
direct. See Section3.4.
driver assistance system Systems that handling diagram Graphical represenassist the driver in his task, a wider
tation of a vehicles steady state
span of VMC including also route
handling characteristics, see Sec-
planing and post-crash activities.
tion 5.2.
drive-by-wire See x-by-wire.
hierarchical Type of coexistence where
some systems adapt to others, see
dynamic inversion By replacing the
Chapter 3.
internal integration in an inverse
model by measurements or estimates, drifting due to integration error can be avoided.
ESP Electronic Stability Program. A
trademark for a type of yaw control through brake intervention.
ICE Internal Combustion Engine.
IDOB Inverse Disturbance Observer, [87]
in-wheel motor Motor that is mounted
inside the wheel, CR. wheel motor.
129
inclination Angle between the tyre ver- None An empty but yet complete model
tical axis and the road normal,
that can be used as a placeholder
also called tyre or wheel cam-
in a conguration of a template.
ber, see Section 2.6.
object An instantiation of a class.
inheritance Term that is used in an
object oriented context meaning
OEM Original Equipment Manufac-
to take over or inherit attributes
turer generally means a company
or behaviours from a base or par-
whose products are sold by a an-
ent class.
other company (reseller). In the
automotive industry OEM status
indirect Term to specify that the ef-
is a legal identication in some
fect on the tyre force is indirect,
countries.
e.g. by change of tyre characteristics, CR. direct. See Section3.4. over-actuation A system that has more
actuators than degrees of free-
instance The realization of a speci-
dom.
cation, e.g. an object is an in-
actuators have inuence on the
stance of a class and a compo-
same degree of freedom.
nent is an instance of a model.
interface A partial model that consists
over-steering Vehicle steering characteristics classication, see Sec-
of connectors and parameters that
tion 5.2.
are common for several models.
internal Type of model that is of no
open-loop System that does not use
state feedback, see Chapter 3.
interest for a normal user of a
Modelica library.
This means that several
package A Modelica keyword for a
class that gathers other classes,
merging Type of control structure clas-
used to form a library structure.
sication, see Chapter 3
Also used to denote sub-libraries.
model A consistent (mathematical) representation, used here for the rep- partial Modelica keyword for a model
that is incomplete and needs to
resentation of any system or part.
be inherited and completed beMSL Modelica Standard Library [56].
fore usage.
face or a template model.
non-minimum phase A system with
zeros in the right half of the complex plane is non-minimum phase,
parallel Coexistence type where sub-
an inversion of such a system would
be unstable since it leads to poles
in the right half plane.
Typically an inter-
systems are unaware of each other,
see Chapter 3.
130
Appendix B
passive Used for parts that are not able
unication Type of control structure
to add energy to the system, see
Chapter 3.
classication, see Chapter 3
VDL The VehicleDynamics Library,
quaternion Quaternions are a non- commutative extension of complex
numbers that are useful to de-
rotations, there are no singular
actuation.
is effected per actuator, see Chapter 3.
wheel motor Motor driving one single wheel, often mounted directly
at the wheel hub, then called an
roll-off Decrease in inclination gain as
in-wheel motor.
the tyre slip angle increases, see
Section 2.6.
WU Wheel Unit. The wheel with actuators that can affect it.
single contact point Tyre model type
that assumes that the road and
tyre surfaces are homogenous
x-by-wire x-by-wire or by-wire refers
enough to allow the contact patch
to be represented by a single contact point.
stability control system Gathering name
for systems like ESP that aim at
stabilizing the yaw motion of the
vehicle.
template A template is a partial model
with placeholders for components
that can be replaced to form another model with the same structure.
UML Unied Modeling Language [66].
under-steering Vehicle steering characteristics classication, see Section 5.2.
to control the vehicles motion.
wheel Term to specify that a single wheel
congurations.
redundancy Robotics term for over-
see Chapter 4.
VMC Vehicle Motion Control, system
scribe three-dimensional rotations,
since unlike a sequence of three
Glossary
to systems that have physical connections such as mechanical links
or hydraulic pipes replaced by
electronics.
Appendix C
Modelica in brief
This appendix gives a very brief introduction to the modelling language Modelica and the simulation tool Dymola. More detailed information can be found
via [56] and [57]. It also outlines the developed guidelines for library and model
structuring and presents an UML notation adapted to Modelica.
Mathematical description
Consider the implementation of a model of a bouncing ball, dened as:
model BouncingBall "Model of a bouncing ball"
import SI = Modelica.SIunits;
SI.Position z(start=1) "Height over ground";
SI.Velocity v "Upwards velocity";
SI.Acceleration a "Upwards acceleration";
SI.Force f "Upwards force";
parameter SI.Mass m=1 "Mass";
parameter SI.Acceleration g = 9.82 "Gravity acceleration";
parameter Real k(quantity="stiffness", unit="N/m") = 1e6;
parameter Real d(quantity="damping", unit="N.s/m") = 140;
equation
m*(a + g) - f = 0 "(1)";
der(z) = v "(2a)";
der(v) = a "(2b)";
if z < 0 then
f = -k*z-d*v;
else
f = 0;
end if "(3)";
end BouncingBall;
131
132
Appendix C
In equation
Modelica in brief
(1) an algebraic relationship is given for the force-acceleration
balance. Note especially that this expression is given without any cardinality
so that any of the variables
a and f could be unknown.
The equations
(2a)
(2b) describe the kinematic relationships between z, v and a where the
der() operator species the time derivative. Equation (3) nally is a logical
condition for how the external force on the ball f is dened. Together these
and
form a system of Differential-Algebraic Equations (DAE) that is the core of
any Modelica model.
In addition to equation sections, it is also possible to
specify algorithms that are executed in sequential order.
Object orientation
The model described above has the advantage of being compact, within only
a few lines the characteristics of a bouncing ball are described. The drawback
is the limited eld of application. It is essentially a description of a mass, a
non-linear spring and a ground when each by itself is much more generic. The
alternative description is seen in Figure C.1 where predened models are connected to form an equivalent model.
To handle this efciently, Modelica is
dened as an object-oriented language with a set of different classes for different needs.
package, model, block, connector,
type but here, only the rst four are discussed.
These classes are
function, record
and
Packages are used as containers much like folders in an operating system and
may only contain declarations of classes and constants. Models are used to dene models of physical components such as springs, masses, resistors, evaporators and pumps. Blocks are used as models but with xed input-output causality
and therefore may not have physical connectors. Inside a block, the causality
must not be explicit though and a block can contain both equations and models.
Connectors dene the interfaces between models and blocks and may themselves not contain any equations. Typically, variables in signal connectors are
dened as either input or output while a physical connector contains a set of
variables representing potential and ow. This can for example be position and
force in a mechanical system. When two or more connectors are connected, the
tool automatically sets the potentials as equal and sums the ows to zero. A
connector should be balanced, meaning that there should be an equal amount
of ows and potentials. This way a tool can automatically check that the model
has equations corresponding to the unknowns. Some examples of connector
denitions from the
MultiBody
Modelica Standard Library
are given in Table C.1. The
connector has more potentials than ows since the rotation is de-
ned as an object containing a 3x3 transformation matrix and the angular velocity. To establish a balance, the rotation object has a function that returns a
133
Figure C.1: The implementation of the bouncing ball model with primitives from the
gravityMass contains the equation to govern the motion of
elastoGap denes the logic for when force is transmitted
and ground is a x point for the ground level and the reaction forces. The connect
statements denes the coupling of the variables between the involved connectors
so that the flow variables (here the forces f) sums to zero and the potentials (here the
positions s) are equal.
Translational library.
a mass in a gravity eld,
three-valued residue used to compare matrices.
Connectors may be both hierarchical meaning that they contain other connectors and
expandable. The latter is intended for lumping signals together
and allow variables to be added to the connector when it is used and not necessarily when it is dened.
Generating executable code
As a Modelica tool, Dymola faces some issues common as any other tool
would.
Here, it is described in brief how Dymola transforms the Modelica
models into simulation code and generates results.
The rst step is to atten the hierarchical model code and as a result, there
will be a huge set of equations and unknowns with a lot of variables describing
the same quantity.
A typical example is the potential of connected connec-
tors. The variables are sorted into an independent and a dependent set where
the independent variables are gathered in both linear and nonlinear systems of
equations that in turn are minimized using e.g. tearing. The symbolic manipulation capabilities are both a strength and weakness of Dymola. It can be used to
form model inverses, resolve algebraic loops and more but at the same time this
134
Appendix C
Modelica in brief
Rotational
SI.Angle phi "Absolute rotation of flange";
flow SI.Torque tau "Cut torque along flange axis";
MultiBody
SI.Position r_0[3] "Position vector from world frame";
Orientation R "Orientation relative world frame";
flow SI.Force f[3] "Cut force resolved in connector frame";
flow SI.Torque t[3] "Cut torque resolved in connector frame";
Eletrical.Analog
SI.Voltage v "Potential at the pin";
flow SI.Current i "Current flowing into the pin";
Table C.1: Standard connector denitions
often makes the tool process models that are erroneously implemented by the
user. In addition to the above, Dymola uses reduction to form index-1 problems
and is also capable of generating code for dynamic state selection.
The result is a sequential structure that is translated into C-code which then
is compiled to an executable. The compiled code is integrated by the chosen
integrator algorithm. The linear systems of equations must be evaluated and
the non linear solved between the integrator steps.
Adapted UML notation
During the development of the VehicleDynamics library, a need for a convenient notation to describe structure of models and packages became obvious
and UML was adapted to suit Modelica according to Table C.2. An application
example is given in Figure 4.2.
Library structuring guidelines
Library is used to denote top level packages (distributions) and packages are
used to denote sub-libraries. The structure are layed-out using the following
package types.
Interfaces
Interfaces
is a package with partial models containing only con-
nectors and parameters. An interface is used to avoid code repetition and
to ensure compatibility.
Interfaces are used as constraining classes in
templates. The interfaces should be named after what the connectors and
parameters are used for, e.g. 'Automatic' or 'Manual' for a transmission.
Standard is used to denote an interface that follow the modeling level of
135
Adapted UML notation
extends).
Inheritance (
Declaration, i.e. instantiation of a model to a component.
No difference has been made between dynamic and static
creation of objects as in UML. This is because the current
version of Modelica does not consider this and adaption
can be made if and when needed.
Declaration as
replaceable. A model that can be re-
placed by any model that share the same constraining
class, i.e.
shares a common set of connectors and pa-
rameters.
The interface symbol is used to emphasize that the
class is an interface. An interface is always incomplete
partial)
(
and may only contain connectors and pa-
rameters.
The template symbol is used to emphasize that the model
is a template.
A template is used to lay out a struc-
ture with replaceable components, often with interfaces
as constraining classes.
The component symbol is used to emphasize that the
model can be instantiated to a component.
A compo-
nent should need no other modications than parameter
settings before usage.
The experiment symbol is used to emphasize that the
model is an experiment. An experiment can always be
simulated on its own.
Relation that is omitted in the diagram.
A double-point indicates that the search path is not fully
written.
Table C.2: Adapted UML notation
136
Appendix C
the library. Thus a
Modelica in brief
StandardManual transmission in VDL means a man-
ual transmission that with connectors that are standard usage in VDL.
An interface package may not contain other sub-packages. An interface
package should always inherit the
InterfacesPackage icon.
Templates A Templates package contains partial models that are built-up from
replaceable interface components that are connected. A template can also
be considered as an architecture or a wizard and use of templates facilitates maintenance, minimizes code repetition and makes it easier for
users to dene own models. To form a complete model, a template is extended and the placeholders are lled with models. The name of the template should describe the structure and not the content. 'Conventional'
is used to denote a template that describes a conventional layout, i.e. a
ConventionalCar
chassis.
A template package may not contain other
sub-packages. A template package should always inherit the
Package icon.
Components A
Components
Templates-
package is the default package type and may
contain any other package. Typically this package contain further Components packages so that the package hierarchy follows that of the models. For example, in
VDL, the Vehicles package contains the Chassis
Suspensions package etc. A Components
package that contains the
package is normally named after its contents, e.g. Suspensions, Bodies
etc. Only when there are a few components these should be gathered in
a package called 'Components'. A
the
Components package should inherit
ComponentsPackage icon or have a descriptive image.
Variants When there are many component models these can be structured using variant packages. These should be considered as a grouping concept
and if there is not enough components to subdivide them according to
variant types there should be no variants package. I.e. there should never
be a package called 'Variants'. To exemplify, the
contains
Suspensions package
Independent, Planar and RigidAxle variants packages.
A Vari-
ants package may only contain other Variants sub-packages. A variant
package should inherit the
VariantsPackage icon.
ExperExperiments package should contain runable
there may be Components and Templates sub-
Experiments Experiments are stored within the package hierarchy in
iments sub-packages.
models. Additionally,
An
packages for convenient conguration of the experiments. No models
137
from within an Examples package may be used outside this package. An
example package should always inherit the
Examples The
ExperimentsPackage icon.
Examples package should be used to highlight the features of
For an application library like VDL, examples are prefer-
the library.
ably stored at the top level. For Composite packages like, Experiments
packages are given under each domain.
Info The Info package should occur in parallel with Examples package. I.e. Info
packages occur on top level on application libraries and under each domain.
Internal
Internal
is a package with components of no normal interest for a
user. Internal packages should extend the
Types A
Types package is used to dene special types such as cornering stiff-
ness.
Only exceptionally should non-type declarations be used.
package should inherit
Tests
InternalPackage icon.
This
TypesPackage icon.
Tests is a package containing test models to be used with e.g. regression
testing. A test package should be on the same level as the corresponding
Components or Experiments package. For example, the suspension tests
Suspensions and the link test are found under Links. A
tests package should always inherit the TestsPackage icon. Tests are
are found under
normally removed before distribution.
138
Appendix C
Modelica in brief
Paper A
J. Andreasson and L. Laine, Driving Dynamics for Hybrid Electric Vehicles
Considering Handling and Control Architecture. Vehicle System Dynamics.
Volume 41, pp. 497-506, 2004.
Paper B
J. Andreasson, L. Laine and J. Fredriksson, Evaluation of a Generic Vehicle
Motion Control Architecture. In Proceedings of World Automotive Congress
FISITA, Barcelona, Spain, 2004.
Paper C
L. Laine, J. Andreasson and J. Fredriksson, Reusable Functional Partitioning
of Tractive Force Actuators Applied on a Parallel Hybrid Electric Vehicle. In
Proceedings of 7th International Symposium on Advanced Vehicle Control,
AVEC'04, HAN University, Netherlands, 2004.
Paper D
J. Andreasson and T. Bünte, Global Chassis Control Based on Inverse Vehicle
Dynamics Models. To be published in Vehicle System Dynamics, Volume 44,
2006.
Paper E
J. Andreasson, C. Knobel and T. Bünte, On Road Vehicle Motion Control Striving towards synergy. In Proceedings of 8th International Symposium on
Advanced Vehicle Control, AVEC'06, Taipei, Taiwan, 2006.
Paper F
M. Jonasson and J. Andreasson, Exploiting Autonomous Corner Modules to
Resolve Force Constraints in the Tyre Contact-Patch. Submitted for publication.
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