Optimal steering control input generation for vehicle's entry speed change manoeuvre

Optimal steering control input generation for vehicle's entry speed change manoeuvre
Optimal steering control input
generation for vehicle's entry speed
maximization in a double-lane
change manoeuvre
Matthias Tidlund
Stavros Angelis
Vehicle Engineering
KTH Royal Institute of Technology
Master Thesis
TRITA-AVE 2013:64
ISSN 1651-7660
Postal address
KTH
Vehicle Dynamics
SE-100 44 Stockholm, Sweden
Visiting Address
Teknikringen 8
Stockholm
Telephone
+46 8 790 6000
Telefax
+46 8 790 9290
Internet
www.kth.se
Acknowledgment
This thesis study was performed between June and November 2013 at Volvo Cars’ Active
Safety CAE department, which provided a really inspiring environment with skilled
colleagues and the opportunity to get an insight of their work. Our Volvo Cars supervisor
Diomidis Katzourakis, CAE Vehicle Dynamics engineer, has constantly provided invaluable
feedback regarding both the content of the thesis as well as our presentations at Volvo. He
has been a great knowledge asset and always available as source of answers and ideas when
the vehicle’s dynamic complexity was decided and modelled.
We also would like to thank our supervisor Mikael Nybacka, Assistant Professor in Vehicle
Dynamics at KTH Royal Institute of Technology, for the support and guidance to reach the
goal of delivering a report of high quality, for the scheduling and the timeline of this work,
and the opportunity to present this work in parts so to have a better overview of its progress
and quality. A special thanks should also be given to Mathias Lidberg, Associate Professor in
Vehicle Dynamics at Chalmers Technical University, for his active participation in the
project, not only saving us a great deal of time in the beginning by helping us understand the
optimization tool Tomlab and the parts within an optimization problem which are most
important, but also for constantly providing input with ideas and feedback.
Field tests would have been impossible without the help of Per Hesslund, who installed the
robot in the vehicle during both of our testing sessions as well as conducted each test and
guided us through the procedure of instrumenting a vehicle and performing a test. We would
also like to thank Georgios Minos, Active Safety CAE department’s manager, who provided
us support with equipment and took care of technicalities like arranging for the vehicle tests
in Hällered as well as giving us the opportunity to take the T1 and T2 test driver courses.
Volvo Cars employees that also provided input to this thesis are: Max Boerboom, Johan
Hultqvist, Egbert Bakker, Mats Jonasson and Kenneth Ekström.
Matthias is particularly grateful for the support from his parents, Stig and Ingela, throughout
the studies as well as Stephanie for her tremendous ability to encourage joy and success.
Stavros would like to specifically thank his family, Nikoleta, Maria and Georgios for their
invaluable support as well as Anastasia for being by his side all of this time.
Stockholm November 22, 2013
Matthias Tidlund
Stavros Angelis
i
Sammanfattning
Under utvecklingsprocessen av nya fordon sker en strävan mot att reducera fysiska tester,
bilindustrier utvecklar därför metoder för att återskapa fysiska testscenarier i virtuella miljöer
med hjälp av simuleringsmjukvara. Denna studie har som målsättning att utveckla en metod,
med vilken fordonets dynamiska egenskaper kan utvärderas utan att utföra fysiska tester.
Målet är att utveckla ett simuleringsverktyg som, i en tidig utvecklingsfas, kan användas av
fordonsindustrin och som skulle införa både modifikations- och kalibreringsmöjligheter i
detta skede.
Såväl en fordonsmodell som ett anti-sladd system är konstruerat och modellens prestanda i
ett dubbelt filbyte, specificerat i ISO3888 del 2, är utvärderad. Då bilens dynamiska prestanda
klassificeras utifrån ingångshastigheten i detta test utfördes en optimeringsprocess där hjulens
styrvinklar reglerades för att uppnå högsta möjliga hastighet vid testets startposition, detta för
att separera fordonets dynamiska klassificering från mänsklig inverkan.
Processen att konstruera fordonsmodellen utfördes med succesivt ökande antal av
fordonsegenskaper, från en enkel implementering av en linjär cykel-modell till en tvåspårsmodell med krängning, transienta däckegenskaper, hjulupphängningsegenskaper samt ett
anti-sladd system. Resultatet av den optimerade styrregleringen testades i motsvarande
fordon på en testbana varefter modellen kunde utvärderas med det verkliga testet som
referens.
Genom en utökad möjlighet till simulering kan detta verktyg ge möjligheten att studera fler
scenarier såväl som alternativa modelleringskonfigurationer; det kan reducera fysiska tester då
fordons dynamiska prestanda ska klassificeras, studeras samt utvärderas.
ii
Abstract
In an effort to reduce physical testing during the development process of a new vehicle, the
automotive industries develop methods that can facilitate the recreation of the physical
testing scenarios in virtual environments using simulation software. This thesis aims to
develop a method which would help evaluate the vehicle’s dynamic properties without it
being subjected to physical testing. The goal is to develop a tool that can be used in an early
development phase by the industry and that would allow for modifications and calibration to
take place.
A vehicle model as well as an electronic stability control implementation is built, and the
model’s performance to an ISO3888 part-2 double lane change test is evaluated. Since the
handling potentials of the vehicle are rated by its entry speed in that test, the model was
subjected to an optimization process where its steering action was controlled in order to
achieve the highest possible entry speed to the test in an effort to isolate the vehicle’s
dynamic potential from the influence of a human driver when conducting this test.
The vehicle modelling procedure is done in steps, from a simple implementation of a linear
bicycle model to a more complex implementation of a four-wheel vehicle including roll, tire
relaxation and suspension compliance properties as well as a simplified ESC implementation.
The results of the steering input optimization process were physically tested on a test track,
where the correspondence of the model to the real vehicle was evaluated.
By further promoting the vehicle dynamics modelling, this tool can facilitate study more
testing scenarios and options and it can serve as a step toward the reduction of the physical
testing when the vehicle’s dynamic and handling performance need to be studied and
evaluated.
iii
Contents
Acknowledgment ............................................................................................................. i
Sammanfattning ............................................................................................................. ii
Abstract .......................................................................................................................... iii
1
2
3
iv
Introduction .............................................................................................................. 1
1.1
Background .............................................................................................................................. 1
1.2
Problem description ............................................................................................................... 2
1.3
Goal of this thesis ................................................................................................................... 2
1.4
Project assumptions ................................................................................................................ 2
Literature study .........................................................................................................3
2.1
Optimization ............................................................................................................................ 3
2.2
Tools for optimization ........................................................................................................... 3
2.3
Track boundary creation ........................................................................................................ 5
2.4
Vehicle parameters and coordinate systems ....................................................................... 5
2.5
Bicycle model ........................................................................................................................... 6
2.6
Four wheel vehicle model ...................................................................................................... 9
2.7
Tire behaviour ....................................................................................................................... 11
2.7.1
Slip angle ......................................................................................................................... 11
2.7.2
Linear tire model ........................................................................................................... 12
2.7.3
Non-linear tire model ................................................................................................... 13
2.7.4
Transient tire properties ............................................................................................... 15
2.7.5
Pneumatic trail ............................................................................................................... 16
2.7.6
Camber influence .......................................................................................................... 16
2.8
Electronic stability control and its functionalities ........................................................... 18
2.9
Vehicle parameter estimation .............................................................................................. 20
Methodology ........................................................................................................... 23
3.1
Double lane-change track modelling ................................................................................. 24
3.2
Objective function ................................................................................................................ 25
3.3
Point mass trajectory and initial inputs ............................................................................. 26
3.4
Bicycle model trajectory optimization ............................................................................... 28
3.4.1
Linear tire model ........................................................................................................... 28
3.4.2
Magic formula tire model............................................................................................. 31
3.5
Car body boundary modelling ............................................................................................ 34
3.6
Four wheel vehicle model trajectory optimization .......................................................... 35
3.7
ESC modelling ....................................................................................................................... 36
3.7.1
Optimal ESC modelling – a torque controller ......................................................... 37
3.7.2
Sideslip angle controlled ESC ..................................................................................... 38
3.7.3
Yaw rate error controlled ESC ................................................................................... 39
3.7.4
Yaw and slip controlled ESC ...................................................................................... 41
3.8
Model refinement .................................................................................................................. 42
3.8.1
Wheel kinematics and compliance ............................................................................. 42
3.8.2
Tire relaxation ................................................................................................................ 46
3.8.3
Camber thrust ................................................................................................................ 47
3.8.4
Fully featured model without ESC ............................................................................. 48
3.9
Parameter study ..................................................................................................................... 51
3.9.1
Vehicle parameter changes .......................................................................................... 51
3.9.2
ESC-characteristics variations ..................................................................................... 53
3.9.3
Influence of discretization ........................................................................................... 54
3.10 Animation of a car along the track..................................................................................... 55
3.11 Steering robot test 1; driving procedure............................................................................ 55
3.11.1
Parameter estimation .................................................................................................... 56
3.12 Steering robot test 2; driving procedure............................................................................ 58
4
Results ..................................................................................................................... 59
4.1
Optimal controller driving inputs....................................................................................... 60
4.2
Optimal trajectory ................................................................................................................. 62
4.3
ESC influence ........................................................................................................................ 63
4.3.1
4.4
Other ESC models investigated .................................................................................. 67
Steering robot test 1.............................................................................................................. 68
4.4.1
Parameter estimation .................................................................................................... 69
v
5
4.5
Steering robot test 2.............................................................................................................. 71
4.6
Parameter study ..................................................................................................................... 75
4.6.1
Mass ................................................................................................................................. 75
4.6.2
Yaw inertia ...................................................................................................................... 76
4.6.3
Body width and length ................................................................................................. 77
4.6.4
Roll stiffness ................................................................................................................... 79
4.6.5
Tire property changes ................................................................................................... 82
4.6.6
ESC-characteristics variations ..................................................................................... 86
4.6.7
Influence of discretization ........................................................................................... 89
Discussion ............................................................................................................... 92
5.1
Vehicle dynamics and ESC modelling ............................................................................... 92
5.2
Results ..................................................................................................................................... 92
5.3
Parameter study and method robustness .......................................................................... 93
6
Recommendations .................................................................................................. 94
7
Future Work ............................................................................................................ 95
8
Nomenclature ......................................................................................................... 97
9
References ............................................................................................................... 99
Appendix
A – ISO 3888 description
B – Test vehicle
C – Measured data part 1
D – Test 2 manoeuvre test list
E – Measured data part 2
F – Code structure
G – Kinematics & compliance
H - Test 1 data illustration
I - Simulink bicycle model
vi
1 Introduction
Traditionally, the car development procedure includes the building of prototypes and
subjecting them to extensive physical testing for assessing overall performance; The double
lane change test constitutes a physical test normally used to rate the car’s dynamic
characteristics. In an effort to reduce development time and promote safety, new
development methods should be employed, which do not require prototype vehicles to be
built before their behaviour can be estimated.
1.1
Background
Rating the handling potentials of a car is a necessary part for the assessment of its safety
performance. Such a rating is conducted by independent organizations such as the
EuroNCAP and among their tests the electronic stability control systems (ESC) are tested by
evaluating the vehicle’s lateral and yaw stability [1].
The ISO 3888 Part 2-Obstacle avoidance is a dynamic test where the vehicle is driven closed
loop in a severe lane change manoeuvre inducing high lateral accelerations [2]. The test is
related to passenger cars 1 and light commercial vehicles up to 3500kg and involves rapidly
driving from an initial lane to another parallel lane and then back to the initial one, so as to
recreate an obstacle avoidance scenario. During the manoeuvre not one of the cones marking
the track should be displaced to assume a valid run. The results of this test serve as a
subjective evaluation of the vehicle’s handling potential. The test track that defines the
manoeuvre can be seen in Figure 1 and more details regarding the test are presented in
Appendix A – ISO 3888 description.
Figure 1. ISO 3888 course, part 2 obstacle avoidance. Definition of the double-lane change track to
be used for the entry speed optimization [3]
1
According to the definition of passenger cars in ISO 3833 [57].
1
Since a driver controls the car, he or she is involved in the control loop, so these methods are
often characterized as unsuitable for objective evaluation. Objective evaluation in an early
development stage of the car would be made possible if the driver could be substituted with
a controller that determines the optimum steering input to maximize the entry speed.
1.2
Problem description
When performing the ISO 3888 part 2 test, objectivity can be achieved if the human driver is
not included in the loop, such that the vehicle’s contribution to the manoeuvre can be
isolated. This substitution would be meaningful if somehow the test drivers are replaced by a
“perfect driver model,” that would always be able to use the full potential of the vehicle
when conducting the test, so that an objective performance metric could be defined. This
“perfect driver” model is the optimal steering controller input generator, whose development
is the fundamental problem of this study. Usually, when performing vehicle simulations, a
driver model controls the steering angle to follow a path. The current problem eliminates
that need to simulate such a driver.
1.3
Goal of this thesis
The goal of this thesis is to develop a vehicle steering controller that generates the optimal
steering inputs that enable maximum vehicle entry speed in a double-lane change manoeuvre.
The vehicle should also be operating in conjunction with its ESC system. The rationale is to
develop an assessment tool to be used in an early development phase, allowing for direct
evaluation of the car’s handling and tuning adaptations. This would reduce development time
and cost and could potentially promote safety.
1.4
Project assumptions
The process of determining the optimal solution requires knowledge of the whole task, with
the solution, dependent on all parameters (vehicle properties, double-lane change track
dimension, etc.). The controlled systems, steering and ESC, behave optimally according to
the objective function.
Model complexity was not limited when the study started; however, when the increase of the
model’s complexity features did not notably affect the results, no more features were added.
Tire model parameters were estimated from tests performed at dry conditions. All the results
from the simulations assumed those parameters to be valid.
Robustness of the optimization algorithm was investigated by changing the ESC torque
characteristics, tire model properties, body length and width, mass, yaw inertia and roll
stiffness in a range which should cover most passenger cars. Other parameters were held in
their nominal values and no more than three parameters were changed at the same time.
2
2 Literature study
Following the aim presented in section 1.3, the interesting areas for this thesis are; theory of
optimal control, vehicle models, tire models as well as different extra features such as
electronic stability control. This chapter condenses the relevant information for this thesis.
2.1
Optimization
If a problem has more than one solution and the goal is to have one single solution as an
answer, that would best fulfil some requirements, it is an optimization problem. The problem
can be constrained or unconstrained and does not necessarily need to be mathematically
described. Often, the problem needs to be solved numerically, meaning that an iterative
procedure is used when solving, and then it is necessary to formulate the problem
mathematically [4].
The algorithm also influences the solving process as cited below.
“…there are numerous algorithms, each of which is tailored to a particular type of
optimization problem. It is often the user’s responsibility to choose an algorithm that is
appropriate for the specific application. This choice is an important one; it may determine
whether the problem is solved rapidly or slowly and, indeed, whether the solution is found
at all.”
(Nocedal and Wright [5])
When a local optimum solution is found, meaning that in an infinitesimal neighbouring
region of the solution a local optimal solution is obtained, it does not ensure that the solution
is the global optimum. There may exist completely different solutions that are better. To
guarantee that the solution is the global optimum one must solve the Hamilton-JacobiBellman equation. However, if different bounds are used and the optimization converges to
the same point for different starting values, it is an indication that the local optimum is the
global optimum [6].
2.2
Tools for optimization
In this section, the term, “tools” refers to software but is not limited to a specific runtime
environment. Optimization can be done with many different tools after the problem has
already been mathematically described; Dyna4 framework [7], VI-carrealtime [8], Mode
Frontier, Optimica [9], Matlab [10] and the Matlab extension Tomlab [11] are examples of
such tools. Tomlab is the tool that was chosen during this study, for reasons described in
chapter 3, and its functionality is described in more detail compared to the other tools. For a
complete description of Tomlab the reader is however referred to the manual from Tomlab
3
[12], and the extensive guide by Stanford [13], which covers the SNOPT solver version 7, is
useful to judge the outcome from a solution.
Dyna4 framework is the latest tool from TESISdynaware, and is a successor to Vedyna and
Endyna. It and performs simulations of vehicle dynamics tests, uses a three dimensional road
description, various manoeuvre controls and a driver model [7].
VI-carrealtime is an environment where vehicle simulations can be done in real time. A real
time model can be exported automatically from the multi-body dynamics vehicle program
ADAMS Car [14]. Optimization studies of vehicle and control system performance can be
facilitated. Matlab can be integrated with VI-carrealtime, making it easier to perform various
post-processing operations. A track can be set up in a virtual environment in the same way as
in Figure 1, but the reference provides poor information about which solvers are actually
being used to solve any optimization problem within the program [8].
Mode frontier has the capability of multi-objective and single-objective optimization as well
as it has built-in functionality for post-processing. The variables can either be discrete or
continuous and the program also has tools to help the user decide if a local optimum, a
global optimum or a point in between should be the goal to achieve the most robust design
[15].
Optimica is an optimization tool which is based on the Modelica language, it enables high
level specification of static and dynamic optimization problems. [9]
Matlab is a high level programming language for analysis of data, algorithms, models and
applications [10]. Different toolboxes are used for different purposes, and for optimization
Matlab has an optimization toolbox with the option to do minimization through linear
programming, binary integer programming, quadratic programming and multi-objective
optimization in serial or parallel [16].
Tomlab, which is a Matlab extension, is an optimization tool with the possibility to define
abruptly changed constraints over the interval, although not recommended for robustness, as
well as it generally solves problems faster than Matlab’s built-in optimization tool. It is built
to simplify optimization of practical problems and gives access to several solvers at the same
time as Matlab can be used for constructing the main program. The Matlab toolboxes for
nonlinear programming, parameter estimation, linear optimization and discrete optimization
are the base for Tomlab and by integrating all these systems together with new solvers the
intention Tomlab has had during the development of the program has been to gather all
optimization tools in one place. This also makes it possible to not only solve the optimization
problem in one environment but also perform extensive analysis with the help of Matlab
[12].
4
A new variable class, called tomsym, is introduced in Matlab with Tomlab; a class that can be
used to generate Matlab code for further processing as well as automatically generate first and
second order derivatives. Tomsym continues to work in two dimensions even when taking
matrix derivatives of matrix functions. That implies that Matlab’s efficient handling of sparse
matrices, matrices with mostly zeros, still can be used, resulting in a faster solution [17].
There also exist a few tools with open source code that everyone can rewrite or adjust; one
example is the tool Openopt, which has several own solvers and can solve both linear and
nonlinear problems with automatic differentiation features [18].
2.3
Track boundary creation
The problem to be optimized, illustrated in Figure 1, is necessary to be formulated in terms
of equations, independent of whether those equations have discontinuous derivatives or not.
Such mathematical description of a track’s boundaries can be done mainly in two ways; either
by a set of equations describing each section that still has a continuous derivative, or by the
use of equations which are approximations of the boundary. These approximations should
then be equal to the real track at least at the positions close to where the optimal trajectory is
expected to be. If an approximate boundary is close to the resulting trajectory at some
positions, the result may be incorrect.
Corners with infinite or undefined derivative can, according to the Tomlab instructions [6],
be described with
1
𝑋𝑋
Eq. 1
𝑌𝑌 = �1 + tanh � ��
2
𝑎𝑎𝑡𝑡𝑡𝑡
where for X ≫ 𝑎𝑎𝑡𝑡𝑡𝑡 gives a behaviour that is the same as the real corner and 𝑎𝑎𝑡𝑡𝑡𝑡 dictates the
corner smoothness.
2.4
Vehicle parameters and coordinate systems
In the vehicle dynamics literature, various coordinate systems are used, with the most
dominant ones being the American SAE and the German DIN system. In this study the
DIN system is used. It is a right hand rule system where the x-axis points forwards, the y-axis
points leftwards and hence the z-axis points upwards. The roll, pitch and yaw motions of the
car, i.e. the rotational motions around the x- y- and z-axis respectively, follow the right hand
grip rule. The steering angle, denoted as 𝛿𝛿, is positive when turning towards the left. A
visualization of the DIN system is shown in the figure below [19].
5
Figure 2 - DIN coordinate system illustration.
The SAE coordinate system, differs from the DIN system in that the z- and y-axis points in
the opposite directions, that is, the y-axis points rightwards while the z-axis points
downwards. The right hand rule for the axes and the right hand grip rule for the rotational
motions apply the same.
2.5
Bicycle model
In the vehicle dynamics literature there exist various models that describe the motion of the
vehicle in space. The simplest model that captures the basic dynamic behaviour of a vehicle is
the bicycle or single-track model, developed by Riekert and Schunck in 1940 [20]. Its simplest
form is a two-degree of freedom model that captures the lateral and the yaw motion of the
vehicle. A three-degree of freedom bicycle model adds the longitudinal motion too, so as to
provide a mathematical description of the full vehicle motion in the Global X-Y plane, as
shown in Figure 3.
6
Figure 3 - Illustrated three-degree-of-freedom bicycle model with notations; it is a simplified vehicle
model that captures the vehicle's motion in the X-Y plane where the degrees of freedom are the
longitudinal, lateral and yaw motion. An index, i , is used to refer to the front and rear part of the
model [19].
The bicycle model is considered to give satisfactory results for lateral accelerations less than 4
[m/s2]. In this case the forces exerted by the tires can be linearly approximated. The
assumptions made by the bicycle model are [21]:
i.
ii.
iii.
iv.
v.
The vehicle’s dynamic behaviour is symmetrical between its right and left wheels.
Therefore, the left and right vehicle tracks are merged into one.
The vehicle’s centre of gravity (CoG) lies on the road level. This implies that there is
no change in the wheel loads during a manoeuvre, and thus no pitch or roll occurs
either.
The lateral forces produced by the tires are linearly dependent on their slip angles.
The lateral forces of the tires act exactly in the centre of their contact patch with the
road, i.e. no pneumatic trail.
The axles’ kinematics and elasto-kinematics are modelled in the tires.
The equations of motion for the bicycle model, with respect to the vehicle coordinate
system, can be derived using small angle approximation as [22] [19]:
and
𝑚𝑚�𝑣𝑣𝑥𝑥̇ − 𝜓𝜓̇𝑣𝑣𝑦𝑦 � = −𝐹𝐹12 sin(𝛿𝛿 ) ≈ −𝐹𝐹12 𝛿𝛿,
Eq. 2
𝐼𝐼𝑧𝑧 𝜓𝜓̈ = 𝑓𝑓𝐹𝐹12 cos(𝛿𝛿 ) − 𝑏𝑏𝐹𝐹34 ≈ 𝑓𝑓𝐹𝐹12 𝛿𝛿 − 𝑏𝑏𝐹𝐹34
Eq. 4
𝑚𝑚�𝑣𝑣𝑦𝑦̇ + 𝜓𝜓̇𝑣𝑣𝑥𝑥 � = 𝐹𝐹34 + 𝐹𝐹12 cos(𝛿𝛿 ) ≈ 𝐹𝐹34 + 𝐹𝐹12
Eq. 3
7
where 𝑚𝑚 and 𝐼𝐼𝑧𝑧 are the vehicle’s mass and yaw inertia respectively and on the right side of
the approximated equal sign small angle approximations, 𝛿𝛿 ≪ 1, have been applied. The
vehicle’s CoG velocities in the global coordinates 𝑋𝑋 and 𝑌𝑌 , as denoted in Figure 3, are
described by [22]:
and
𝑋𝑋̇ = 𝑣𝑣𝑥𝑥 cos(𝜓𝜓) − 𝑣𝑣𝑦𝑦 sin(𝜓𝜓)
Eq. 5
𝑌𝑌̇ = 𝑣𝑣𝑥𝑥 sin(𝜓𝜓) + 𝑣𝑣𝑦𝑦 cos(𝜓𝜓).
Eq. 6
𝐹𝐹12 = −𝐶𝐶12 𝛼𝛼12
Eq. 7
𝐹𝐹34 = −𝐶𝐶34 𝛼𝛼34
Eq. 8
As mentioned in the bicycle model assumptions, the lateral forces generated by the tires are
linearly dependent on their slip angles. This means they can be described as [22] [23] [19]:
and
where 𝛼𝛼12 and 𝛼𝛼34 , the front and rear slip angle respectively, are given by [22] [19]:
and
𝛼𝛼12 = tan−1
𝑣𝑣𝑦𝑦 +𝑓𝑓𝜓𝜓̇
𝑣𝑣𝑥𝑥
𝛼𝛼34 = tan−1
− 𝛿𝛿 ≈
𝑣𝑣𝑦𝑦 −𝑏𝑏𝜓𝜓̇
𝑣𝑣𝑥𝑥
≈
𝑣𝑣𝑦𝑦 +𝑓𝑓𝜓𝜓̇
𝑣𝑣𝑥𝑥
𝑣𝑣𝑦𝑦 −𝑏𝑏𝜓𝜓̇
𝑣𝑣𝑥𝑥
− 𝛿𝛿
Eq. 9
.
Eq. 10
When Eq. 9 and Eq. 10 are written in the form of the right side of the approximated equal
sign, small angle approximations have been applied. 𝐶𝐶12 and 𝐶𝐶34 are the front and rear
cornering stiffness respectively. The minus sign in Eq. 9 and Eq. 10 is added for convention
purposes, such that the cornering stiffness is defined as a positive value [23].
The bicycle model is a simple mathematical description but its practicality is more than just
academic, where it is used to introduce the basic vehicle dynamic concepts. It is found also in
practical applications determining vehicle attributes like under- and over-steering, stability,
peak response time and even for calculating the vehicle’s desired yaw rate during a
manoeuvre. This desired yaw rate can be compared to the vehicle’s actual/measured yaw rate
and then corrections can be applied by a stability system.
8
2.6
Four wheel vehicle model
The theory presented in this chapter is based on “Dynamik der Kraftfahrzeuge” [21]. When a
more detailed study of the vehicle’s behaviour is desired, than that possible with the bicycle
model, then a more detailed model needs to be utilized, with as many degrees of freedom as
necessary to capture the desired behaviour. The two-track model is such a model and was the
one used in the current study. The two-track model extends the bicycle model in the
following basic ways:
i.
ii.
There is no symmetry between the left and right wheels of the vehicle regarding its
dynamic behaviour. This allows for the modelling of the different steering angles
between the left and right wheel (normally invoked by the Ackermann geometry) and
therefore the modelling of different slip angles between the right and left wheels.
The vehicle’s centre of gravity lies at a certain height above the ground. This
introduces changes in the wheel loads during a manoeuvre (load transfer), as well as
pitch and roll motions. Also, the centre of gravity can be modelled and positioned at a
certain lateral position with respect to the vehicle’s longitudinal symmetry axis. This
introduces a difference in the static load between the left and right wheels and the
normal load on each wheel can then be calculated by [19] [24]
𝐹𝐹𝑧𝑧𝑧𝑧𝑧𝑧 =
𝐹𝐹𝑧𝑧𝑧𝑧𝑧𝑧 =
and
𝐹𝐹𝑧𝑧𝑧𝑧𝑧𝑧 =
𝐹𝐹𝑧𝑧𝑧𝑧𝑧𝑧 =
𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚2
𝐿𝐿𝐿𝐿
−
𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚 2
𝐿𝐿𝐿𝐿
𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚2
𝐿𝐿𝐿𝐿
+
𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚 2
𝐿𝐿𝐿𝐿
−
𝑡𝑡
𝑚𝑚ℎ2
𝐿𝐿𝐿𝐿
𝑡𝑡
𝑚𝑚ℎ2
𝐿𝐿𝐿𝐿
𝑡𝑡
𝑚𝑚ℎ2
+
𝐿𝐿𝐿𝐿
𝑎𝑎𝑥𝑥 −
𝑎𝑎𝑥𝑥 −
𝑡𝑡
𝑚𝑚ℎ2
𝐿𝐿𝐿𝐿
𝑎𝑎𝑥𝑥 +
𝑚𝑚
𝑡𝑡
𝑚𝑚
𝑎𝑎𝑥𝑥 +
𝑡𝑡
𝐾𝐾𝑓𝑓 +𝐾𝐾𝑟𝑟 −𝑚𝑚𝑚𝑚ℎ𝑒𝑒
𝑚𝑚
𝑡𝑡
ℎ𝑒𝑒 𝐾𝐾𝑟𝑟
�
𝐾𝐾𝑓𝑓 +𝐾𝐾𝑟𝑟 −𝑚𝑚𝑚𝑚ℎ𝑒𝑒
�
ℎ𝑒𝑒 𝐾𝐾𝑟𝑟
Eq. 11
𝑏𝑏
Eq. 12
𝑓𝑓
Eq. 13
𝐿𝐿
ℎ𝑒𝑒 𝐾𝐾𝑓𝑓
�
𝑏𝑏
+ 𝑒𝑒𝑓𝑓 � 𝑎𝑎𝑦𝑦 ,
𝐾𝐾𝑓𝑓 +𝐾𝐾𝑟𝑟 −𝑚𝑚𝑚𝑚ℎ𝑒𝑒
𝑡𝑡
𝑚𝑚
ℎ𝑒𝑒 𝐾𝐾𝑓𝑓
�
+ 𝑒𝑒𝑓𝑓 � 𝑎𝑎𝑦𝑦 ,
𝐿𝐿
+ 𝑒𝑒𝑟𝑟 � 𝑎𝑎𝑦𝑦 ,
𝐾𝐾𝑓𝑓 +𝐾𝐾𝑟𝑟 −𝑚𝑚𝑚𝑚ℎ𝑒𝑒
𝐿𝐿
𝑓𝑓
+ 𝑒𝑒𝑟𝑟 � 𝑎𝑎𝑦𝑦
𝐿𝐿
Eq. 14
where 𝐾𝐾𝑓𝑓 is the roll stiffness at the front, 𝐾𝐾𝑟𝑟 is the roll stiffness at the rear 2, 𝑒𝑒𝑓𝑓 is the height
of the roll centre at the front, 𝑒𝑒𝑟𝑟 the height of the roll centre at the rear. ℎ𝑒𝑒 is the distance of
the centre of gravity from the roll axis, given by
ℎ𝑒𝑒 = ℎ −
𝑓𝑓∙𝑒𝑒𝑏𝑏 +𝑏𝑏∙𝑒𝑒𝑓𝑓
𝐿𝐿
.
Eq. 15
The term “roll stiffness” includes not only the stiffness imposed by antiroll bars, but also from the suspension
geometry, springs, frame, and all the factors that contribute to the axle’s total roll stiffness in general.
2
9
Figure 4. Two track model with notations [25]; index i is front or rear as for the bicycle model and
index j is left or right when referring to one of the vehicle’s wheels or corners.
The equations of motion in the case of the two-track front wheel steer vehicle, Figure 4, are
given by
𝑚𝑚�𝑣𝑣𝑥𝑥̇ − 𝜓𝜓̇𝑣𝑣𝑦𝑦 � =
Eq. 16
𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑜𝑜 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑖𝑖 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑜𝑜 ,
𝑚𝑚�𝑣𝑣𝑦𝑦̇ + 𝜓𝜓̇𝑣𝑣𝑥𝑥 � =
𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 sin 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 sin 𝛿𝛿𝑜𝑜 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 cos 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 cos 𝛿𝛿𝑜𝑜 ,
𝐼𝐼𝑧𝑧 𝜓𝜓̈ =
𝑓𝑓�𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 cos 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 cos 𝛿𝛿𝑜𝑜 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 sin 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 sin 𝛿𝛿𝑜𝑜 � −
𝑏𝑏(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) + 𝑊𝑊𝑙𝑙 (𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑖𝑖 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑖𝑖 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) +
𝑊𝑊𝑟𝑟 (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑜𝑜 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑜𝑜 )
and the wheel rotation dynamics by
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑖𝑖𝑖𝑖 = 𝑇𝑇𝑖𝑖𝑖𝑖 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 𝑟𝑟.
Eq. 17
Eq. 18
Eq. 19
Translation between the local and global coordinate systems can still be performed with Eq.
5 and Eq. 6. The calculation of the forces generated by the tires depends on the tire model
that is used. If a linear approximation is used, then the calculation is the same as in the case
10
of the bicycle model described in sections 2.5 and 2.7.2. In the case of a magic formula tire
model the force calculation is conducted as described in section 2.7.3. Other details can be
added if desired, as:
i.
ii.
More detailed tire modelling: The tires can be modelled by utilizing models like the
Dugoff model, the Brush model or the Magic Formula model [23]. These models
capture the tire behaviour also in the nonlinear tire region, which makes them suitable
for cases where the lateral accelerations exceed 0.4g. Also, the lateral force can be
modelled to act at a point different than the centre of the tire-road contact patch, i.e.
the pneumatic trail.
Suspension kinematics and compliance phenomena can also be modelled, and not
included in the tire modelling, like in the bicycle model case.
As the details captured by the vehicle model increase, a significant computational cost is also
introduced, and this is the main disadvantage of the two-track model. This model is most
often used when performing suspension analysis/design or ride and comfort studies since it
captures load transfer, roll and pitch motions that primarily affect those factors.
2.7
Tire behaviour
In this chapter the most fundamental concepts and principles of tire behaviour are presented.
It is intended to serve as an introduction to the tire related modelling details that have been
used in this study rather than a complete description of all existing models, which could have
been used.
2.7.1
Slip angle
When a pneumatic tire is subjected to a lateral load while rolling, its sidewalls deform, and it
deflects along a direction different than that defined by the wheel plane [23]. This difference
in the directions between the wheel’s plane and the velocity vector of the wheel’s centre is
called “slip angle 3 ”, see Figure 5. The slip angle is intertwined with the lateral tire force
generation that is acting on the wheel, called “cornering force”, which causes changes in the
vehicle direction during the manoeuvre.
The tire’s slip angle is different than the vehicle’s slip angle (also called sideslip angle or float angle), which is the angle
between the longitudinal axis of the vehicle and the velocity vector acting on its centre of gravity, or 𝛽𝛽 = tan−1 (𝑣𝑣𝑦𝑦 /𝑣𝑣𝑥𝑥 ),
where 𝑣𝑣𝑦𝑦 and 𝑣𝑣𝑥𝑥 are the vehicle’s lateral and longitudinal velocity respectively [20] [52].
3
11
Figure 5. Slip angle definition [23].
A non-zero slip angle arises because of deformation in the tire. As the tire rotates, the
friction between the contact patch and the road results in individual tread elements remaining
stationary with respect to the road. This tire deflection gives rise to the slip angle and the
cornering force [26].
2.7.2
Linear tire model
For small slip angles the lateral force increases approximately linearly for the first few degrees
of slip, called the elastic region, and then increases non-linearly to a maximum before
beginning to decrease. A typical diagram relating the lateral force to the wheel slip angle is
presented below in Figure 6.
Figure 6. For small slip angles the lateral force is linearly proportional to the slip angle. The figure
shows a typical example of this relationship. The numbers and regions noted above depend on
factors like tire size, tire pressure and normal load [23].
12
This linear relationship can be formulated as
Eq. 20
𝐹𝐹𝑦𝑦 = −𝐶𝐶𝑎𝑎 ∙ 𝑎𝑎
where 𝐶𝐶𝑎𝑎 is called the cornering stiffness of the tire, defined as [23]
𝜕𝜕𝐹𝐹𝑦𝑦
𝐶𝐶𝑎𝑎 = − �
�
𝜕𝜕𝜕𝜕 𝑎𝑎=0
.
Eq. 21
It is noteworthy to emphasize that this relationship holds only for small slip angles, as shown
also in Figure 6.
2.7.3
Non-linear tire model
While the linear tire model gives satisfactory results for small slip angles, where the tire is
considered to be in its linear region 4, it needs to be either improved or replaced for larger slip
angles. Many tire models can be found in the literature, and one of the most well-known is
the “magic formula” model, which is an empirically derived curve fit. It utilizes a
combination of trigonometric functions rather than a polynomial, which is a more usual
fitting method, and it is the result of studies carried by Egbert Bakker and Lars Lidner and
Hans B. Pacejka [27]. Largely owing its popularity to the fact that its equations were made
public, the Magic Formula has very quickly been adopted from the industry as a standard tire
model for vehicle handling simulations. Since its first appearance in 1987 various
modifications have been made to improve the accuracy and to extend the capabilities of the
model, and include factors like the camber angle, the tire inflation pressure, the rolling
resistance and the overturning moment, and depending on the vehicle type or the
phenomena that one wishes to study, each version may better apply to some certain
situations [20][21].
The most used version of the formula is
𝑓𝑓(𝑢𝑢) = 𝐷𝐷 sin�𝐶𝐶 tan−1 �𝐵𝐵𝐵𝐵 − 𝐸𝐸 (𝐵𝐵𝐵𝐵 − tan−1 (𝐵𝐵𝐵𝐵))��
𝐹𝐹 (𝑈𝑈) = 𝑓𝑓(𝑢𝑢) + 𝑆𝑆𝑣𝑣
Eq. 22
𝑢𝑢 = 𝑈𝑈 + 𝑆𝑆ℎ
where F(U) represents the output, that is the lateral or longitudinal force, or the self-aligning
torque and U denotes the input, that is the slip angle 𝛼𝛼 or the longitudinal slip 𝜅𝜅 [26].
4
The linear region is usually considered for up to 0.4𝑔𝑔 [22] [43].
13
A simplified version of Eq. 22 also exists, which is an early version of it, and gives a relation
between the friction coefficient, 𝜇𝜇 , and the resultant tire slip, 𝑠𝑠 , at a given tire. The
expression holds as follows
Eq. 23
𝜇𝜇(𝑠𝑠) = 𝐷𝐷 sin(𝐶𝐶 tan−1 (𝐵𝐵𝐵𝐵))
where 𝐷𝐷, 𝐶𝐶 and 𝐵𝐵 are the peak value factor, shape factor and stiffness factor respectively.
The product 𝐵𝐵 ∙ 𝐶𝐶 ∙ 𝐷𝐷 ∙ 𝐹𝐹𝑧𝑧 , where 𝐹𝐹𝑧𝑧 is the normal load on the tire, represents the cornering
stiffness 𝐶𝐶𝑖𝑖𝑖𝑖 of the tire with index ij [28] [29].
The resultant slip, 𝑠𝑠, is given by
𝑠𝑠 = �𝑠𝑠𝑥𝑥 2 + 𝑠𝑠𝑦𝑦 2
Eq. 24
where the indices x and y denote the longitudinal and lateral slip respectively [28] [22] [23].
From the friction coefficient 𝜇𝜇, assuming linear dependence of the tire friction forces on the
tire vertical force, one can calculate the resultant friction force on the plane of the road
surface as
𝑓𝑓 = �𝑓𝑓𝑥𝑥 2 + 𝑓𝑓𝑦𝑦 2 = 𝜇𝜇𝑓𝑓𝑧𝑧
Eq. 25
𝜇𝜇 = �𝜇𝜇𝑥𝑥 2 + 𝜇𝜇𝑦𝑦 2
Eq. 26
where the friction coefficient is related to its longitudinal and lateral component according to
and the lateral and longitudinal forces are then obtained from Eq. 27 [28] [22] [23].
𝑠𝑠𝑥𝑥
𝑠𝑠
𝜇𝜇𝑓𝑓𝑧𝑧 = − 𝑥𝑥 𝑓𝑓
𝑠𝑠
𝑠𝑠
𝑠𝑠𝑦𝑦
𝑠𝑠𝑦𝑦
𝑓𝑓𝑦𝑦 = 𝜇𝜇𝑦𝑦 𝑓𝑓𝑧𝑧 = − 𝜇𝜇𝑓𝑓𝑧𝑧 = − 𝑓𝑓
𝑠𝑠
𝑠𝑠
𝑓𝑓𝑥𝑥 = 𝜇𝜇𝑥𝑥 𝑓𝑓𝑧𝑧 = −
Eq. 27
Assuming the above behaviour, the tire’s force potentials move inside a so called “friction
circle”, or in general a “friction ellipse”, as indicated by Eq. 26 and illustrated in Figure 7.
14
Figure 7. Illustration [30] of the friction circle, the total force can never exceed a certain limit [24]
[31].
This means that a tire has a limit to the lateral force that it can produce for a given value of
the longitudinal force and vice versa, such that the resultant force never reaches out the
friction circle, which is a characteristic of each tire [28] [22] [23].
2.7.4
Transient tire properties
As mentioned above, the deformation of the tire during a cornering manoeuvre gives rise to
a slip angle which results in a lateral force generation on the tire. This lateral force will not
appear instantly on the tire though. A delay between the slip angle application and the lateral
force generation is always present. After a steering angle is imposed, the wheel needs to travel
a certain amount of length, during which the tire will gradually begin to deform and build up
the lateral force up to its steady state value. This length, 𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 , is called relaxation length and it
is a characteristic property of the pneumatic tires. Half to one wheel turn is a typical amount
of roll needed for the force to build up to its steady state value [22] [23] [31].
A way to define this transient behaviour is through the first order differential equation
𝜏𝜏𝑓𝑓𝑦𝑦̇ (𝑎𝑎, 𝑡𝑡) + 𝑓𝑓𝑦𝑦 (𝑎𝑎, 𝑡𝑡) = 𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎)
Eq. 28
where 𝜏𝜏 is a time constant and 𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦 is the steady state value of the lateral force for a given slip
angle 𝑎𝑎 . The time constant is related to the relaxation length as
𝜏𝜏 =
𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝑉𝑉𝑥𝑥
Eq. 29
15
where 𝑉𝑉𝑥𝑥 is the tire’s longitudinal velocity. When reading equation Eq. 29 the relaxation
length is taken as a constant value, a characteristic of the tire. But in reality the relaxation
length is dependent on the level of slip, more specifically, the higher the slip angle, the
shorter the relaxation length becomes [26].
2.7.5
Pneumatic trail
When a pneumatic tire is subjected to a lateral load while rolling, as it is in the case of
cornering, it will produce lateral forces throughout the whole length of its contact patch. Due
to the asymmetrical deformation of the tire, these tire forces will also be asymmetrically
distributed and thus the resultant lateral force will be a force applied to some distance behind
the centre of the contact patch 5. This distance is called the pneumatic trail and for low lateral
accelerations, linear tire region, the pneumatic trail is almost constant 6. For higher lateral
accelerations (large slip angles/nonlinear tire region) the tire deformation becomes more
symmetric, and thus the pneumatic trail becomes smaller reducing to almost zero, and in
some cases it might even change sign [23] [24] [32] [26].
Figure 8. The asymmetrical deformation of the tire in low slip angles results in a lateral force F y
acting at a distance tp behind the centre of the contact patch. This distance is called the pneumatic
trail [23].
2.7.6
Camber influence
The angle between the vertical plane and the plane defined by the wheel is termed as camber
angle; it can be measured either with respect to the vehicle body or with respect to the
ground as shown in Figure 9. In the first case the camber angle is denoted as 𝜀𝜀 and it is
considered positive when the top of the wheel tilts outwards from the vehicle body.
This phenomenon, of a force applied at a certain distance, gives rise to a moment. This moment tends to steer the
wheel back to its straight position, and is therefore called aligning moment. The aligning moment is an important concept
of the steering design [17] [19] [22].
6 In this model, a typical value for the pneumatic trail at low lateral accelerations was used, equal to approximately 30mm
[30].
5
16
Figure 9. Positive camber angle definitions with respect to the vehicle body and the ground, as seen
from behind. The lateral force generation due to camber is also shown. This force acts toward the
direction of the apex of the virtual cone defined by the wheel [19].
This means that when both the left and right wheel have positive camber 𝜀𝜀 they are tilted
towards opposite directions. In the second case, when the camber angle is defined with
respect to the ground – also termed as inclination angle of the wheel, it is denoted as 𝛾𝛾 and it
is considered positive when the wheel is tilted in the clockwise direction as seen from behind
[23] [24] [31] [19] [26]. This time, when the wheels of an axle have positive inclination angle,
they both tilt toward the same direction. It is often that the wheels on a vehicle have a pre-set
camber angle so that some desired ride and handling characteristics, like for example
performance and safety, are achieved [26] [33]. This camber pre-set, which the wheels have
when at rest, is called static camber.
While manoeuvring the vehicle will be subjected to wheel steer, body roll and suspension
jounce/rebound movements. These factors affect the camber angle. This change to the
camber angle due to wheel steer and suspension travel (body roll), which depends on the
suspension design, is called camber gain. The camber of the wheel is important since it
generates a force towards the direction of the tilt when the wheel rolls, as shown also in
Figure 9 above, termed as camber thrust [24]. The lateral force due to camber is much lower
than the one due to the slip angle, but it does nevertheless add to the lateral force due to slip
and can play an important role for handling characteristics like understeer and oversteer [19].
It is therefore important that camber relative to the ground is always as near as possible to its
optimal value (taken care by the camber gain of the suspension), so that the desired handling
behaviour is achieved, while also other factors like braking ability or tire wear are not
compromised. An interesting study on this camber optimization can be found in [33].
The lateral force due to camber can be calculated as [19] [31]
𝐹𝐹𝛾𝛾 = −𝐶𝐶𝛾𝛾 ∙ 𝛾𝛾
Eq. 30
17
where 𝐶𝐶𝛾𝛾 is called the camber stiffness and the minus sign is a convention such that 𝐶𝐶𝛾𝛾 is
positive. The contribution of the camber angle to the lateral force is complex; in the linear
region the camber thrust and lateral force due to slip can be considered as separate and
additive effects, but when the tire enters its nonlinear region the camber thrust’s additive
effect decreases, a behaviour termed as “camber roll-off” [24]. In recent improvements in the
magic formula equation, camber is one of the factors that can be taken into consideration by
the formula [34] [26].
2.8
Electronic stability control and its functionalities
The electronic stability control (ESC) is a system used to prevent the vehicle’s path to deviate
from the desired path [22]. ESC may also be referred to as yaw stability control systems and
automotive manufacturers use numerous other branded names. The yaw rate relationship to
the steering wheel angle is very different for small and large slip angles. Manoeuvrability is
lost at different steering angles for different surfaces [22]. The vehicle’s yaw rate is controlled
by the steering wheel angle, and could pose difficulty for the driver to utilize the maximum
available physical adhesion between the tires and the road [35] [36]. At the same time the
need for driving stability motivates the ESC principle [22].
The ESC can influence the yaw rate by adapting the braking torque and steering angle that is
applied to each wheel. ESC can apply differential braking to generate yaw moment; those
systems can also utilize torque vectoring to independently control the drive torque. In Steerby-wire systems the steering angle can be controlled by the ESC system [22].
To employ such control, the vehicle needs to be equipped with a yaw rate sensor, a lateral
acceleration sensor, wheel speed sensors and a steering wheel angle sensor. Assuming that
certain vehicle and environment properties are known (friction coefficient, tire cornering
stiffness, etc...), the sensors make it possible to calculate the desired slip angle as described by
Eq. 31 [22].
𝑓𝑓𝑓𝑓𝑉𝑉 2
2𝐶𝐶𝛼𝛼𝛼𝛼 (𝑓𝑓 + 𝑏𝑏)
𝛽𝛽𝑑𝑑 =
𝛿𝛿𝑠𝑠𝑠𝑠
𝑚𝑚𝑉𝑉 2 �𝑓𝑓𝐶𝐶𝛼𝛼𝛼𝛼 − 𝑏𝑏𝐶𝐶𝛼𝛼𝛼𝛼 �
(𝑓𝑓 + 𝑏𝑏) +
2𝐶𝐶𝛼𝛼𝛼𝛼 𝐶𝐶𝛼𝛼𝛼𝛼 (𝑓𝑓 + 𝑏𝑏)
𝑏𝑏 −
Eq. 31
It is also possible to calculate the steady state relation between the steering angle and the
radius of the vehicle trajectory according to
1
=
𝑅𝑅
18
(𝑓𝑓 + 𝑏𝑏) +
𝛿𝛿𝑠𝑠𝑠𝑠
𝑚𝑚𝑉𝑉 2 �𝑓𝑓𝐶𝐶𝛼𝛼𝛼𝛼 − 𝑏𝑏𝐶𝐶𝛼𝛼𝛼𝛼 �
2𝐶𝐶𝛼𝛼𝛼𝛼 𝐶𝐶𝛼𝛼𝛼𝛼 (𝑓𝑓 + 𝑏𝑏)
Eq. 32
which implies that the desired yaw rate is given by Eq. 33 [22]
𝜓𝜓𝑑𝑑 =
𝑥𝑥̇
=
𝑅𝑅
(𝑓𝑓 + 𝑏𝑏) +
and the yaw acceleration is given by Eq. 18.
𝑥𝑥̇ 𝛿𝛿𝑠𝑠𝑠𝑠
𝑚𝑚𝑥𝑥̇ 2 �𝑓𝑓𝐶𝐶𝛼𝛼𝛼𝛼 − 𝑏𝑏𝐶𝐶𝛼𝛼𝛼𝛼 �
2𝐶𝐶𝛼𝛼𝛼𝛼 𝐶𝐶𝛼𝛼𝛼𝛼 (𝑓𝑓 + 𝑏𝑏)
Eq. 33
The friction coefficient of the road influences the yaw rate that the vehicle can develop. An
upper limit for the yaw rate that the controller can achieve is given by
ẋ Ψ̇ − tan(𝛽𝛽 ) 𝑥𝑥̈ +
𝑥𝑥̇ 𝛽𝛽̇
�1 + 𝑡𝑡𝑡𝑡𝑡𝑡2 (𝛽𝛽)
≤ 𝜇𝜇𝜇𝜇
Eq. 34
according to [22]. The next step for the controller is to determine the desired yaw torque to
track the target yaw rate and slip angle [22]. The objective of tracking yaw rate and slip angle
can be done with sliding mode control design, where the sliding surface is chosen so either
the yaw rate, the slip angle or a combination of them is tracked [37] [38] [39] [40]. Rajamani
[22] suggests the use of the formula given in Eq. 35, and also suggests the reader who wants
an introduction to the subject to look further in the text by Slotine and Li [41].
Eq. 35
𝑠𝑠 = Ψ̇ − Ψ̇𝑑𝑑 + 𝜉𝜉(𝛽𝛽 − 𝛽𝛽𝑑𝑑 )
In Eq. 35, 𝜉𝜉 is a weighting factor for the slip angle contribution, 𝛽𝛽 is the sideslip angle, 𝛽𝛽𝑑𝑑 is
the desired sideslip angle and 𝑠𝑠 can be seen as a surface. If one can ensure that the vehicle
response converges to 𝑠𝑠 = 0 the desired yaw rate and slip angle are obtained. The
differentiation of Eq. 35 is then given by Eq. 36 [22].
𝑠𝑠̇ = Ψ̈ − Ψ̈𝑑𝑑 + 𝜉𝜉(𝛽𝛽̇ − 𝛽𝛽𝑑𝑑̇ )
Eq. 36
By assuming a small steering angle, a fixed brake ratio, 𝜌𝜌, between the front and rear wheel
on each side of the car, the yaw acceleration is given by Eq. 38 with the yaw torque generated
by the brakes defined by Eq. 37.
𝑙𝑙𝑤𝑤
Eq. 37
𝑀𝑀Ψ𝑏𝑏 = �𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 �
2
Ψ̈ =
1
�𝑓𝑓�𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � cos(𝛿𝛿 ) − 𝑏𝑏�𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � + (cos(𝛿𝛿 ) + 𝜌𝜌)𝑀𝑀Ψ𝑏𝑏 �
𝐼𝐼𝑧𝑧
Eq. 38
and setting 𝑠𝑠̇ = −𝜂𝜂𝜂𝜂 when substituting for Ψ̈ it yields the control law in Eq. 39 [22].
19
𝑀𝑀Ψ𝑏𝑏
𝑏𝑏(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) 𝑓𝑓(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 )
−
− 𝜂𝜂𝜂𝜂 + Ψ̈𝑑𝑑 − 𝜉𝜉(𝛽𝛽̇ − 𝛽𝛽𝑑𝑑̇ )
𝐼𝐼𝑧𝑧
𝐼𝐼𝑧𝑧
=
𝜌𝜌 + cos(𝛿𝛿)
𝐼𝐼𝑧𝑧
Eq. 39
Estimations for the sideslip angle, sideslip angle derivative and each of the lateral tire forces
are needed, and for the interested reader some literature [42] [36] [43] [44] is suggested. After
the desired torque, 𝑀𝑀Ψ𝑏𝑏 , produced by the differential braking has been calculated, the brake
pressure can be calculated as the torque produced by differential braking is directly coupled
to the dynamics of the wheels. If only the front wheels are used for braking the resulting
equations for left, 𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏 , and right, 𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏 , brake pressure can be seen in Eq. 40 and Eq. 41
respectively [22].
𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏
𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏
2𝑀𝑀
� Ψ𝑏𝑏 � 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒
𝑙𝑙𝑤𝑤
= 𝑃𝑃0 − 𝑎𝑎
𝐴𝐴𝑤𝑤 𝜇𝜇𝑏𝑏 𝑅𝑅𝑏𝑏
2𝑀𝑀
� Ψ𝑏𝑏 � 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒
𝑙𝑙𝑤𝑤
= 𝑃𝑃0 + (1 − 𝑎𝑎)
𝐴𝐴𝑤𝑤 𝜇𝜇𝑏𝑏 𝑅𝑅𝑏𝑏
Eq. 40
Eq. 41
If small steering angles cannot be assumed, the yaw torque that should be generated by the
brakes needs to be calculated from each force for each wheel in Eq. 18.
2.9
Vehicle parameter estimation
The procedure of performing a parameter estimation of the tire properties can be performed
in three steps which are described in this chapter and are all based on the paper “An
enhanced generic single track vehicle model and its parameter identification for 15 different
passenger cars” [45].
Circular driving test – Effective cornering stiffness estimation
The vehicle’s effective cornering stiffness 7, for the front and rear axle, can be calculated from
the results of circular driving tests. The necessary data for calculating the effective cornering
stiffness are the vehicle’s lateral acceleration, forward velocity, lateral velocity, yaw rate and
steering wheel angle. The bicycle model, presented in section 2.5, is used for simplicity in the
calculations.
By the term “effective” cornering stiffness the contribution of additional phenomena to the axle’s cornering stiffness is
described. An axle’s cornering stiffness does not depend only on the tires but also on phenomena like camber forces, tire
load sensitivity, steering elasticity and compliance steer, all of which contribute to the reduction of the cornering stiffness
imposed by the tires alone [26] [17].
7
20
As a first step, equation Eq. 60 and Eq. 61 are used to calculate the slip angle of the front
and rear wheel. The semi steady-state test assumes 𝜓𝜓̈ = 0 , since the turning radius is
approximately the same. Then, from Eq. 3 and Eq. 4, the lateral forces on the front and rear
axle are calculated as
and
𝐹𝐹12 = 𝑚𝑚𝛼𝛼𝑦𝑦
𝑏𝑏
Eq. 42
𝑓𝑓
Eq. 43
𝐿𝐿
𝐹𝐹34 = 𝑚𝑚𝛼𝛼𝑦𝑦 .
𝐿𝐿
Figure 10. Lateral forces during cornering [19].
Next, the lateral forces, as can be seen in Figure 10, at the front and rear are plotted against
the respective slip angles, resulting in a graph similar to Figure 6. The gradient of the curve
for zero slip angle is the axle’s effective cornering stiffness [23] [24], that is
𝜕𝜕𝐹𝐹𝑦𝑦
as also mentioned in Eq. 21.
𝐶𝐶𝑎𝑎 = �
�
𝜕𝜕𝜕𝜕 𝑎𝑎=0
Eq. 44
Pseudorandom steering test – Yaw inertia calculation
The yaw inertia of the vehicle can be calculated by the results of a pseudorandom steering
test. From this test the transfer functions from the measured steering angle to measured
lateral acceleration and yaw rate can be obtained by calculating the fraction of the fast
Fourier transform (FFT) of the output signal (lateral acceleration or yaw rate) over the FFT
of the input signal (steering angle). From the bicycle model the expressions in Eq. 45 and
Eq. 46, for analytically calculating these transfer functions, are obtained [19]:
21
𝛹𝛹̇
𝛥𝛥
and
𝐴𝐴𝑌𝑌
𝛥𝛥
=
=
𝐿𝐿
𝑓𝑓𝐶𝐶12 𝑚𝑚𝑚𝑚+𝑣𝑣 𝐶𝐶12 𝐶𝐶34
𝑥𝑥
�𝑓𝑓2 𝐶𝐶12 +𝑏𝑏2 𝐶𝐶34 �𝑚𝑚+𝐼𝐼𝑧𝑧 (𝐶𝐶12 +𝐶𝐶34 ) 𝐿𝐿2 𝐶𝐶12 𝐶𝐶34
2
𝑚𝑚𝐼𝐼𝑧𝑧 𝑠𝑠 +
𝑠𝑠+
+𝑚𝑚(𝑏𝑏𝐶𝐶34 −𝑓𝑓𝐶𝐶12 )
𝑣𝑣𝑥𝑥
𝑣𝑣2
𝑥𝑥
𝑣𝑣𝑥𝑥2
𝐿𝐿+𝐾𝐾𝑢𝑢𝑢𝑢 𝑣𝑣𝑥𝑥2
with the understeer gradient
∙
𝑏𝑏
𝐼𝐼
1+𝑣𝑣 𝑠𝑠+𝐿𝐿𝐶𝐶𝑧𝑧 𝑠𝑠 2
𝑥𝑥
34
�𝑓𝑓2 𝐶𝐶12 +𝑏𝑏2 𝐶𝐶34 �𝑚𝑚+𝐼𝐼𝑧𝑧 (𝐶𝐶12 +𝐶𝐶34 )
𝑚𝑚𝐼𝐼𝑧𝑧 𝑣𝑣2
𝑥𝑥
1+𝑣𝑣𝑥𝑥
𝑠𝑠+
𝑠𝑠 2
2
𝐿𝐿𝐶𝐶12 𝐶𝐶34 (𝐾𝐾𝑢𝑢𝑢𝑢 𝑣𝑣𝑥𝑥 +𝐿𝐿)
𝐿𝐿𝐶𝐶12 𝐶𝐶34 (𝐾𝐾𝑢𝑢𝑢𝑢 𝑣𝑣2
𝑥𝑥 +𝐿𝐿)
𝐾𝐾𝑢𝑢𝑢𝑢 =
𝑚𝑚(𝑏𝑏𝐶𝐶34 −𝑓𝑓𝐶𝐶12 )
𝐿𝐿𝐶𝐶12 𝐶𝐶34
.
Eq. 45
Eq. 46
Eq. 47
The yaw inertia, 𝐼𝐼𝑧𝑧 , value is then calculated by an optimization, where the error between the
transfer functions calculated from the measurements and the transfer functions calculated
analytically from the bicycle model is minimized by varying the 𝐼𝐼𝑧𝑧 value accordingly.
Magic formula coefficients
Next, the D value of the magic formula model for the tires can be calculated from a severe
lane change manoeuvre test, like the double lane change. A number of lane change
manoeuvres are conducted and recorded. For the coefficients calculation the manoeuvre with
the highest achieved lateral acceleration is selected. From that manoeuvre, the peak lateral
force of the tires is obtained and therefore the D value of the magic formula is found (since
D defines the peak value of the magic formula curve). The C value can be assumed to be 1 8,
which means that the curve of the magic formula does not fall for large slip angles (i.e. the
peak value force is not decreasing for large slip angles) which then leads to the calculation of
the B value as
where 𝐶𝐶𝐹𝐹𝐹𝐹 is the tire cornering stiffness.
𝐵𝐵 =
𝐶𝐶𝐹𝐹𝐹𝐹
𝐶𝐶𝐶𝐶
.
Eq. 48
The C = 1 assumption facilitates an approximation of the magic formula curve for the tire. In general, C can be
determined by the use of regression procedures [51].
8
22
3 Methodology
The problem of optimizing the steering control input generation for a vehicle's entry speed
maximization in a double-lane change manoeuvre can be divided into the following smaller
problems:
•
Double-lane change track modelling
•
Point mass trajectory optimization
•
Bicycle model trajectory optimization
•
Four wheel vehicle model trajectory optimization
•
Electronic stability control implementation
•
Model refinement
•
Complete vehicle model optimal iteration process procedure
One of the first steps was to select the optimization tool to use. Matlab [10] was selected for
pre- and post-processing and Tomlab [11] was selected for the optimization process. Tomlab
was seen as an appropriate choice, since it offered its functionalities within Matlab. A general
coding layout to be used in all models used in this study can be seen in Appendix F – Code
structure.
A parameter study, in which “key” vehicle properties were altered, offered a study on the
robustness of the method and demonstrated the way the parameters can influence the entry
speed, as well as how close to the final result the initial guess needs to be. Visualization was
also made by animating the vehicle, making it possible to examine the vehicle’s movement at
any position and take a closer look at some parts. This type of evaluation is important since
the car body should be discretized in as few points as possible to make the numerical
solution converge faster, but between two discrete points it is possible that the car hits a
corner of the track.
Further evaluation of the results with a driving robot in a real car was also done; a first test to
improve the realism of the vehicle dynamics model, and a second test to investigate the
performance of the method. When problems were detected, some of them could be
corrected directly during the testing, while other problems led to knowledge about what to
improve in the model.
23
3.1
Double lane-change track modelling
Initially, the track was modelled as an exact representation of Figure 1, with the global
coordinates X, representing the length of the track, and Y, describing its width, the vehicle
must then be constrained by Eq. 49.
𝐴𝐴
⎧ − ,
2
⎪
𝐴𝐴
+ 1,
⎨2
⎪ 𝐴𝐴
⎩ 2 − 3,
𝐴𝐴
𝑋𝑋 < 25.5 ⎫
,
⎧
2
⎪
⎪
𝐴𝐴
25.5 ≤ 𝑋𝑋 ≤ 36.5 ≤ 𝑌𝑌 ≤
+ 1 + 𝐵𝐵,
⎬
⎨2
𝐴𝐴
⎪
⎪
36.5 < 𝑋𝑋 ≤ 61 ⎭
,
⎩
2
𝑋𝑋 ≤ 12
⎫
⎪
Eq. 49
12 < 𝑋𝑋 < 49
⎬
⎪
49 ≤ 𝑋𝑋 ≤ 61 ⎭
Constraints with discontinuous derivatives often need more iteration with a numerical solver
compared to if they had continuous derivatives, and the problem may not be solved at all. To
avoid this problem Eq. 1 was implemented in Eq. 49 around each of the abrupt changes with
atr = 0.1 to create a large curvature on each corner, see Figure 11. The new constraints to
limit the car’s movements are given by Eq. 50 and the rounded corners were placed so that
the original corner point was still covered by the constraint.
1 + 𝐴𝐴
𝑋𝑋 − 25.2
𝐴𝐴 1 + 𝐶𝐶
𝑋𝑋 − 36.8
�1 + tanh �
�� − +
�1 − tanh �
�� − 1 + 𝐶𝐶
2
𝑎𝑎𝑡𝑡𝑡𝑡
2
2
𝑎𝑎𝑡𝑡𝑡𝑡
≤ 𝑌𝑌 ≤
𝑋𝑋 − 12.3
𝐴𝐴 1 + 𝐵𝐵
𝑋𝑋 − 48.7
1 + 𝐵𝐵
�1 + tanh �
�� + +
�1 − tanh �
�� − 1 − 𝐵𝐵
2
𝑎𝑎𝑡𝑡𝑡𝑡
2
2
𝑎𝑎𝑡𝑡𝑡𝑡
24
Eq. 50
Figure 11. Real track boundaries are shown as solid lines, estimated boundaries as dashed lines with
continuous derivatives and illustration of cone positions with circles. One of the corners has been
magnified in the bottom of the figure to make the differences between the two types of boundaries
more clear.
Figure 11 illustrates the difference between the two different types of constraints. The main
parts are equal while the corners are different as well as two vertical constraints are within the
track but at positions where the vehicle is not expected to drive when achieving its maximum
entry speed.
3.2
Objective function
Section 1.3 describes that the objective was to maximize the entry speed; using Tomlab’s [11]
solver standard mode, that was the minimization of the cost function, one could set the cost
function, J, according to
Eq. 51
𝐽𝐽 = −𝑉𝑉𝑥𝑥 (𝑋𝑋 = 0)
as
max 𝑣𝑣𝑥𝑥 = min −𝑣𝑣𝑥𝑥 .
𝑡𝑡=𝑡𝑡0
𝑡𝑡=𝑡𝑡0
Eq. 52
The above objective function does not include the control variable, the steering angle, which
implies that a solution may have bang-bang behaviour of the steering angles [46]. Such
behaviour was not favourable but could be avoided by introducing a second part in the
25
objective, which grew larger when the steering angle rate became higher. From section 1.3 it
occurs that the only objective should be the maximization of the entry speed, therefore a
weighting factor for limiting the steering angle rate contribution to the result was also
introduced. By doing a stepwise reduction of this weight factor, 𝑊𝑊𝛿𝛿 , while the model got
solved with more details, the most detailed model also got very little influence from the
steering angle rate, thus the final objective used is given by Eq. 53.
𝑡𝑡=𝑡𝑡𝑓𝑓𝑖𝑖𝑛𝑛𝑛𝑛𝑛𝑛
𝐽𝐽 = −𝑉𝑉𝑥𝑥 (𝑡𝑡 = 0) + 𝑊𝑊𝛿𝛿 �
𝑡𝑡=0
𝛿𝛿̇ 2
Eq. 53
In the first iteration, in the simplest model, 𝑊𝑊𝛿𝛿 was set to 1.67 while it was set to just 0.05 in
the most advanced model. If the first and second term in Eq. 53 are compared when 𝑊𝑊𝛿𝛿 =
0.05, it can be seen that the influence from the second term is less than 1 %. A secondary
weight factor can also be used for the first term to optimize J to be used with the numerical
solver; in this case such a factor would just be set equal to 1.
3.3
Point mass trajectory and initial inputs
A simple guess, to the first optimization problem was crucial if it would be desired to easily
change parameters and still have a solution which could converge without major changes to
the input information. Therefore, the first optimization problem started from a guess of how
much time, 𝑡𝑡𝑔𝑔 , the run would need as well as how much the speed would drop in percentage
of the entry speed, 𝑣𝑣𝑑𝑑 , and the states of the vehicle were then estimated according to Eq. 54Eq. 59.
𝑌𝑌𝑔𝑔 =
−
𝑋𝑋𝑔𝑔 =
𝐴𝐴+𝐵𝐵+2
4
𝐵𝐵+𝐶𝐶+2
4
61
𝑡𝑡𝑔𝑔
𝑡𝑡
Eq. 54
�1 + tanh�(𝑋𝑋𝑔𝑔 − 20) /𝑎𝑎𝑔𝑔 ��
�1 + tanh�(𝑋𝑋𝑔𝑔 − 43) /𝑎𝑎𝑔𝑔 ��
𝑣𝑣𝑥𝑥 𝑔𝑔 =
61
𝑡𝑡𝑔𝑔
�1 +
𝑣𝑣𝑑𝑑
100
𝑣𝑣𝑦𝑦 𝑔𝑔 =
𝑡𝑡
�1 − ��
𝑡𝑡𝑔𝑔
̇
𝑌𝑌𝑔𝑔
3
̇
𝑌𝑌𝑔𝑔
ψ𝑔𝑔 = 𝑡𝑡𝑡𝑡𝑡𝑡−1 � ̇ �
𝑋𝑋𝑔𝑔
26
Eq. 55
Eq. 56
Eq. 57
Eq. 58
δ𝑔𝑔 =
ψ̇𝑔𝑔
2
Eq. 59
Time, t, is continuous and present in all of the guess equations and it was found to be time
efficient to provide a guessed total time, 𝑡𝑡𝑔𝑔 , equal to 2.2 seconds as well as a speed decrease,
𝑣𝑣𝑑𝑑 , of 15 %. Such a guess has a higher entry speed than was expected for the solution to
find; the guessed position, yaw angle and speed can also be seen in Figure 12, Figure 13 and
Figure 14 respectively.
Figure 12. Guessed optimal position along the track.
Figure 13. Guessed optimal yaw angle along the track.
27
Figure 14. Guessed longitudinal vehicle speed over time during the manoeuvre.
The aforementioned guess was unrealistic but still provided information about the general
pattern of the solution regarding the vehicle global coordinates, yaw angle and speed decrease
over time.
3.4
Bicycle model trajectory optimization
Initially, the bicycle model, described in section 2.5, was used with a linear tire model,
described in section 2.7.2. The results were used as a guess in a bicycle model with magic
formula tire model as described in section 2.7.3. Specific details, such as the objective
formulation and constrains, for the two models can be seen in the following sections.
3.4.1
Linear tire model
As a first step, the bicycle model, shown in Figure 3, was studied. The side forces on the tires
were calculated as linear functions of the slip angles and the cornering stiffness of the tires.
In this step, no longitudinal forces were modelled.
Small angle approximations were used to keep this initial problem simple. This means that
the steering angle 𝛿𝛿 and the slip angles of the front and rear tire, 𝛼𝛼12 and 𝛼𝛼34 respectively,
were considered to be small, 𝛼𝛼12 , 𝛼𝛼34 ≪ 1 . From Eq. 9 and Eq. 10 the slip angles were then
given by
and
𝛼𝛼12 ≈
𝑣𝑣𝑦𝑦 +𝑓𝑓𝜓𝜓̇
𝛼𝛼34 ≈
28
𝑣𝑣𝑥𝑥
− 𝛿𝛿
𝑣𝑣𝑦𝑦 −𝑏𝑏𝜓𝜓̇
𝑣𝑣𝑥𝑥
.
Eq. 60
Eq. 61
As shown in section 2.5 the front and the right side force are then given by Eq. 62, Eq. 63
with the assumption to be linear functions of the slip angles.
𝐹𝐹12 = −𝐶𝐶12 𝛼𝛼12
𝐹𝐹34 = −𝐶𝐶34 𝛼𝛼34
Eq. 62
Eq. 63
The model was extended to incorporate air drag along the vehicle’s longitudinal direction.
With the extra term added, the equations of motion described by Eq. 2-Eq. 4 became:
1
𝑚𝑚�𝑣𝑣𝑥𝑥̇ − 𝜓𝜓̇𝑣𝑣𝑦𝑦 � ≈ −𝐹𝐹12 𝛿𝛿 − 𝜌𝜌 ∙ 𝐴𝐴 ∙ 𝐶𝐶𝑑𝑑 ∙ 𝑉𝑉𝑥𝑥 2 ,
2
𝑚𝑚�𝑣𝑣𝑦𝑦̇ + 𝜓𝜓̇𝑣𝑣𝑥𝑥 � = 𝐹𝐹34 + 𝐹𝐹12 cos(𝛿𝛿 ) ≈ 𝐹𝐹34 + 𝐹𝐹12
and
1
𝐼𝐼𝑧𝑧 𝜓𝜓̈ = 𝑓𝑓𝐹𝐹12 cos(𝛿𝛿 ) − 𝑏𝑏𝐹𝐹34 ≈ 𝑓𝑓𝐹𝐹12 𝛿𝛿 − 𝑏𝑏𝐹𝐹34
Eq. 64
Eq. 65
Eq. 66
with the term ∙ 𝜌𝜌 ∙ 𝐴𝐴 ∙ 𝐶𝐶𝑑𝑑 ∙ 𝑉𝑉𝑥𝑥 2 representing the vehicle air drag [22] [23]. This air drag was
2
also considered to be independent of the vehicle’s sideslip angle. A solution was searched for
the movement of the CoG position in the global coordinate system shown in Figure 3, thus
Eq. 5 and Eq. 6 could be used.
An implementation of Eq. 60-Eq. 66 and the coordinate translation Eq. 5-Eq. 6 was
programmed in TOMLAB and the solutions were constrained to be paths travelling inside
the double lane change manoeuvre, as described in the ISO3888 Part 2 [2], according to
Table 1. The state variables were 𝑣𝑣𝑥𝑥 , 𝑣𝑣𝑦𝑦 , 𝜓𝜓̇9, 𝜓𝜓, 𝑋𝑋 and 𝑌𝑌 while the control variable was the
steering angle, 𝛿𝛿 , alone. This meant that during the manoeuvre the only way for the
controller to control the vehicle was by changing the steering wheel angle, with no throttle,
brakes, or engine braking at all.
The choice of 𝜓𝜓̇ as a Tomlab state variable was done in order to avoid the double derivation of 𝜓𝜓 required in Eq. 4,
such that only a single derivation of the state 𝜓𝜓̇ was required instead. This was done since in Tomlab double derivations
are advised against, and an inclusion of an additional state is preferred [16].
9
29
Table 1. Constraints for the linear bicycle model. No vehicle dimensions in the model.
Variable constraints
𝑉𝑉𝑥𝑥 ≥ 10(i)
−20 ≤ 𝑉𝑉𝑦𝑦 ≤ 20(i)
−𝜋𝜋/2 ≤ 𝜓𝜓 ≤ 𝜋𝜋/2(i)
−4 ≤ 𝜓𝜓̇ ≤ 4(i)
−31° ≤ 𝛿𝛿 ≤ 31°
̇ ≤ 4𝜋𝜋(ii)
−4𝜋𝜋 ≤ 𝛿𝛿𝑠𝑠𝑠𝑠
𝑋𝑋(𝑡𝑡 = 0) = 0(iii)
𝑋𝑋�𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 � = 61(iii)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 − 𝐴𝐴/2 ≤ 𝑌𝑌(𝑡𝑡 = 0) ≤ 𝐴𝐴/2 −
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 + 𝐴𝐴/2 − 𝐶𝐶 ≤ 𝑌𝑌(𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎 ) ≤
𝐴𝐴/2 − 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
𝜓𝜓(𝑡𝑡 = 0) = 0(iv)
𝜓𝜓̇(𝑡𝑡 = 0) = 0(iv)
𝛿𝛿 (𝑡𝑡 = 0) = 0(iv)
𝑉𝑉𝑦𝑦 (𝑡𝑡 = 0) = 0(iv)
(i)
(ii)
(iii)
(iv)
(v)
𝐶𝐶𝑓𝑓 =
𝐶𝐶𝑟𝑟 =
𝐵𝐵𝐵𝐵𝐵𝐵
2
𝐵𝐵𝐵𝐵𝐵𝐵
𝑁𝑁𝑓𝑓 (v)
Description
The car’s longitudinal speed in m/s
The car’s lateral speed in m/s
Yaw angle in rad
Yaw rate in rad/s
Steering angle in degrees
Steering wheel rate in rad/s
Initial longitudinal position of CoG in m
Final longitudinal position of CoG in m
Initial vertical position of the CoG in m
Final vertical position of the CoG in m
Initial yaw angle in rad
Initial yaw rate in rad/s
Initial steering angle in rad
Initial lateral velocity in m/s
Cornering stiffness on the front in N/rad
Cornering stiffness on the rear in N/rad
𝑁𝑁𝑟𝑟
2
Constraints to help limit the search space. In order to help the optimizer converge to a solution this limitation should
be neither too loose nor too strict [6].
Constraint that prevents the steering output from being faster than a human driver or steering robot [47]. This
constraint was added to the model only after the first visit to Hällered test track took place. Initially, only the steering
angle 𝛿𝛿 was restricted, and the resulted steering rate, being unrestricted, was too high. As a result, the steering torque
request from the steering robot was also too high and the robot could not perform the manoeuvre. In a later step, this
restriction to the steering rate was posed.
Constraints that limit the vehicle movement outer boundaries.
Constraints that define the entrance of the vehicle to be in a straight driving manner.
The values of the cornering stiffness for the front and rear tire were set to half of the magic formula stiffness, division
by 2 was necessary in order not to produce unrealistic high forces with the linear model when performing a manoeuvre
outside of the linear range. This facilitated the use of the result as a guess when the magic formula also got
implemented.
(v)
Furthermore, the solution was done in two steps with 𝑛𝑛 equals to 20 and 80 collocation
points 10 respectively where the cost function was given by Eq. 53, and the weight factor, 𝑊𝑊𝛿𝛿 ,
was set according to
𝑊𝑊𝛿𝛿 =
10
3000
90𝑛𝑛
Collocation points as used by the collocation method described in the Tomlab manual [6].
30
Eq. 67
which implies its value was 1.67 during the first iteration and 0.04 for the result given by this
model. The high initial value of the cost function weight factor, 𝑊𝑊𝛿𝛿 , was selected as an
arbitrary number. After the first iteration it was lowered in order to reduce its effect on the
output without getting a bang-bang behaviour of the steering angle input.
3.4.2
Magic formula tire model
After the above linear model produced an optimal path the model was augmented by
increasing the complexity, and thus the accuracy, of the equations of motion. While the linear
tire model can describe lateral dynamics that involve accelerations of up to 40% of the road
friction potential, that is up to 4 𝑚𝑚/𝑠𝑠 2 or 0.4 𝑔𝑔 for dry road conditions, it is inadequate for
higher lateral accelerations, which imposes the need for a non-linear model, since the
manoeuvre that was studied was highly dynamical [24] [45]. The non-linear model that was
used here was the Magic Formula tire model where the vehicle’s tire forces were calculated
by their slip, according to the simplified Magic Formula [28] as described in 2.7.3 Non-linear
tire model and given by Eq. 23.
This simplified expression was chosen since the modelling in this study focused on the whole
vehicle, rather than the tires alone, and thus it was considered that this version captured a
satisfactory amount of information, which was reasonably detailed, yet simple enough for use
in an optimization study like this. The resultant tire slip 𝑠𝑠 for each tire was defined as
Eq. 68
𝑠𝑠𝑖𝑖 = �𝑠𝑠𝑖𝑖𝑖𝑖 2 + 𝑠𝑠𝑖𝑖𝑖𝑖 2
with
𝑠𝑠𝑖𝑖𝑖𝑖 =
𝑉𝑉𝑖𝑖𝑖𝑖
𝑉𝑉𝑖𝑖𝑖𝑖 − 𝜔𝜔𝑖𝑖 𝑟𝑟𝑖𝑖
, 𝑠𝑠𝑖𝑖𝑖𝑖 =
𝜔𝜔𝑖𝑖 𝑟𝑟𝑖𝑖
𝜔𝜔𝑖𝑖 𝑟𝑟𝑖𝑖
Eq. 69
as the theoretical slip quantities [28]. 𝑉𝑉𝑖𝑖𝑖𝑖 and 𝑉𝑉𝑖𝑖𝑖𝑖 are the tire frame components of the vehicle
velocity vector at the centres of the wheels, such that 𝑠𝑠𝑖𝑖𝑖𝑖 and 𝑠𝑠𝑖𝑖𝑖𝑖 are the tire slips in the tire
frame’s x and y direction respectively. With the slip values calculated, the friction coefficients
in the tire’s x and y direction are defined as
𝜇𝜇𝑖𝑖𝑖𝑖 = −
𝑠𝑠𝑖𝑖𝑖𝑖
𝑠𝑠𝑖𝑖
𝜇𝜇𝑖𝑖 , 𝜇𝜇𝑖𝑖𝑖𝑖 = −
𝑠𝑠𝑖𝑖𝑖𝑖
𝑠𝑠𝑖𝑖
𝜇𝜇𝑖𝑖 .
Eq. 70
The normal forces at the tires, 𝐹𝐹𝑓𝑓𝑓𝑓 and 𝐹𝐹𝑟𝑟𝑟𝑟 for the front and rear tire respectively, was then
obtained from
𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚ℎ
𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚ℎ
Eq. 71
𝐹𝐹𝑓𝑓𝑧𝑧 =
−
𝑎𝑎𝑥𝑥 , 𝐹𝐹𝑟𝑟𝑟𝑟 =
+
𝑎𝑎
𝐿𝐿
𝐿𝐿
𝐿𝐿
𝐿𝐿 𝑥𝑥
31
which resulted in the tire forces at each tire frame as
Eq. 72
𝐹𝐹𝑖𝑖𝑖𝑖 = 𝜇𝜇𝑖𝑖𝑖𝑖 𝐹𝐹𝑖𝑖𝑖𝑖 , 𝐹𝐹𝑖𝑖𝑖𝑖 = 𝜇𝜇𝑖𝑖𝑖𝑖 𝐹𝐹𝑖𝑖𝑖𝑖 .
It is noteworthy that in this model also the longitudinal force, at the tire frame of each tire,
𝐹𝐹𝑖𝑖𝑖𝑖 was included, in contrast to the linear tire bicycle mode. The term
𝑚𝑚ℎ
𝐿𝐿
𝑎𝑎𝑥𝑥 in Eq. 71 is the
load transfer in the longitudinal direction, that is, between the front and the rear wheels.
Load transfer occurs since the extended version of the bicycle model used here also included
the height of the CoG. Therefore, when the vehicle accelerates, the normal force on the
front will become lower as the term
𝑚𝑚ℎ
𝐿𝐿
𝑎𝑎𝑥𝑥 , introduced by the acceleration, is subtracted
from its static load, while the normal force on the rear will become larger since this load will
be now be added to its static load value. The ODE’s describing this model and the global
velocities of the CoG were then formulated by Eq. 5, Eq. 6, Eq. 19 and Eq. 64-Eq. 66; with
indices corresponding to this model they are given by
𝑚𝑚�𝑣𝑣𝑥𝑥̇ − 𝜓𝜓̇𝑣𝑣𝑦𝑦 � = 𝐹𝐹𝑓𝑓𝑓𝑓 cos 𝛿𝛿 − 𝐹𝐹𝑓𝑓𝑓𝑓 sin 𝛿𝛿 + 𝐹𝐹𝑟𝑟𝑟𝑟 −
1
2
𝜌𝜌 ∙ 𝐴𝐴 ∙ 𝐶𝐶𝑑𝑑 ∙ 𝑉𝑉𝑥𝑥 2 ,
𝑚𝑚�𝑣𝑣𝑦𝑦̇ + 𝜓𝜓̇𝑣𝑣𝑥𝑥 � = 𝐹𝐹𝑟𝑟𝑟𝑟 + 𝐹𝐹𝑓𝑓𝑓𝑓 sin 𝛿𝛿 + 𝐹𝐹𝑓𝑓𝑓𝑓 cos 𝛿𝛿,
Eq. 73
Eq. 74
𝐼𝐼𝑧𝑧 𝜓𝜓̈ = 𝑓𝑓�𝐹𝐹𝑓𝑓𝑓𝑓 cos 𝛿𝛿 + 𝐹𝐹𝑓𝑓𝑓𝑓 sin 𝛿𝛿� − 𝑏𝑏𝐹𝐹𝑟𝑟𝑟𝑟 ,
Eq. 75
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑓𝑓 = 𝑇𝑇𝑓𝑓 − 𝐹𝐹𝑓𝑓𝑓𝑓 𝑟𝑟,
Eq. 76
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑟𝑟 = 𝑇𝑇𝑟𝑟 − 𝐹𝐹𝑟𝑟𝑟𝑟 𝑟𝑟 ,
Eq. 77
𝑋𝑋̇ = 𝑣𝑣𝑥𝑥 cos(𝜓𝜓) − 𝑣𝑣𝑦𝑦 sin(𝜓𝜓)
𝑌𝑌̇ = 𝑣𝑣𝑥𝑥 sin(𝜓𝜓) + 𝑣𝑣𝑦𝑦 cos(𝜓𝜓)
Eq. 78
Eq. 79
where 𝑚𝑚 is the vehicle’s mass, 𝐼𝐼𝑧𝑧 its moment of inertia around the vertical axis (yaw inertia),
𝐼𝐼𝑤𝑤 the moment of inertia of each wheel and 𝑇𝑇𝑓𝑓 , 𝑇𝑇𝑟𝑟 the applied torque in the forward
movement direction on the front and rear wheel respectively 11 [28] [22]. As shown in Eq. 76
The drive/brake torques are considered as zero in this case, but will be used later where the ESC implementation will
also be included in the model.
11
32
and Eq. 77, in addition to the nonlinear tire model, the wheel rotation dynamics were also
introduced in this step.
Eq. 23 and Eq. 68-Eq. 72 were used to calculate the relevant slips and forces and Eq. 73-Eq.
79 were inserted in the TOMLAB solver as the model’s ODEs. The constraints used for the
state and control variables for this problem are shown in Table 2.
Table 2. Constraints for the nonlinear bicycle model.
Variable constraints
𝑉𝑉𝑥𝑥 ≥ 10(i)
−20 ≤ 𝑉𝑉𝑦𝑦 ≤ 20(i)
−𝜋𝜋/2 ≤ 𝜓𝜓 ≤ 𝜋𝜋/2(i)
−4 ≤ 𝜓𝜓̇ ≤ 4(i)
−31° ≤ 𝛿𝛿 ≤ 31°
̇ ≤ 4𝜋𝜋(ii)
−4𝜋𝜋 ≤ 𝛿𝛿𝑠𝑠𝑠𝑠
𝑋𝑋(𝑡𝑡 = 0) = 0(iii)
𝑋𝑋�𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 � = 61(iii)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 − 𝐴𝐴/2 ≤ 𝑌𝑌(𝑡𝑡 = 0) ≤ 𝐴𝐴/2 −
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 + 𝐴𝐴/2 − 𝐶𝐶 ≤ 𝑌𝑌(𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑎𝑎𝑙𝑙 ) ≤
𝐴𝐴/2 − 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
𝜓𝜓(𝑡𝑡 = 0) = 0(iv)
𝜓𝜓̇(𝑡𝑡 = 0) = 0(iv)
𝛿𝛿 (𝑡𝑡 = 0) = 0(iv)
𝑉𝑉𝑦𝑦 (𝑡𝑡 = 0) = 0(iv)
10/𝑟𝑟 ≤ 𝜔𝜔𝑓𝑓 ≤ 180/𝑟𝑟(i)
10/𝑟𝑟 ≤ 𝜔𝜔𝑟𝑟 ≤ 180/𝑟𝑟(i)
𝜔𝜔𝑓𝑓 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥 (i)
𝜔𝜔𝑟𝑟 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥 (i)
(i)
(ii)
(iii)
(iv)
Description
The car’s longitudinal speed in m/s
The car’s lateral speed in m/s
Yaw angle in rad
Yaw rate in rad/s
Steering angle in degrees
Steering wheel rate in rad/s
Initial longitudinal position of the CoG in
m
Final longitudinal position of the CoG in
m
Initial vertical position of the CoG in m
Final vertical position of the CoG in m
Initial yaw angle in rad
Initial yaw rate in rad/s
Initial steering angle in rad
Initial lateral velocity in m/s
The front wheel’s rotational speed in rad/s
The rear wheel’s rotational speed in rad/s
Indirect limitation on front wheel slip
Indirect limitation on rear wheel slip
Constraints to help limit the search space. In order to help the optimizer converge to a solution this limitation should
be neither too loose nor too strict [6].
Constraint that prevents the steering output from being faster than a human driver or steering robot [47]. This
constraint was added to the model only after the first visit to Hällered test track took place. Initially, only the steering
angle 𝛿𝛿 was restricted, and the resulted steering rate, being unrestricted, was too high. As a result, the steering torque
request from the steering robot was also too high and the robot could not perform the manoeuvre. In a later step, this
restriction to the steering rate was posed.
Constraints that limit the vehicle movement into some specific boundaries dictated by the manoeuvre.
Constraints that define the entrance of the vehicle to be in a straight driving manner.
The cost function was the same as in the linear tire bicycle model, given by Eq. 53, but with
the weight factor, 𝑊𝑊𝛿𝛿 = 0.0521, which is slightly higher than the value used for the start
33
guess’ last iteration. Once again an iterative solution strategy with increasing collocation
points; 20, 30, 35, 40, and 80 collocation points were used. A long iterative procedure was
used in this case, as the guess was the result from the linear solution, thus the solution was
expected to change for all states.
3.5
Car body boundary modelling
While the dimensionless car model moves along the track the global position of all points on
the car body can be calculated from the information about its CoG position, the yaw angle
and the points’ coordinates in the car’s local coordinate system. The numerical solver cannot
check an infinite number of points on the car body boundary, which means the boundary has
to be discretized into a few points which then represent the whole car body. As the corners
of the car body are describing the car body extension, those were selected as the first points.
The right and left side of the car could still hit a corner when turning, posing an additional
problem, which was solved by describing the sides of the car with eight points each; the
number of points needed will depend on the yaw rate, the length of the car and how sharp
the corner is.
Figure 15. Illustration of the discretized car travelling from left to right. Each filled circle is a discrete
point, and a corner is passing through the car body without violating any discrete point. It is also
shown how the car is divided into the four quadrants Q1, Q2, Q3 and Q4.
The situation in Figure 15 can exist if the yaw rate is high enough to place the car away from
the corner before the next discrete point reaches it, and therefore it is important to check if
that is the case after the solution has been found. In section 3.10 a visual verification method
is described which gives an indication of whether the car hit the corner or not.
34
Each discrete point on the left side of the car was given the constraints to not cross the left
side of the track, corresponding to the left side of Eq. 50, while each point on the right side
of the car was given the constraints to not cross the right side of the track, corresponding to
the right side of Eq. 50. It was possible to use one side of Eq. 50 at a time since the points
also are constrained to each other and Eq. 50 then must be fulfilled if the points do not cross
the boundary on their own side of the car. It would be possible to manually specify the
constraints for each point but then it would also be necessary to redo that process for cars
with different sizes; instead the car was divided into four quadrants according to Figure 15.
Each quadrant contains evenly spaced discrete points which are generated with each corner
as base and a number specifying the number of the points that should be used. This method
makes it possible to create a model, which decides how many points should be used based on
the dimensions of the car.
3.6
Four wheel vehicle model trajectory optimization
In Figure 16 a vehicle model known as the two track model is shown. This is the vehicle
model that was used for this study as a basis for drawing some first conclusions, and is also
studied in [28] [48] [47] [22]. It is an actual extension of the bicycle model, in order to
differentiate between the left and right wheels of the vehicle, and allows the vehicle’s
movement to be described in a more detailed manner. This model was the one that produced
the first results to test and compare with a real vehicle in a test track, with no ESC activated
yet.
Figure 16. Illustration of the four wheel vehicle model with notation [25].
35
As depicted in Figure 16, during a turn, the inner and outer wheels steer by a different
angle, 𝛿𝛿𝑖𝑖 and 𝛿𝛿𝜊𝜊 respectively, since the outer wheel is required to travel at a longer path
(lower curvature) than the inner one 12 . This applies to actual vehicles. During this study
though, a common simplification was followed and the steering angle 𝛿𝛿 was considered to be
the same for both of the wheels and equal to their mean value, that is [22]
𝛿𝛿 =
𝛿𝛿𝑖𝑖 + 𝛿𝛿𝑜𝑜
2
Eq. 80
The details that were modelled and were possible to use in the two track model, are described
in section 3.8. The equations used for this case are presented in 3.8.4 Fully featured model
without ESC.
3.7
ESC modelling
As Appendix A – ISO 3888 description involves releasing the throttle before the first turn
when entering the double-lane change track, as seen in Figure 1, the ESC can only choose to
brake each wheel or change the steering angle of the front wheels, no extra torque can be
transferred to any wheel. However, in this study it was only the brakes which were
considered to be a part of the ESC as the steering angle input was optimized no matter if the
ESC was used or not.
Four types of ESC systems will be described below, of which the three first are slip angle
controlled ESC, yaw rate error controlled ESC as well as a combined slip angle error and yaw
rate error controlled ESC. An optimal controller, for the studied scenario, has also been
defined and modelled to be used as a reference; by definition it should outperform any ESC
implementation.
By using a threshold for activating the system and a logic for determining which wheel to
brake the ESC system got described by discontinuous functions, which is not recommended
to be used within the Tomlab solver. Such discontinuous functions behave according to
Eq. 81; they were approximated by using Eq. 1 as shown in Eq. 82.
𝑂𝑂 < 𝑄𝑄 → 𝑃𝑃 = 0
𝑂𝑂 ≥ 𝑄𝑄 → 𝑃𝑃 = 1
1
𝑂𝑂 − 𝑄𝑄
𝑃𝑃 = �1 + tanh �
��
2
𝑎𝑎
Eq. 81
Eq. 82
For cars operating in low speed and lateral acceleration conditions (street cars) usage of Ackermann geometry is most
commonly the case. With this design the wheels are traveling along concentric paths, and that ensures that all the wheels
roll freely with no slip angles. In vehicles designed for other purposes, like for example racing, other steering geometries
apply [12][18].
12
36
In Eq. 82 𝑎𝑎 is a constant determining how small the transition region from the lower to the
higher value of the curve is, in which the equation does not behave as Eq. 81, as illustrated in
Figure 17 below.
1
0.8
P
0.6
0.4
a=3
a=1
0.2
lim a -> 0
Q=-2
0
-10
-8
-6
-4
-2
0
O
2
4
6
8
10
Figure 17. Illustration of Eq. 82 for different values of a . When a approaches zero the resulting value
of P approaches the value which is given by Eq. 81.
The value of a in Eq. 82 should always be much smaller than the output value, C, of the
equation but always greater than zero.
3.7.1
Optimal ESC modelling – a torque controller
Optimality in this case means everything that results in a higher entry speed without hitting
any cone when driving through the track. For an ESC this results in controlling exactly how
much braking torque should be applied on each wheel at all times. This was implemented in
the optimal controller by adding a torque control variable to each wheel’s equations of
motion with the only restriction that the variable had to be negative. The tire-road friction
limit serves as the upper bound for how much braking torque can be applied, resulting in
maximum utilization of friction forces.
As it was desired to know how well an ESC could perform, and use that knowledge when
creating another type of ESC, two versions of the torque model were modelled with different
limitations.
• Optimal braking torque applied only after the vehicle reached the first turn
• Optimal braking torque applied only if a threshold was reached, e.g. yaw rate error or
slip angle error. The torque rate was then limited to one rate for pressure build up and
another rate for releasing the brakes.
37
The first of these limited optimal torque models was just intended to avoid braking during
the initial straight path while the second one should determine how well an optimal ESC with
a certain threshold would perform. The second limited model then could be used as a
reference, and an upper limit, for any ESC-model as long as the same threshold was used.
3.7.2
Sideslip angle controlled ESC
When using the Tomlab [11] optimization environment it was not recommended by the
solver instructions to use discontinuous functions (although it was allowed). As the test
vehicle 13 had an ESC with discontinuous behaviour, a simple ESC, but with discontinuous
behaviour, was implemented to study its robustness.
Control of when to activate each brake was done by only activating the ESC after the
magnitude of the sideslip angle, which was defined as
𝑣𝑣𝑦𝑦
𝛽𝛽 = 𝑡𝑡𝑡𝑡𝑡𝑡−1 � � ,
𝑣𝑣𝑥𝑥
Eq. 83
was higher than a certain level – the case of a 2 ° threshold, 𝛽𝛽𝑡𝑡ℎ , was investigated in this
study.
When the ESC had been activated, braking torque was applied to the right front wheel when
oversteering to the left, to the left front wheel when oversteering to the right, to the right
rear wheel when understeering right and to the left rear wheel when understeering to the left.
Whether the vehicle understeered or not was determined by calculating the understeer
gradient, Kus, according to [22]
where R is given by
𝐾𝐾𝑢𝑢𝑢𝑢 =
𝑅𝑅 =
𝐿𝐿
�𝛿𝛿−𝑅𝑅�
Eq. 84
𝑎𝑎𝑦𝑦
�𝑣𝑣𝑥𝑥 2 +𝑣𝑣𝑦𝑦 2
�𝜓𝜓̇�
.
Eq. 85
Table 3 shows the SlipESC𝑖𝑖𝑖𝑖 value for each wheel during each of the above described
situations. If 𝛽𝛽 < 𝛽𝛽𝑡𝑡ℎ then SlipESC𝑖𝑖𝑗𝑗 was always set to 0.
13
The test vehicle was a Volvo S60 with specifications according to Appendix B.
38
Table 3. Binary logic of the sideslip controlled ESC.
SlipESC𝑓𝑓𝑓𝑓
SlipESC𝑓𝑓𝑓𝑓
SlipESC𝑟𝑟𝑟𝑟
SlipESC𝑟𝑟𝑟𝑟
𝐾𝐾𝑢𝑢𝑢𝑢
1
0
0
0
Oversteer
< 0, 𝛿𝛿 > 0 𝐾𝐾𝑢𝑢𝑢𝑢 < 0, 𝛿𝛿 < 0
0
1
0
0
𝐾𝐾𝑢𝑢𝑢𝑢
0
0
1
0
Understeer
> 0, 𝛿𝛿 > 0 𝐾𝐾𝑢𝑢𝑢𝑢 > 0, 𝛿𝛿 < 0
0
0
0
1
The magnitude of the torque was then set to increase proportionally with the slip angle,
which resulted in two selectable variables for this type of ESC; the slip angle threshold and
the proportional constant between the braking torque and the slip angle. The proportional
constant could even be different for each wheel but in this study the ESC was activated over
a slip angle magnitude of 2 ° and with each wheel torque, Tfr, Tfl, Trr and Trl, according to
𝑇𝑇 𝑖𝑖𝑖𝑖 = 5000(𝛽𝛽 − 𝛽𝛽𝑡𝑡ℎ )SlipESC𝑖𝑖𝑖𝑖
Eq. 86
Note that if both the slip angle threshold and proportional constants for the braking torque
at each wheel are set to be control variables within the optimization, this method becomes
equal with the optimal torque control method.
3.7.3
Yaw rate error controlled ESC
Control of when to activate each brake was done by only activating the ESC after the
magnitude of the yaw rate error was higher than a certain level. A desired yaw rate was
calculated with Eq. 33, which describes the vehicle’s steady state yaw rate.
The actual yaw rate was a part of the solution of the states, thus it was possible to use directly
without any extra calculations. It was then compared to the desired yaw rate and the error
was defined as the difference between them,.
𝜓𝜓̇𝑒𝑒 = 𝜓𝜓̇ − 𝜓𝜓̇𝑑𝑑
Eq. 87
When the yaw rate error was obtained, it was possible to know when the ESC should be
activated, but a logic braking scheme is also needed for which wheel should brake.
39
The following four scenarios describe the situation when each wheel needs to brake:
• Brake rear left wheel 𝐸𝐸𝐸𝐸𝐸𝐸𝑟𝑟𝑟𝑟 (understeer): Positive desired yaw rate, a left turn, but
negative yaw rate error – the vehicle does not turn left as much as desired.
• Brake rear right wheel 𝐸𝐸𝐸𝐸𝐸𝐸𝑟𝑟𝑟𝑟 (understeer): Negative desired yaw rate, a right turn,
and positive yaw rate error – the vehicle does not turn right as much as desired.
• Brake front left wheel 𝐸𝐸𝐸𝐸𝐸𝐸𝑓𝑓𝑓𝑓 (oversteer): Negative desired yaw rate, a right turn, and
negative yaw rate error – the vehicle turns right more than desired.
• Brake front right wheel 𝐸𝐸𝐸𝐸𝐸𝐸𝑓𝑓𝑓𝑓 (oversteer): Positive desired yaw rate, a left turn, and
positive yaw rate error – the vehicle turns left more than desired.
The logic described above was implemented using continuous functions according to Eq. 82;
thus the resulting logic is described by
1
−𝜓𝜓̇𝑒𝑒
1
𝜓𝜓̇
𝐸𝐸𝐸𝐸𝐸𝐸𝑟𝑟𝑟𝑟 = �1 + tanh �
4
𝑎𝑎
𝜓𝜓̇
�� �1 + tanh � 𝑑𝑑��,
−𝜓𝜓̇𝑑𝑑
Eq. 88
��,
Eq. 89
𝐸𝐸𝐸𝐸𝐸𝐸𝑓𝑓𝑓𝑓 = �1 + tanh � 𝑒𝑒�� �1 + tanh � 𝑑𝑑��.
Eq. 91
𝐸𝐸𝐸𝐸𝐸𝐸𝑟𝑟𝑟𝑟 = �1 + tanh � 𝑒𝑒�� �1 + tanh �
4
1
and
𝑎𝑎
𝐸𝐸𝐸𝐸𝐸𝐸𝑓𝑓𝑓𝑓 = �1 + tanh �
4
1
4
𝑎𝑎
−𝜓𝜓̇𝑒𝑒
𝑎𝑎
𝜓𝜓̇
𝑎𝑎
𝑎𝑎
−𝜓𝜓̇𝑑𝑑
�� �1 + tanh �
𝑎𝑎
𝜓𝜓̇
𝑎𝑎
��,
Eq. 90
When the logic scheme given by Eq. 88-Eq. 91 determined which wheel should brake, the
approach was to apply a braking torque as function of the yaw rate error as long as the error
is larger than the threshold, 𝜓𝜓̇𝑡𝑡 . By formulating the threshold according to Eq. 82 and
multiplying it with the desired initial braking torque, Tinit, as well as a linear increase as
function of the yaw rate error, the ESC behaviour could be adapted while always being
continuous. The torque for the rear left wheel, 𝑇𝑇𝑟𝑟𝑟𝑟 , the rear right wheel, 𝑇𝑇𝑟𝑟𝑟𝑟 , the front left
wheel, 𝑇𝑇𝑓𝑓𝑓𝑓 , and the front right wheel, 𝑇𝑇𝑓𝑓𝑓𝑓 , is then given by
40
𝑇𝑇𝑖𝑖𝑖𝑖 = 𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 �−
−
𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜓𝜓̇𝑒𝑒 − 𝜓𝜓̇𝑡𝑡
�1 + tanh �
�� �1 + 𝑇𝑇𝑖𝑖𝑖𝑖 �𝜓𝜓̇𝑒𝑒 − 𝜓𝜓̇𝑡𝑡 ��
2
𝑎𝑎
𝑇𝑇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜓𝜓̇𝑒𝑒 + 𝜓𝜓̇𝑡𝑡
�1 + tanh �
�� �1 − 𝑇𝑇𝑖𝑖𝑖𝑖 �𝜓𝜓̇𝑒𝑒 + 𝜓𝜓̇𝑡𝑡 ���
2
𝑎𝑎
Eq. 92
The substantial torque increase, Tinit, at the yaw rate error threshold as well as the increase
factor, Tif, for the torque relative to its value at the threshold, could, in the ESC torque
model, Eq. 92, be arbitrary positive numbers.
Figure 18. ESC torque characteristics; scheme for the magnitude of the braking torque applied on
one of the wheels for a given yaw rate error, which here is shown as vertical dashed lines. In this case
the figure show a yaw rate error of 2°, an initial torque of 200 Nm, a torque increase factor of 5 and a
smoothness factor of 0.005.
The ESC torque characteristics shown in Figure 18 was seen to be one of the characteristic
setups which was producing the highest entry speed and was used in all comparisons with an
ESC except for in the ESC characteristic parameter study.
3.7.4
Yaw and slip controlled ESC
All solutions without any ESC suggested that the steering angle was not small and hence it
was not possible to use the simplified equation, Eq. 39, to determine how much yaw torque
the brakes needed to generate. Instead Eq. 18 was used without simplification in Eq. 36 and
with 𝑠𝑠̇ = −𝜂𝜂𝜂𝜂. When the contact forces between the road and tire were rearranged to the left
hand side, Eq. 93 was obtained.
41
−𝑙𝑙𝑓𝑓 �𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 � sin(𝛿𝛿 ) − 𝑙𝑙𝑓𝑓 �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � cos(𝛿𝛿 )
𝑙𝑙𝑤𝑤𝑤𝑤
+ 𝑙𝑙𝑟𝑟 �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � −
�𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 � cos(𝛿𝛿 )
2
𝑙𝑙𝑤𝑤𝑤𝑤
𝑙𝑙𝑤𝑤𝑤𝑤
(𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) −
�𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � sin(𝛿𝛿 )
−
2
2
= 𝜂𝜂𝜂𝜂𝐼𝐼𝑧𝑧 − Ψ̈𝑑𝑑 + 𝜉𝜉(𝛽𝛽̇ − 𝛽𝛽𝑑𝑑̇ )
Eq. 93
As the numerical solver in Tomlab [11] solves the problem in all points along the track at
once Eq. 93 can now be a part of the iterative procedure. The solution to each wheel’s
dynamic equation, Eq. 19, then also contains the torque needed and as no propulsion is
allowed during the manoeuvre the torque was limited to never be positive. But fulfilling Eq.
93 by only controlling the steering wheel angle in such a way 𝑠𝑠̇ = −𝜂𝜂𝜂𝜂 became an
optimization problem itself, thus not suitable to use in another optimization problem.
The yaw rate error and the slip angle error, which is part of the sliding surface s, can still be
used to control the ESC if the relation to the braking torque, Tij, is a function of the form
𝑇𝑇𝑖𝑖𝑖𝑖 = 𝑓𝑓(𝜓𝜓̇, 𝜓𝜓̇𝑒𝑒 , 𝛽𝛽, 𝛽𝛽𝑒𝑒 )
Eq. 94
as the braking torque then does not introduce another optimization problem. A threshold for
the activation of the ESC could be placed on the sliding surface s. This type of ESC is
however not covered by this study.
3.8
Model refinement
It is almost impossible to provide a guess that is close to the solution for many different
choices of parameters if the model has all of its modelling features implemented at once,
hence the need of a stepwise refinement of it. The following modelling features and their
corresponding implementation is depicted in this chapter:
1. Wheel kinematics and compliance
2. Tire relaxation
3. Wheel camber effects
3.8.1
Wheel kinematics and compliance
A phenomenon modelled is the change of the toe angle due to two important factors: body
roll and lateral force application on the wheels. The first case is termed “roll steer,” which is a
change in the wheel angle that results from body roll motion 14, and the second one is the
Even though this is undesirable, it is a very common characteristic of most of the suspension and steering systems,
which depends on their geometry [17] [22].
14
42
lateral force compliance steer, which is a change in the wheel angle due to compliance of the
suspension by lateral forces applied at the tire-road contact [24] [31] [19] [21].
Roll steer
From the Kinematics and Compliance (K&C) datasheet for the Volvo S60, more specifically
from the “Toe-in angle at double side roll movement” test case, the values in the following Table 4
were extracted. The values are the ratio of the induced toe-in angle over the corresponding
roll angle, and would be translated to what is commonly called roll steer coefficient
𝜕𝜕𝛿𝛿
𝜕𝜕𝜕𝜕
, such that
the steering angle 𝛿𝛿 was described as a function of roll angle 𝜑𝜑 [19]. The roll angle was
positive when the vehicle was leaning to the right as seen from behind (i.e. the right side is at
jounce while the left at rebound). [Appendix G – Kinematics & compliance]
Table 4. Toe-in versus roll angle,
𝝏𝝏(𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻)
Front axle
Rear axle
𝝏𝝏𝝋𝝋
[𝒅𝒅𝒅𝒅𝒅𝒅/𝒅𝒅𝒅𝒅𝒅𝒅] for the Volvo S60.
Left wheel
+0.14
+0.00
Right wheel
-0.11
-0.01
The values above depict that during cornering the front wheels steer outwards with respect
to the curve. In other words, the wheels toe out when compressed and toe in when
expanded. For example, in a left turn the roll angle is positive. According to Table 4 then the
front left wheel (in rebound) will toe in and the front right wheel (in jounce) will toe out. The
rear wheels have negligible roll steer. 15 For the front axle, a negative roll steer coefficient then
applies, and results in an understeer effect. When occurring on the rear axle an oversteer
effect would occur [31] [19]. If the toe-in angle of each wheel is translated to steer angle 𝛿𝛿
according to ISO standard [19], Table 4 is equivalent to Table 5 below.
Table 5. Roll steer coefficients,
Front axle
Rear axle
𝝏𝝏𝝏𝝏
𝝏𝝏𝝋𝝋
[𝒅𝒅𝒅𝒅𝒅𝒅/𝒅𝒅𝒅𝒅𝒅𝒅] for the Volvo S60.
Left wheel
-0.135
-0.00
Right wheel
-0.111
-0.01
This result came from the interpretation of the S60 K&C datasheet that can be found in the Appendix G. According to
Volvo Cars’ Vehicle Dynamics engineer Johan Hultqvist: “In most passenger cars a roll steer understeer effect is desired. This means
that the front wheels will tend to steer outwards with respect to the curve. The rear wheels are usually neutral, the change in toe is negligible, but
in some cars they can be designed to give roll steer understeer too – that would mean that the rear wheels tend to steer inwards with respect to the
curve. In this case, we see some very low values for the rear axle but the measurement is obviously very noisy. From the shape of the curve we can
assume a neutral rear axle behaviour.
With roll steer oversteer tuning we can achieve some better “turn in” to the curve, but the vehicle might be unstable to higher speeds and the
driver has to make corrections, so it is generally undesirable.”
15
43
The different sign for the case of the left wheels results from the fact that the steering angle
𝛿𝛿 is positive in the opposite direction towards which the left wheel toes in. The opposite is
true for the right wheels, so the signs for the right wheels are the same between Table 4 and
Table 5.
Since the steering angle was considered the same for the left and right wheels, as mentioned
in Eq. 80, a common roll steer coefficient can be used for the left and rear wheels at each
axle. This value was the mean value of the right and left wheel roll steer coefficient. This
means that for the front axle the value to be used would be
𝜕𝜕𝛿𝛿
�
𝜕𝜕𝜕𝜕 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
=
−0.135+(−0.111)
2
adding an understeer effect, and for the rear axle
= −0.123
𝜕𝜕𝛿𝛿
�
≈0
𝜕𝜕𝜕𝜕 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
Eq. 95
Eq. 96
The rear axle was considered to have a neutral behaviour to roll movement.
Modelled in this study was roll steer kinematics rather than roll steer dynamics, since the
change to the steering angle as a function to roll angle was not inserted in a differential
equation form, but rather as a steady state calculation (i.e. the change to steer angle induced
by roll occurs instantaneously rather than gradually). In reality though, roll steer effects lag
the steer input, appearing after the sprung mass has begun rolling [31]. The respective
equations that were used in the model was
and
𝜕𝜕𝛿𝛿
𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟 =
𝜕𝜕𝜕𝜕 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
�
𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟 =
𝜕𝜕𝜕𝜕 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝜕𝜕𝛿𝛿
�
∙ 𝜑𝜑
Eq. 97
∙ 𝜑𝜑
Eq. 98
for the change in the steering angle due to roll steer on the front and rear wheel respectively.
The second equation, the roll steer at the rear wheels, could be regarded here as trivial, since
the roll steer coefficient of the rear axle was zero as mentioned above. But for consistency
and generality purposes it was inserted in the model nonetheless.
Lateral force compliance steer
For modelling the lateral force compliance steer two test cases from the K&C datasheet were
studied. The first one is the “Toe-in angle at lateral force, unidirectional LH RH, at X = 0 mm”
case, which implies the change in the toe-in angle of a wheel when a lateral force is applied to
44
it at the centre of its contact patch with the ground (X = 0). The second one is the “Toe-in
angle at lateral force, unidirectional LH RH, at X = 30 mm” case, which implies the change in the
toe-in angle of a wheel when a lateral force is applied at a distance of X = 30 mm behind 16
the centre of the tire-ground contact patch [Appendix G – Kinematics & compliance]. The
distance X = 30mm behind the centre of the contact patch is an approximate value for a
typical tire’s pneumatic trail in low slip angles [32]. Table 6 and Table 7 show extracted toein angle values.
Table 6. Toe-in angle for positive lateral force applied at X = 0mm
𝝏𝝏(𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻)
Left wheel
-0.029
+0.07
Front axle
Rear axle
Table 7. Toe-in angle for positive lateral force applied at X = 30mm
Left wheel
-0.108
+0.016
Front axle
Rear axle
𝝏𝝏𝑭𝑭
[𝒅𝒅𝒅𝒅𝒅𝒅/𝒌𝒌𝒌𝒌].
Right wheel
-0.046
+0.065
𝝏𝝏(𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻)
𝝏𝝏𝑭𝑭
[𝒅𝒅𝒅𝒅𝒅𝒅/𝒌𝒌𝒌𝒌].
Right wheel
-0.143
+0.018
The values in Table 6 and Table 7 refer to the change in a wheel’s toe-in angle with respect to
the force acting on a given point on it, but they also describe the change in the steering angle
for that given force. This time, no sign change was needed in order to match the wheel toe-in
with the positive direction of the wheel steer (like in the case of the roll steer). The reason for
that is the direction towards which the force was applied for each wheel when the
measurement was taken. In the K&C tests a positive force indicates a force acting inwards to
the wheel, and this means that while a positive toe-in angle at the left wheels translates to a
negative respective steering angle, a positive force to the left wheels also translates to a
negative force in the coordinate system that was used (where a lateral force is positive when
it acts toward the direction of a positive abscissa – positive y axis), which means that for the
left wheels
𝜕𝜕(𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇)
𝜕𝜕𝐹𝐹
�
𝐾𝐾&𝐶𝐶 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.
=
𝜕𝜕𝜕𝜕
�
𝜕𝜕𝐹𝐹 𝑣𝑣𝑣𝑣ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
.
Eq. 99
Following the same argumentation it results to that the signs for the right wheels also needed
no modification. The reason that these two sets of values were extracted from the K&C
datasheet is the dynamic change of the pneumatic trail which results in a non-constant lateral
force compliance steer coefficient. For small slip angles/linear tire region the pneumatic trail
16 The word behind here indicates the direction that is opposite to the tire’s longitudinal travelling direction at the tire
frame’s coordinate system.
45
is almost constant. But in larger slip angles/nonlinear tire region the pneumatic trail reduces
(see also section for pneumatic trail in section 2.7.5) [23] [24] [32] [26]. The change in the toein angle of a wheel depends on the distance from its contact patch centre where the lateral
force will be applied, which is the pneumatic trail, as Table 6 and Table 7 also indicate. The
change in this lateral force compliance steer coefficient was approximated by linearly
interpolating between its values for X = 0 mm (nonlinear region pneumatic trail) and X = 30
mm (linear region pneumatic trail). For example, the straight line between the lateral force
compliance coefficients at X = 0 mm and X = 30 mm for the front left wheel would be
𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 =
0.029−0.108
0.03
∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 − 0.029 = −2.633 ∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 − 0.029 .
Eq. 100
and in a similar manner Table 8 was obtained.
Table 8. Lateral force compliance steer coefficient as a function of the pneumatic trail when linearly
interpolating between the K&C values [deg/kN]. LF cij = lateral force compliance steer coefficient for
the i,j wheel(i = f,r j = l,r). tpij = pneumatic trail for the i,j wheel.
Front axle
Rear axle
Left wheel
𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 = −2.633 ∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 − 0.029
𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 = −1.8 ∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 + 0.07
Right wheel
𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 = −3.233 ∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 − 0.046
𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 = −1.5667 ∗ 𝑡𝑡𝑝𝑝𝑝𝑝𝑝𝑝 + 0.065
The change in the steering angle due to a lateral force could then be calculated as
Eq. 101
𝛿𝛿𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐿𝐿𝐿𝐿𝑐𝑐𝑐𝑐𝑐𝑐 ∗ 𝐹𝐹𝑖𝑖𝑖𝑖𝑖𝑖 .
where 𝐹𝐹𝑖𝑖𝑖𝑖𝑖𝑖 is the applied lateral force to the respective wheel. Again, as mentioned above, a
mean induced steering angle due to lateral force compliance steer was used since the model
did not include the difference in the right and left wheel steering angles but rather their mean
value. This yields
𝛿𝛿𝑐𝑐𝑐𝑐 =
𝛿𝛿𝑐𝑐𝑐𝑐𝑐𝑐 +𝛿𝛿𝑐𝑐𝑐𝑐𝑐𝑐
2
, 𝛿𝛿𝑐𝑐𝑐𝑐 =
𝛿𝛿𝑐𝑐𝑐𝑐𝑐𝑐 +𝛿𝛿𝑐𝑐𝑐𝑐𝑐𝑐
2
Eq. 102
for the induced steering angles on the front and rear axle. The tire’s pneumatic trail also
needed to be calculated at each instant, since this double lane change test is a manoeuvre
where the tires largely operate in their nonlinear/high slip angle region, and thus their
pneumatic trail cannot be considered constant. Its calculation was done using the
methodology in [49].
3.8.2
Tire relaxation
Tire relaxation phenomena were modelled according to the methodology described in section
2.7.4. The dynamic build-up of the lateral forces on the tires was described by a first order
46
differential equation, which can also be seen as a first order low pass filter. At each time
instant, each tire’s lateral force should satisfy
𝜏𝜏𝑓𝑓𝑦𝑦̇ (𝑎𝑎, 𝑡𝑡) + 𝑓𝑓𝑦𝑦 (𝑎𝑎, 𝑡𝑡) = 𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎)
Eq. 103
with a typical value of 𝐿𝐿 = 0.3𝑚𝑚 chosen as the relaxation length in Eq. 29 [32].
3.8.3
Camber thrust
The lateral force due to camber, the camber thrust, was modelled as an additive force to the
lateral force due to slip angle. Although this is not the case in the tires’ nonlinear region, it is
a simplification used in this work. From the K&C tests “Camber angle at double side roll
movement,” as can be seen in Appendix G – Kinematics & compliance, where the camber of
the wheel with respect to the vehicle body is measured for a given body roll value, Table 9 as
well as all other tables in this chapter were obtained:
Table 9. Camber gain per roll angle
Front axle
Rear axle
𝝏𝝏𝜺𝜺
𝝏𝝏𝝋𝝋
[𝒅𝒅𝒅𝒅𝒅𝒅/𝒅𝒅𝒅𝒅𝒅𝒅].
Left wheel
+0.243
+0.452
Right wheel
-0.264
+0.434
The roll angle is considered positive when the vehicle leans right as seen from behind (the
right side of the car is in jounce while the left in rebound). The decisive parameter for the
camber thrust is the camber relative to the ground, the wheel’s inclination angle, and
therefore a translation from the camber relative to the body, given on the table above, is
needed. This translation holds as [19]
and
𝛾𝛾𝑙𝑙 = 𝛾𝛾0 + 𝜑𝜑 − 𝜀𝜀𝑙𝑙 .
Eq. 104
𝛾𝛾𝑟𝑟 = 𝛾𝛾0 + 𝜑𝜑 + 𝜀𝜀𝑟𝑟 .
Eq. 105
𝐹𝐹𝛾𝛾𝛾𝛾𝛾𝛾 = −𝐶𝐶𝛾𝛾 ∙ 𝛾𝛾𝑖𝑖𝑖𝑖 .
Eq. 106
for the left and right wheel respectively as seen from behind. The factor 𝛾𝛾0 is the static
camber of the wheels. For the S60’s front wheels the static camber is taken as 𝛾𝛾0𝑓𝑓 = 𝜀𝜀0𝑓𝑓 =
−0.7° and for the rear wheels 𝛾𝛾0𝑟𝑟 = 𝜀𝜀0𝑟𝑟 = −1.3°, which are typical static camber values for
modern passenger vehicles. Also, for the camber stiffness a typical value of 𝐶𝐶𝛾𝛾 =
2000 [𝑁𝑁/𝑟𝑟𝑟𝑟𝑟𝑟] was used [19]. The camber thrust for each tire was then calculated from
which then was added to the relevant calculated lateral force produced by the tire’s slip angle.
47
3.8.4
Fully featured model without ESC
All features described in sections 3.8.1 - 3.8.3 were implemented in a model without ESC.
The equations to be inserted into the solver, considering front-wheel steering and travelling
on a horizontal plane, with indices corresponding to this model were:
Eq. 107
𝛿𝛿𝑓𝑓 = 𝛿𝛿 + 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟 + 𝛿𝛿𝑐𝑐𝑐𝑐 ,
Eq. 108
𝛿𝛿𝑟𝑟 = 𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟 + 𝛿𝛿𝑐𝑐𝑐𝑐 ,
̇ (𝑎𝑎, 𝑡𝑡) = �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦_𝑠𝑠𝑠𝑠 (𝑎𝑎) − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎, 𝑡𝑡)� ∙
𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦
̇ (𝑎𝑎, 𝑡𝑡) = �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦_𝑠𝑠𝑠𝑠 (𝑎𝑎) − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎, 𝑡𝑡)� ∙
𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦
̇ (𝑎𝑎, 𝑡𝑡) = �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦_𝑠𝑠𝑠𝑠 (𝑎𝑎) − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎, 𝑡𝑡)� ∙
𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦
̇ (𝑎𝑎, 𝑡𝑡) = �𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦_𝑠𝑠𝑠𝑠 (𝑎𝑎) − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 (𝑎𝑎, 𝑡𝑡)� ∙
𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦
𝑚𝑚�𝑣𝑣𝑥𝑥̇ − 𝜓𝜓̇𝑣𝑣𝑦𝑦 � =
𝑉𝑉𝑥𝑥𝑥𝑥𝑥𝑥
,
Eq. 109
𝑉𝑉𝑥𝑥𝑥𝑥𝑥𝑥
,
Eq. 110
𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
𝑉𝑉𝑥𝑥𝑥𝑥𝑥𝑥
,
Eq. 111
𝑉𝑉𝑥𝑥𝑥𝑥𝑥𝑥
,
Eq. 112
𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑥𝑥
𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
(𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) cos 𝛿𝛿𝑓𝑓 + (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) cos 𝛿𝛿𝑟𝑟 − (𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) sin 𝛿𝛿𝑓𝑓 −
(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) sin 𝛿𝛿𝑟𝑟 −
1
2
Eq. 113
𝜌𝜌 ∙ 𝐴𝐴 ∙ 𝐶𝐶𝑑𝑑 ∙ 𝑉𝑉𝑥𝑥 2 ,
𝑚𝑚�𝑣𝑣𝑦𝑦̇ + 𝜓𝜓̇𝑣𝑣𝑥𝑥 � =
�𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � cos 𝛿𝛿𝑟𝑟 + (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) sin 𝛿𝛿𝑟𝑟 + (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) sin 𝛿𝛿𝑓𝑓 +
(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) cos 𝛿𝛿𝑓𝑓 ,
Eq. 114
𝐼𝐼𝑧𝑧 𝜓𝜓̈ =
𝑓𝑓�(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 ) cos 𝛿𝛿𝑓𝑓 + (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 + 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) sin 𝛿𝛿𝑓𝑓 � − 𝑏𝑏�(𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 � cos 𝛿𝛿𝑟𝑟 + Eq. 115
(𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 ) sin 𝛿𝛿𝑟𝑟 ] + 𝑊𝑊𝑙𝑙 (𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑓𝑓 + 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑟𝑟 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑓𝑓 −
𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑟𝑟 ) + 𝑊𝑊𝑟𝑟 (𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑓𝑓 +𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 cos 𝛿𝛿𝑟𝑟 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑓𝑓 − 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦 sin 𝛿𝛿𝑟𝑟 ) ,
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑓𝑓𝑓𝑓 = 𝑇𝑇𝑓𝑓𝑓𝑓 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 𝑟𝑟 ,
Eq. 116
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑟𝑟𝑟𝑟 = 𝑇𝑇𝑟𝑟𝑟𝑟 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 𝑟𝑟 ,
Eq. 118
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑓𝑓𝑓𝑓 = 𝑇𝑇𝑓𝑓𝑓𝑓 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 𝑟𝑟 ,
48
Eq. 117
and
𝐼𝐼𝑤𝑤 𝜔𝜔̇ 𝑟𝑟𝑟𝑟 = 𝑇𝑇𝑟𝑟𝑟𝑟 − 𝐹𝐹𝑥𝑥𝑥𝑥𝑥𝑥 𝑟𝑟 ,
Eq. 119
𝑌𝑌̇ = 𝑣𝑣𝑥𝑥 sin(𝜓𝜓) + 𝑣𝑣𝑦𝑦 cos(𝜓𝜓)
Eq. 121
𝑋𝑋̇ = 𝑣𝑣𝑥𝑥 cos(𝜓𝜓) − 𝑣𝑣𝑦𝑦 sin(𝜓𝜓)
Eq. 120
where 𝑚𝑚 is the vehicle’s mass, 𝐼𝐼𝑤𝑤 the moment of inertia of each wheel (considered the same
for all wheels), 𝐹𝐹𝑦𝑦𝑦𝑦𝑦𝑦_𝑠𝑠𝑠𝑠 (𝑎𝑎) the steady state lateral force of the i,j wheel for a given slip angle 𝑎𝑎,
𝑊𝑊𝑟𝑟 the distance of the right wheel from the axle’s centreline, 𝑊𝑊𝑙𝑙 the distance of the left wheel
from the axle’s centreline the and 𝑇𝑇𝑓𝑓𝑓𝑓 , 𝑇𝑇𝑓𝑓𝑓𝑓 , 𝑇𝑇𝑟𝑟𝑟𝑟 , 𝑇𝑇𝑟𝑟𝑟𝑟 the drive/brake torque on the front right,
front left, rear right and rear left wheel respectively. The constraints used for the state and
control variables are shown in Table 10.
49
Table 10. Constraints for the full vehicle model.
Variable constraints
𝑉𝑉𝑥𝑥 ≥ 10(i)
−10 ≤ 𝑉𝑉𝑦𝑦 ≤ 10(i)
−𝜋𝜋/2 ≤ 𝜓𝜓 ≤ 𝜋𝜋/2(i)
−4 ≤ 𝜓𝜓̇ ≤ 4(i)
−31.5° ≤ 𝛿𝛿 ≤ 31.5°
−4𝜋𝜋/𝑖𝑖 ≤ 𝛿𝛿̇ ≤ 4𝜋𝜋/𝑖𝑖 (ii)
𝑋𝑋(𝑡𝑡 = 0) = 0(iii)
Description
The car’s longitudinal speed in m/s
The car’s lateral speed in m/s
Yaw angle in rad
Yaw rate in rad/s
Steering angle in degrees
Steering angle rate in rad/s
Initial longitudinal position of the CoG
in m
Final longitudinal position of the CoG
𝑋𝑋�𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 � = 61(iii)
in m
Initial
vertical
position
of the CoG in
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 − 𝐴𝐴/2 ≤ 𝑌𝑌(𝑡𝑡 = 0) ≤ 𝐴𝐴/2 −
m
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2 + 𝐴𝐴/2 − 𝐶𝐶 ≤ 𝑌𝑌(𝑡𝑡 = 𝑡𝑡𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ) ≤ 𝐴𝐴/2 − Final vertical position of the CoG in m
𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ/2(iii)
Initial yaw angle in rad
𝜓𝜓(𝑡𝑡 = 0) = 0(iv)
Initial yaw rate in rad/s
𝜓𝜓̇(𝑡𝑡 = 0) = 0(iv)
Initial steering angle in rad
𝛿𝛿 (𝑡𝑡 = 0) = 0(iv)
Initial lateral velocity in m/s
𝑉𝑉𝑦𝑦 (𝑡𝑡 = 0) = 0(iv)
Front left wheel rotational speed rad/s
10/𝑟𝑟 ≤ 𝜔𝜔𝑓𝑓𝑓𝑓 ≤ 80/𝑟𝑟(i)
Front right wheel’s rotational speed
10/𝑟𝑟 ≤ 𝜔𝜔𝑓𝑓𝑓𝑓 ≤ 80/𝑟𝑟(i)
rad/s
Rear left wheel’s rotational speed rad/s
10/𝑟𝑟 ≤ 𝜔𝜔𝑟𝑟𝑟𝑟 ≤ 80/𝑟𝑟(i)
Rear right wheel’s rotational speed
10/𝑟𝑟 ≤ 𝜔𝜔𝑟𝑟𝑟𝑟 ≤ 80/𝑟𝑟(i)
rad/s
(v)
Indirect limitation on front left wheel
𝜔𝜔𝑓𝑓𝑓𝑓 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥
slip
Indirect limitation on front right wheel
𝜔𝜔𝑓𝑓𝑓𝑓 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥 (v)
slip
(v)
Indirect limitation on rear left wheel
𝜔𝜔𝑟𝑟𝑟𝑟 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥
slip
Indirect limitation on rear right wheel
𝜔𝜔𝑟𝑟𝑟𝑟 ∙ 𝑟𝑟 ≤ 1.2 ∙ 𝑉𝑉𝑥𝑥 (v)
slip
(i)
(ii)
(iii)
(iv)
(v)
50
Constraints to help limit the search space. In order to help the optimizer converge to a solution this limitation should
be neither too loose nor too strict [6].
Constraint that prevents the steering output from being faster than a human driver or steering robot [47]. This
constraint was added to the model only after the first visit to Hällered test track took place. Initially, only the steering
angle 𝛿𝛿 was restricted, and the resulted steering rate, being unrestricted, was too high. As a result, the steering torque
request from the steering robot was also too high and the robot could not perform the manoeuvre. In a later step, this
restriction to the steering rate was posed.
Constraints that limit the vehicle movement into some specific boundaries dictated by the manoeuvre.
Constraints that define the entrance of the vehicle to be in a straight driving manner.
The wheel speed was limited to never exceed 120 % of the vehicle longitudinal speed to reduce the search space.
The cost function was kept the same as in Eq. 53 with the weight factor, 𝑊𝑊𝛿𝛿 , set to 0.111 in
the first iteration step and then 0.042 in the solution.
3.9
Parameter study
This parameter study was in primarily intended to investigate the influence of the various
vehicle parameters on the maximum entry speed and study the robustness of the method,
especially for vehicles as the core study vehicle, the Volvo S60 2013. Robustness, R, was here
defined by
𝑅𝑅 =
100𝑛𝑛𝑠𝑠
𝑛𝑛𝑡𝑡𝑡𝑡𝑡𝑡
Eq. 122
where 𝑛𝑛𝑠𝑠 is the number of solved 17 cases, 𝑛𝑛𝑡𝑡𝑡𝑡𝑡𝑡 is the number of tested cases and R is then
given in percentage of the total number of tested cases. This robustness metric can then be
used to give an indication of whether the controller can be used as a generic controller to
find a path. The differences between the resulting steering angle inputs, when comparing the
four-wheel vehicle model with different features implemented, were very small (section 4.1).
However, the simulation time of the model with most features implemented was up to 100
times longer compared to the model with least features 18. For that reason the parameter
study was based on the four-wheel vehicle model with yaw rate controlled ESC and roll
kinematics but no tire relaxation and no wheel kinematics. 80 collocation 19 points as solution
for each variable were used for all parameter studies and the influence of the number of
discrete points was also studied, which was of interest to determine the minimum number of
points needed.
3.9.1
Vehicle parameter changes
Parameters that describe the vehicle geometry, as well as its mass and the tire model, were
selected for performing changes of the vehicle properties. Those parameters have an impact
on both the shape of the double-lane change track as well as on the vehicle response. Tire
forces dictate what is possible to do in terms of aggressive manoeuvres, thus it was also of
interest to study the tire model effects.
A case was considered ”solved” when Tomlab gave the output message ”Optimality conditions satisfied” as defined by
the Tomlab manual [6] and described, together with outer outputs, in the Tomlab guide [12].
18 The most complex model includes all features as defined by section 3.7.3 and chapter 3.8 while the model with least
features refers to the linear bicycle model as described in section 3.4.1.
19 The solution takes the form of a polynomial function that satisfies all collocation points [4].
17
51
Table 11. Setup scheme for all vehicle parameter changes. When one of the parameters was changed
the rest were set according to their base values while non-specified parameters were set according to
Appendix B – Test vehicle.
Parameter
Weight
Yaw inertia
Body length
Body width 20
Magic formula tire stiffness,
(B*constant, C*constant, D*constant)
Magic formula, B
Magic formula, C
Magic formula, D
Nominal value
1823 kg
3500 kgm2
4.635 m
1.865 m
12.6118
Variation
± 500 kg
± 500 kgm2
± 0.5 m
± 0.5 m
± 50 %
Step
50 kg
50 kgm2
0.05 m
0.05 m
10 %
7.5418
1.4897
1.1233
± 50 %
± 50 %
± 50 %
10 %
10 %
10 %
All parameter variations that were used to study the influence from individual parameters can
be seen in Table 11. When changing the tire specifications the same changes were always
used for all tires. But a special case was also investigated, namely the case of keeping the
product B*C*D constant while changing D according to Table 12.
Table 12. Special case of the tire parameter study, constant tire stiffness but variations of the
parameter D.
Parameter
Magic formula, D
Nominal value Variation
1.1233
± 50 %
Step
10 %
The product B*C*D was kept constant by using the relations given by Eq. 123, Eq. 124 and
Eq. 125.
𝐷𝐷 = 𝐷𝐷𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ∗ 𝐷𝐷𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
1
𝐵𝐵 = 𝐵𝐵𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ∗ �
𝐷𝐷𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
1
𝐶𝐶 = 𝐶𝐶𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ∗ �
𝐷𝐷𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
Eq. 123
Eq. 124
Eq. 125
This corresponds to a tire that always has the same cornering stiffness but different peak
force.
20
The ISO-3888 track sections width depends on the vehicle body width [1].
52
3.9.2
ESC-characteristics variations
The way the ESC applies the braking torque influences the entry speed; this influence was
studied. An upper limit that cannot be exceeded with any ESC is posed by the optimal torque
controller described in section 3.7.1; this optimal controller cannot be changed but can serve
as a reference for various ESC configurations. No threshold and an optimal ESC algorithm
would result in the same entry speed as the optimal torque controller, but not higher, since
the ESC then is always activated.
With a constant ESC activation threshold, it was investigated how its braking torque
characteristics influenced the entry speed; first by keeping the torque increase factor constant
while changing the initial torque at the activation point and secondly by keeping the initial
torque constant and changing the torque increase factor.
Table 13. ESC-torque characteristic variation; all combinations of the specified parameters were used
while non-specified parameters were set according to Appendix B – Test vehicle.
Parameter
Initial torque at the
ESC activation yaw
error
Torque increase factor
Yaw error threshold
Base value
200 Nm
Variation
- 200 Nm
+ 100 Nm
Step
50 Nm
5
2°
±5
±2°
0.5
1°
Figure 19. ESC characteristics when changing only the initial torque.
In Figure 19 the torque characteristics can be seen for the lowest tested initial torque as well
as for the highest; for the base value of the torque increase factor and yaw rate error
threshold. All tested values used during the study are given in Table 13. No initial torque at
all represents a deactivated ESC system.
53
Figure 20. ESC characteristics when changing only the torque increase factor.
In Figure 20 the torque characteristics can be seen for the lowest tested torque increase
factor as well as for the highest, for the base value of the initial torque and yaw error
threshold, while all tested values used during the study are given by Table 13. A torque
increase factor of zero represents a constant applied torque after the threshold limit has been
reached. The smoothness factor, which determines the transition between the inactive ESC
and activated ESC, was not changed during the parameter study.
An optimal ESC implementation derived an entry speed to be used as a reference; such an
ESC would only brake when the specified yaw rate error threshold had been exceeded. The
torque magnitude would not be limited while the torque application rate was limited to
−25000 ≤ 𝑇𝑇̇𝑖𝑖𝑖𝑖 ≤ 7500 𝑁𝑁𝑁𝑁/𝑠𝑠
Eq. 126
according to brake specifications given by [50].
3.9.3
Influence of discretization
An optimal solution was found in a finite number of points along the track, hence it was
possible to obtain solutions which are not necessarily optimal between two consecutive
collocation points. However, if the distance between two consecutive points approaches
zero, the difference to the last result should also approach zero.
To study the effect of the collocation points, the entry speed was studied as a function of
their number during the optimization. This was done in steps of 5 points from 20 points up
to 150 points for the bicycle model with a linear tire model. For the models with more
features this variation of discrete points was not possible to perform within this study due to
the very long time to perform one single optimization.
54
3.10 Animation of a car along the track
Animation of the result facilitates subjective judgement of the result, and hence identify the
parts of the model that need a more detailed and objective approach. During this study the
vehicle, as well as the wheels, were visualized as rectangles and seen from above following
the path of a polynomial function going through all of the discrete points. In each wheel
arrows were plotted to show the braking torque as well as the lateral tire force. This
information made it possible to gain understanding about the vehicle behaviour as it
provided help to determine the validity of all the geometrical constraints. Another important
benefit from the animation was that the controller could also be visualized, in Figure 21 an
example of such an animation can be seen.
Figure 21. Vehicle animation along the double-lane change track with illustrated lateral forces as well
as braking torque.
3.11 Steering robot test 1; driving procedure
After the vehicle dynamics model and the ESC implementation were built, physical tests
were performed for estimating the vehicle’s parameters as well as assessing the model’s
realism. The model featured four-wheel dynamics, roll kinematics and an optimal torque
controller that was activated after 12 m. The late activation limitation was included to get
characteristics closer to a real ESC, which would not brake as long as the vehicle is driving
straight, while an optimal torque controller can brake even when travelling straight. During
all manoeuvres, which will be described in this chapter, the recorded data consisted of the
parameters listed in Table 14.
55
Table 14. Recorded parameters during testing.
Parameter
Time
Steering robot angle
Steering robot Torque
Steering robot Column Torque
X position
Y position
Forward velocity
Lateral velocity
Forward acceleration
Lateral acceleration
Vertical acceleration
Roll angle
Pitch angle
Yaw angle
Roll velocity
Pitch velocity
Yaw velocity
Slip angle
Yaw acceleration
Accuracy
Unknown
0.0008 °
±2%
±2%
3 cm
3 cm
0.05 km/h
2%
0.01 m/s2
0.01 m/s2
0.01 m/s2
0.03 °
0.03 °
0.03 °
0.01 °/s
0.01 °/s
0.01 °/s
0.15 °
Unknown
Steady state cornering was performed both in a left and right turn. The calculated optimal
path though the ISO-3888 track was then intended to be driven both with and without ESC
activated. As a last step the car was also driven manually through the track; an indication of
the validity of the optimal controller would be the comparison between the entry speed it
achieved and the maximum entry speed achieved by a human driver. The controller would be
expected to always achieve higher entry speeds.
The generated optimal steering angles were imported to the steering robot and by running
the test at a lower speed it was possible to verify that the robot could perform the requested
operations. In this case the steering angle rate showed to be faster than the robot could
handle; the steering angle rate limit was then introduced in the model, but the new optimal
solution then had larger steering angles than the car could handle which led to a correction of
the model steering angle limit too.
3.11.1 Parameter estimation
As mentioned earlier, four circular driving tests with constant radius, and pseudorandom
steering at constant speed were conducted during the first test run. These additional tests
were performed for acquisition of data that would allow some post process as well as some
vehicle parameter evaluation. The tests were performed both by a human driver and a
56
steering robot 21. The parameters that needed to be calculated were the effective cornering
stiffness for the front and rear axle, so that the B, C and D parameters of the magic formula
can be eventually calculated, and the vehicle’s yaw moment of inertia [45].
Despite the performance of such tests though, the vehicle’s parameters evaluation could not
be conducted in the usual systematic manner described in section 2.9 in the case of this
thesis. The reason for this is that the recorded lateral velocity and sideslip angle data from the
circular driving tests could not be used as the IMU had not been set up correctly. For the
interested reader Appendix H - Test 1 data illustration provides an illustration of the data
that were recorded.
As the recorded steering wheel angle comes directly from the measurement of the steering
wheel position while the recorded lateral velocity and sideslip angle come after a calculation
that the IMU performs 22 , it becomes apparent that the lateral velocity and sideslip angle
values could not be used. Since the recordings of the lateral velocity hindered the calculation
of the axles’ effective cornering stiffness another method was used to approximate those
values. For the yaw inertia 𝐼𝐼𝑧𝑧 , the empirical formula
𝐼𝐼𝑧𝑧 = 0.46 ∗ 𝑚𝑚 ∗ 𝐿𝐿 ∗ 𝑡𝑡 ≅ 3500 𝑘𝑘𝑘𝑘𝑚𝑚2
Eq. 127
was used [51] with 𝑚𝑚 , 𝐿𝐿 and 𝑡𝑡 being the vehicle mass, wheelbase and track width
respectively, with a 68% confidence level.
The magic formula’s B, C and D values for the S60’s tires would be attempted to be found
via curve fitting using the recordings from the double lane change manoeuvres.
Nine double lane change tests were conducted and recorded. The idea was to build a model,
where the B, C and D values could be modified. This model would be subjected to the
longitudinal velocity 𝑉𝑉𝑥𝑥 and steering angle 𝛿𝛿 according to the recordings from the real vehicle
tests. The model would output some results for the yaw rate and the lateral acceleration and
these results would be compared with the recorded/real vehicle results. The B,C,D values of
the model would then be modified accordingly such that the curves from the model and
from the recordings fitted as close as possible.
Volvo’s test engineer Per Hesslund performed the tests. The steering robot used was an SR60 standard robot by
Anthony Best Dynamics.
22 The IMU used in the test records the output from accelerometers and gyroscopes and with the use of observers
(Kalman filters) it calculates also other values, like the velocities [50]. This design though is sensitive to accumulating
error from the integral action of the observer, which makes it unsuitable for this kind of test (circular driving). Instead, a
twin-antenna GPS or a slip angle optical sensor could be used to obtain reliable values for the vehicle’s slip angle and
lateral velocity [51].
21
57
The model that was used for the fitting was a bicycle model with nonlinear tires and was built
in Simulink. For simplicity only the lateral tire slip was included in the modelling. The input
to the model was, as mentioned, the recorded longitudinal velocity and steering angle from
the test. The output of the model, the yaw rate and lateral acceleration, was used in the curve
fitting. The Simulink model is shown in Appendix I - Simulink bicycle model.
Before comparing the model results with the recorded results some filtering was needed for
the recorded ones, since the measurements were noisy and as pure a curve as possible was
desired 23 . The fitting process was approached as a nonlinear programming problem and
Matlab’s “fmincon” function was used. In the objective function the error between the model’s
output and the respective measured data was calculated and minimized. In order to avoid
over fitting the total error of all the nine double lane change tests, rather than the error of an
individual test, was calculated and used as a minimization objective. The total error for each
iteration was the sum of the individual errors from each model.
3.12 Steering robot test 2; driving procedure
During the second test the main goal was to study the influence of the model features that
were used in the prediction as well as to determine the accuracy of the estimated result. It
was also of interest to assess the extent to which the real ESC could be approximated with
the simple ESC implementation of this thesis. The vehicle was therefore driven through the
track with optimized steering angles according to a vehicle model with ESC models that had
different characteristics. Then it was also driven with optimized steering angles according to
vehicle models of varying detail as well as the standard subjective test was performed.
Characteristic values for each tested ESC as well as entry speed can be seen in the manoeuvre
test list in Appendix D – Test 2 manoeuvre test list.
While the vehicle parameters were estimated from driving data during the first test (section
3.11), the second test was performed two months after the first one, and thus environmental
changes were expected. In a case of a wet test track it was assumed that such conditions
would lower the tire peak force capability by 30 %, though the tire cornering stiffness
remained the same. Each tire setting as well as features that were implemented in the model
during each test can be seen in Appendix D – Test 2 manoeuvre test list. During the testing
day the asphalt was wet and the optimization results from both tire settings were used to
perform the tests.
For filtering the yaw rate and lateral acceleration a second order Butterworth low pass filter was used, with a
normalized cut-off frequency of 0.2. The signal was filtered both in the forward and reverse direction for avoidance of
phase shifting introduced by one-directional filtering.
23
58
4 Results
Except for the parameter study all results shown within this chapter are related to the Volvo
S60 2013 given by Appendix B – Test vehicle, which was also used during the tests with a
steering robot. The results from the parameter study show how robust the method used in
this study is and hence give an indication of how much different a vehicle can be, compared
to a Volvo S60 2013, in order to still be able to calculate the optimal steering angles. If not
stated differently the nominal parameters for the vehicle model are assumed as:
• Two track model with yaw rate error controlled ESC
• No tire relaxation, no wheel kinematics and no suspension compliance
(Negligible influence to the steering output)
• ESC yaw rate threshold of 2 °
• ESC activation torque of 200 Nm
• ESC torque increase factor of 5
• Vehicle parameters according to Appendix B – Test vehicle
59
4.1
Optimal controller driving inputs
The optimal steering angles achieved when using different features in the model can be seen
in Figure 22, making it obvious that the modelled features play a significant role in this kind
of manoeuvre as the steering angle input difference between the cases can be 10 °.
Figure 22. Steering angles input calculated with different vehicle modelling features implemented.
The graph illustrate the scattering of the result.
The largest magnitude in Figure 22 derives from the linear bicycle model, since that model
allows higher lateral force generation than the other models just by increasing the steering
angle. The bicycle model with a magic formula tire model is however among the four lines in
the figures that are closest to each other, being thus already a significantly improved model.
Between X = 40 and X = 50 the differences are biggest between the different models, Figure
22 shows that part in detail.
60
Figure 23. Optimal steering angles calculated with different models. In this figure non-linear tires
refer to the magic formula and are used for all two track models. The ESC refer to the yaw rate error
controlled ESC. It should be noted that the linear model also has half of the equivalent tire stiffness
compared to the non-linear tires to make it behave more realistic as described in 3.4.1 Linear tire
model.
It is seen that the bicycle model with non-linear tires is closer to the two track model, with
tire relaxation, than the usual two track model at the position of the peak steering angles, but
it is the opposite situation when not at the peak. That makes the tire relaxation an important
part of the model when the ESC is not activated. When the ESC is present in the model
there is however almost no difference between a model with or without the tire relaxation
effect, thus it is not an important component of the model in that situation. If one also
compares the maximum entry speeds that each model gives as a result, Table 15 is obtained.
Table 15. Maximum entry speed with different models. In this table non-linear tires refer to the
magic formula tire model and are used for all two track models. ESC refers to the yaw rate error
controlled ESC.
Model
Bicycle model, linear tires
Bicycle model, non-linear tires
Two track model
Two track model, tire relaxation
Two track model, ESC
Two track model, tire relaxation, ESC
(i)
Entry speed
69.1 km/h (90.7 km/h)(i)
68.5 km/h
70.8 km/h
69.6 km/h
73.6 km/h
72.5 km/h
If the tire cornering stiffness is equivalent to the non-linear tire model cornering stiffness.
61
When comparing the entry speeds it is obvious that the linear bicycle model was unrealistic;
thus the half equivalent tire cornering stiffness (resulting from the B, C, D product) was used.
It can also be seen that the bicycle model with non-linear tires results in a speed closer to the
two track model with tire relaxation, than the usual two track model, in a similar manner to
how the peak steering angle was closer for the bicycle model. Tire relaxation lowers the entry
speed with approximately 1 km/h in both the case of a model with ESC as well as without,
something which was also seen for other specifications of the tires.
10
Steering angle [°]
5
0
-5
-10
-15
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 24. Optimized steering angle input in the case of a nominal set-up.
In Figure 24 the steering angles along the double-lane change track can be studied for the
case of a nominal set-up.
4.2
Optimal trajectory
The case of implementing a bicycle model with linear tires, a bicycle model with Magic
formula tires, a two track model with and without tire relaxation as well as a two track model
with ESC with and without tire relaxation was studied in the same plot in Figure 25.
Global Y-coordinate [m]
10
5
0
-5
0
10
20
30
Global X-coordinate [m]
40
Figure 25. Trajectory with different modelling implementations.
62
50
60
As was seen in Figure 25 the optimal trajectory had the same general shape Independent of
the modelled features. In Figure 26 it can be seen for the case of a nominal set-up.
Figure 26. Vehicle trail when the vehicle was set up with nominal values.
The trajectory can be seen passing close to three of the cones in the track. Those parts were
investigated by using the animation tool described in section 3.10 in which no boundary was
violated.
4.3
ESC influence
Influence here refers to the difference between the results when an ESC model is used in the
optimization algorithm and when it is not. Different ESC models had different influence, as
the model was constructed in such a way that it is possible to use some ESC settings which
would result in no influence at all; for this comparison the yaw rate error controlled ESC was
used with a yaw rate error threshold of 2 °, an initial torque upon activation of 200 Nm and a
torque increase factor of 5 while the vehicle parameters were set according to Appendix B –
Test vehicle. The reason to use that type of ESC is that it solves its equation set quickly with
continuous functions. As discussed in section 4.2, there was almost no difference in the
trajectory when using an ESC, compared to when not using it. The difference there was is
therefore studied as the absolute lateral deviation between the two trajectories in Figure 27.
63
0.35
Absolute lateral difference [m]
0.3
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 27. Absolute lateral difference between optimization with and without ESC present in the
model.
It is shown in Figure 27 that the absolute lateral difference, when adding the ESC to the
system, is normally less than 0.27 m along the track. The difference is largest at the same
position as where the steering angles deviate the most, as seen in Figure 23. As there are two
peaks, between which the difference almost reaches zero, the difference does not tend to
increase by accumulating errors. The steering angles, which were shown to be more
dependent upon the features that were modelled in section 4.1, can be seen with and without
ESC present in the model in Figure 28.
15
Tw o track model
Tw o track model ESC
Steering angle [°]
10
5
0
-5
-10
-15
0
10
20
30
Global X-coordinate [m]
40
50
Figure 28. Steering angles with and without ESC present in the two track model.
64
60
From Figure 28 it is obvious that all steering angle peaks are lowered when the ESC is used,
something which could be expected as steering motion is not only generated by the steering
of the wheels but also from the braking torque on each wheel (which produces a yaw torque).
Figure 29 shows the same comparison for the steering angle rate.
50
Tw o track model
40
Tw o track model ESC
30
Steering rate [°/s]
20
10
0
-10
-20
-30
-40
-50
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 29. Steering angle rate with and without ESC present in the model.
One can see that the steering angle rate transition between different values is smoother when
the ESC is used while the case without ESC has an abrupt change between a positive and
negative rate at several positions along the track.
65
74
Tw o track model
Tw o track model ESC
72
Longitudinal speed [km/h]
70
68
66
64
62
60
58
56
54
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 30. Longitudinal speed comparison between the two track model with and without ESC
included in the optimization.
The slower changing steering angles and their lower amplitudes due to braking allow for
higher entry speed, as seen in Table 15, but also a lower speed along a large part of the track
compared to the model which does not include ESC as can be seen in Figure 30.
For the case of an ESC which would be activated at the yaw rate threshold specified, using
the optimal braking torque for each wheel, the steering angle can be seen in Figure 31.
15
Steering angle [°]
10
5
0
-5
-10
-15
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 31. Steering angles resulting from an optimization with a Yaw rate error activated optimal
torque controller.
Comparing the steering angles between Figure 31 and the nominal case (Figure 24) a
smoother behaviour is observed for the former. The corresponding braking torque for each
wheel is shown in Figure 32 for the front wheel and in Figure 33 for the rear wheels.
66
Tfr as function of X
500
0
0
-500
-500
Tfr [Nm]
Tfl [Nm]
Tfl as function of X
500
-1000
-1500
-1000
-1500
-2000
-2000
-2500
-2500
-3000
0
10
20
30
40
50
Global X-coordinate [m]
60
-3000
70
0
10
50
40
30
20
Global X-coordinate [m]
60
70
Figure 32. Braking torque in the case of a Yaw rate error activated optimal torque controller, to the
left for the front left wheel and to the right for the front right wheel.
Trr as function of X
Trl as function of X
200
200
0
0
-200
-400
Trr [Nm]
Trl [Nm]
-200
-400
-600
-600
-800
-1000
-800
-1200
-1000
-1200
-1400
0
10
20
30
40
50
Global X-coordinate [m]
60
70
-1600
0
10
50
30
40
20
Global X-coordinate [m]
60
70
Figure 33. Braking torque in the case of a Yaw rate error activated optimal torque controller, to the
left for the rear left wheel and to the right for the rear right wheel.
In Figure 32 and Figure 33 it can be seen that braking only occurs in the entry part of the
track.
4.3.1
Other ESC models investigated
For the sideslip angle controlled ESC a solution was found, despite inclusion of
discontinuous functions, but even minor changes to the vehicle specifications, wheel torque
proportional constants or ESC activation threshold made it impossible for the solver to
converge towards one solution. For this reason it was regarded as unsuitable to use, as the
optimization procedure is intended to be fully automated.
In the case of yaw rate error and sideslip angle controlled ESC a second optimization
problem needs to be solved within the ESC equation set, as each contact force due to ESC
67
interaction is optimized individually in such a case. Thus it would require many more
iterations for a numerical solver and within the time frame of this study it was not possible to
investigate the solution of this further.
4.4
Steering robot test 1
From the robot driven double-lane change tests the measured lateral acceleration, yaw rate
and longitudinal speed were used; the estimated front magic formula coefficients (B, C and
D), were found as 7.5318, 1.4897 and 1.1233 respectively with a yaw inertia of 3500 kgm2.
The generated optimal steering angles from the simulation were inserted to the steering robot
and by running the test at a lower speed it was possible to verify that the robot could
perform the requested operations. In this case the steering angle rate showed to be faster
than the robot could handle; the steering angle rate limit was then introduced in the model,
but the new optimal solution had larger steering angles than the car could handle which led
to a correction of the model’s steering angle limits as well. The performance of the controller
during the first driving test can be seen in Appendix C – Measured data part 1. An illustration
of the steering robot set-up can be seen in Figure 34 while Figure 35 and Figure 36 show the
vehicle during testing with that robot. The vehicle is also shown during a manual driven test
through the double-lane change cone track in Figure 37.
Figure 34. SR60 standard robot by Anthony Best Dynamics mounted on the vehicle.
68
Figure 35. Entry of the double-lane change track where the vehicle is driven by the steering robot.
Figure 36. Middle section of the double-lane change track where the vehicle is driven by the steering
robot.
Figure 37. Manual driven vehicle through the double-lane change track.
4.4.1
Parameter estimation
Starting from different initial guesses, the curve fitting procedure using the double-lane
change recordings would converge to some different set of values for B, C and D, though all
of these values were close. This indicates that several local optima could be found with this
69
approach. But, since it is an approximate rather than a systematic and accurate method, such
a result was expected.
10
8
2
Lateral acceleration [m/s ]
6
4
2
0
-2
-4
-6
Measurement
-8
-10
Model
0
0.5
1
1.5
2.5
Time [s]
2
3.5
3
4.5
4
5
Figure 38 - Fitting between the measured and the calculated lateral acceleration.
40
30
Yaw rate [deg/s]
20
10
0
-10
-20
Measurement
-30
Model
-40
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
5
Figure 39 - Fitting between the measured and the calculated yaw rate.
An example of the results is shown in Figure 38 and Figure 39; the output of the model is
plotted along the respective recorded values. Also, the curve of the tire model, that this
approach resulted in, can be seen in Figure 40.
70
1.4
1.2
Friction coefficient µ
1
0.8
0.6
0.4
0.2
0
0
5
10
15
25
20
Slip angle α [deg]
30
35
40
45
Figure 40 - Tire curve for the values found by curve fitting. D=1.1233, C =1.4897, B =7.5318
4.5
Steering robot test 2
During all of the tests the asphalt was wet and during a few tests it was raining. Each test was
carried out with two different tire models; the tire model that was estimated with a dry
surface from the first test and is called “dry model,” and the model with reduced peak force
capability, called “wet model”. Figure 41 and Figure 42 show the vehicle during testing with
the robot, which already has been shown in Figure 34 above.
Figure 41. Entry of the double-lane change track where the vehicle is driven by the steering robot.
71
Figure 42. Middle section of the double-lane change track where the vehicle is driven by the steering
robot.
The results from the second test vary significantly when outputs from cases, with various
implemented features in the model, were tested. The input was the steering angle calculated
by the optimization process. Input steering angles, simulated positions and positions at the
double-lane change test track are shown for the nominal vehicle set-up for wet conditions in
Figure 43-Figure 45 while the same is shown for dry conditions in Figure 46-Figure 48. All of
the test cases can be found in the Appendix E – Measured data part 2. The legend in each
figure denotes the code name of each test. As every model was tested twice, i.e. every test
was run two times, two plots can be seen in each figure.
72
10
Steering angle [°]
5
0
-5
-10
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 43. Steering angle simulation output along the double-lane change track for a model with
nominal parameters in wet conditions.
6
Global Y-coordinate [m]
5
4
3
2
1
0
-1
-2
-3
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 44. Simulation trajectory in the double-lane change track for a model with nominal
parameters in wet conditions, also corresponding to the steering angles shown in Figure 43.
6
Global Y-coordinate [m]
5
02500401
4
02500402
3
2
1
0
-1
-2
-3
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 45. Real vehicle trajectory in the double-lane change track for a model with nominal
parameters in wet conditions, also corresponding to the steering angles shown in Figure 43.
73
10
Steering angle [°]
5
0
-5
-10
-15
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 46. Steering angle simulation output along the double-lane change track for a model with
nominal parameters in dry conditions.
6
Global Y-coordinate [m]
5
4
3
2
1
0
-1
-2
-3
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 47. Simulation trajectory in the double-lane change track for a model with nominal
parameters in dry conditions, also corresponding to the steering angles shown in Figure 45.
6
Global Y-coordinate [m]
5
02501201
4
02501202
3
2
1
0
-1
-2
-3
0
10
20
30
Global X-coordinate [m]
40
50
60
Figure 48. Real vehicle trajectory in the double-lane change track for a model with nominal
parameters in dry conditions, also corresponding to the steering angles shown in Figure 45.
74
Since each figure depicts the same test run twice, it would be expected that the two curves,
originating from the same control inputs (steering and initial travelling speed), would
coincide; this is not the case. While the same steering inputs and travelling speed were fed to
the robot, the vehicle was not on the same position on the track in every run. This indicates
that the friction between the tires and the ground was varying. As it is shown in the figures,
all of the passes regarded as most successful manage to clear the entry and the exit lane of the
track, while crossing the lower part of lane B. A possible reason for such a phenomenon can
be inappropriately modelled tires as well as an erroneous estimation of the yaw inertia. The
vehicle does not turn as much as the model predicts both entering as well as exiting the
second lane. This can be the case when the vehicle model assumed tires with higher
cornering stiffness, and thus higher lateral force generation capability for the same steering
angles, than the real ones. In addition, too low a yaw inertia in the model can also be
responsible for such a behaviour.
In all of the cases of the most successful tests though, the fact that the vehicle passes the first
and last lane of the track indicates a symmetry to the vehicle’s trajectory, meaning that an
appropriate correction to the steering angle’s magnitude will approach the desired trajectory.
This correction can result both from an improved tire model and/or by a more exact yaw
inertia value.
4.6
Parameter study
Steering angles, possible maximum entry speeds and robustness, as it was defined in 3.9
Parameter study, for different scenarios are the main results that are presented for each
parameter variation.
4.6.1
Mass
In Figure 49 below the dependency of the entry speed on the vehicle mass is shown. This
study was performed by increasing the mass in steps while keeping every other parameter
constant. It is clear that the entry speed tends to increase the larger the mass gets. At first,
this might seem counterintuitive, since the mass is known to negatively affect the handling of
a vehicle in general. The mass is the only parameter altered here, with everything else held
constant; larger mass means that the normal forces on the wheels are greater and hence their
ability to generate lateral forces is also increased. As the lateral force generation ability of the
tires is of crucial importance in this double lane change manoeuvre, the increased mass has a
positive effect on the vehicle’s entry speed in the manoeuvre.
75
74
73
Entry speed [km/h]
72
71
70
69
68
67
1200
1400
1600
1800
Vehicle mass [kg]
2000
2200
2400
Figure 49. The effect of the vehicle's mass on its entry speed to the DLC test. The actual mass of the
S60 was 1823kg.
In reality, vehicles with higher mass have usually higher moments of inertia, which practically
means less agility. This happens because the increased mass is distributed among different
parts of the vehicle. But in the case of this study, the mass, being the only parameter altered,
is considered as if it is added only at the vehicle’s centre of gravity (or it is distributed in such
a way that the vehicle’s moments of inertia remain unaffected), which means that the vehicle
does not actually become less agile. The increased mass, with everything else being constant,
has a positive effect on the vehicle’s entry speed in the manoeuvre.
4.6.2
Yaw inertia
The effect of the vehicle’s yaw inertia on its entry speed at the manoeuvre is depicted in the
Figure 50. In a similar manner with the mass study, the yaw inertia study was performed by
increasing the yaw inertia value while keeping every other parameter at its nominal value. The
tendency of the entry speed to get lower with increased yaw inertia is clear from the figure.
Higher inertia means in practice that the vehicle tends to be less agile, and thus its ability to
violently change direction is decreased. This results in a lower maximum possible entry
speed.
76
74.5
74
Entry speed [km/h]
73.5
73
72.5
72
71.5
71
70.5
2400
2600
2800
3000
3200
3400
3600
3800
4000
4200
2
Yaw inertia [kgm ]
Figure 50. The effect of the vehicle's yaw inertia on its entry speed to the DLC test. The actual yaw
inertia of the S60 was 3500kgm2.
Altering the yaw inertia without affecting any other vehicle parameter means that the mass,
though remaining the same in total, is distributed in different ways. A lower value for the yaw
inertia would mean that the mass concentrates more toward the vehicle’s centre of gravity,
while a higher value would mean the opposite. An example of different mass distributions as
a means to alter the vehicle’s moment of inertia is the various powertrain layouts. The most
common layout, used in the majority of passenger cars, is the front-engine and front wheel
drive vehicle that in general has a high yaw inertia value. Another layout, used often by more
sporty cars, is the mid-engine rear wheel drive one that has a lower inertia value, which adds
to the vehicle’s agility.
Mass distribution is of crucial importance for the vehicle’s handling characteristics, since,
except for its effect on the vehicle’s agility, it dictates its centre of gravity position, its
understeer or oversteer behaviour, and hence its predictability and safety.
4.6.3
Body width and length
The study of the effect of the body width and length of the vehicle on its entry speed rsulted
in entry speeds given by Figure 51 and Figure 52. From the graphs it is apparent that an
increased vehicle width or length has a negative effect on its entry speed in the track.
77
82
80
Entry speed [km/h]
78
76
74
72
70
68
1.6
1.4
1.8
2
Vehicle w idth [m]
2.2
2.4
2.6
Figure 51. The effect of the overall vehicle width on the entry speed. The actual overall width for the
S60 is 1.865m, without the rear view mirrors.
74.4
74.2
Entry speed [km/h]
74
73.8
73.6
73.4
73.2
73
72.8
72.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Vehicle length [m]
4.8
4.9
5
5.1
Figure 52. The effect of the overall vehicle length on the entry speed. The actual overall length for the
S60 is 4.635m.
This result can be explained by the fact that a wider or longer vehicle has less space to move
through when performing the manoeuvre. While this is clear for the case of the length, a
closer look might be needed for the case of width, since a change in the track geometry takes
place according to the vehicle’s width (specifically the change in the first and second lane
widths).
78
The lanes that are affected by the vehicle’s body width are the A and B lane as named in
Figure 53. Even though these lanes become wider for a wider vehicle, their in-between
distance remains the same, which makes the transition from lane A to lane B and from lane B
to lane C harder for a wider vehicle. In addition, lane C is always 3 meters wide, posing
another restraint to the wider vehicles.
Figure 53. Wider and longer vehicles perform worse in regards of entry speed. Even though the width
of lanes A and B becomes larger for wider vehicles, their in-between spacing remains the same. This
also holds true for the spacing between lanes B and C. Additionally, lane C is always kept at 3 meters
wide. The parts that pose those restrictions are marked in circles.
4.6.4
Roll stiffness
The effect of roll stiffness was also studied by altering the front roll stiffness and the rear roll
stiffness while keeping everything else in their nominal values. In Figure 54 the effect of the
front roll stiffness on the entry speed can be seen and in Figure 55 the related steering wheel
angles can be seen.
79
72.7
72.6
Entry speed [km/h]
72.5
72.4
72.3
72.2
72.1
72
71.9
5
4
3
6
Roll stiffness front [Nm/rad]
9
8
7
4
x 10
Figure 54. The effect of the front roll stiffness on the entry speed. The actual front roll stiffness used
in this study was 45kNm/rad.
150
Steering wheel angle [deg]
100
50
0
-50
Kf = 30 kNm/rad
Kf = 45 kNm/rad
-100
Kf = 70 kNm/rad
Kf = 80 kNm/rad
-150
0
10
20
40
30
50
60
70
X [m]
Figure 55. Steering wheel angle profiles for different front roll stiffness values.
Altering the front or rear roll stiffness does not seem to have a significant effect to the entry
speed of the vehicle as one also studies Figure 56, which shows the effect of the rear roll
stiffness on the entry speed, as well as Figure 57, which shows the corresponding steering
wheel angles.
80
72.4
72.35
Entry speed [km/h]
72.3
72.25
72.2
72.15
72.1
72.05
2
4
3
5
Roll stiffness rear [Nm/rad]
6
7
8
4
x 10
Figure 56. The effect of the rear roll stiffness on the entry speed. The actual rear roll stiffness used in
this study was 37.5kNm/rad.
150
Steering wheel angle [deg]
100
50
0
-50
Kf = 30 kNm/rad
Kf = 45 kNm/rad
-100
Kf = 70 kNm/rad
Kf = 80 kNm/rad
-150
0
10
20
40
30
50
60
70
X [m]
Figure 57. Steering wheel angle profiles for different rear roll stiffness values.
For reference, the corresponding steering wheel angle profiles are also plotted for some
values of the roll stiffness. The biggest differences in the generated steering wheel angle
profiles are observed around X = 0-2 m, X = 10 m, X = 30-35 m for both the cases of the
front and rear roll stiffness studies. These areas define the entry to the double-lane change
81
track, the preparation for exit from the first lane (part A), and the middle/preparation for
exit from the second lane (part B) respectively.
The entry part of the double-lane change track allows for some small steering action that can
be utilized for braking by steering and this is a possible reason for which every steering angle
profile begins with steering at the beginning. But the direction towards which this initial
steering occurs affects the dynamics of the vehicle and therefore its future behaviour. The
exit-from-the-first-lane and the middle/exit-from-the-second-lane parts are the parts where
the vehicle is subjected to high lateral accelerations and corrections towards the desired path
afterwards. During these parts the vehicle’s roll stiffness distribution front/rear plays an
important role since it affects its understeer or oversteer behaviour and therefore it should be
handled differently, hence the differences in the steering wheel angle profiles.
As mentioned, the roll stiffness distribution has an effect on the vehicle’s oversteer or
understeer behaviour. In general, the end (front or rear) with the higher roll stiffness tends to
lose traction earlier than the end with the lower roll stiffness. A vehicle with much higher roll
stiffness on the rear will be oversteer while a vehicle with much higher roll stiffness on the
front will be understeer. As an oversteer vehicle is unstable it is considered more unsafe for
the driver and thus is generally undesired [24] [19]. According to the results of this roll
stiffness study though, the vehicle’s entry speed in the manoeuvre remains almost unaffected
whether it is configured to be understeer or oversteer. This indicates that the controller is
able to compensate for understeering or oversteering behaviour.
4.6.5
Tire property changes
These results refer to changes of the magic formula’s B, C and D components and in Figure
58 all possible maximal entry speeds can be seen for all 47 successfully tested cases. In total
54 different cases were tested which implies that the optimization algorithm robustness for
tire changes is 87 %.
82
Figure 58. Maximal entry speed for all tested tire properties. Tire setup numbers 1-10 correspond to
an equal relative increase of each tire parameter (B,C,D), 11-20 correspond to a relative increase of B ,
21-29 correspond to a relative increase of C , 30-39 correspond to a relative increase of D but with
constant product B*C*D and 40-47 correspond to a relative increase of D. The range for each test is
between 50-150 % of the parameter’s nominal value.
It can be seen that all test cases except setup number 40 and 46 are in the same interval,
meaning that an increase of the product B*C*D has approximately the same influence on the
maximum entry speed that can be achieved as when changing only parameter B or C. Setups
40 and 46 both had a change of only the parameter D which implies that D has a larger
impact on the result, something which could be expected as the vehicle should drive on its
friction limits and D dictates the peak value for the tires. With a 50 % increase of D the entry
speed increased 17.5 %, resulting in an entry speed of 86.2 km/h. If instead D decreased with
50 % the entry speed decreased with 27 %, resulting in an entry speed of 53.6 km/h. Steering
angles corresponding to changes of parameter D can be seen in Figure 59.
83
Figure 59. Steering angles in the case of the minimal, maximal and normal value for the parameter D
in the tire magic formula.
At the parts of the track where the largest steering angles were present in the nominal case
and in the case with extra high D value it can be seen that the low D value results in much
lower steering magnitudes. A possible explanation for these lower magnitudes is the fact that
the vehicle is also forced to drive slower and the whole manoeuvre becomes violent. It
should also be noted that the steering angles are very equal between the normal case and a
case of a 50% higher D value, thus for high enough D values the entry speed increases but
the steering angles remain. That is also still the case if the total B*C*D product is held
constant while D is changed, as can be seen in Figure 60.
84
Figure 60. Constant B*C*D product while changing D.
There is even less difference between the maximum D value and the nominal case when the
product B*C*D is held constant. No coupling can be seen between the D value and the
magnitude of the steering angle as the line, that has the largest magnitude, is not always the
same for all X-coordinates.
85
4.6.6
ESC-characteristics variations
In total 600 different ESC characteristics were used, of which 52 cases could not be solved
without lowering the increase of collocation points between two consecutive iterations,
resulting in an ESC model robustness of 91 %. The entry speed, which could be achieved
with each of these cases, can be seen in Figure 61.
Figure 61. Illustration of the scattering in the maximum entry speeds that were achieved by the
simulation for each of the five different groups. Each group corresponds to a different yaw rate error
threshold for the ESC activation. For every group the activation torque and torque increase factor
were varied and the result above was obtained.
In Figure 61 the resulting entry speeds are grouped; the difference between those groups is
the yaw rate error threshold and although it does not have a high influence in the tested
interval some differences can be seen. The threshold is 0° to the left and increasing 1° per
group to the right; the 2° group is then found to be both the best and the worst 24. The best,
as well as the worst, ESC characteristics can be seen in Figure 62.
24
“Best” is defined as the case where the highest entry speed is achieved while ”worst” is defined as the opposite.
86
Figure 62. Best and least efficient ESC torque characteristics of those tested in this study. The
optimal settings are a yaw rate error threshold of 2°, an activation torque of 300 Nm and a torque
increase factor of 4. The least efficient ESC settings are a yaw rate error threshold of 2°, an activation
torque of 50 Nm and a torque increase factor of 1. An infinite amount of combinations are possible to
test.
An entry speed in the case of the best ESC settings, shown in Figure 62, was 74.1 km/h
while the least efficient ESC settings implied that the entry speed could not be higher than
70.7 km/h which is 0.1 km/h less than without ESC. One may argue that the accuracy is not
high enough to consider 0.1 km/h as a lower speed but it should be noted that there is a
possibility as well that an ESC with the wrong settings is worse than not having it at all. Also
the influence on the optimal steering angles can be seen by studying Figure 63. It can be seen
in which sections the ESC was influencing the most by detecting the position where the
difference is largest between the lowest and highest steering angles in the Figure 63.
Figure 63. Resulting optimal steering angles for the 548 successful ESC combinations.
Note that some solutions have negative steering angles while other solutions have positive
steering angles right after the start, but within the same magnitude interval. As the vehicle
87
just drives straight through that section it would be useful to reduce the speed before the
cornering starts and as steering manoeuvres reduce the speed a steering angle in any direction
of the same magnitude would produce a braking force. Depending on the ESC characteristics
that were used, different yaw rate errors may be resulting in optimal braking torque on each
wheel; the controller then compensates the steering angles in such a way that this yaw rate
error is generated. Other ESC characteristic parameters were held at their nominal values
while changing the torque increase factor for a few different initial torque values which
resulted in the maximal entry speeds as given by Figure 64.
Figure 64. Maximal entry speed as function of torque increase factor for three different values of
initial torque at a yaw rate threshold of 2°.
One can see the connection between the maximum entry speed that can be achieved and the
ESC initial torque for most values of torque increase factor. The same was done for a few
different yaw rate error threshold values instead of initial torque values in Figure 65.
88
Figure 65. Maximum entry speed as function of torque increase factor for yaw rate error thresholds
between 0-5° at an initial torque of 200 Nm.
It is clear that a yaw rate error threshold of 2° is the least efficient choice when the ESC is
setup with an initial torque of 200 Nm but as was seen in Figure 61 one will be able to
achieve the highest entry speed with a yaw rate threshold of 2°. That speed corresponds to
the best ESC, which can be seen in detail in Figure 62.
4.6.7
Influence of discretization
Optimal steering angles were calculated for a bicycle model with non-linear tires for different
resolutions of the discretization, resulting in Figure 66.
89
Figure 66. Optimized steering angle inputs with different number of collocation points. The
magnification window shows a part of the result where the difference between the solutions is
biggest.
One can see that the result coming from the discretization with the lowest resolution, 20
collocation points, is furthest away from the solution with 150 collocation points. Too few
collocation points result in a solution which may miss to place a point where a steering angle
change should occur in an ideal and continuous solution; in the beginning of the figure, at
X = 2 m, the 20 collocation points solution deviates from the others because there is no
collocation point there. If the 150 collocation points’ solution (𝛿𝛿150 ) is set as reference the
relative accumulated absolute error (Err), given by Eq. 127, for each of the other solutions
can be studied in Figure 67.
w
90
𝐸𝐸𝐸𝐸𝐸𝐸 = 1 −
∫|𝛿𝛿|
|𝛿𝛿
∫ 150 |
Eq. 128
Figure 67. Relative accumulated absolute steering angle error for solutions with 20-150 collocation
points in step of 5 collocation points. The 150 collocation point solution was set as the reference.
A decrease of the error can be seen the more collocation points are used, but the error is not
always lower for each added collocation point. In the same manner as before, this can be
explained by the distribution of the collocation points along the X-position in space. In the
X-direction there might be some points where a change in the steering angle is critical, like
for example the X = 2 m position in Figure 66. If the distribution of the collocation points by
the solver does not include these critical points, then the relevant important change in the
steering angle will not be predicted, and therefore the error would be negatively affected.
That “miss” of the important instants is declining the more collocation points are used, since
the probability to miss the crucial parts decreases with more points, and will tend to zero as
the collocation points tend to infinity, i.e. a continuous rather than a discrete solution. While
the increase in the collocation points is therefore expected in general to decrease the error, it
cannot serve as an absolute metric for the quality of the solution.
91
5 Discussion
This thesis investigated the generation of the optimal steering control input that facilitates the
maximization of a vehicle’s entry speed in the double-lane change manoeuvre. A two-track
vehicle model was built in Matlab, where also various vehicle dynamics features were
implemented. Also an ESC implementation was presented and was integrated into the
vehicle model. The model, including the ESC, was subjected to an optimization process to
obtain the corresponding optimal steering angles, using the optimization tool Tomlab.
During this study simplifications were made to a few of the modelled parts of the vehicle as
well as the result was studied with various features implemented in the model. The possibility
to use the suggested approach to solve the problem with other vehicles than the test vehicle
was also investigated. These aspects of the study are discussed briefly in this chapter,
providing our view on the process of creating a tool for an early development phase of a
vehicle.
5.1
Vehicle dynamics and ESC modelling
The vehicle’s dynamics were modelled by a two-track vehicle model which included features
as roll stiffness, roll steer and camber gain. The effect of those features was studied and it
was found that the controller is able to compensate for various stiffness and suspension
compliance setups. For the tire modelling a simplified Magic formula was used and tire
relaxation phenomena were also implemented. Obtaining accurate values for those features’
parameters can be a cumbersome task, but a proper implementation of the tire model is of
crucial importance when performing vehicle dynamics studies, especially on the vehicle’s
handling limits.
An ESC logic was constructed, which was demonstrating discontinuous behaviour that was
needed to be approximated by a continuous one. A method for such an approximation was
suggested and this method has also the potential to be used for the real vehicle’s ESC. Such
an application of the method would be an interesting continuation of this work. It would also
be possible to investigate the ESC characteristics that are the optimal to use, as the short
investigation in the parameter study shows. It is also of interest to optimize the ESC
characteristics, not just for the case of this double-lane change manoeuvre, but also consider
other scenarios as well.
5.2
Results
The trajectory deriving from the optimization process implies that the vehicle travels close to
the first, second and fourth inner corner while being further out from the third corner. In
sections A and C, as noted in Figure 53, the car travels as close as possible to the lower track
boundary while the rear part of the car also travels as close to the upper track boundary as
92
possible in section B as was seen in Figure 26. Consequently, a simple model as the bicycle
model might be enough if the trajectory is the searched result. Such a simple model can be
solved fast and may even be used to calculate a new optimal trajectory even while the car is
driving, implying potential real-time usage. However, Figure 23 showed that the bicycle
model would be insufficient to predict the correct steering wheel angle inputs. An interesting
fact is that an increasing number of collocation points can increase the accuracy but does not
necessarily do so when comparing the accuracy done with two arbitrary numbers of points.
5.3
Parameter study and method robustness
The thesis evaluated the influence of different vehicle settings on the maximum entry speed
for the double-lane change test. Parameters as the mass, the tires’ cornering stiffness and the
D value of the magic formula have a direct positive effect on the entry speed. On the other
hand, parameters like the vehicle’s body width and length and its yaw inertia have the
opposite effect. The vehicle’s roll stiffness on the front and rear axle did not seem to
influence its ability to perform in the test, as the controller was able to compensate for
understeer or oversteer behaviour.
ESC parameters that were studied were the yaw rate error activation threshold, the torque
increase factor and the activation torque. No direct influence on the entry speed from one
parameter alone, while keeping the others unchanged, was found. An increase or decrease to
one parameter would invoke an increase or decrease to the entry speed depending on the
values of the other two parameters.
Robustness of the optimization algorithm has been tested by performing the optimization
with more than 780 different parameter settings with a total robustness of 92.3 %, without
decreasing iteration steps manually. The method of manually changing the increase of the
collocation points between two consecutive iterations was tested for solving a few random
chosen cases out of the ones which did not converge, so as to also reach to an optimal
solution. The confidence level to always solve the problem, for passenger cars and light
commercial vehicles, is high if the solver is allowed to start with one of the solved cases as a
guess and then make small changes to each parameter to reach the solution. The optimal
method has though limitations; it cannot be guaranteed that it will converge for any vehicle,
as there are more than 1016 cases to solve if the parameters describing the vehicle are only
changed between 10 values.
93
6 Recommendations
A few recommendations can be considered, deriving from the results from the optimization
and from physical testing. Those are:
• Vehicle dynamics model complexity; can differ depending on the situation; if one
would only be interested in the optimal trajectory it is recommended to use the
bicycle model with non-linear tires. If the steering angle inputs are of interest the most
time efficient solution without losing accuracy has been found to be the use of the
two track model with an ESC present but without tire relaxation and instead lower the
entry speed with 1 km/h due to the effect is was supposed to have.
• New parameter estimation; it was performed by curve fitting. More accurate
methods, where standardized tests are performed for this purpose, exist. It is
important to retrieve reliable information about the tire properties, preferably for
various friction conditions (dry, wet, etc…). The tires play a significant role in the
behaviour of the vehicle and a proper implementation is crucial for the model to have
validity.
• Appropriate physical testing; should be done in conditions where the tire
properties are estimated. The new results could then be used to either prove the
model to be accurate enough or provide information about what happened when the
model prediction was incorrect.
• ESC implementation; a more sophisticated ESC model which captures a “more”
realistic ESC logic could be studied. The current ESC logic is a simple implementation
of a basic ESC algorithm, where only one wheel at a time is braked. Production ESC
systems can brake more than one wheel simultaneously in order to achieve some
desired result. Also, it is possible to implement other control methods such as sideslip
control into the ESC model [22].
94
7 Future Work
The current study depicted interesting areas for future research. A few suggestions for future
work and continuation of this study are:
• Improved vehicle model with constrained trajectory; it was found that the CoG
trajectory could be regarded as solved with only a bicycle model. Subsequently, it may
be possible to solve the more complex model with that predetermined path, which
would reduce the boundaries for the problem. The solution time may get shorter but
more important; the robustness can increase if there would be less local minima in the
solution space.
• Real ESC integration; two alternatives to optimize the problem with a real ESC are
proposed.
o Change the ESC algorithm in the vehicle to the same as in the optimal
controller.
o Use the method developed in this study to convert the real, discontinuous,
ESC equations to equivalent continuous functions.
• Close the loop; the way the vehicle is driven in this thesis is open loop. The steering
inputs are predetermined by an optimization process and then are fed to a steering
robot to perform the manoeuvre. This method is prone to accumulating error.
Discrepancies between the model and the real vehicle, as well as the testing surface,
induce some errors at some points and these errors are transmitted to the rest of the
future points in the trajectory. Any error induced accumulates, and the final result
might be far from the model output. This kind of error can be dealt with by feeding
back the output to a controller, which would then act appropriately in order to reduce
it. This controller could be either a driver model included in the loop, or a software
controller, which would act on top of the optimized steering input and add
corrections to the steering angles in real time.
• Expand the tool’s functionalities; upon successful extension of the vehicle model,
to a degree where it actually produces high fidelity results, other dynamic manoeuvres
recreating testing scenarios can be included. This way, a more complete view of the
vehicle’s dynamic and handling characteristics can be obtained, since the number of
different tests gives a higher likelihood to capture the vehicle’s capabilities (and mainly
weaknesses). Such manoeuvres could be the ISO 3888 Part 1 test, slalom tests, or
95
manoeuvres where the objective would be the minimization of the traveling time in a
track.
• Reduction of solution time; depending on the situation the user will use the
optimization algorithm in, it may be necessary to improve the speed of the
calculations. Since the same solver, SNOPT, was always called by the Tomlab
optimization environment, one may call the solver manually to avoid the automated
symbolic processing. Other programming languages, such as C/C++, could be used
but require programming of the optimization algorithm used by Tomlab.
96
8 Nomenclature
Symbol
a
T
m
Y
X
𝜓𝜓
𝜑𝜑
𝜃𝜃
x
y
z
v
ay
𝜔𝜔
W
C
µ
𝜏𝜏
𝜀𝜀
𝐶𝐶𝛾𝛾
M
Kus
t
Description
Smoothness
torque
mass
Global lateral coordinate
Global longitudinal coordinate
Yaw angle
Roll angle
Pitch angle
Local longitudinal coordinate
Local lateral coordinate
Local and global vertical
coordinate
Local velocity
Lateral acceleration
Rotational speed
Distance from CoG to side
Magic formula shape factor
Friction coefficient
Relaxation time constant
Camber angle
Camber stiffness
Yaw torque
Understeer gradient
Time
Symbol
F
b
CoG
f
I
Cα
α
β
L
K
e
Description
Force
Distance of rear axle from CoG
Centre of gravity
Distance of front axle from CoG
Moment of inertia
Cornering stiffness
Slip angle (For the tires)
Sideslip angle
Wheelbase
Roll stiffness
Roll centre height
he
ax
r
B
D
s
CoG height over roll axis
Longitudinal acceleration
Wheel radius
Magic formula stiffness factor
Magic formula peak factor
Resultant tire slip
Relaxation length
Inclination angle
Turn radius
Area
Objective function
Weighting factor for steering rate
influence on objective function
Lrelax
𝛾𝛾
R
A
J
𝑊𝑊𝛿𝛿
97
Index
tb
fr
fl
rr
rl
init
if
tr
d
g
98
Description
Track boundary
Front right
Front left
Rear right
Rear left
Initial value
Increase factor
track
desired
Initial guess value
Index
i
j
x
y
z
r
l
eff
w
Description
Front and Rear
Left and Right
Longitudinal direction
Lateral direction
Vertical direction
right
left
effective
Brake disc
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103
A – ISO 3888 description
The ISO 3888 Part 2-Obstacle avoidance is a dynamic process during which the vehicle is
driven closed loop in a severe lane change manoeuvre such that high alternate lateral
accelerations will be produced. It concerns passenger cars and light commercial vehicles up
to 3500kg and involves rapidly driving from an initial lane to another parallel lane and then
back to the initial one, so to recreate an obstacle avoidance scenario. During the manoeuvre
no cone marking the track should be displaced for the pass to be considered as valid. The
results of this test serve as a subjective evaluation of the vehicle’s lateral dynamics and road
holding ability. The test track that defines the manoeuvre can be seen below.
Figure A 1. ISO 3888 part 2 double-lane change track illustration.
The track sizing is vehicle dependent as can be seen above where the width of the initial lane
A and the second lane B is calculated, as noted in the figure’s legend. The value of
VehicleWidth is the tested vehicle’s overall width without the rear-view mirrors. The geometry
differs only in regards of the lane widths though, as the length of the lanes and their inbetween spacing remains the same.
Driving should be performed with the highest gear that guarantees a minimum of 2000rpm
of engine speed and vehicles with automatic gearbox should have the selector placed in the
drive (D) position. The vehicle enters the track in lane A and after a distance of 2m the
throttle is released 25 where after the remainder of the track is driven in the throttle release
While the ISO 3888 Part 2 document does not make any direct reference to whether the clutch should be disengaged
or not, one may argue that since there is a reference to the engine speed at the entry point, it is implied that the clutch
should remain engaged. In that case some engine braking is expected. In this thesis though, no such phenomenon is
modelled, and the test is simulated as if the clutch is disengaged, i.e. no powertrain effects at all. This allows a clearer
picture of the vehicle’s dynamic behaviour that comes from its chassis and ESC setup since it isolates it from external (i.e.
powertrain) influences.
25
A1
position, without any braking action. Only the steering wheel is being controlled by the driver
and the vehicle’s longitudinal speed is measured at the end of section A, that is at X = 12 m
in the track.
It should be noted that this is a test that best suits subjective rather than objective evaluation
of the vehicle’s lateral dynamics characteristics. This is due to the fact that different drivers,
with different skills, choose different paths and strategies to drive through the track so the
measured speeds are considerably scattered, in spite of the velocity measurement taking place
at the end of section A as a means of reducing this effect.
A2
B – Test vehicle
The vehicle used during all physical testing and as an example when performing the
optimization of the steering angles is described below.
Table B 1. Test vehicle nominal parameters.
Parameter description
Value
Model name
Volvo S60 T5 automat, year model 2009
Weight during testing (driver + equipment)
1823 kg
Curb weight (Standard equipment, full tank
1635 kg
of fuel and no driver)
Gross weight
2060 kg
Yaw moment of inertia
2500 kgm2
Width (excluding mirrors)
1865 mm
Length
4635 mm
Distance of front wheel from mass centre
1854 mm
Distance of rear wheel from mass centre
2781 mm
Distance to the right track from mass centre
793.5 mm (Estimated to mass in middle)
Body length in front of the front wheels
929.5 mm (Estimated to equal as in the rear)
Front wheels track width
1588 mm
Rear wheels track width
1586 mm
Height of centre of mass
500 mm
Front roll centre height
120 mm
Rear roll centre height
240 mm
Front roll stiffness
45 000 Nm/rad
Rear roll stiffness
37 500 Nm/rad
Rolling friction coefficient
0.013
Air drag coefficient
0.28
Frontal area
2.27 m2
Maximum steering angle
31 °
Maximum steering angle rate
720 °/s
Steering ratio
14.95
Front roll steer coefficient
0.123
Rear roll steer coefficient
0.007
Pneumatic trail a linear range
1/30 m
Wheel radius
316 mm
Wheel width
200 mm
Magic formula, B
7.5418
Magic formula, C
1.4887
Magic formula, D
1.1233
Tire relaxation length
300 mm
B1
C – Measured data part 1
Despite that the robot couldn’t perform the manoeuvre exactly in the way specified it
performed a part of the manoeuvre at an entry speed of 74.3 km/h when the ESC was
activated, which can be seen in Figure C 1.
Figure C 1. Resulting coordinates for the vehicle's centre of gravity along the track when the model
demanded larger steering angles as well as higher steering angle rates than was physically possible
with the test vehicle. The solid line use the start of the measurement as zero position while the
dashed line has a corrected coordinate system with -2.8° and 2.11 m shifted X-position zero point; a
correction which was based on where the steering wheel started to move compared to the input
signal.
Below follows measured steering angles, yaw angles and speed for the first physical testing,
which was carried out to validate the model. The position of the reference cones was not
recorded during the measurement, for this reason all the results are presented both with the
start of the recording as reference position as well as a correction based on a comparison of
the steering angle input and the actual steering angle as shown in Figure C 2.
C1
Figure C 2. Input steering wheel angle and actual performed steering wheel angle.
The corrections made for Figure C 3 and Figure C 4 are a shift of -2.11 m of the X-position
as well as turning the coordinate system -2.8° to make the actual yaw angle match the
expected yaw angle at the initial position.
Figure C 3. Measured and corrected yaw angle.
Figure C 4. Longitudinal velocity and corrected longitudinal velocity.
C2
D – Test 2 manoeuvre test list
These testing scenarios were performed, in total 24 robot driven scenarios and 2 human
driven scenarios. Each robot driven test was performed twice and was carried out at
approximately the same position at Volvo Hällered proving ground brake and handling track.
1. Measured vehicle weight, 1836 kg.
2. Activated ESC tests with steering robot
a.
𝜓𝜓̇𝑡𝑡 = 2, Tgain = 200Nm, Tif = 5, Entry speed= 72.8 km/h
Entry speed = 73.6 km/h
b. 𝜓𝜓̇𝑡𝑡 = 2 , Tgain = 200Nm, Tif = 10,
No tire relaxation, Entry speed = 73.6 km/h
c. 𝜓𝜓̇𝑡𝑡 = 2 , Tgain = 100Nm, Tif =
No tire relaxation, Entry speed = 72.3 km/h
d. 𝜓𝜓̇𝑡𝑡 = 0 , Tgain = 200Nm, Tif = 5,
No tire relaxation, Entry speed = 74.1 km/h
Entry
5,
Entry
No tire relaxation,
speed=
Entry
73.6
km/h
speed=
72.3
73.8
km/h
speed=
2.1 Wet alternative (In case of wet asphalt D is reduced by 30% but BCD is the same)
a.
No tire relaxation, Entry speed = 65.9 km/h
b. No tire relaxation, Entry speed = 66.0 km/h
c. No tire relaxation, Entry speed = 64.9 km/h
d. Yaw-threshold 0 degree model, Tgain = 200Nm, Tif = 5, Entry speed= 66.5 km/h
No tire relaxation, Entry speed = 67.0 km/h
3. Deactivated ESC tests with steering robot
a.
Most detailed model, Entry speed = 69.5 km/h
b. Skip tire relaxation and wheel kinematics, Entry speed = 70.8 km/h
c. Bicycle model with magic formula, Entry speed = 68.5 km/h
3.1 Wet alternative (In case of wet asphalt D is reduced by 30% but BCD is the same)
a.
Most detailed model, Entry speed = 62.7 km/h
D1
b. Skip tire relaxation and wheel kinematics, Entry speed = 63.1 km/h
c. Bicycle model with magic formula, Entry speed = 60.4 km/h
4. Human driver, deactivated ESC, cone track
a.
Record the first left and last left cone, the distance should be long between the
cones in order to get high track angle accuracy
b. Measure the track with in each section
c. Note in which maximum entry speed a driver could go through the track without
hitting any cone.
5. Human driver, activated ESC, cone track
a.
D2
Note in which maximum entry speed a driver could go through the track without
hitting any cone.
E – Measured data part 2
In this appendix a figure for the resulting position along the track for each of the tests can be
seen. Figure E 1-Figure E 8 show results with the approximated wet tire model as described
in 4.5 Steering robot test 2 while Figure E 9-Figure E 18 show results when using the normal
model.
Wet tests
6
02500401
02500402
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 1. Two track model with ESC implemented; threshold of 2 °, activation torque of 200 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
E1
6
02500501
02500502
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 2. Two track model with ESC implemented; threshold of 2 °, activation torque of 200 Nm
and torque increase factor of 10. The vehicle parameters were set to their nominal values.
6
02500601
02500602
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 3. Two track model with ESC implemented; threshold of 2 °, activation torque of 100 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
E2
6
02500701
02500702
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 4. Two track model with ESC implemented; threshold of 0 °, activation torque of 200 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
6
02500801
02500802
4
2
Y position
m
0
-2
-4
-6
-8
-10
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 5. Two track model with wheel kinematics, tire relaxation and no ESC, the vehicle
parameters were set to their nominal values.
E3
6
02500901
02500902
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 6. Two track model with wheel kinematics and no ESC, the vehicle parameters were set to
their nominal values.
6
02501001
02501002
4
Y position
m
2
0
-2
-4
-6
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 7. Bicycle model with Magic formula tires, the vehicle parameters were set to their nominal
values.
E4
6
02501101
02501102
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 8. Two track model with tire relaxation and ESC implemented; threshold of 0 °, activation
torque of 200 Nm and torque increase factor of 5. The vehicle parameters were set to their nominal
values.
Dry tests
6
02501201
02501202
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 9. Two track model with ESC implemented; threshold of 2 °, activation torque of 200 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
E5
p
6
02501301
02501302
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 10. Two track model with tire relaxation and ESC implemented; threshold of 2 °, activation
torque of 200 Nm and torque increase factor of 5. The vehicle parameters were set to their nominal
values.
6
02501401
02501402
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 11. Two track model with ESC implemented; threshold of 2 °, activation torque of 200 Nm
and torque increase factor of 10. The vehicle parameters were set to their nominal values.
E6
6
02501501
02501502
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 12. Two track model with tire relaxation and ESC implemented; threshold of 2 °, activation
torque of 200 Nm and torque increase factor of 10. The vehicle parameters were set to their nominal
values.
6
02501601
02501602
4
Y position
m
2
0
-2
-4
-6
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 13. Two track model with ESC implemented; threshold of 2 °, activation torque of 100 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
E7
6
02501801
02501802
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 14. Two track model with ESC implemented; threshold of 0 °, activation torque of 200 Nm
and torque increase factor of 5. The vehicle parameters were set to their nominal values.
6
02501901
02501902
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 15. Two track model with tire relaxation and ESC implemented; threshold of 0 °, activation
torque of 200 Nm and torque increase factor of 5. The vehicle parameters were set to their nominal
values.
E8
6
02502001
02502002
5
4
Y position
m
3
2
1
0
-1
-2
-3
-4
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 16. Two track model with tire relaxation, wheel kinematics and no ESC implemented. The
vehicle parameters were set to their nominal values.
6
02502101
02502102
4
Y position
m
2
0
-2
-4
-6
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 17. Two track model with no ESC implemented. The vehicle parameters were set to their
nominal values.
E9
6
02502201
02502202
5
4
Y position
m
3
2
1
0
-1
-2
-3
-10
0
10
20
30
X position
m
40
50
60
70
Figure E 18. Bicycle model with Magic formula tires. The vehicle parameters were set to their
nominal values.
E10
F – Code structure
Before the Tomlab solver is called the following need to be coded:
• Phases to use in the solver with start and end time as well as number of collocation
points
• Define state variables
• Define control variables
• Initial guess (All states need a guess)
• General boundary conditions (The solution will be within these), optional but
facilitates convergence
• Initial boundary conditions (in this case at t = 0), optional but facilitates convergence
• Final boundary conditions (in this case at t = final time), optional but facilitates
convergence
• Differential equations
• Additional constraints (A selectable margin for error is allowed)
• Objective function definition
The solver outputs one result variable and optionally a structure in which information about
the solver used can be found.
F1
G – Kinematics & compliance
From Kinematics and Compliance, Volvo S60, Y283 datasheet
Roll steer phenomena
Figure G 1. Toe-in angle at double-sided roll movement. Left Front and Right Front wheel shown in
the left and right picture respectively.
G1
Figure G 2. Toe-in angle at double-sided roll movement. Left Rear and Right Rear wheel shown in
the left and right picture respectively.
G2
Lateral force compliance steer phenomena
Lateral force applied at ground level (GL), at the middle of the tire-road contact patch (X=0m):
Figure G 3. Toe-in angle at lateral force, GL. Force LH+RH unidirectional at X = 0mm. Left Front
and Right Front wheel shown at the left and right picture respectively.
G3
Figure G 4. Toe-in angle at lateral force, GL. Force LH+RH unidirectional at X = 0mm. Left Rear
and Right Rear wheel shown at the left and right picture respectively.
G4
Lateral force applied at ground level (GL), at approximately the distance of the pneumatic trail for low slip
angles (X=30mm):
Figure G 5. Toe-in angle at lateral force, GL. Force LH+RH unidirectional at X = 30mm. Left Front
and Right Front wheel shown at the left and right picture respectively.
G5
Figure G 6. Toe-in angle at lateral force, GL. Force LH+RH unidirectional at X = 0mm. Left Rear
and Right Rear wheel shown at the left and right picture respectively.
G6
Camber gain due to roll
Figure G 7. Camber angle at double-sided roll movement. Left Front and Right Front wheels are
shown in the left and right picture respectively.
G7
Figure G 8. Camber angle at double-sided roll movement. Left Rear and Right Rear wheels are
shown in the left and right picture respectively.
G8
G9
H - Test 1 data illustration
As an example to illustrate the erroneously recorded data, the recorded lateral velocity from
one counter clockwise circular test is plotted in Figure H 2 26 , as well as the respective
steering wheel angle in Figure H 1. Those two figures describe two contradictory vehicle
behaviours, which prove this data corruption that occurred. In Figure H 1 it can be seen that
the steering wheel angle starts from a positive value (that is a counter clockwise turn of the
steering wheel) and constantly increases as the longitudinal speed also increases, up to the
point that the constant radius cannot be maintained anymore.
180
160
Steering wheel angle [deg]
140
120
100
80
60
40
20
0
0
20
40
60
Time [s]
80
100
120
Figure H 1. Steering wheel angle measured by the steering robot.
This implies an understeering behaviour. Next, Figure H 2 is red. The lateral velocity of the
vehicle, starting from a positive value, initially increases and after a while it starts decreasing
and even changing sign. With the help of Figure H 3 it becomes apparent what this lateral
velocity profile indicates: Initially, when the lateral velocity has a positive sign, the vehicle
understeers. Up to about 42 seconds the lateral velocity component increases, which means
the vehicle tends to understeer even more as the time passes and the longitudinal velocity
increases. After the first 42 seconds, though, the vehicle’s lateral speed starts decreasing,
which means that the vehicle starts to understeer less with an ever increasing longitudinal
speed, it then becomes neutral steer at around 90 seconds when the lateral velocity is zero,
and then it starts oversteering after that time. This behaviour contradicts the one occurring
26
The lateral velocity graph is filtered with a low pass filter – see the filtering at the curve fitting part below.
H1
from the graph of the steering wheel angle, which is only increasing, implying constant
understeer as mentioned before.
0.3
0.2
Lateral velocity [m/s]
0.1
0
-0.1
-0.2
-0.3
-0.4
0
20
40
60
Time [s]
80
100
120
Figure H 2. Lateral velocity calculated and recorded by the steering robot.
Figure H 3. The left case shows a vehicle understeering (Vy>0) during a counter clockwise circular
drive while the right case shows vehicle oversteering (Vy<0) during a counter clockwise circular
drive.
H2
I - Simulink bicycle model
Figure I 1. Bicycle model with nonlinear tire model to be used for curve fitting.
I1
In Figure I 1 Vx is the CoG longitudinal velocity, delta_w is the steering angle, Vfy is the
front wheel lateral velocity, Vfx is the front wheel longitudinal velocity, Vry is the rear wheel
lateral velocity, Vrx is the rear wheel longitudinal velocity, sfy is the front wheel lateral slip,
sry is the rear wheel lateral slip, mufy is the front wheel lateral friction coefficient, mury is the
rear wheel lateral friction coefficient, Ffy is the front lateral force, Fry is the rear wheel lateral
force, y_r is the CoG lateral velocity, psi_r is the yaw rate, psi_rr is the yaw acceleration,
YawR is the calculated yaw rate, LatAcc is the calculated lateral acceleration, psi_rf is the
measured and filtered yaw rate, ayf is the measured and filtered lateral acceleration and ax is
the longitudinal acceleration.
I1
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