Mathematical Optimisation of the Suspension System

Mathematical Optimisation of the Suspension System
University of Pretoria etd – Thoresson, M J (2005)
Mathematical Optimisation of the Suspension System
of an Off-Road Vehicle for Ride Comfort and Handling
by
Michael John Thoresson
Submitted in partial fulfilment of the requirements for the degree
Master of Engineering
in the Faculty of
Engineering, Built Environment and Information Technology
University of Pretoria, Pretoria
November 2003
University of Pretoria etd – Thoresson, M J (2005)
i
Mathematical Optimisation of the Suspension System of an
Off-Road Vehicle for Ride Comfort and Handling
Michael John Thoresson
Supervisor:
Mr. P.S. Els
Co-Supervisors:
Prof. J.A. Snyman and Dr. P.E. Uys
Department:
Mechanical and Aeronautical Engineering
Degree:
Master of Engineering
Abstract
This study aims to evaluate the use of mathematical optimisation algorithms for the optimisation of a vehicle’s spring and damper characteristics,
with respect to ride comfort and handling.
Traditionally the design of a vehicle’s suspension spring and damper characteristics are determined by a few simple planar model calculations, followed
by extensive trial-and-error simulation or track testing. With the current advanced multi-body dynamics computer software packages available to the design engineer, the integration of traditional mathematical optimisation techniques with these packages, can lead to much faster product development.
This, in turn results in a reduction of development costs.
A sports utility vehicle is modelled by means of a general-purpose computer programme for the dynamic analysis of a multi-body mechanical system. This model is validated against measurements from road tests. The
mathematical model is coupled to two gradient-based mathematical optimisation algorithms. The performance of the recently proposed Dynamic-Q
optimisation algorithm, is compared with that of the industry-standard gradient based Sequential Quadratic Programming method. The use of different
University of Pretoria etd – Thoresson, M J (2005)
ii
finite difference approximations for the gradient vector evaluation is also investigated.
The results of this study indicate that gradient-based mathematical optimisation methods may indeed be successfully integrated with a multi-body
dynamics analysis computer program for the optimisation of a vehicle’s suspension system. The results in a significant improvement in the ride comfort
as well as handling of the vehicle.
Keywords :
mathematical optimisation, vehicle suspension,
spring and damper characteristics, SQP,
Dynamic-Q, ride comfort, handling.
University of Pretoria etd – Thoresson, M J (2005)
iii
Wiskundige Optimering van die Suspensiestelsel van ’n
Veldvoertuig vir Ritgemak en Hantering
Michael John Thoresson
Studieleier:
Mnr. P.S. Els
Mede-leiers:
Prof. J.A. Snyman en Dr. P.E. Uys
Departement:
Meganiese en Lugvaartkundige Ingenieurswese
Graad:
Magister in Ingenieurswese
Opsomming
Die doel van hierdie studie is om wiskundige optimeringsalgoritmes te
evalueer met die oog op die optimering van ’n voertuigsuspensiestelsel se
veer- en demperkarakteristieke vir beide ritgemak en hantering.
Die praktyk by ontwerp van voertuigsuspensies, is om veer- en demperkarakteristieke te bepaal aan die hand van vereenvoudigde twee-dimensionele
modelberekenings gevolg deur intensiewe probeer-en-tref simulasies en/of
padtoetse. Integrasie van bestaande gevorderde multi-liggaam dimanika rekenaar pakette met beskikbare wiskundige optimeringstegnieke, kan produkontwikkeling versnel en baie koste bespaar.
Vir die doeleindes van hierdie ondersoek word ’n ontspanningsvoertuig
gemodelleer met behulp van ’n veeldoelige rekenaarprogram vir die dinamiese
analise van meganiese stelsels. Die simulasieresultate van die model word
gevalideer aan die hand van padtoetse. Die rekenaarmodel word daarna
gekoppel aan gradiënt -gebaseerde wiskundige optimeringsalgoritmes.
Om die effektiwiteit van optimeringsalgoritmes vir die optimering van
suspensiekarakteristieke te evalueer, word die voorgestelde Dynamic-Q algoritme vergelyk met die standaard Opeenvolgende Kwadratiese Programmering (SQP)- metode. Die gebruik van verskillende benaderings vir die
berekening van die gradiëntvektor in Dynamic-Q word ook ondersoek.
Uit die ondersoek blyk dat gradiëntgebaseerde wiskundige optimeringsmetodes suksesvol met multi-liggaam dinamika pakette geı̈ntegreer kan word vir
University of Pretoria etd – Thoresson, M J (2005)
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die optimering van ’n voertuig se suspensiestelsel. Dit het ’n aansienlike verbetering in ritgemak en hantering van die voertuig tot gevolg.
Sleutelwoorde :
wiskundige optimering, voertuigsuspensie,
veer- en demperkarakteristieke, SQP,
Dynamic-Q, ritgemak, hantering.
University of Pretoria etd – Thoresson, M J (2005)
v
Acknowledgments
• Optimisation related investigations were performed under the auspices
of the Multi-disciplinary Optimisation Group (MDOG) of the Department of Mechanical and Aeronautical Engineering of the University of
Pretoria.
• The vehicle dynamics simulation for the design of the controllable suspension system is based upon work supported by the European Research Office of the US Army under Contract No. N68171-01-M-5852.
• National Research Foundation (NRF) grant for funding of the optimisation related investigations.
• Land Mobility Technologies (LMT) for vehicle testing assistance and
supplying tyre data.
• Land Rover South Africa for supplying test vehicles.
• My study leaders Prof Snyman, Dr Uys, and Mr Els for their continual
support and enthusiasm throughout this research.
• My parents for their continual support and encouragement throughout
my studies.
University of Pretoria etd – Thoresson, M J (2005)
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. Literature Study . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1
2.2
2.3
Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1.1
Road input . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.3
Suspension . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.4
Ride . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.5
Handling . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Optimisation Algorithms . . . . . . . . . . . . . . . . . . . . . 19
2.2.1
Gradient Approximation Methods . . . . . . . . . . . . 19
2.2.2
Sequential Quadratic Programming . . . . . . . . . . . 23
2.2.3
Leap-Frog Algorithm LFOPC . . . . . . . . . . . . . . 24
2.2.4
Dynamic-Q . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.5
Genetic Algorithms . . . . . . . . . . . . . . . . . . . . 26
2.2.6
Nelder-Mead . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.7
Sequential Linear Programming . . . . . . . . . . . . . 28
Vehicle Suspension Optimisation . . . . . . . . . . . . . . . . . 28
2.3.1
Scania Bus . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2
Neural Network Approximation Approach . . . . . . . 29
2.3.3
LFOPC and Damper Optimisation . . . . . . . . . . . 30
2.3.4
Dynamic-Q Two Variable Optimisation . . . . . . . . . 31
2.3.5
SLP and Damper Optimisation . . . . . . . . . . . . . 31
2.3.6
SQP and Damper Optimisation . . . . . . . . . . . . . 32
2.3.7
BMW Vehicle Optimisation Procedure . . . . . . . . . 33
University of Pretoria etd – Thoresson, M J (2005)
Contents
vii
3. The Optimisation Problem . . . . . . . . . . . . . . . . . . . . . . . 34
3.1
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3
Mathematical Model of Vehicle . . . . . . . . . . . . . . . . . 35
3.3.1
General Background . . . . . . . . . . . . . . . . . . . 35
3.3.2
Vehicle Body . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.3
Front Suspension . . . . . . . . . . . . . . . . . . . . . 36
3.3.4
Rear Suspension . . . . . . . . . . . . . . . . . . . . . 38
3.3.5
Anti-Roll Bar . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.6
Force Elements . . . . . . . . . . . . . . . . . . . . . . 40
3.3.7
Tyres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.8
Driver Implementation . . . . . . . . . . . . . . . . . . 42
3.4
Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5
Optimisation Algorithms . . . . . . . . . . . . . . . . . . . . . 46
3.6
Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7
3.8
3.6.1
Two Variable Case . . . . . . . . . . . . . . . . . . . . 48
3.6.2
Four Variable Case . . . . . . . . . . . . . . . . . . . . 50
3.6.3
Seven Variable Case . . . . . . . . . . . . . . . . . . . 50
Definition of Objective Functions . . . . . . . . . . . . . . . . 53
3.7.1
Ride Comfort . . . . . . . . . . . . . . . . . . . . . . . 53
3.7.2
Handling . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Integration of Mathematical Vehicle Model and Optimisation
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9
Preliminary Sensitivity Investigations . . . . . . . . . . . . . . 55
3.9.1
Design Space . . . . . . . . . . . . . . . . . . . . . . . 55
3.9.2
Gradient Sensitivity
. . . . . . . . . . . . . . . . . . . 56
4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1
Handling Results . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1
Two Design Variables . . . . . . . . . . . . . . . . . . . 58
4.1.2
Four Design Variables . . . . . . . . . . . . . . . . . . 59
4.1.3
Seven Design Variables . . . . . . . . . . . . . . . . . . 60
University of Pretoria etd – Thoresson, M J (2005)
Contents
4.2
viii
Ride Comfort Results . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1
Two Design Variables . . . . . . . . . . . . . . . . . . . 66
4.2.2
Four Design Variables . . . . . . . . . . . . . . . . . . 67
4.2.3
Seven Design Variables . . . . . . . . . . . . . . . . . . 76
5. Discussion of Conclusions . . . . . . . . . . . . . . . . . . . . . . . 80
6. Discussion of Future Work . . . . . . . . . . . . . . . . . . . . . . . 81
Appendix
88
A. Evaluation of The Handling Objective Function . . . . . . . . . . . 89
A.1 The Concern
. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.2 Tests Performed . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.3 Results From Study . . . . . . . . . . . . . . . . . . . . . . . . 90
University of Pretoria etd – Thoresson, M J (2005)
LIST OF TABLES
2.1
Current spring force system design variables . . . . . . . . . . 12
3.1
ADAMS model degrees of freedom
3.2
Front suspension model degrees of freedom . . . . . . . . . . . 38
3.3
Rear suspension model degrees of freedom . . . . . . . . . . . 41
3.4
Land Rover 110 test points . . . . . . . . . . . . . . . . . . . . 46
3.5
Definition of damper variables . . . . . . . . . . . . . . . . . . 52
4.1
Two design variable handling optimisation results . . . . . . . 58
4.2
Four design variable handling optimisation results . . . . . . . 60
4.3
Seven design variable handling optimisation results . . . . . . 62
4.4
Two design variable ride comfort optimisation results . . . . . 66
4.5
Effect of changing the gas volume range for four variable ride
. . . . . . . . . . . . . . . 35
comfort optimisation . . . . . . . . . . . . . . . . . . . . . . . 69
4.6
Seven variable ride comfort optimisation results . . . . . . . . 77
A.1 Summarized vehicle measurements . . . . . . . . . . . . . . . 90
A.2 Test track specifications . . . . . . . . . . . . . . . . . . . . . 91
University of Pretoria etd – Thoresson, M J (2005)
LIST OF FIGURES
2.1
Vehicle response due to road and steering input . . . . . . . .
3
2.2
ISO 8608 classification PSD of Belgium paving test track . . .
5
2.3
Tyre models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Hermite spline approximate damper characteristics [10] . . . .
8
2.5
Etman damper model characteristics [11]
9
2.6
Six piece-wise linear damper model as used by Naude and
Snyman [12]
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7
Schematic diagram of the hydro-pneumatic suspension unit . . 12
2.8
Hydro-pneumatic spring characteristics . . . . . . . . . . . . . 14
2.9
Hydro-pneumatic strut on test rig . . . . . . . . . . . . . . . . 15
2.10 BS 6842 weighting curves for ride comfort . . . . . . . . . . . 16
2.11 Finite difference gradient approximation methods . . . . . . . 21
3.1
Land Rover front suspension schematic diagram . . . . . . . . 37
3.2
Land Rover front suspension modelling . . . . . . . . . . . . . 37
3.3
Land Rover rear suspension schematic diagram . . . . . . . . . 39
3.4
Land Rover rear suspension modelling . . . . . . . . . . . . . 40
3.5
Tyre side force properties . . . . . . . . . . . . . . . . . . . . . 42
3.6
Implementation of driver model . . . . . . . . . . . . . . . . . 43
3.7
Full Land Rover model . . . . . . . . . . . . . . . . . . . . . . 44
3.8
Test vehicle indicating measurement positions . . . . . . . . . 45
3.9
apg, 25km/h, Model validation results . . . . . . . . . . . . . 47
3.10 Double lane change, 80 km/h, model validation results . . . . 48
3.11 Definition of spring characteristics for various gas volumes . . 49
3.12 Definition of damper characteristics for various damper scale
factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.13 Definition of damper variables . . . . . . . . . . . . . . . . . . 51
University of Pretoria etd – Thoresson, M J (2005)
List of Figures
xi
3.14 Body roll angle of a standard Land Rover Defender 110 while
performing the double lane change manœuvre . . . . . . . . . 54
3.15 Optimisation process flow diagram . . . . . . . . . . . . . . . 55
3.16 Vehicle roll angle, double lane change at 80 km/h for the two
variable design space . . . . . . . . . . . . . . . . . . . . . . . 56
3.17 Vehicle ride comfort, Belgian paving at 60 km/h for the two
variable design space . . . . . . . . . . . . . . . . . . . . . . . 57
4.1
Optimisation histories of handling for two design variables . . 59
4.2
Optimisation histories of handling for four design variables . . 61
4.3
Optimisation histories of handling for seven design variables . 64
4.4
Optimum damper characteristics for handling . . . . . . . . . 65
4.5
Existence of many local minima away from current optimum,
for the seven design variable, handling optimisation . . . . . . 65
4.6
Optimisation histories of ride comfort for two design variables
4.7
Dynamic-Q ffd ride comfort, 4 design variables, 10 percent
67
move limit, gas volume range from 0.1 to 3 litres . . . . . . . . 68
4.8
Optimisation histories of ride comfort for four design variables
(gas volume range from 0.1 to 3 litres) . . . . . . . . . . . . . 70
4.9
SQP optimisation movements . . . . . . . . . . . . . . . . . . 72
4.10 Optimisation histories of ride comfort for four design variables
(gas volume ranges from 0.008 to 0.5 litres, with inf - being
an infeasible starting point) . . . . . . . . . . . . . . . . . . . 74
4.11 Optimisation histories of ride comfort for four design variables
(gas volume range from 1.03 to 3 litres) . . . . . . . . . . . . . 75
4.12 Convergence to optimum, seven design variables . . . . . . . . 77
4.13 Optimisation histories of ride comfort for seven design variables 78
4.14 Optimum damper characteristics for ride comfort . . . . . . . 79
A.1 Different drivers in Ford Courier on dynamic handling track . 91
A.2 Different drivers in Ford Courier on ride and handling track . 92
A.3 Different drivers in VW Golf 4 GTi on dynamic handling track 92
A.4 Different drivers in VW Golf 4 GTi on ride and handling track 93
University of Pretoria etd – Thoresson, M J (2005)
List of Figures
xii
A.5 Standard Land Rover Defender in double lane change manœuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
University of Pretoria etd – Thoresson, M J (2005)
LIST OF ABBREVIATIONS
AAP
Average Absorbed Power
ANN
Artificial Neural Network
apg
Abberdene Proving Ground
BFGS
Broyden-Fletcher-Goldfarb-Shanno
BS
British Standard
cfd
Central Finite Difference
CFD
Computational Fluid Dynamics
cg
Center of Gravity
DFP
Davidon-Fletcher-Powell
FEM
Finite Element Method
ffd
Forward Finite Difference
GA’s
Genetic Algorithms
inf
Infeasible Starting Point
ISO
The International Organisation for Standardisation
LFOP
Leap-Frog Optimiser
LFOPC
Leap-Frog Optimiser for Constrained Problems
LMT
Land Mobility Technologies
MDOG
Multi-disciplinary Optimisation Group
NRF
National Research Foundation
PSD
Power Spectral Density
RMS
Root Mean Square
SLP
Sequential Linear Programming
SQP
Sequential Quadratic Programming
SUV
Sports Utility Vehicle
VDV
Vibration Dose Value
University of Pretoria etd – Thoresson, M J (2005)
LIST OF SYMBOLS
A
Piston Area
A
Dynamic-Q Hessian Matrix
a
Curvature
aRM S
Root Mean Square Acceleration
aw
Weighted Acceleration
ay
Lateral Acceleration
d
Displacement Error
dpsf
Damper Scale Factor
dpsf f
Damper Scale Factor Front
dpsf r
Damper Scale Factor Rear
dxk
Perturbation in Design Variable k
f (x)
Objective Function
f (x∗ )
Optimum Objective Function Value
F
Force
F (v)
Damper Force
F (x)
Multi-Variable Function
Fc
Current Force
Fs
Static Force
G ay
Lateral Acceleration Gain
Gdo
Road Roughness Coefficient
Gdr
Road Displacement Power Spectral Density
gj (x)
Inequality Constraint Functions
GYw
Yaw Velocity Gain
gvol
Static Gas Volume
gvolf
Static Gas Volume Front
gvolr
Static Gas Volume Rear
University of Pretoria etd – Thoresson, M J (2005)
List of Symbols
hj (x)
Equality Constraint Functions
Hk
Hessian Matrix at Iteration k
I
Identity Matrix
k
Constant
K
Steering Angle Gain
l
Driver Preview Distance
m
Number of Inequality Constraints
n
Number of Design Variables
N
Population Size for Genetic Algorithms
n
Spatial Frequency
P
Current Pressure
Po
Static Gas Pressure
r
Number of Equality Constraints
R
Piston Radius
S
Total Strut Stroke
s
Vehicle Speed
t
Time
T
Total Sample Time
V
Current Volume
v
Velocity
V1−8
Vo
Damper Definition Variables
Static Gas Volume
V DV
Vibration Dose Value
x
Relative Displacement
xk
Design Variable k
xo
Initial Piston Height
xs
Static Spring Displacement
x
Design Variables Vector
xv
University of Pretoria etd – Thoresson, M J (2005)
List of Symbols
x∗
Optimum Design Variables Vector
Yw
Yaw Velocity
β0−3
Characteristic Damper Coefficients
δ
Steering Angle
γ
Gas Constant
ω
Terrain Index
τ
Driver Preview Time
xvi
University of Pretoria etd – Thoresson, M J (2005)
1. INTRODUCTION
One of the fastest growing sectors in the vehicle market is currently the
Sports Utility Vehicle (SUV) sector. This sector includes vehicles ranging
from baseline to luxury units, with manufacturers such as Toyota, MercedesBenz and Porsche offering SUV’s. The biggest problem with these vehicles
is that the owners expect the luxury and comfort associated with a large
luxury vehicle, while still demanding good off-road tractability and sports
car on-road handling performance.
Since the invention of the motor vehicle, the suspensions used have always represented a compromise between ride comfort and handling. Good
ride comfort requires a supple (soft) suspension, whilst good handling requires a stiff (hard) suspension. This compromise has been reduced in newer
passenger vehicles by the addition of anti-roll bars, to stiffen the suspension
for handling manœuvres, while keeping a soft suspension for ride comfort.
With the SUV’s becoming luxury orientated, more emphasis is being placed
on comfort, resulting in a vehicle that has good off-road tractability and good
ride comfort, but poor on-road handling behaviour. With this configuration
the driver and passengers can comfortably be transported at high speeds
over poor road surfaces. An accident avoidance type manœuvre, can however
not be performed safely, leading to vehicle roll-over in handling situations.
The performance orientated SUV’s have good ride comfort and reasonable
on-road handling, but poor off-road capability. This thesis reports on the
investigation of a novel suspension system, currently under development at
the University of Pretoria, which aims to avoid these traditional compromises
in the design of suspension systems.
Traditionally suspension spring and damper characteristics are determined by a few simple planar model calculations and many trial-and-error
University of Pretoria etd – Thoresson, M J (2005)
1. Introduction
2
simulations or road tests of a prototype vehicle. The spring and damper
characteristics undergo many changes before the final configuration is put
into production. In today’s competitive world, this type of time consuming
design work is no longer acceptable, as it adds unnecessary development costs
to the vehicle. Using modern advanced multi-body dynamics simulation software, designers have the ability to model the vehicle driving under almost all
possible road conditions. The integration of an optimisation procedure with
such simulation software, will enable the design engineer to determine the desired suspension damper and spring characteristics, with limited prototype
testing.
A brief literature overview concerning vehicle dynamics, suspension optimisation and optimisation algorithms, is presented in Chapter 2. Chapter
3 deals with the optimisation problem at hand, with details of the mathematical model and its validation being given. In Chapter 4 the results are
presented and discussed. Conclusions drawn from the study together with
suggestions for future work to be performed are presented in the final two
chapters. Appendix A provides a summary of an investigation into the handling objective function.
University of Pretoria etd – Thoresson, M J (2005)
2. LITERATURE STUDY
2.1 Vehicle Dynamics
The vehicle under investigation is a very complex dynamical system, into
which much thought must go, in order to develop a marketable and competitive vehicle. As illustrated in Figure 2.1 the response of the vehicle body
depends on the vehicle’s suspension response, and that in turn depends on
the road input conditions. These road input conditions can be further complicated by the steering action of the driver.
Figure 2.1: Vehicle response due to road and steering input
University of Pretoria etd – Thoresson, M J (2005)
2. Literature Study
4
2.1.1 Road input
The road is the primary input to the vehicle system. Its condition affects
the vehicle in a number of ways, ranging from ride comfort experienced by
the vehicle passengers and driver, to roll-over, trip-up, and road banking
affecting the vehicle’s cornering performance. It is therefore important to
take cognizance that not all roads have smooth surfaces. Rather they are
complex randomly irregular three dimensional profiles, that may exhibit a
certain degree of harmonic profiles, like the sinusoidal corrugations commonly
found on gravel roads or traffic speed bumps. In this study the road will be
considered as a smooth plane for the handling requirements, and as a socalled Belgian paving for ride comfort requirements. The Belgian paving is
a test track specially designed to excite all the vehicle’s vibration modes for
the purpose of ride comfort and endurance evaluations. The Belgian paving
test track used is classified by using the ISO 8608 [1] standard. In terms of
this standard it has a roughness coefficient Gdo of 1e-4 m2 /(cycles/m), and
a terrain index ω of 4. These are obtained from the linear approximation
that is applied to all road profiles where the power spectral density (PSD)
function is defined as follows:
Gdr = Gdo n−ω
(2.1)
where n is the spatial frequency. The power spectral density of the Belgian
paving is included in Figure 2.2 with a photograph of the vehicle driving on
the Belgian paving in Figure 2.1 top left corner.
2.1.2 Tyres
The tyre of a vehicle is one of the most important components of the vehicle
model, as it serves as the interface between the vehicle and the road. Ideally
it must maintain traction at all times, while absorbing most of the road
irregularities. The tyre generates a lateral force which is necessary to keep the
vehicle on track when travelling along a curve. It also generates longitudinal
force which is the driving force that ensures that the vehicle can propel or
brake when required. Self-aligning moments are also generated to ensure it
University of Pretoria etd – Thoresson, M J (2005)
2. Literature Study
5
Figure 2.2: ISO 8608 classification PSD of Belgium paving test track
keeps following the desired path when road disturbances are encountered.
Because of the wide range of functions the tyre must perform, it is equally
difficult to model. In most vehicle dynamics simulations, the tyre model is
the most limiting factor in achieving correlation with measured results. The
tyre would filter out road irregularities shorter than the tyre contact patch,
while the simple tyre models do not. The magic formula tyre model [3] is
regarded as the industry standard for handling simulation, and in its latest
form the swift tyre model [4, 5] is considered appropriate for rough road
simulation. However, it has been difficult to obtain reliable coefficients, the
determination of which require many tyre tests to be performed, to accurately
model the real tyre [6]. The swift tyre makes use of the empirical magic
formula model for handling, and a rigid ring model (Figure 2.3) for ride
comfort. The recently proposed FTire [7] makes use of a flexible ring model,
to more accurately describe the tyre road enveloping effects, for ride comfort.
With computational time being more than 20 times real time, it becomes too
expensive to use for optimisation. On the other hand, the Fiala tyre model
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in ADAMS [4, 5] is a very simple model based on a linear approximation to
the tyre data, but is normally not sufficiently accurate. It makes use of a
sector type model. The middle of the way tyre model is the ADAMS 521
tyre, which makes use of lookup tables with the tyre characteristics, and has
the option of a point follower tyre model (Figure 2.3) for ride comfort. This
model is best for limited tyre test data, and does not require the fitting of
complex coefficients.
Figure 2.3: Tyre models
2.1.3 Suspension
Damper Characteristics
Although most textbook approaches to vehicle dynamics assume a linear
relation between damper force and relative velocity, this is hardly the characteristic that a vehicle damper assumes. The vehicle damper is a very
complicated non-linear element, thus being difficult to model mathematically. This can mainly be ascribed to the complex fluid dynamics occuring
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inside the automotive damper. The characteristics of dampers are normally
designed with switch over points, to try and minimize the traditional compromise between ride comfort and handling. The automotive damper is thus
designed to have a high damping rate for low velocities associated with handling, and a low damping rate for higher velocities associated with vehicle
ride comfort. A switch between these two settings is normally gradual in
form. The fact that a damper effectively consists of various orifices, that
restrict the flow of oil, and in doing so, generates a force, also needs to be
considered. Oil flow through an orifice is normally described by a quadratic
relationship. When optimising a vehicle required to perform various manœuvres, it becomes necessary to describe this non-linear damper behaviour in
terms of a mathematical expression. It must also be kept in mind that the
method used to evaluate the mathematical model, must be computationally
inexpensive.
Complex Models:
There are many detailed damper models available that all aim to accurately describe the damper characteristics, but most of these models require
data that are normally not available at the design stage, such as working oil
temperature and pressure [8]. These models are normally time-consuming to
fit to current damper data, and they will therefore not be further considered
in this study.
Spline Approximation:
The Hermite spline has been used by Eberhard et al. [10] for the description of the damping characteristics of an automotive damper, in their
optimisation process. They made use of five definition points and four gradient curves. A Hermite spline was then defined between two consecutive
points, exhibiting the same gradient at these points. This was done for all
the defined points of the damper characteristics as illustrated in Figure 2.4.
Eberhard et al. [10], however, stated that:
The constraints which have to be formulated to ensure physical
feasibility are hard to handle and prevent complete automation
of the approach.
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Figure 2.4: Hermite spline approximate damper characteristics [10]
Etman Damper Model:
Etman et al. [11] proposed an eight variable empirical curve fit method
to specify the damper characteristics. This model consists of a function of
four variables for compression, and four for rebound. The same function is
used for both compression and rebound. The damper force is described as :
F (v) =
β0 v 2 (β1 − β2 v)
+ β3 v 2
β0 v 2 + β1 − β2 v
(2.2)
where F is the damper force, v the velocity and βi , i = 0, 1, 2, 3. are the
characteristic coefficients. Etman et al. suggests the following ranges for the
characteristic coefficients:
Variable
Compression Rebound
Units
β0
-22 - -0.3
0.3 - 22
106 N s2 m−2
β1
-200 - -2
2 - 200
103 N
β2
0.7 - 70
0.7 - 70
103 N sm−1
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They found that β3 has a limited contribution to the curve shapes, and thus
kept it constant for optimisation. Attempts were made in this study to fit
this model to the standard Land Rover damper characteristics. However,
as this turned out to be troublesome, it was decided not to use this model
for the optimisation process. The resulting characteristic for this model is
illustrated in Figure 2.5.
Figure 2.5: Etman damper model characteristics [11]
Piece-wise Linear Approximation:
Naude and Snyman [12, 13] and Naude [14] proposed a six piece-wise
linear approximation (Figure 2.6) for the description of the damper characteristics for the optimisation of a pitch plane ride comfort vehicle model.
Their study, however, indicates that a four piece-wise linear approximation
may be sufficient. For this reason, the four piece-wise linear model is used in
Section 3.6.3. where the optimisation of the hydro-pneumatic spring-damper
is dicussed.
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Figure 2.6: Six piece-wise linear damper model as used by Naude and Snyman
[12]
Semi-active Hydro-pneumatic Suspension Unit
The design and implementation of a semi-active suspension system on a four
wheel drive off-road leisure vehicle has been an ongoing research activity at
the University of Pretoria. The suspension system consists of a two state
semi-active damper and two state hydro-pneumatic spring.
Principle of Operation:
The semi-active spring-damper (shown schematically in Figure 2.7) incorporates two damper packs (fitted with bypass valves), and two gas accumulators, effectively giving two damper characteristics and two spring characteristics in a single suspension unit [15]. Switching between the two spring
and damper characteristics is achieved by solenoid valves, as illustrated in
Figure 2.7. Valve switching times vary between 50 and 100 milliseconds depending on system pressure. This means that spring and damper characteristics can be taken as design variables, to be optimised for both ride comfort
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and handling respectively, by switching the suspension system to either the
ride comfort or handling option, depending on the vehicle’s operating conditions. Each operating setting is expected to have different optimum values
for the spring and damper characteristics. This approach eliminates the traditional ride comfort versus handling compromise. With this capability the
optimisation is done by treating the suspension unit as two passive systems.
One for handling and one for ride comfort.
Refering to Figure 2.7 the handling condition of the suspension unit, is obtained when accumulator A and damper x are working, with all the solenoid
valves (a, b and c) closed. If lower damping is required for the particular
driving condition, solenoid valve a is opened resulting in a larger flow area
in turn reducing the damper force. For the ride comfort setting, solenoid
valves a and b would be closed and solenoid valve c would be open, thus
creating a larger effective accumulator volume with both accumulator A and
B working. This condition would result in a lower spring stiffness. Dampers
x and y would be generating forces that can be lowered by opening solenoid
valves a and b should the operating conditions require lower damping.
Spring Characteristics:
The spring force is generated by the compression and expansion of the
nitrogen filled gas accumulator volume, resulting in very non-linear spring
characteristics. The spring characteristics can be defined in terms of a simple
relation between displacement and force, if the hysteresis effects are ignored.
The system design variables are defined in Table 2.1. The piston area is
defined as:
A = πR2
(2.3)
The static accumulator gas pressure is:
Po = Fs /A
(2.4)
xo = Vo /A
(2.5)
for an initial piston height of:
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Figure 2.7: Schematic diagram of the hydro-pneumatic suspension unit
Table 2.1: Current spring force system design variables
Variable
Value
Name
xs
316 mm
Static Spring Displacement
S
300 mm
Total strut stroke
Fs
5500 N
Static Spring load
R
25 mm
Accumulator piston radius
γ
1.3
Gas constant
Vo
1 liter
Static gas volume
and the ideal gas law constraint:
Po Voγ = k
(2.6)
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The current volume for the relative displacement x is defined as:
V = A(xo + x)
(2.7)
with the current accumulator conditions also conforming to the ideal gas law
constraint:
PV γ = k
(2.8)
Thus using the current volume (equation 2.7) in equation 2.8 gives:
P (A(xo + x))γ = k
(2.9)
where the current pressure P can be defined in terms of the current force Fc
as:
P = Fc /A
(2.10)
resulting in the current force:
Fc =
Ak
(A(xo + x))γ
(2.11)
For ease of comparison this force is measured relative to the static force to
give:
F = Fc − Fs =
Ak
− Fs
(A(xo + x))γ
(2.12)
Figure 2.8 illustrates the resulting spring force vs. displacement characteristics that can be achieved for various static gas volumes, zeroed around
the static suspension height and static vertical load of the Land Rover test
vehicle.
Damper Characteristics:
For damping, the suspension unit has two separate damper packs that
can be designed for the required damping characteristics. These damper
packs can be completely bypassed by the solenoid valves, when demanded by
the vehicles operating conditions. Currently the damper packs are standard
Land Rover Defender rear damper packs. However, one of the purposes of this
study is to determine exactly what characteristics are required. A prototype
unit, of the proposed suspension unit, was tested in the Sasol Laboratory
(Figure 2.9), using a Schenck hydropulse actuator in the test rig designed for
the testing of the prototype.
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Figure 2.8: Hydro-pneumatic spring characteristics
2.1.4 Ride
The ride comfort of the vehicle depends on the motion of the whole vehicle and is a very subjective quantity, depending on the individual driver or
passenger. Ride comfort is very important, as it is this characteristic that
influences the potential purchaser of a vehicle. The ride comfort is defined in
terms of the range of excitation frequencies between 0 and 25 Hz according
to Gillespie [16], and 0 to 80 Hz according to the BS 6841 standard [17] and
Reimpell and Stoll [18]. Noise is classified as frequencies above these levels.
In this study, the emphasis is on ride comfort, rather than vehicle noise.
The vehicles suspension system is the primary device used to minimize the
discomfort, while engine vibration mounts and body materials generally affect the noise quality of a vehicle. There are many standards that relate to
ride comfort measurement and acceptable levels of ride comfort of a vehicle
[19, 17]. Different countries use different standards and measurement units.
The most commonly used measurements are the frequency weighted RMS
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Figure 2.9: Hydro-pneumatic strut on test rig
acceleration, vibration dose value and average absorbed power.
The BS 6841 weighting curves for vertical ride comfort are illustrated in
figure 2.10. The Root Mean Square (RMS) acceleration is defined as:
s
aRM S =
1ZT 2
a (t).dt
T 0 w
(2.13)
with T being the total sample time, aw the weighted acceleration, and t the
time. For values of aRM S above 1 m/s2 the ride is rated as uncomfortable,
and above 2.5 m/s2 as extremely uncomfortable.
The vibration dose value is defined as:
V DV =
s
Z
4
0
T
a4w (t).dt
(2.14)
The vibration dose value takes the duration of the event into account and
thus can only be compared with other events, if the same time span is taken
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into consideration. Four hour VDV’s with values above 16 m/s1.75 are rated
as uncomfortable, and above 26 m/s1.75 as extremely uncomfortable. VDV
also places more emphasis on the magnitude of the peaks of the acceleration
in the frequency domain. The RMS acceleration and VDV are the most
commonly used measures of ride comfort. The U.S.A. makes use of the
Average Absorbed Power (AAP), where a value above 6 Watts is regarded
as uncomfortable and above 12 Watts as extremely uncomfortable.
Figure 2.10: BS 6842 weighting curves for ride comfort
However, in a study conducted by Els [20] it was found that VDV and
aRM S exhibited a linear relationship between subjective and objective measurements, with AAP having a non-linear, but predictable, relationship between subjective and objective measurements. It was also found that the
vertical acceleration values are of greater importance than the accelerations
experienced in other directions by passengers. The conclusion reached is that
the consistent application of one of the standards to vertical accelerations at
the seat is sufficient for ride comfort evaluation. The BS 6841 weighted vertical acceleration is therefore used as the measure of ride comfort for the
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optimisation of the spring-damper as discussed in Section 3.7.1.
2.1.5 Handling
The vehicle’s suspension system is the main contributor to the vehicle’s handling performance. The suspension kinematics, and dynamics, affect the vehicle’s handling. The main focus of this study is on the effect of the dynamic
elements (springs and dampers) on the vehicle’s handling performance. Unlike ride comfort, no standard exists that provides the design engineer with
guidelines, on the level of certain key parameters, that will provide good
handling behaviour in a vehicle. This makes the definition of a handling
objective function more challenging than for ride comfort. Many vehicle
handling test manœuvres are used to measure vehicle handling. There are
open loop manœuvres where the outcome of the manœuvre does not depend
heavily on the driver-vehicle interaction. Typical manœuvres falling into this
category are the steady state steering tests (constant radius, constant steer
angle), and transient tests (J-Turn, Fishhook turn). There are also closed
loop manœuvres that better describe real world handling manœuvres, such
as the ISO 3888 [21] severe double lane change manœuvre that heavily depends on the driver-vehicle interaction. The important question that needs
to be answered is : what measure must be used to evaluate handling? The
results from a study (presented in Appendix A) indicate that there is a linear relationship between the lateral acceleration, body roll angle and yaw
velocity.
Many researchers make use of yaw velocity gain as one of the measures of
vehicle handling. The yaw velocity gain [16] is defined as the ratio of vehicle
body yaw velocity Yw to mean steering angle δ of the steered wheels, being:
GYw =
Yw
δ
(2.15)
The lateral acceleration gain [16] is defined as the ratio of vehicle lateral
acceleration ay to mean steering angle of the steered wheels, being:
Gay =
ay
δ
(2.16)
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Because of the linear relationship between the yaw velocity and lateral acceleration, these two gain values should exhibit a linear relationship as well.
Crolla et al. [22] also observed that for transient cornering on a smooth road
surface, the body roll angle correlates with the lateral acceleration phase, yaw
velocity, lateral acceleration gain and yaw velocity gain. In studies conducted
by Dahlberg [23] he states that:
During steady state cornering on an even road under no influence of external forces, the level of lateral acceleration determines
whether the vehicle rolls over or not.
Vehicle steering properties are normally tested by performing a constant
radius test. From this the vehicle’s speed and steering angle are compared. If
the vehicle requires an increasing steering input as the vehicle speed increases,
the vehicle exhibits under-steer. If the vehicle requires less steering input with
increasing vehicle speed, the vehicle over-steers. If no steering adjustment is
required with increasing vehicle speed, then it is a neutral-steer vehicle. The
vehicle’s spring stiffnesses can affect the steering behaviour of the vehicle. If
the front roll stiffness is high (i.e. stiff springs in front), this induces understeer characteristics, while a high rear roll stiffness will induce over-steer
characteristics.
The vehicle body roll can result in an improvement in the vehicle’s steering characteristics, refered to as roll-steer. This is especially true on trailing
arm solid axle suspension systems, as in the Land Rover Defender. The
trailing arm angle of the suspension at the rear, can result in over-steer,
neutral-steer, or under-steer characteristics. If the trailing arm is horizontal
with the ground, then it has a neutral-steer contribution. If the trailing arm
is angled downwards, from the body to the rear axle, it has an over-steering
contribution, as the inside wheel is pulled forward while the outside wheel
is pushed outward in a corner. With an upward angle from the body to the
axle, the trailing arm will have an under-steering effect, as with body roll the
inside wheel is pushed outwards while the outside wheel is pushed forwards.
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2.2 Optimisation Algorithms
Mathematical optimisation algorithms are being used more frequently in the
product development phase, to obtain a more cost effective and improved
design. There is an increasing amount of optimisation algorithms available to
the designer. However, in spite of this proliferation of optimisation methods,
there is no universal method for solving all possible optimisation problems.
Each method seems to have its limitations. A short review of the optimisation
algorithms applicable to this research is presented below.
Mathematical optimisation is the minimization of an objective function
(design objective) subject to design constraints, in order to obtain an improved design configuration. The general optimisation problem, which optimisation algorithms aim to solve, is defined as:
minimize
w.r.t.x
f (x), x = [x1 , x2 , .., xn ]T ∈ Rn
(2.17)
subject to the inequality constraints:
gj (x) ≤ 0,
j = 1, 2, .., m
(2.18)
j = 1, 2, .., r
(2.19)
and the equality constraints:
hj (x) = 0,
where f (x), gj (x) and hj (x) are scalar functions of x. In this formulation
x is the vector of design variables, f (x) is the objective function, gj (x) the
inequality constraint functions, and hj (x) the equality constraint functions.
The optimum solution is denoted by x∗ .
2.2.1 Gradient Approximation Methods
Most continuous optimisation methods require first order and/or second order gradient information of the objective and constraint functions with respect to the design variables. In most engineering optimisation problems
this gradient information is not analytically available. The only information
available to the designer is the values of the objective and constraint functions obtained via expensive simulations. The optimisation algorithm must
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then approximate the gradients at each iteration step by using function values obtained from simulations. This is normally done by making use of finite
differencing methods to approximate the gradient. The most common finite
differencing method uses forward finite differences.
Forward Finite Difference (ffd)
This is the simplest and most economic method for approximating the gradients of the objective and constraint functions, required by gradient-based
mathematical optimisation algorithms. This method approximates the first
order gradient information of a multi-variable function F (x), by evaluating
the change in the function F (x) for a small change dxk in each of the design
variables xk , k = 1, 2, ..., n, as illustrated in Figure 2.11. Thus, in order to
carry out the full gradient vector evaluation, a total number of n + 1 function evaluations are required for each iteration, where n is the total number
of design variables. The forward finite difference approximation to the k th
component of the gradient at x is defined as follows:
F (x1 , x2 , ..., xk + dxk , ..., xn ) − F (x)
∂F
=
∂xk
dxk
(2.20)
for k = 1, 2, ..., n. Noisy objective functions, however, severely limit the accuracy of the forward finite difference gradient approximation, as is apparent
from Figure 2.11. This can be partly overcome by using larger stepsizes dxk
or by considering instead, central finite differences.
Central Finite Difference (cfd)
This study also looks into the viability of using the central finite difference
gradient evaluation procedure. Although this method requires 2n+1 function
evaluations per gradient vector evaluation, it may result in fewer optimisation
iterations to obtain a minimum because of its greater accuracy. Central
differences make use of a function evaluation on either side of the current
iteration point x, giving a more accurate approximation to the gradient of
the underlying smooth function in the presence of noise. The central finite
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Figure 2.11: Finite difference gradient approximation methods
21
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difference procedure is defined as follows:
∂F
F (x1 , x2 , ..., xk + dxk , ..., xn ) − F (x1 , x2 , ..., xk − dxk , ..., xn )
=
(2.21)
∂xk
2dxk
for k = 1, 2, ..., n. In this way the gradient is evaluated by looking at what is
happening behind and ahead of the current iteration point, while the forward
finite difference only looks ahead of the current iteration point. Central differencing should therefore give a more accurate approximation to the function
gradient as illustrated for the case depicted in Figure 2.11.
The effects of noise cannot be completely eliminated by this method, but
it certainly yields gradient approximations that are superior to that given by
forward finite differences. As can be deduced from Figure 2.11 the greater
the perturbation dxk , in the presence of noise, the more accurate the approximation of the gradient of the function becomes, no matter which finite
differencing technique is used. However, too large a perturbation dxk will
result in the local optima being completely missed by the optimisation algorithm. Thus the correct selection of perturbations dxk for the function at
hand, is very important.
Second Order Curvature Approximation
The Sequential Quadratic Programming (SQP) method [25, 31] and other
Quasi-Newton optimisation algorithms such as the Davidon-Fletcher-Powell
(DFP) method uses, in addition to first order gradient approximations, also
second order curvature information. This information is very costly to obtain, as it corresponds to a partial derivative of a partial derivative. This
information is stored in a n x n square matrix, commonly known as the Hessian matrix. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) approximation
to the Hessian matrix is used in Matlab’s implementation of SQP. The Hessian matrix is approximated and updated at iteration k + 1, k = 0, 1, 2, ...
by:
Hk+1
HkT sTk sk Hk
qk qTk
= Hk + T − T
qk sk
sk Hk sk
(2.22)
where
sk = xk+1 − xk
(2.23)
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and
qk = ∇f (xk+1 ) − ∇f (xk )
(2.24)
and
∇f (xk ) = [
∂f ∂f
∂f
,
, ...,
]
∂x1 ∂x2
∂xn
(2.25)
At the start of the optimisation procedure, (i.e. at iteration k = 0) most
algorithms set H0 equal to any positive definite symmetric matrix, normally
the identity matrix I. Thereafter the approximation is updated at every
iteration via equations 2.22-2.25.
2.2.2 Sequential Quadratic Programming
The Sequential Quadratic Programming (SQP) optimisation algorithm mentioned above is considered the industry-standard method for constrained optimisation problems if the number of variables is not too large. The version
used in this study is found in Matlab’s Optimisation Toolbox [25]. It finds the
solution by minimizing successive quadratic approximations of the objective
function. The quadratic objective function of the sub-problem at iteration
k, then takes on the form :
minimize
w.r.t.s
1
F (s) = f (xk ) + ∇T f (xk )s + sT H(xk )s
2
(2.26)
where s is defined in terms of the next and current x values as:
xk+1 = xk + s
(2.27)
In constructing these approximations, second order curvature information
is also required in the form of the Hessian matrix. The Hessian matrix
is approximated by making use of the Broyden-Fletcher-Goldfarb-Shanno
(BFGS) approximation described by equations 2.22-2.25. The Hessian matrix
does require updating, which means an extra n + 1 function evaluations
per iteration. The successive quadratic problems are solved iteratively for s
until s = 0. If an optimum of an approximate sub-problem lies outside the
bounds, the violating variables are set to the boundary values. A line search
is performed using s as the search direction, until a variable set is found that
gives an objective function value equal to or less than the function value of
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the previous iteration. For the line searching Matlab makes use of the merit
function approach [26, 27].
2.2.3 Leap-Frog Algorithm LFOPC
The dynamic trajectory or Leap-Frog OPtimiser, LFOP, has been an ongoing development at the University of Pretoria since 1982 [28] when the first
paper on this method was published. The approximation method DynamicQ [30], currently uses this method to solve spherical quadratic approximate
sub-problems created at each iteration. In its current version Leap-Frog
OPtimiser for Constrained problems, LFOPC [29], is designed to handle
optimisation problems with constraints. The algorithm has the following
characteristics :
• No explicit line searches are performed.
• No explicit function evaluations are required.
• Gradient information is the only explicit function information used.
• Two convergence tolerances and a maximum step size need to be specified.
• The algorithm is suitable for objective functions which exhibit noise.
The penalty function solution to the constrained problem is carried out in
three phases.
• Phase 1 : Moderate penalty parameter value is used which gives fast
and smooth progression to the region of x∗ .
• Phase 2 : A greatly increased penalty parameter is then applied to give
a better approximation to x∗ , and the active constraints at this point
are identified.
• Phase 3 : Determination of least squares solution of active set of constraints is carried out, with the shortest path from the phase 2 solution,
to the actual x∗ being taken.
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LFOP Algorithm
The Leap-Frog OPtimiser (LFOP) for unconstrained optimisation is used for
the optimisation of each phase in LFOPC. The algorithm seeks the minimum
of a n variable function by considering the problem of the motion of a particle
of unit mass in a n dimensional conservative force field. The potential energy
of the particle is the function f (x) to be minimized. The solution of the
equations of motion of the particle, given an initial velocity and position,
is required by the algorithm. The trajectory is approximated by using the
leap-frog (Euler-forward-Euler-backward) method. An interfering strategy is
applied to reduce the kinetic energy of the particle whenever it moves uphill.
The particle is thus forced to follow the path to the local minimum x∗ . For
more information on the algorithm the reader is refered to [28, 29, 30].
2.2.4 Dynamic-Q
The Dynamic-Q algorithm is defined as: ‘Applying a Dynamic trajectory optimisation algorithm to successive spherical Quadratic approximations of the
actual optimisation problem ’[30]. This algorithm has the major advantage
that it only needs to do relatively few function evaluations (simulations) of
the original expensive objective and constraint functions to construct a simple quadratic approximate sub-problem. These approximate functions can
then be cheaply evaluated and the optimum point of the approximate problem may be found economically, using the robust dynamic trajectory method
LFOPC discussed in the previous sub-section. At this new approximate optimum point, new quadratic approximate objective and constraint functions
are constructed, and the associated sub-problem solved. This procedure is
iteratively repeated until convergence is obtained. This method is very efficient for optimisation problems with functions that require an expensive
computer simulation for their evaluation. Dynamic-Q usually may make use
of forward finite differences to obtain the gradient information required for
the generation of the approximations. The basic details of the method are as
set out below. A sequence of sub-problems P[i] i = 0,1,2,... is generated by
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constructing successive spherically quadratic approximations to the objective
and constraint functions, at successive points xi . The approximation to the
objective function, for example, is as follows :
1
f˜(x) = f (xi ) + ∇T f (xi )(x − xi ) + (x − xi )T A(x − xi )
2
(2.28)
The Hessian matrix A takes on a simple diagonal matrix form :
A = aI;
(2.29)
This form of Hessian matrices indicates that the approximate sub-problems
are spherically quadratic in nature. The curvature a takes on a value of zero
for the first sub-problem i = 0. Thereafter it is defined by :
a=
2[f (xi−1 ) − f (xi ) − ∇T f (xi )(xi−1 − xi )]
kxi−1 − xi k2
(2.30)
The approximate constraint functions are constructed in a similar manner.
If the gradient vectors ∇f , ∇gj , and ∇hj are not known analytically they
may be approximated by first order finite differences.
Additional side constraints of the form k̂i ≤ xi ≤ ǩi are normally imposed
on the design variables. Because these constraints do not exhibit curvature
properties they are treated as linear inequality constraints.
To obtain stable and controlled convergence of the solutions of successive
sub-problems, a move limit is set which takes on the form of an inequality :
gδ (x) = kx − xi k2 − δ 2 ≤ 0
(2.31)
where δ corresponds to a specified move limit. The sub-problem at xi can now
be solved using the dynamic trajectory ’Leap-Frog’ optimisation algorithm
for constrained optimisation LFOPC. This solution is taken as xi , the point
at which the next approximate sub-problem is constructed. This process is
continued until convergence is obtained.
2.2.5 Genetic Algorithms
Genetic algorithms (GA’s) do not use gradient information but rather perform stochastic searches of the design space in order to minimize the objective
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function. The stochastic searching is, however, improved by the application
of reproductive theory. The reproductive theory is implemented by the selection of the fittest members (possible solutions xi , i = 1, 2, ..., N ) of the
population (with N members) and allowing them to reproduce, mutate and
crossover, creating a stronger population (lower objective function values)
with time(iterations). The fact that no gradient information is required, is
the main argument for the use of genetic algorithms. However, a substantially large population (N ) is required to approach the minimum, if the design
space is to be comprehensively covered. This results in the need to perform
many function evaluations from the beginning. GA’s are normally terminated on objective function value or variable changes are within tolerances.
Because GA’s do not require gradient information they fall into the class
of zero-order optimisation algorithms together with the Nelder-Mead and
simulated annealing methods.
2.2.6 Nelder-Mead
Nelder-Mead like the GA’s is a zero-order optimisation algorithm. The main
advantage of the Nelder-Mead [31] algorithm is that like genetic algorithms
it does not require gradient information, and is easy to code, resulting in
its more wide spread use. Because gradient information is not required, the
Nelder-Mead algorithm is a good choice if the objective function has discontinuities over the design space. The Nelder-Mead, or simplex method as
it is often called, creates a generalized triangle or simplex in n dimensions.
The simplex points are evaluated with the point having the highest function
value being replaced with a new point. With every iteration the size of the
simplex is increased or reduced. The new point is normally reflected opposite to the poor point by the axis between the best two points. If the new
point exhibits no improvement in the objective function value a new point is
selected at twice the distance from the previous point. If this is still unsuccessful the worst point is replaced by one in the middle of the triangle. The
points are again evaluated and the worst point is changed as before, until
termination occurs. Termination normally occurs when the function value
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28
or variable changes are within prescribed tolerances. Although this method
was originally designed for unconstrained optimisation, the application of the
penalty function process makes it possible to use it for constrained optimisation. The disadvantage with the Nelder-Mead optimisation procedure is that
it is slower than most first order methods. As a result it requires many more
function evaluations, resulting in a computationally more expensive method.
2.2.7 Sequential Linear Programming
Sequential linear programming uses linear approximations of the objective
and constraint functions obtained from truncated Taylor series expansions.
The inequality constraints are transformed to equality constraints. This is
done as the optimum will lie on the boundary. Only the critical constraints
are considered for the current iteration. These linear approximations are
then cheaply solved for the minimum, subject to move limits imposed on the
design variables. This then limits the variables to a gradual move towards the
global optimum. The optimum will normally lie at the intersection of constraints. This point is then used for the linear approximation of the objective
and constraint functions, for the next iteration. This process is continued
iteratively until a minimum is found that conforms to the constraints and
termination criteria.
2.3 Vehicle Suspension Optimisation
2.3.1
Scania Bus
Eriksson and Friberg [32] investigated the use of mathematical optimisation
of the engine mounts of a city bus for ride comfort enhancement. The engine
mounts were assumed to exhibit a linear characteristic for both spring and
damper. There were thus two spring constants and two damper constants
equating to a four design variable optimisation problem. The body of the
bus was modelled as a flexible finite element method (FEM) model, with
spring and damper characteristics assumed linear. A measured random road
profile was inserted as the input to the four wheels. The normalised sum of
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29
the weighted vertical accelerations at three points in the bus was used as the
objective function. The optimisation algorithm used was the IDESIGN [33]
recursive quadratic programming algorithm (similar to SQP). The results
showed that convergence was achieved in 12 iterations using 63 function
evaluations. The objective function exhibited a few local minima points,
with an improvement of seven percent in the ride comfort being obtained.
Eriksson stated [32] :
A problem with the current gradient based algorithm is the result
dependency on the choice of starting design.
Recently Andersson and Eriksson [34] presented a procedure for the optimisation of the handling and ride comfort of a bus. The ride comfort was again
evaluated by observing the nature of the vertical accelerations in three positions in the bus, but also looked at the weighted vibration dose values (VDV)
for the bus travelling over double and single sided obstacles in the road. For
handling optimisation the bus performed a single lane change manœuvre at
80 and 40 km/h. They reported that the minimization of the maximum yaw
velocity gain provides better optimisation results than the minimization of
the minimum yaw velocity time lag, with a constraint on the maximum body
roll angle, so as to ensure sufficient roll stiffness. Individual and combined
optimisation was performed, and for the individual case an 18 percent improvement in ride comfort was obtained, while for the combined optimisation
a 12 percent improvement was achieved. For the ride comfort optimisation
it is stated that six out of the eight variables ran to their bounds. The built
in Sequential Quadratic Programming method in ADAMS 12 [35] was used.
The variables were the front and rear gas spring stiffnesses, front and rear
anti-roll bar diameters, and the front and rear multiplication factors of the
damper force for both compression and rebound.
2.3.2 Neural Network Approximation Approach
Gobbi et al. [36, 37, 38, 39] have done extensive work in the field of robust vehicle suspension optimisation. The models use linear characteristics for both
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30
springs and dampers with up to as many as 38 design variables being considered for 38 performance indices describing both ride comfort and handling
for ranging vehicle speeds [37]. The variables correspond to characteristics
relating to tyre pressures, springs and dampers, rubber suspension bushes
and bump stops. For such an extensive optimisation problem the vehicle
multi-body dynamics model was used to train an artificial neural network
(ANN) [36, 37]. The ANN makes use of 38 state equations and uses the back
propagation method. The Spearman rank correlation coefficient [40, 41] was
used to reduce the number of performance indices, by investigating correlations between indices. Multi-objective programming techniques have been
employed for the description of the overall objective function. The ANN is
then optimised using Genetic Algorithms. Separate ANN’s have been trained
for each prescribed manœuvre of the three dimensional vehicle model, taking
about 1000 function evaluations per ANN trained. An overall mean error
of less than 2 percent was achieved between ANN and the mathematical
multi-body dynamics model, which was also correlated with experimental
data.
2.3.3 LFOPC and Damper Optimisation
Naude and Snyman [12, 13] and Naude [14] made use of the LFOPC algorithm to optimise a six-wheeled military vehicle’s ride comfort by using a
six piece-wise linear damper characteristic. A problem specific program was
written to enable speedy solution of the objective functions. The vehicle was
modelled as a pitch plane model driving over a dirt road, Belgian paving and
200mm ditch bump profile. The objective function was the average of four
vibration dose values. These are the vertical accelerations of the driver, the
center of gravity, and a point at the rear, as well as the pitch acceleration of
the vehicle body. The LFOPC algorithm required about 40 iterations to reach
an optimum. This was acceptable since the custom written objective function evaluator required only a few seconds to perform the necessary function
evaluations. The optimised damper characteristics took on a four piece-wise
linear relationship. Of interest was the existence of many local minima of
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31
the objective function that gave the same objective function value.
2.3.4 Dynamic-Q Two Variable Optimisation
Els and Uys [24] investigated the applicability of using the Dynamic-Q optimisation algorithm for the optimisation of a SUV’s spring and damper
characteristics. Only two variables, corresponding to the spring and damper
characteristics, were used in the optimisation process. The spring variable
is the static gas volume while the damper variable is a force multiplication
factor for the damper characteristics. A full three dimensional DADS vehicle
model was used in simulating driving over the Belgian paving and the driver’s
weighted vertical RMS acceleration was used as a measure of ride comfort.
For handling the vehicle was simulated performing the double lane change
manœuvre at 60 km/h, with the first peak value of the body roll angle being used as the objective function. The optimisation process was performed
manually, and not integrated with the simulation program DADS. The combined optimisation of the vehicle performing the double lane change over the
Belgian paving was also performed with promising results.
2.3.5 SLP and Damper Optimisation
Etman et al. [11] proposed the optimisation of a stroke dependent damper
characteristic for the front suspension of a truck using Sequential Linear
Programming (SLP) with a move limit strategy. The goal was to achieve the
best compromise between the suspension system’s working space and driver
comfort. First a quarter car model was considered for the optimisation,
thereafter a full three dimensional model of the truck and semi-trailer was
built. The damper characteristics were described in terms of the empirical
relationship proposed by Etman et al. and discussed earlier in Section 2.1.3.
Three road disturbances were considered, namely a 250mm traffic hump,
500mm sine wave hump, and a typical railway crossing. It was found that
consideration of the wave and hump were the critical road conditions with
regards to finding an acceptable design. It was found that multiple local
minima of the objective function occurred. For the three dimensional model
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the following simplifications were carried out:
• The bump stops were not modelled, as contact with the bump stops,
leads to a series of high frequency accelerations that makes numerical
computation expensive.
• The vehicle makes contact with the obstacles with both front tyres at
exactly the same time.
• Design variables found to run to their bounds for the quarter car analysis, were kept at their respective bounds.
Etman et al. concluded that the shape of the damper curve was very dependent on the vertical accelerations. They proposed the following design
rules:
The blow-off stiffness can remain small. The bleed and blow-off
pre-load are important variables. To reduce inward suspension
deflection, the compression side of the damping curve is of main
interest.
They express concern about the large step taken in progressing from a quarter
car model to full scale vehicle model, and point out the necessity of a model
of intermediate complexity for optimisation. Such an intermediate model
was constructed and optimised by Naude [12]. Also, some difficulties in the
full scale optimisation were attributed to inaccurate finite difference gradient
approximations, and a multi point finite difference approach is suggested.
2.3.6 SQP and Damper Optimisation
Eberhard et al. [10] made use of a pitch plane vehicle model to optimise for
driver comfort and safe driving over a single obstacle. The weighted vertical acceleration was used as a measure of comfort, and the normalised tyre
vertical force as a measure of safety. The nonlinear damper characteristics
were modelled by Hermite splines as described earlier in Section 2.1.3. The
Sequential Quadratic Programming (SQP) algorithm was used for the optimisation. Eberhard et al. conclude that the Hermite splines were difficult to
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implement and required substantial user input, also that the problem should
be expanded to two dimensional damper characteristics where velocity and
displacement are taken into account.
2.3.7 BMW Vehicle Optimisation Procedure
Schuller et al. [42] proposed modelling of the vehicle components and behaviour by making use of transfer functions in a Simulink environment. This
allows the vehicle model to be evaluated faster than real-time. A total of 150
design variables are considered, where the suspension dynamics and kinematics are described by non-linear characteristic curves. These curves are
approximated using Hermite splines, where the design variables are the coefficients of the splines. For handling, only open loop manœuvres were considered, being the J-Turn and steady-state constant radius test. Many measures were used for the handling objective function, some being peak lateral
acceleration, maximum body roll angle, yaw velocity gain, and yaw response
time. The ISO 15037 [43] rough road test was used for the ride comfort
manœuvre. The vehicle’s center of gravity (cg) deviation from the straight
path, the vertical RMS deviation of the vehicle body from the design level,
and the maximum yaw velocity were used as the ride comfort objective function. Genetic Algorithms were used for the multi criteria optimisation, with
improvements achieved over the baseline vehicle in all manœuvres. Only side
constraints were set on the design variables, being 50 percent change from
the baseline vehicle, with no consideration made for feasibility of the optimised design in terms of kinematics. The most significant changes suggested
by the optimisation study, were to increase the mass of the vehicle body and
wheelbase, while decreasing the yaw moment of inertia. Schuller et al. [42]
stated that the integration of a driver model in the optimisation procedure,
enabling closed loop manœuvres, would lead to:
a new quality in the optimisation of vehicle handling behaviour
in simulation.
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3. THE OPTIMISATION PROBLEM
3.1 Problem Statement
The optimisation of the spring and damper characteristics of a Land Rover
Defender 110 Wagon, for ride comfort and handling, is to be investigated.
The vehicle is fitted with the semi-active hydro-pneumatic suspension unit,
described in Section 2.1.3.
3.2 Introduction
Because of the increasing use of SUV’s and the current research focus at
the University of Pretoria on suspension systems, a Land Rover Defender
110 Wagon is being investigated for the application of the semi-active hydropneumatic suspension system. The suspension has to be tuned for the appropriate spring and damper characteristics for all road conditions. The
suspension characteristics will therefore be optimised for both ride comfort
and handling.
The published works on vehicle suspension optimisation, described in
Section 2.3, all concentrate on the use of only one gradient-based optimisation
algorithm, namely the SQP method, or GA’s, without benchmarking against
other algorithms. In a few studies [44, 45, 46] gradient-based algorithms, like
SQP, have been compared to the performance of stochastic methods, like
GA’s. These studies, however, make use of a simplified pitch plane model
to describe the dynamics. The stochastic methods, were found to be very
expensive in terms of number of function evaluations. This study aims to
provide the reader with more information regarding the use of the DynamicQ algorithm, as an alternative to the well-established and industry-standard
gradient based SQP method.
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3.3 Mathematical Model of Vehicle
3.3.1 General Background
A model of the Land Rover Defender 110 Wagon was built using the multibody dynamics package ADAMS View [47], based on measurements of the
vehicle’s hard points. The model was made as simple as possible, while
complex enough to realistically model the essential features.
The vehicle model consists of 23 rigid bodies excluding ground, 11 revolute
joints, 10 spherical joints, 9 Hooke’s joints, and one driving motion. This
represents a system with 16 degrees of freedom (Table 3.1). The vehicle in
question has leading arms, and a Panhard rod in front with trailing arms,
an A-arm, and an anti-roll bar fitted to the rear suspension. A simplified
steering system was also modelled to enable steering of the vehicle during
handling simulations.
Table 3.1: ADAMS model degrees of freedom
Body
Vehicle Body
Degrees of Freedom Associated Motions
6
longitudinal
lateral
vertical
roll
pitch
yaw
Front axle
2
vertical
roll
Rear axle
2
vertical
roll
Tyres
4
rotation
Anti-Roll Bar
2
rotation (2x)
Total
16
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3.3.2 Vehicle Body
The vehicle body and chassis are assumed to be one rigid body with a specified mass and inertia. The inertias are obtained by scaling down data available for an armoured prototype Land Rover 110 Wagon, to correspond to
our vehicle’s lighter weight. These values are considered to be sufficiently
representative.
3.3.3 Front Suspension
The front suspension (as schematically shown in Figure 3.1, and modelled
in Figure 3.2) consists of 11 rigid bodies, and 14 joints, as detailed in Table
3.2. Suspension links that appear similar to spherical-spherical joints on the
vehicle, are modelled by a connecting body, a spherical and Hooke’s joint
(Figures 3.1 and 3.2). This was required to prevent unnecessary instability
that may arise when solving for the motion of a rotating link. On the actual
vehicle, friction in the suspension joints prevents the suspension links from
rotating. In the ADAMS model it is very inefficient, from a solution point of
view, to include joint friction. Adding joint friction results in more degrees
of freedom, as well as additional spring and damper connection forces. This
results in longer solution times. The solution of the kinematic constraints
is much faster than dynamic constraints. The inclusion of these bushing
stiffnesses add unnecessary high frequency (noise) accelerations to the solution. As this study is only concerned with the ride comfort and not the
noise aspect of the vehicle, all joints are modelled as kinematic constraints.
An additional body has to be added to prevent the suspension from rolling
forward with the vehicle. This was done to keep the model simple and to
avoid modelling of joint friction. The prop-shaft connections to the axle are
not modelled directly, although such modelling would have eliminated the
need for the additional link that was introduced in the current model. The
front suspension has four resulting degrees of freedom, being roll, vertical
freedom, and two rotational degrees of freedom associated with the tyres.
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3. The Optimisation Problem
Figure 3.1: Land Rover front suspension schematic diagram
Figure 3.2: Land Rover front suspension modelling
37
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Table 3.2: Front suspension model degrees of freedom
Quantity
Body
Degrees of Freedom Result
Rigid Bodies
1
Front Axle
6
6
2
Leading Arms
6
12
1
Guide Arm
6
6
2
Wheels
6
12
1
Panhard rod
6
6
2
Steering Arm
6
12
1
Steering Link
6
6
Sub Total
60
Joints
5
Hooke’s
-4
-20
5
Spherical
-3
-15
4
Revolute
-5
-20
Sub total
-55
-1
-1
Total
4
Motions
1
Steering driver
1
Vertical
1
Roll
2
Wheels
Rotation
3.3.4 Rear Suspension
The rear suspension (as schematically shown in Figure 3.3, and modelled in
Figure 3.4) consists of 4 rigid bodies, 2 revolute, 3 spherical, and 2 Hooke’s
joints, detailed in Table 3.3. The suspension is a live axle with two trailing
arms at the bottom, and one A-arm above the rear axle (Figure 3.4). The
springs are mounted vertically to the body, but the separate dampers are
at a trailing angle between the body and rear axle. The A-arm is modelled
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using a rigid link with revolute and spherical joints at the ends. The trailing
arms are modelled with a combination of Hooke’s and spherical joints. The
rear suspension has four resulting degrees of freedom, being roll, vertical
translation, and two rotational degrees of freedom associated with the tyres.
Figure 3.3: Land Rover rear suspension schematic diagram
3.3.5 Anti-Roll Bar
Attached to the rear suspension is the anti-roll bar system modelled with 6
rigid bars, 4 revolute, 2 spherical , and 2 Hooke’s joints (Figure 3.3). The
two bars at the rear are connected with a torsion spring with a stiffness of
22Nm/degree, which was determined from a finite element model of the antiroll bar by Stipinovich [48]. These bars are connected to each other and the
body by two revolute joints. Each of these bars are then connected to two
side link bars, and the rear axle with one revolute, one spherical and one
Hooke’s joint (Figure 3.4). The rear anti-roll bar has two resulting rotational
degrees of freedom, being the rotation of the rear bars in relation to the
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3. The Optimisation Problem
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Figure 3.4: Land Rover rear suspension modelling
vehicle body.
3.3.6 Force Elements
The springs and dampers were first modelled using the standard ADAMS
spring/damper element with a spline for force versus displacement and velocity. The new hydro-pneumatic suspension strut was modelled as a force
between the two moving bodies. The gas spring characteristics were calculated in Matlab and imported as a spline into ADAMS. The displacement
axis was scaled for the gas volume. This entails the multiplication of the
axes by the gas volume. The damper characteristics are modelled by the
non-linear function of 8 variables defining switch points and slope. An effort
was made to fit the function proposed by Etman et al. to these characteristics [11]. This function, however, proved to be very difficult to fit to our
current damper data, thus making the function non-viable. The details of
the function are discussed in Section 2.1.3. For the initial investigation the
rear damper graph was used and the force scaled with a design variable.
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Table 3.3: Rear suspension model degrees of freedom
Quantity
Body
Degrees of Freedom Result
Rigid Bodies
1
Rear Axle
6
6
1
A Arm
6
6
2
Trailing Arms
6
12
2
Wheels
6
12
2
Anti-roll Bar
6
12
2
Anti-roll Bar arms
6
12
2
Anti-roll Bar links
6
12
Sub Total
72
Joints
4
Hooke’s
-4
-16
5
Spherical
-3
-15
7
Revolute
-5
-35
Sub Total
-66
Total
6
1
Vertical
1
Roll
2
Wheels
Rotation
2
Anti-roll Bar
Rotation
3.3.7 Tyres
The tyres were created using the built-in ADAMS tyre module. Initially a
Fiala tyre was selected because of its simplicity, and ADAMS license constraints. It was, however, found that the correlation with measurements
could be improved if a load sensitive tyre was used. The 521 tyre was used
for the handling and ride comfort simulations. This tyre is a simple lookup
table interpolation tyre element for side-force and self-aligning torque, with
ride comfort evaluated by a point follower model. Tyre data (Figure 3.5)
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3. The Optimisation Problem
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from tests conducted by LMT is used in the 521 interpolate tyre model.
The four tyres are connected to the front and rear suspensions with revolute
joints.
Figure 3.5: Tyre side force properties
3.3.8 Driver Implementation
For the vehicle to follow a specified path, a driver must be implemented. The
steering driver implemented uses a marker at a preview distance in front of
the vehicle (Figure 3.6) and adjusts the error according to the desired path
to be travelled. This distance in front of the vehicle is defined as the driver
preview distance, which can be calculated by multiplying the speed with the
driver preview time. The differential equation describing the drivers action
[49] with time t is:
τ δ̇(t) + δ(t) = −K · d
(3.1)
where τ is the driver preview time:
τ=
l
s
(3.2)
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3. The Optimisation Problem
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with l the driver preview distance, s the vehicle speed, d the error, δ(t) the
steering angle, and K the steering angle gain.
Figure 3.6: Implementation of driver model
For the vehicle to maintain a constant speed throughout the simulation,
a speed driver must be implemented. A torque and force are applied to
the wheels to ensure that the vehicle keeps a constant speed. This forcing
value is calculated from the difference between the desired speed and actual
vehicle speed, multiplied by a gain. Fifty percent of the force is applied as a
horizontal force at the wheel centre, and the other 50 percent, multiplied by
the wheel radius, as a torque on the wheel. The complete force was not added
to the wheel as a torque, due to difficulties that were encountered, associated
with the longitudinal dynamics of the tyre. The ratio used was found to be a
good compromise. This is permitted, as this research is not concerned with
the longitudinal dynamics of the vehicle, but rather the lateral dynamics
(handling) and the vertical dynamics (ride comfort). With the full vehicle
model illustrated in Figure 3.7.
3.4 Model Validation
Every mathematical model needs to be evaluated for accuracy against measured test data. A Land Rover Defender 110 Wagon was fitted with test
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3. The Optimisation Problem
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Figure 3.7: Full Land Rover model
equipment as described in Table 3.4 and illustrated in Figure 3.8. The vehicle then performed the double lane change manœuvre at 80 km/h, with
half crew (2340 kg) and fully laden (2640 kg). Then it drove over the local
Aberdeen Proving Grounds (apg) 100mm bumps at various speeds, and also
crossed the Belgian paving at various speeds. The results for the comparison between the ADAMS model and the measured results for the apg are
presented in Figure 3.9. Good agreement between the measured results and
the ADAMS model were achieved. In Figure 3.10 the results for the vehicle performing the double lane change manœuvre at 80 km/h are presented.
Although the handling agreement of the measured and the ADAMS model
results is acceptable, for the purpose of the current study, the agreement is
not as good as for the vertical dynamics. This is attributed to the single point
preview driver model used in this study for the execution of the double lane
change. It was suggested [50] that a three-point preview controller would
perform better at executing the double lane change manœuvre than a single
point preview controller. This, however, is still not the ideal driver model
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3. The Optimisation Problem
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for transient closed loop manœuvres. The three-point preview controller was
recently implemented and found to be much smoother. The original single
point controller model was, however, used consistently throughout this research. As can be seen from Figure 3.10 the steering input provided by the
driver model contains more high frequency corrections than the measured
steering response. This results in the very noisy nature of the front lateral
acceleration. The rear lateral acceleration is much cleaner than the front
lateral acceleration. This is attributed to the smaller influence the steering
input has on the rear of the vehicle. This research, however, only uses the
first four seconds of the simulated data of the double lane change manœuvre, which is in reasonable agreement with the measured results, of the same
order of magnitude, and exhibits the same tendencies.
Table 3.4: Land Rover 110 test points
channel
point
position
measure
axis
1
B
center of gravity
velocity
longitudinal
2
G
left front bumper
acceleration
longitudinal
3
lateral
4
vertical
5
C
rear passenger
acceleration
longitudinal
6
lateral
7
vertical
8
I
right front bumper
acceleration
vertical
9
A
steering arm
displacement
relative arm/body
10
D
left rear spring
displacement
relative body/axle
11
E
right rear spring
12
F
left front spring
13
H
right front spring
14
B
center of gravity
angular velocity
roll
15
pitch
16
yaw
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3. The Optimisation Problem
Figure 3.8: Test vehicle indicating measurement positions
Figure 3.9: apg, 25km/h, Model validation results
46
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3. The Optimisation Problem
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Figure 3.10: Double lane change, 80 km/h, model validation results
3.5 Optimisation Algorithms
The optimisation algorithms chosen for this investigation are the DynamicQ method and the industry-standard SQP method. The SQP method is
chosen because of its wide acceptance as a superior optimisation method.
The Matlab implementation of SQP was preferred to the built-in ADAMS
SQP. As the model had to be made compatible with Matlab, this provided an
easy comparison of results. Also ADAMS has a new SQP algorithm available
in the latest version that makes future comparisons difficult, as the user does
not know what the next built-in optimisation algorithm will be. This is due
to uncertainties as to license agreements between the optimisation company
and Mechanical Dynamics. The University of Pretoria currently has a licence
for the Matlab optimisation toolbox, making it more convenient for future
comparison purposes.
The Dynamic-Q method was selected for its proven applicability to suspension design [24], and in the solution of computational fluid dynamics
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3. The Optimisation Problem
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problems [51, 52, 53]. These problems exhibit noise and discontinuities for
which Dynamic-Q appears to be suitable. Naude [14] also suggests in his
thesis that the coupling of Dynamic-Q to a multi-body simulation package
should be considered for future work.
3.6 Design Variables
In choosing the design variables for optimisation, the assumption is made
that the left hand and right hand suspension settings will be the same, but
that front and rear settings may differ. The design variables chosen for
optimisation are therefore the static gas volume (Figure 3.11), and damper
scale factors (Figure 3.12), on both the front and rear axles.
Figure 3.11: Definition of spring characteristics for various gas volumes
For the initial study the standard damper force characteristic is multiplied
by a factor which constitutes the damping design variable (Figure 3.12). The
general shape and switch velocities of the damper are thus kept the same.
This research considers the cases of two and four design variables, which
respectively corresponds to the cases where the spring and damper characteristics are identical for the front and rear axles (two design variables), and
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Figure 3.12: Definition of damper characteristics for various damper scale factors
where they may differ for front and rear (four design variables). The study
is then extended to include seven design variables where the characteristics
are identical front and rear, but the damper characteristics are defined by
six design variables.
3.6.1 Two Variable Case
The two design variable study is an important starting point in the optimisation procedure, as it gives the necessary insight into the problem. For this
two design variable study, it was decided to use the same design variables
as those considered by Els and Uys [24] in their preliminary study, namely
the static gas volume and the damper force scale factor. Figure 3.11 illustrates the spring characteristics for various static gas volumes. Figure 3.12
illustrates the damper characteristics for various damper scale factors.
The static gas volume is denoted by gvol, and the damper force scale
factor by dpsf . These variables are allowed to range from 0.1 to 3 in magnitude, which are accordingly chosen as upper and lower bounds. The design
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3. The Optimisation Problem
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variables are explicitly defined as follows :
x1 = dpsf,
x2 = gvol
(3.3)
i = 1, 2
(3.4)
with bounds
0.1 ≤ xi ≤ 3,
3.6.2 Four Variable Case
For the four design variable problem the front and rear settings are uncoupled. This means that there are separate front and rear damper scale factors
and front and rear spring static gas volumes. This results in two design
variables describing the front and two describing the rear, giving four design
variables in total.
The front damper scale factor is denoted by dpsf f , the front static gas
volume by gvolf , the rear damper scale factor by dpsf r, and the rear static
gas volume by gvolr. These variables are also allowed to range from 0.1 to 3
in magnitude. Thus the design variables are defined explicitly as follows:
x1 = dpsf f,
x2 = gvolf,
x3 = dpsf r,
x4 = gvolr
(3.5)
i = 1, ..., 4
(3.6)
with bounds
0.1 ≤ xi ≤ 3,
3.6.3 Seven Variable Case
For the seven design variable case the front and rear suspension characteristics are again identical as for the previous case with two design variables,
however, the definition of the damper characteristic curve is given in terms of
six variables. Although not considered in this study, the next step would be
14 variables allowing for the front and rear suspension to be uncoupled. The
damper variables are defined in Figure 3.13. The variables for the problem
are as listed in Table 3.5.
These variables vary over a range from 0.1 to 5000 in magnitude, so the
problem needs to be scaled. For the purposes of scaling the current damper
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Figure 3.13: Definition of damper variables
Table 3.5: Definition of damper variables
gvol Static Gas volume
V1
High point negative velocity on damper graph
V2
Change point negative velocity on damper graph
V3
Change point positive velocity on damper graph
V4
High point positive velocity on damper graph
V5
Damper force corresponding to V1
V6
Damper force corresponding to V2
V7
Damper force corresponding to V3
V8
Damper force corresponding to V4
setting values are used as scale factors. Variables V1 and V4 are kept at
their original values so as to avoid convergence problems in the optimisation
process. Problems were encountered when the value of V4 was less than
V3 resulting in two possible forces for the same velocity. The ADAMS solver
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cannot solve the dynamics of the system, as there exists two possible solutions
for the same relative velocity between the vehicle body and axles. This leads
to many discontinuities in the objective function. The design variables are
thus defined as follows:
x1 = gvol
x2 = −
V2
0.125
(3.7)
x3 =
x4 = −
V5
3500
x5 = −
x6 =
V7
2000
x7 =
V3
0.1
(3.8)
V6
1000
(3.9)
V8
5000
(3.10)
with bounds
0.1 ≤ xi ≤ 3,
i = 1, ..., 7
(3.11)
V4 = 1
(3.12)
where
V1 = −1,
According to the above scaling a design variable value of one for the damper
settings will result in the current damper value. All the design variables have
a range of 0.1 to 3, which defines the side constraints. For the purposes of
this investigation it is assumed that V8 can take on a value lower in magnitude than V7 . This is, however, very difficult to implement in practice, so
for a passive damper, additional inequality constraints should be set. However, with the availability of the bypass valves on the semi-active unit, it is
envisaged that this condition can be achieved.
3.7 Definition of Objective Functions
3.7.1 Ride Comfort
For ride comfort the motion of the vehicle is simulated for travelling in a
straight line over the Belgian paving. The sum of driver and rear passengers
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British Standard 6841 weighted root mean square (RMS) vertical accelerations [17], are used for the objective function. The motion sickness component was ignored as it requires long run times and the Belgian paving test
track is not long enough to evaluate motion sickness. Both rear passenger
and front driver RMS accelerations were considered for the ride comfort objective function, so as not to improve a single seat’s comfort while severely
decreasing the other seat’s comfort.
3.7.2 Handling
For handling, the vehicle performs the ISO3888 [21] severe double lane change
manœuvre at 80 km/h and the maximum body roll angle of the first peak [24],
coinciding with the first steering input for the first lane change, is used as the
objective function (Figure 3.14). This first peak is taken as the simulation
is less likely to fail (vehicle roll over) due to the incorrect suspension setup,
than looking at the last peak which is normally the most severe. This limits
discontinuities in the design space. Roll angle is used as a measure of handling
as suggested by Uys et al. [54] (Summarized in Appendix A), due to the
linear relation observed between lateral acceleration and body roll angle.
The minimization of vehicle body roll angle is a visual parameter that can
easily be validated when the vehicles are tested.
The optimisation is performed with limited constraints, so as to allow for
a better understanding of the performance requirements of the semi-active
unit in attempting to achieve the optimisation objectives. The bounds on
the design variables were the only constraints considered in this study.
3.8 Integration of Mathematical Vehicle Model and
Optimisation Algorithms
In order to perform the optimisation process, it is desirable to have the
optimisation algorithms and the mathematical simulation model working independently together, without user interaction. This is achieved by making
use of the ADAMS Controls package which allows the user to define input
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Figure 3.14: Body roll angle of a standard Land Rover Defender 110 while performing the double lane change manœuvre
and output data. This model can then be run from the Matlab Simulink
environment. Figure 3.15 shows the vehicle model in Simulink with 9 input
variables and the driver and passenger output results bins. The model can be
run by the Matlab based optimisation codes by using the sim function. This
model is then used for optimisation with respect to seven design variables.
3.9 Preliminary Sensitivity Investigations
3.9.1 Design Space
For the two design variable optimisation, surface plots of the objective function over the complete design space were generated. However, with an increasing number of variables added, this is not possible. These objective
function surfaces were generated when the optimisation of handling (Figure
3.16) and ride comfort (Figure 3.17) were performed separately. From the
figures it can be seen that for excellent handling capability, high damping
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Figure 3.15: Optimisation process flow diagram
and high spring stiffness is required. The damping does not, however, contribute greatly to the improvement if the spring stiffness is high (a small gas
volume)(Figure 3.16).
For ride comfort (Figure 3.17) on the other hand the opposite holds.
Medium spring stiffness and low damping is required. The damper scale factor has a more noticeable effect on the ride comfort, as established previously
by Els and Uys for the heavier version of this vehicle [24].
3.9.2 Gradient Sensitivity
Due to the complexity of the problem to be optimised, a few preliminary
sensitivity investigations were required. The algorithms used in this study
require gradient information for the optimisation process. This gradient information must be calculated using finite differencing methods. Gradient
sensitivity studies were carried out in order to determine suitable values for
the perturbations (dxi ) to be used for forward and central finite difference
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Figure 3.16: Vehicle roll angle, double lane change at 80 km/h for the two variable design space
gradients. The effects of the size of dxi , for each variable, on the different gradient components, were evaluated and appropriate dxi determined. From the
gradient sensitivity studies it was observed that perturbations dxi of around
0.1 were most appropriate for the optimisation problem.
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Figure 3.17: Vehicle ride comfort, Belgian paving at 60 km/h for the two variable
design space
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4. RESULTS
4.1 Handling Results
4.1.1 Two Design Variables
No serious problems were encountered in applying the algorithms to the optimisation of handling. For handling optimisation with two design variables
no substantial difference can be reported between SQP and Dynamic-Q. The
SQP convergence history for handling optimisation (Figure 4.1) indicates two
local minimum solution sets with the same objective function value. The use
of Dynamic-Q with 10 percent move limit (Figure 4.1) re-iterates the fact that
design variable one (damper multiplication factor) has a limited effect on the
handling objective function value as has already been established in Figure
3.16. Using a 20 percent move limit (Figure 4.1) Dynamic-Q progresses faster
to a minimum. Because of the excellent performance of the forward finite
difference method the use of central finite differences at additional cost was
not necessary. The optima found by the optimisation algorithms are detailed
in Table 4.1.
Table 4.1: Two design variable handling optimisation results
algorithm
func evals
f (x∗ )
x∗1
x∗2
DynQ 10 o /o
9
0.78
1.35
0.05
DynQ 20 o /o
9
0.81
1.99
0.05
SQP
21
0.78
1.60
0.05
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Figure 4.1: Optimisation histories of handling for two design variables
4.1.2 Four Design Variables
The handling optimisation results for four design variables (Figure 4.2) were
not really different to that for two variables. This is as to be expected as the
dynamics of the system has not changed substantially. It is interesting to
note that a move limit of as big as 30 percent of the variables range may be
used in Dynamic-Q using forward finite differences. It can also be seen from
Figure 4.2 that the optimisation histories are very well behaved. Figure 4.2
again indicates the definite existence of more than one local minimum with
the same objective function value. The SQP algorithm performed similarly
to Dynamic-Q, and also found two different local minima, with the same
objective function value. SQP converged in 9 iterations (49 function evaluations), compared to Dynamic-Q’s 5 iterations (25 function evaluations).
Table 4.2 compares the optimum values obtained using Dynamic-Q (with
different move limits) to SQP. It can be seen that the gas volumes (x2 and
x4 ) all ran to their respective lower bounds, while for the best roll angle, the
damper scale factors (x1 and x3 ) went to their upper bounds, as happened
in the case of the two design variable optimisation. The two different optima
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correspond to the cases where the damper scale factors are respectively the
same and different at the front and rear. The results here reinforce the initial
conclusion from the results for two variables: that the damper scale factor
has a negligible effect on the vehicle’s handling performance (body roll angle)
through the double lane change manœuvre at the optimum (stiff) spring rate.
Although an improvement for the four design variable optimisation over the
two design variable optimisation is expected, it was not realised as the lower
limit set for the gas volume, in the two design variable case, is not physically
feasible.
Table 4.2: Four design variable handling optimisation results
algorithm
func evals
f (x∗ )
x∗1
x∗2
x∗3
x∗4
DynQ 10 o /o
20
0.96
1.79
0.1
1.47
0.1
DynQ 20 o /o
25
0.95
3
0.1 2.90 0.1
DynQ 30 o /o
15
0.95
3
0.1
DynQ 45 o /o
15
0.95
3
0.1 2.97 0.1
SQP
17
0.96
1.49
0.1
3
1.44
0.1
0.1
4.1.3 Seven Design Variables
For the handling optimisation 9 design variables were initially used but it
was found that this allowed too much freedom in the damper characteristics, resulting in two different forces for the same velocity. It was decided
that the end velocities should be kept constant and only the interior velocity points moved by the optimisation algorithm, as defined in Section 3.6.3.
Again the algorithms progressed quickly towards a local optimum. The best
local optimum design states found by the algorithms are presented in Table
4.3. It is observed that the gas volume must be at the lower limit resulting
in a very stiff spring characteristic. It can be observed that variables two
and three must be at the lower bound, while variable five must be at the
higher bound. The other damper variables do not appear to affect the performance of the vehicle substantially. A move limit in Dynamic Q up to 30
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4. Results
Figure 4.2: Optimisation histories of handling for four design variables
61
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4. Results
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percent was found to arrive at the same optimum objective function value,
with slight differences in the corresponding optimum design variables. The
move limit was kept at 30 percent and a random starting point (Figure 4.3)
tried with the same optimum area being reached. Figure 4.4 depicts the different optimum damper characteristics for handling obtained by the different
optimisation algorithms. It is interesting to note that significantly different
damper characteristics result in similar optimum behaviour. In the study by
Naude [14] significantly different damper characteristics were found to result
in similar optimum behaviour for the ride comfort optimisation. From Table
4.3 it can be seen that the SQP optimum found is significantly higher than
that for Dynamic-Q, the results of this can be clearly seen by the completely
different damper characteristic obtained in Figure 4.4. The SQP damper
characteristic exhibits almost no damping force in the low speed region. It is
generally accepted that high damping is required in the low speed region of
the damper characteristic, for good handling. The optimum values are also
lower than for the four design variables, but it must be remembered that for
the seven design variables the front and rear suspension characteristics are
the same, unlike for four design variables where the front and rear suspension
characteristics may differ.
Table 4.3: Seven design variable handling optimisation results
algorithm
func evals
f (x∗ )
x∗1
x∗2
x∗3
x∗4
x∗5
x∗6
x∗7
DynQ ffd 10 o /o
48
1.04
0.1
0.1
0.1
0.63
3
1.40
1.45
DynQ ffd 20 o /o
40
1.02
0.1
0.1
0.1
2.07
3
1.83
1.24
DynQ ffd 30 o /o
64
1.05
0.1
0.1
0.96
2.68
3
3
2.06
DynQ rand 30 o /o
48
1.05
0.1
0.1
0.1
0.1
3
0.90
0.1
SQP
52
1.34
0.1 1.78
1.38
1.85
3
0.1
2.21
Figure 4.5 takes all the points obtained in the optimisation history and
evaluates the Euclidean-norm distance of the points from the found minimum. This distance is then plotted against its corresponding objective func-
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tion value. This gives a general indication of the robustness of the minimum
point. From this it can be seen that for all the minima found by the algorithms, the objective function value does not change significantly when a
move limit of 1.5 from the optimum, subject to design variable x1 (gas volume) being at its lower bound. Above 1.5 the Dynamic-Q with 10 percent
move limit becomes difficult to predict. However, there exists many local
minima at a Euclidean-norm distance of more than 2 away from the current
optimum points. This is, however, not a complete picture of the situation,
as not all the variables can change as much as others due to the insensitivity
of some variables, such as x6 and x7 , on the objective function.
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Figure 4.3: Optimisation histories of handling for seven design variables
64
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Figure 4.4: Optimum damper characteristics for handling
Figure 4.5: Existence of many local minima away from current optimum, for the
seven design variable, handling optimisation
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4.2 Ride Comfort Results
4.2.1 Two Design Variables
The two design variable ride comfort optimisation encountered problems associated with a noisy objective function. The SQP method (Figure 4.6)
took eight iterations (33 function evaluations) to stabilize at the minimum,
which corresponds to the lowest possible damping and stiffness values, as expected from Figure 3.17, and also corresponds to the prescribed lower bounds
of the variables. The Dynamic-Q method, on the other hand, experienced
greater difficulty in reaching a stable minimum. The central finite difference
method was introduced to evaluate the gradient of the objective function in
an attempt to obtain stability in the optimisation process. The Dynamic-Q
method with central finite differences, with a ten percent move limit (Figure
4.6) took nine iterations (50 function evaluations) to find a minimum. Inspection of the results shows that this minimum is effectively reached after
only four iterations (25 function evaluations). The vertical acceleration at
this point is, however, significantly higher than that found with SQP, indicating the existence of a separate interior local minimum. A 20 percent move
limit (Figure 4.6) took six iterations (30 function evaluations), finding a local
minimum not far off the SQP minimum, but still lying within the bounds of
the feasible region. It should be noted that the Dynamic-Q minimum design
variable values found in this case are not at the extrema found by the SQP
method. This reinforces the fact that the ride comfort design space has a
flat plateau with multiple local minima.
Table 4.4: Two design variable ride comfort optimisation results
algorithm
func evals
f (x∗ )
x∗1
x∗2
DynQ cfd 10 o /o
25
4.14
0.37
1.44
DynQ cfd 10 o /o
50
4.07
0.39
1.60
DynQ cfd 20 o /o
30
3.22
0.25
2.50
SQP
33
2.70
0.11
3
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Figure 4.6: Optimisation histories of ride comfort for two design variables
4.2.2 Four Design Variables
For the four design variable optimisation, Dynamic-Q was modified so that
the move limit for each iteration is 90 percent of the move limit of the previous iteration. This was done to stabilize the convergence behaviour of the
algorithm, and to try and prevent high spikes in the optimisation process.
These spikes are caused by a poor approximation to the objective function
close to the minimum. This results in the LFOPC algorithm finding a minimum of the approximate problem, on the slope of the steep valley, very close
to the actual minimum. Indicative of the close proximity to the minimum
is the fact that, in spite of the large spike at iteration six, the corresponding changes in the variables are very small. However, Dynamic-Q quickly
recovers within a single iteration (5 function evaluations) as can be seen at
iteration seven in Figure 4.7. When looking at the Euclidean-norm distance
from the optimum, it is observed that although the objective function experiences an increase in value at iteration 6, the Euclidean-norm distance from
the optimum is decreasing, proving that Dynamic-Q is converging towards
the optimum.
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Figure 4.7: Dynamic-Q ffd ride comfort, 4 design variables, 10 percent move
limit, gas volume range from 0.1 to 3 litres
The results of the optimisation are presented in Figure 4.8 for both central finite differences and forward finite differences used for the gradient approximations. From Figures 4.7 and 4.8 for the forward finite difference
Dynamic-Q implementation, it can be seen that the smaller move limit of 5
percent is more stable, reaching a minimum within 6 iterations (35 function
evaluations), while a 10 percent move limit takes 12 iterations (65 function
evaluations). The algorithm, however, does not converge due to the noisy
objective function with steep valley. The convergence behaviour for central
finite differences coupled to Dynamic-Q is shown in Figure 4.8 requiring four
iterations (45 function evaluations). Again it has been determined that the
smaller move limit improves convergence to the minimum. The central finite
difference gradient evaluation builds into the system a level of robustness.
From the results in Table 4.5 it can be seen that a value of around 1.5 litre
gas volume (x2 and x4 ), and limited damping (x1 and x3 ), returns the best
results. From the central finite difference results, it can be seen that an in-
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creased rear gas volume x4 , with minimal damping x3 , compared to the front,
results in a better overall ride comfort (Figure 4.8).
Table 4.5: Effect of changing the gas volume range for four variable ride comfort
optimisation
algorithm
Func evals
f (x∗ )
x∗1
x∗2
x∗3
x∗4
0.1 ≤ gvol ≤ 3
DynQ ffd 10 o /o
65
3.77
0.18
1.36
0.10 1.70
DynQ ffd 5 o /o
35
3.85
0.48
1.44
0.14 1.60
DynQ cfd 10 o /o
45
3.61
0.31
1.32
0.24 2.04
SQP
65
3.49
0.19
2.08
0.35 2.18
0.008 ≤ gvol ≤ 0.5
DynQ cfd 10 o /o
108
7.24
0.60
2.98
0.72
3
DynQ cfd 5 o /o
144
7.40
0.53
2.68
0.74
3
DynQ ffd 10 o /o
65
7.19
0.90
3
0.16
3
DynQ ffd 5 o /o
80
8.71
0.64
1.28
0.56 2.37
SQP
114
9.32
0.14
0.94
0.99 1.90
infeasible starting point for 0.008 ≤ gvol ≤ 0.5
DynQ ffd 10 o /o
65
8.10
2.62
1.76
0.30
3
SQP
94
7.80
2.96
2.08
0.08
3
3
1.03 ≤ gvol ≤ 3.0
DynQ cfd 10 o /o
99
2.86
0.33
1.81
0.05
DynQ cfd 5 o /o
117
3.39
0.33
1.15
0.19 1.36
DynQ ffd 10 o /o
70
3.46
0.24
1.06
0.15 1.61
DynQ ffd 5 o /o
50
3.49
0.36
0.89
0.25 1.48
SQP
111
3.59
0.29
1.06
0.28 1.10
started at predicted minimum from 2 variable optimisation
DynQ cfd 10
126
2.67
0.19
2.83
0.05 2.96
SQP also found similar improved results within 8 iterations (65 function
evaluations) (Figure 4.8). From Table 4.5 it is concluded that DynamicQ, with forward finite differences, does not reach the same minimum as
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Figure 4.8: Optimisation histories of ride comfort for four design variables (gas
volume range from 0.1 to 3 litres)
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Dynamic-Q with central finite differences. The performance of Dynamic-Q
with central finite differences is also relatively economic, compared to that of
SQP, finding a minimum within five percent of the SQP minimum objective
function value. Again it was found that SQP has a tendency to run to the
boundaries. The reason for this is because the optimum of the approximated
sub-problem may lie outside the boundary. SQP then sets the violating variables to their boundary values and finds the optimum from the boundary
by performing line searches. Quite often it finds that the previous minimum
is not at the boundary for a specific variable, but because the gradient indicates that going in the direction of the boundary decreases the function
value, it returns to the boundary point for the next iteration. If found that
this new point has a higher function value than the previous function value,
SQP performs a few line searches until a minimum lower than previously is
found. SQP thus gets stuck in a region which is not the case with Dynamic
Q, where the move is more rigorously controlled by the move limit setting.
Dynamic-Q will thus move more gradually to the optimum, while SQP jumps
faster to a minimum, at the boundary, and such a jump may, as in this case,
fortuitously give the lowest function value.
The SQP minimization is better understood with reference to Figure 4.9
giving a schematic representation of the optimisation, with respect to one
variable. Consider starting point a. At this point the objective function is
approximated using a quadratic approximation. The optimum of this function is at a*, resulting in b being the next iteration point. The approximate
optimum b* now lies outside the feasible region, thus the variable is set to
the boundary value, resulting in iteration point c with minimum point c*. In
turn c* gives d as a minimum, and thus the approximation leads back to the
boundary at point d*. The next iteration point would be e1, however, here
the objective function value is significantly higher than that for the previous
iteration value d. A line search is done to obtain point e in the region of iteration d, resulting in another boundary approximated iteration point. The
SQP algorithm thus gets stuck in the region of d and e.
Because of the boundary clinging nature of SQP, and Dynamic-Q’s dif-
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Figure 4.9: SQP optimisation movements
ficulty in handling this noisy objective function space, the design space for
the gas volumes was split. This was done by imposing the 0.05 to 3 side
constraints on the variables and by changing the definition of the gas volume
variables as follows:
1
gvolf = x2
6
(4.1)
1
gvolr = x4
6
(4.2)
for the front, and
for the rear. The gas volume now has a range between 0.008 and 0.5 litres,
which is practically more realizable than the initial upper boundary of 3 litres.
From Figure 4.10 it is observed that there is little difference in the convergence behaviour of Dynamic-Q with central finite differencing and forward
finite differencing. It is also observed, that the convergence histories are much
smoother, not exhibiting the spikes previously observed. The main reason for
this, is attributed to the fact that the gas volume variable is not permitted
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to reach into the flat plateau region. The optimum objective function values
(Table 4.5) are significantly higher than previously obtained, indica-ting that
the gas volume range is too limiting. Optimisation runs were performed from
the infeasible region, with the SQP method finding a better optimum than
when started in the feasible region. Dynamic-Q, however, needed more iterations than the limited 15 iterations that it was permitted to perform. This is
probably because the SQP method sets all initial starting design values, outside the feasible region, to their boundary values, while Dynamic-Q gradually
progresses from its given starting design variable values. Although some of
the starting points were outside the boundaries, none of the boundaries were
violated for the optimum design conditions obtained. However, almost all
algorithms found optima where the rear gas volume design variable x4 , was
lying on the upper boundary, resulting in a gas volume of 0.5 litres.
Because of the improved behaviour obtained, the gas volume was modified
again to look only at the flat plateau region of the design space. The design
variables relating to the gas volume are defined as follows:
2
gvolf = x2 + 1
3
(4.3)
2
gvolr = x4 + 1
3
(4.4)
for the front, and
for the rear. This means that the gas volume can range between 1 and 3
litres for a variable range of 0.05 to 3. As was to be expected (Figure 4.11)
many problems were experienced with convergence, due to the noisy nature
of the objective function. Table 4.5 summarizes the results for the different
gas volume ranges. Again, none of the boundary constraints were violated. It
can be seen that Dynamic-Q with central finite differences and a 10 percent
move limit, obtained the most optimum objective function values, with the
rear gas volume at its upper bound and damper scale factor at its lower
bound. It is also interesting to note that even when started at the predicted
minimum for the two design variable optimisation, the Dynamic-Q method
with central finite differences took as many as 13 iterations (126 function
evaluations) to obtain the minimum.
University of Pretoria etd – Thoresson, M J (2005)
4. Results
74
Figure 4.10: Optimisation histories of ride comfort for four design variables (gas
volume ranges from 0.008 to 0.5 litres, with inf - being an infeasible
starting point)
University of Pretoria etd – Thoresson, M J (2005)
4. Results
75
Figure 4.11: Optimisation histories of ride comfort for four design variables (gas
volume range from 1.03 to 3 litres)
University of Pretoria etd – Thoresson, M J (2005)
4. Results
76
4.2.3 Seven Design Variables
Figure 4.12 illustrates the gradual convergence of Dynamic-Q towards the
local minimum, while it can be clearly seen how SQP jumps and gets stuck
in converging to a minimum. While gradual convergence can be seen as both
advantageous and disadvantageous for the problem at hand, the gradual
movement into and out of the local minimum is viewed as more favourable.
It gives more insight into the robustness of the design corresponding to the
local minimum, and also has a greater probability of finding a better minimum, that would otherwise have been the case. It is interesting to note that
Dynamic-Q with central finite differences and a ten percent move limit, obtained the same optimum objective function value (Table 4.6) as Dynamic-Q
with forward finite differences and a five percent move limit, but not corresponding to the same combination of variable values. This again reinforces
the belief that the ride comfort objective function exhibits a region with various local minima. It is also of interest to note that a gas volume of 1.77
litres is found as an optimum value by using SQP. Although this value differs
significantly from that found by Dynamic-Q, the objective function value is
more or less the same. The damper characteristics are vastly different for the
different local optima found (Figure 4.14). It is important to note that for
negative damper velocities the ideal damper characteristics mostly take on a
negative gradient, corresponding to a damping coefficient of -1250 N s/m, or
positive gradient corresponding to 830 N s/m . Again, a number of design
variables (x2 , x3 , x4 ) assume their lower boundary values in the case of SQP.
University of Pretoria etd – Thoresson, M J (2005)
4. Results
77
Figure 4.12: Convergence to optimum, seven design variables
Table 4.6: Seven variable ride comfort optimisation results
algorithm
Func evals
f (x∗ )
x∗1
DynQ cfd 10 o /o
120
3.55
2.23
DynQ ffd 10 o /o
150
4.24
DynQ ffd 5 o /o
120
DynQ ffd 3.5 o /o
SQP
x∗2
x∗3
x∗4
x∗5
x∗6
x∗7
1.28 1.49 0.23
0.10
0.66
0.38
1.56
0.49 0.88 0.31
1.78
0.55
0.42
3.55
2.48
2.26 1.15 0.11
0.86
0.20
0.38
112
4.99
1.07
1.06 0.75 0.49
0.90
0.82
0.50
74
3.57
1.77
0.1
1.14
0.84
0.40
0.1
0.1
University of Pretoria etd – Thoresson, M J (2005)
4. Results
78
Figure 4.13: Optimisation histories of ride comfort for seven design variables
University of Pretoria etd – Thoresson, M J (2005)
4. Results
Figure 4.14: Optimum damper characteristics for ride comfort
79
University of Pretoria etd – Thoresson, M J (2005)
5. DISCUSSION OF CONCLUSIONS
The integration of the optimisation algorithms with the full multidimensional
vehicle model has been successfully achieved, allowing for an automated optimisation process. It has also been shown that fundamental gradient-based
optimisation algorithms can be successfully applied to the optimisation of
the vehicle’s suspension system for both ride comfort and handling.
The use of central finite differences has been shown to offer substantial
benefits, if noise is present in the objective and constraint functions, as is the
case with the ride comfort optimisation. The central finite differences allows
for better approximations to the gradients of the objective and constraint
functions, and leads to more promising results, without too high a penalty
being paid in terms of number of objective functions needed to obtain convergence.
Although for the ride comfort optimisation Dynamic-Q generally took
more function evaluations than SQP, it found the local minima by moving
within the whole design space. In most cases SQP jumped from boundary
to boundary, finding the local minimum from the boundaries. SQP might
therefore miss local minima that do not lie close to the design space boundaries. Thus it can be concluded that Dynamic-Q is a competitive and reliable
alternative to SQP for vehicle suspension design.
University of Pretoria etd – Thoresson, M J (2005)
6. DISCUSSION OF FUTURE WORK
Only a limited number of design variables have been considered here. It is
believed that Dynamic-Q will come into its own when more variables are
considered. The next step in the optimisation will be to change the front
and rear spring and damper characteristics independently. This will result
in fourteen design variables.
On the algorithmic side the study shows that, a better and desirable
convergence behaviour is obtained for Dynamic-Q if smaller move limits are
specified. It would be interesting to see how much of an improvement the
use of central finite differences would give when used in SQP, as opposed to
the forward finite differences used in this study.
A greater variety of road conditions need to be considered over varying vehicle speeds and loading conditions, before a decision can be made regarding
the final overall optimum design. The ultimate test will be the optimisation
of the vehicle’s performance under severe handling manœuvres on a rough
track. More efficient tyre models should ideally also be considered, in order
to achieve better correlation with measured track results. The incorporation
of the complex model describing the hydro-pneumatic suspensions characteristics as proposed by Theron [55], should be included in the next optimisation
phase.
The final optimised spring and damper characteristics should be investigated for robustness. This should be done in terms of the effect normal
manufacturing tolerances will have on the vehicle’s handling and ride comfort.
University of Pretoria etd – Thoresson, M J (2005)
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University of Pretoria etd – Thoresson, M J (2005)
APPENDIX
University of Pretoria etd – Thoresson, M J (2005)
A. EVALUATION OF THE HANDLING OBJECTIVE
FUNCTION
A.1 The Concern
From the beginning of this research there was mounting concern as to what
parameter must be used for the handling objective function on a smooth
surface. Must it be a combination of variables or just one single variable. If
so which ones should be considered? In the preliminary study by Els and
Uys [24], the roll angle was minimized in order to achieve optimum handling.
A.2 Tests Performed
Due to this uncertainty it was decided to do some basic driving tests. The
tests consisted of 3 vehicles and 4 drivers. The vehicles were a Ford Courier
LDV, a VW CitiGolf 1 Chico, a VW Golf 4 GTi. These vehicles were chosen due to their availability and their handling characteristics ranging from
almost nonexistent to excellent. The vehicles were instrumented to measure
lateral acceleration front and rear, roll, pitch and yaw velocity, roll and pitch
angle, vertical acceleration front and rear, longitudinal acceleration front and
rear, as well as vehicle speed and steering wheel angle. Table A.1 summarizes
the instrumentation fitted to the vehicles. The pack of drivers consisted of
a student, a lady, and two men. The tests were performed on two tracks at
Gerotek being the ride and handling track and the dynamic handling track.
The ride and handling track simulates typical tarred mountain pass driving.
The ride and handling track simulates typical high speed manœuvring. The
track specifications are summarized in Table A.2.
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A. Evaluation of The Handling Objective Function
90
Table A.1: Summarized vehicle measurements
Instrument
Position
Measurement
Accelerometer
Front Center
Lateral, Longitudinal, Vertical Accelerations
Accelerometer
Right Rear
Lateral, Longitudinal, Vertical Accelerations
Accelerometer
Left Rear
Lateral, Longitudinal, Vertical Accelerations
Angle sensor
Front Center
Roll, Pitch Angles
Gyro
Left Center
Roll, Yaw, Pitch Velocities
Displacement
Steering System
Steering Wheel Angle
Speed
Rear
Longitudinal Speed
A.3 Results From Study
The following trends relating to the handling can be observed in Figures
A.1-A.4:
• Non-linear relation between vehicle speed and lateral acceleration
• Linear relation between roll angle and yaw velocity
• Linear relation between yaw velocity and lateral acceleration
• Linear relation between roll angle and lateral acceleration
These trends were the same for all drivers on both tracks in all the vehicles. The absolute values differed from driver to driver and vehicle to vehicle.
The most important is the linear relation observed between roll angle, yaw
velocity and lateral acceleration. This means for the purpose of optimisation
of handling on a smooth surface either roll angle, yaw velocity or lateral acceleration may be used for the objective function. Roll angle was chosen as
it is a visible improvement. Figure A.5 illustrates the standard Land Rover
Defender while completing the double lane change, note the amount of body
roll.
University of Pretoria etd – Thoresson, M J (2005)
A. Evaluation of The Handling Objective Function
91
Table A.2: Test track specifications
Ride and Handling Track
Designed to evaluate ride and handling characteristics and driveline endurance
of wheeled vehicles. Simulating typical tarred mountain passes.
Distance: 4.2 km
Turns: 13 left 15 right
Max gradient: 15 percent
Dynamic Handling Track
Designed to evaluate the high speed handling characteristics of light vehicles.
Distance: 1.68 km
Track surface: Asphalt
Coefficient of Friction: 0.7 average
Consisted of trapezium curve, spiral curve, kink/hairpin combination,
high speed sweep
Figure A.1: Different drivers in Ford Courier on dynamic handling track
University of Pretoria etd – Thoresson, M J (2005)
A. Evaluation of The Handling Objective Function
92
Figure A.2: Different drivers in Ford Courier on ride and handling track
Figure A.3: Different drivers in VW Golf 4 GTi on dynamic handling track
University of Pretoria etd – Thoresson, M J (2005)
A. Evaluation of The Handling Objective Function
93
Figure A.4: Different drivers in VW Golf 4 GTi on ride and handling track
Figure A.5: Standard Land Rover Defender in double lane change manœuvre
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