Design of V ehicle System Dynamics
using Solution Spaces
v orgelegt v on
Dipl.-Ing.
Markus Eic hstetter
geb. in Münc hen
v on der F akultät V – V erk ehrs- und Masc hinensysteme
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des akademisc hen Grades
Doktor der Ingenieurwissensc haften
-Dr.-Ing.-
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr.-Ing. Dietmar Göhlic h
Gutac h ter: Prof. Dr.-Ing. Steffen Müller
Gutac h ter: Prof. Dr.-Ing. Marcus Jautze
Gutac h ter: Prof. Dr.-Ing. F abian Duddec k
T ag der wissensc haftlic hen A ussprac he: 19. Septem b er 2017
Berlin 2018

Dissertation, T ec hnisc he Univ ersität Berlin, 2018

Preface
This w ork w as created from 2011 till 2014 during m y time as PhD studen t in the division
of functional design and analysis of v ehicle dynamics in the departmen t of dev elopmen t
driving dynamics at BMW A G in Munic h. It w as sup ervised b y the institute for motor
v ehicles of the T ec hnical Univ ersit y of Berlin.
I w ould esp ecially lik e to thank m y do ctoral sup ervisor Prof. Dr.-Ing. Steffen Müller for
imp ortan t suggestions and pieces of advice whic h led to the success of this w ork.
F urthermore I w ould lik e to thank sincerely m y previous sup ervisor at the BMW A G
and rapp orteur Prof. Dr.-Ing. Marcus Jautze for initiating the w ork and enabling an
ideal w orking en vironmen t. In sev eral con v ersations and discussions he ga v e me significan t
suggestions and man y pieces of advice. F urther thanks for initiating and encouragemen t
go es to m y previous c hief engineer Dr.-Ing. Martin Sc hleic h and m y previous division c hief
engineer Dipl.-Ing. Reinhard Mühlbauer. I w ould lik e to thank m y in terim sup ervisor at
the BMW A G Dipl.-Ing. Oliv er P ogun tk e for supp orting and encouraging of the w ork.
Sp ecial thanks go es also to Dr.-Ing. P a v el K v asnic ka as m y c hief engineer for the p er-
sonal and tec hnical sup ervision. In sev eral con v ersations and discussions he ga v e me go o d
suggestions and tec hnical pieces of advice. I am grateful for an ideal w orking en vironmen t
and framew ork for the success of the w ork.
I w ould esp ecially lik e to thank m y sup ervisor at the BMW A G Dr. rer. nat. Markus
Zimmermann, who sup ervised the w ork from the in termediate part till the end. In man y
con v ersations and discussions he ga v e me critical reviews and v ery useful hin ts on tec hnical
and p ersonal lev el. His other w a y of thinking inspired and encouraged me. My thanks
go es also to Dr.-Ing. Dominik Mäder, Dr.-Ing. Johan Berote and Dipl.-Ph ys. Matthias
F uc hs for pro of reading and their v aluable con ten t-related commen ts. Thanks also to m y
do ctoral colleagues Dipl.-Ing. Florian Niedermeier, Dipl.-Ing. Ingo Sc harfen baum and
Dipl.-Ing. Martin Münster for v aluable discussions and pieces of advice. Without studen t
assistance this w ork w ould not b e successful, therefore also thanks to Dipl.-Ing. Minh-T ri
Nguy en, M.Sc. Mic ha W egener, Dipl.-Math. Kathrin Sc hlenk er, M.Sc. Changyi Ge and
M.Sc. Christian Redek er.
Munic h, Jan uary 2017 Markus Eic hstetter
I I I

IV

"The b est w a y to predict y our future is to create it."
Peter Drucker

Contents
List of Symb ols VI I I
Kurzfassung X
Abstract XI I
1 Intro duction 1
1.1 Motiv ation and Aim of the W ork . . . . . . . . . . . . . . . . . . . . . . . 1
1 . 2 S t a t e o f t h e A r t ................................. 2
1.2.1 Ev aluation of V ehicle Beha viour . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Conceptual La y out Strategies . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Robust and Reliable Design . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Pro duct F amily Design . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.5 Ev aluation of Metho ds and Need for A ction . . . . . . . . . . . . . 16
1 . 3 C o n c e p t o f R e s e a r c h .............................. 1 8
1 . 3 . 1 A i m s o f t h e W o r k ............................ 1 8
1 . 3 . 2 C o n c e p t o f M e t h o d s .......................... 1 9
1 . 4 S t r u c t u r e o f t h e W o r k ............................. 2 1
2 Deriving Quantitative Goals fo r Objective Quantities 23
2.1 V ehicle Kno wledge Managemen t System . . . . . . . . . . . . . . . . . . . 24
2.2 Generation of Ob jectiv e Kno wledge . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Design of Exp erimen ts Metho d . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Sensitivit y Analysis Metho ds . . . . . . . . . . . . . . . . . . . . . 28
2.3 Generation of Ob jectified Kno wledge . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Ob jectification Kno wledge . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Ob jectified Kno wledge . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.3 Comparison of Sub jectiv e and Ob jectified Kno wledge . . . . . . . . 33
2.4 Generation of P erception Thresholds . . . . . . . . . . . . . . . . . . . . . 34
3 Robust System Design using Solution Space 38
3.1 Principles of Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Example shap es of Solution Spaces (2-D) . . . . . . . . . . . . . . . . . . . 42
3.3 Robustness and Solution Space . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Robustness against Design P arameter Uncertain t y . . . . . . . . . . 45
3.3.2 Robustness against Uncon trollable P arameter Uncertain t y . . . . . 45
3 . 4 C o n c u r r e n t D e s i g n ............................... 4 7
3.5 Solution Space for High-dimensional Design . . . . . . . . . . . . . . . . . 48
3.5.1 Seeking Bo x-shap ed Solution Space . . . . . . . . . . . . . . . . . . 48
3.5.2 Review of the Underlying Algorithm . . . . . . . . . . . . . . . . . 50
VI

Con ten ts
4 Pro duct F amily Design using Solution Spaces 52
4.1 Problem Statemen t for Pro duct F amily Design . . . . . . . . . . . . . . . . 52
4.2 Com binatorics of Common P arameters . . . . . . . . . . . . . . . . . . . . 53
4.3 Sharing V alues of Design P arameters using Solution Spaces . . . . . . . . . 54
4.4 Deriving Pro duct F amilies from Bo x-shap ed Solution Spaces . . . . . . . . 55
5 Application 57
5 . 1 P r o b l e m D e s c r i p t i o n .............................. 5 7
5 . 1 . 1 D e s i g n C a t e g o r i e s ............................ 5 7
5 . 1 . 2 D e s i g n O b j e c t i v e s ............................ 5 8
5 . 1 . 3 D e s i g n P a r a m e t e r s ........................... 6 0
5.2 Generation of P erception Thresholds for Middle-class V ehicles . . . . . . . 62
5.2.1 Sub jectiv e Kno wledge for Middle-class V ehicles . . . . . . . . . . . 62
5.2.2 Generation of Ob jectiv e Kno wledge for Middle-class V ehicles . . . . 64
5.2.3 Comparison of Sub jectiv e and Ob jectified Kno wledge of Middle-class
V e h i c l e s ................................. 6 7
5.2.4 P erception Thresholds of Middle-class V ehicles . . . . . . . . . . . . 69
5.3 Robust Design Strategy of An ti-roll Bars . . . . . . . . . . . . . . . . . . . 70
5.3.1 Shap e of Solution Space of An ti-roll Bars . . . . . . . . . . . . . . . 71
5.3.2 Concurren t Engineering of a Solution Space of An ti-roll Bars . . . . 72
5.3.3 Classical Approac h: Robust Design Optimization . . . . . . . . . . 73
5.3.4 New Approac h: Robust Design with Solution Space . . . . . . . . . 74
5.4 Pro duct F amily Design for V ehicle System Dynamics Design . . . . . . . . 77
5.4.1 Classical Approac h: Optimizing V ehicles Separately . . . . . . . . 77
5.4.2 New Approac h: Deriving Common P arameter V alues from Solution
S p a c e s .................................. 7 8
5.5 High Dimensional Design Problems . . . . . . . . . . . . . . . . . . . . . . 79
5.5.1 Robust Design for High Dimensional Design Problems . . . . . . . . 80
5.5.2 Pro duct F amily Design for High Dimensional Design Problems . . . 81
6 Discussion 84
7 Conclusion and Outlo ok 88
7 . 1 C o n c l u s i o n .................................... 8 8
7 . 2 O u t l o o k ..................................... 9 0
A App endix 93
A.1 Histogram of Ob jectiv e Quan tities . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 P attern Matc hing of Sub jectiv e and Ob jectified Kno wledge . . . . . . . . . 94
A.3 Details of the Solution Space Shap e of An ti-roll Bars . . . . . . . . . . . . 94
A.4 A classical Approac h constructing Platform Arc hitectures . . . . . . . . . . 96
List of Figures 98
List of T ables 100
Bibliography 103
VI I

List of Symb ols
Indices
sym b ol unit description
< medium impact
# high impact
* optim um
0 op erating or start p oin t
b b ound stop
c critical v alue
f fron t axle
lb lo w er b ound
up upp er b ound
min minim um
max maxim um
norm normalized
r rear axle or reb ound stop
stat stationary
x in x direction
y in y direction
z in z direction
Greek letters
sym b ol unit description
α index of systems
β systems index or side-slip angle in ◦
δ h
◦ steering angle
γ system index of subset
ν total n um b er of systems
φ ◦ roll angle
˙
φ ◦ /s roll angle v elo cit y
ψ ◦ y a w angle
˙
ψ ◦ /s y a w angle v elo cit y
˙
ψ
δ h 1/s y a w v elo cit y amplification
σ standard deviation
Ω b o x in design space
Ψ subset of systems
Φ system set
VI I I

List of Sym b ols
Latin letters
sym b ol unit description
a y m/s 2 lateral acceleration
c a N/mm an ti-roll bar stiffness
c b N/mm bump stop
c r N/mm reb ound stop
c s N/mm susp ension spring stiffness
d Ns/m damp er c haracteristic in dep endence of deflection v elo cit y
ds design space
f function
g inequalit y constrain t
i index of design parameters
j index of ob jectiv e quan tities
k index of inequalit y constrain ts
l index for n um b er of subsets of Ψ
l t m trac k
l w m wheelbase
l x m cen tre of gra vit y in longitudinal direction
l z m cen tre of gra vit y in v ertical direction
m index of design categories or b o dy mass of v ehicle in kg
n n um b er of differen t parameter v alues
p uncon trollable parameter
q ob jectiv e quan tit y
s 0 ,r mm free length under reb ound
s 0 ,b mm free length under compression
u index of uncon trollable parameters
w w eigh ting
x design parameter
y general system output
z system p erformance
B Bell’s n um b er
D total n um b er of design parameters
E mean v alue
F ty re N t yre force
K total n um b er of inequalit y constraints
I in terv al of parameter
J total num b er of ob jectiv e quan tities
J xx k g /m 2 roll momen t of inertia of the v ehicle b o dy
J y y k g /m 2 pitc h momen t of inertia of the v ehicle b o dy
J z z k g /m 2 y a w momen t of inertia of the vehicle bo dy
M total n um b er of design categories
RM R % roll momen t ratio
RMS ro ot mean square v alue
S sensitivit y index
ˆ
S aggregated sensitivit y index
U total num b er of uncon trollable parameters
V high-dimensional volume
V ar v ariance
IX

Kurzfassung
Die A utomobilindustrie forsc h t an einem effizien ten Pro dukten tsteh ungsprozess, denn n ur
so kann eine Vielzahl an un tersc hiedlic hen und qualitativ ho c h w ertigen F ahrzeugen an-
geb oten und zugleic h deren En t wic klungsk osten reduziert w erden. Ein möglic her Ansatz
hierzu ist, viele Asp ekte der A uslegung in die virtuelle W elt zu v erlagern. Diese Arb eit
b efasst sic h mit drei Beiträgen zu einem systematisc hen, virtuellen Grundauslegungspro-
zess in der F ahrdynamik. Der erste Beitrag b esc hreibt eine neue Metho de, um sub jektiv e
in ob jektiv e Ziele zu üb ersetzen, der zw eite b efasst sich mit der robusten A uslegung k om-
plexer Systeme und im letzten Beitrag wird eine effizien te Metho de zur Gestaltung v on
Pro duktfamilien v orgestellt.
Die Grundv oraussetzung für einen zielführenden virtuellen A uslegungsprozess ist die
Üb ertragung eines sub jektiv en Zielsystems auf quan titativ e, ob jektiv e Bew ertungsgrößen.
Dieser Prozess ist k omplex, da die Sub jektivurteile stark v on F ahrzeugklasse, F ahrer und
Um w elt abhängen. In dieser Arb eit wird ein neuer systematisc her Ansatz zur Bestimm ung
v on Sc h w ellw erten erläutert, der es erlaubt Relativb ezieh ungen zwisc hen sub jektiv en Zie-
len und ob jektiv en Bew ertungsgrößen herzustellen. Die Metho de basiert auf einem Wis-
senssystem, das relativ e Sub jektivb eurteilungen zwisc hen K omp onen ten- und Systemei-
gensc haften b einhaltet. Diese Beurteilungen k önnen mit Hilfe der Sim ulation und einem
v orhandenen qualitativ en Ob jektivierungswissen v erifiziert w erden, indem eine Vielzahl
v on Exp erimen ten und eine globale Sensitivitätsanalyse durc hgeführt w erden. A us dem
Abgleic h k önnen dann die gesuc h ten W ahrnehm ungssc h w ellen iden tifiziert w erden. Diese
erlaub en in K om bination mit einer absoluten sub jektiv en und ob jektiv en Bew ertung eines
Referenzfahrzeugs, den Üb ertrag eines neuen sub jektiv en in ein quan titativ es Zielsystem.
Um flexib el im En t wic klungsprozess auf Anforderungsänderungen zu reagieren und ei-
ne hohe Pro duktqualität zu gew ährleisten, m üssen Unsic herheiten frühzeitig im En t wic k-
lungsprozess aufgenommen w erden. Unsic herheiten treten in der frühen En t wic klungsphase
aufgrund eines reduzierten Detailwissens o der aufgrund v on Anforderungsänderungen auf.
Eine Herausforderung stellt auc h die hohe Anzahl an k onkurrierenden Anforderungen b ei
der Gestaltung k omplexer nic h t-linearer Systeme dar. Die V erw endung v on Lösungsräu-
men wird als neue Metho de v orgesc hlagen, um eine große Anzahl an Anforderungen zu
b eherrsc hen und Robustheit auf eine einfac he Art und W eise in den Gestaltungsprozess
zu in tegrieren. Ein Lösungsraum b esteh t aus einem Set v on A uslegungspunkten, die alle
Anforderungen an das System v erhalten erfüllen. Ein zen traler A uslegungspunkt innerhalb
dieses Raums gilt als b esonders robust, da b ei kleinen P arameteränderungen immer no c h
alle Anforderungen erfüllt sind. F ür ho c hdimensionale Problemstellungen wird ein b o xför-
miger Lösungsraum v erw endet, der die K omplexität vieler Dimensionen reduziert, indem
die A uslegungsgrößen en tk opp elt w erden. Ein b o xförmiger Lösungsraum lässt sic h als en t-
k opp elte In terv alle v on A uslegungsgrößen darstellen und kann zwisc hen mehreren En t wic k-
lungsteams für w eiterführende En t wic klungsaufgab en ohne großen K o ordinationsaufw and
aufgeteilt w erden.
Die Gestaltung v on möglic hst vielen gemeinsam v erw endeten Bauteilen innerhalb einer
X

Kurzfassung
Pro duktfamilie un tersc heidet sic h v on der klassisc hen Optimierung v on Zielw erten mehre-
rer Systeme insofern, dass zur Erlangung der b estmöglic hen, individuellen Zielw erte jedes
einzelnen Systems nic h t alle A uslegungsgrößen einzeln optimiert w erden m üssen. F ür eine
optimale K omm unalität wird ein b estimm tes V erteilungssc hema der gemeinsam v erw ende-
ten Bauteile für alle F ahrzeuge b enötigt. Die Anzahl an Möglic hk eiten, Bauteile v ersc hie-
dener F ahrzeuge zu teilen, kann durc h die Bellsc he Zahl b esc hrieb en w erden und wird mit
ansteigender Menge v on Systemen sehr sc hnell groß. Dadurch wird die Anzahl an Opti-
mierungen mit k om binierten Systemen sehr groß. Diese Problemstellung wird erstmals mit
Lösungsräumen gelöst, die zugleic h eine neue Metho de zur Gestaltung v on Pro duktfamilien
darstellt. Anstatt k om binierte Systeme zu b ew erten, wird für jedes System ein Lösungs-
raum erstellt, der n ur A uslegungspunkte en thält, die zur systemeigenen Zielerfüllung füh-
ren. Indem Lösungsräume v on mehreren Systemen üb erlagert w erden, w erden Räume der
gemeinsamen Zielerreic h ung iden tifiziert. Zur V ereinfac h ung b ei ho c hdimensionalen Pro-
blemstellungen w erden erneut b o xförmige Lösungsräume v erw endet. Dadurc h k önnen für
alle A uslegungsparameter möglic he In terv alle gefunden w erden. Dies ist V oraussetzung für
einen neu en t wic k elten Algorithm us, um die Systemk om bination mit der kleinsten Anzahl
an geteilten A uslegungsparametern zu iden tifizieren. Diese Metho de gelingt auc h dann,
w enn nic h t b ekann t ist, w elc he Systeme w elc he A uslegungsparameter teilen sollen.
Alle drei Metho den w erden auf ein k omplexes A uslegungsproblem aus der F ahrdy-
namikauslegung angew andt. Eigenlenkv erhalten, querdynamisc he Stabilität, Kippsic her-
heit, stationäres und dynamisc hes W ankv erhalten sollen für eine neue Mittelklasse-
F ahrzeugfamilie ausgelegt w erden. F ür diese Eigensc haften liegt eine Sub jektiv- und Ob-
jektivb ew ertung der V orgänger-Limousine v or. Darauf basierend wird ein neues sub jektiv es
Zielsystem erstellt und mit W ahrnehm ungssc h w ellen in relativ e quan titativ e Ziele transfor-
miert. Die A uslegungsziele w erden mit Hilfe v on zehn A uslegungsgrößen realisiert: jew eils
die Stabilisatorensteifigk eit so wie die Steifigk eit und der Einsatzpunkt v on Zusatzfeder und
Zugansc hlagsfeder der V order- und Hin terac hse. In einem einführenden Beispiel wird der
robuste A uslegungsen t wurf mit Lösungsräumen zunäc hst n ur auf die Stabilisatorsteifigk eit
der V order- und Hin terac hse angew andt. Dieser En t wurf wird um unsic here Größen in der
frühen Phase, wie das F ahrzeuggewic h t o der die Dämpferc harakteristik, erw eitert und mit
einem pareto-optimalen robusten En t wurf v erglic hen. Es zeigt sic h, dass die A uslegung mit
Lösungsräumen die In tegration v on Robustheit v ereinfac h t, da k ein zusätzlic hes Robust-
heitsmaß optimiert w erden m uss. Die En t wurfsmetho dik wird ansc hließend auf alle zehn
A uslegungsgrößen angew andt, indem b o x-förmige Lösungsräume v erw endet w erden, w el-
c he die Handhabung und die Visualisierung der möglic hen In terv alle der A uslegungsgrößen
v ereinfac hen. Dieses A uslegungsproblem wird ansc hließend auf mehrere F ahrzeuge üb er-
tragen, um eine hohe Vielfalt an F ahrzeugen anzubieten. In einem einführenden Beispiel
wird zunäc hst eine Pro duktfamilie v on drei F ahrzeugen mit der Stabilisatorsteifigk eit der
V order- und Hin terac hse ausgelegt und mit einem klassisc hen Ansatz v erglic hen. Daraus
resultiert, dass die A uslegung v on Pro duktfamilien mit Lösungsräumen die Bestimm ung
gemeinsamer K omp onen ten v ereinfac h t, da die Systeme nic h t in K om bination optimiert
w erden m üssen. Die A uslegungsmetho de wird auf ein praxisrelev an tes A uslegungsb eispiel
angew andt: Zehn A uslegungsgrößen w erden für eine F ahrzeugfamilie v on 13 Mittelklasse-
fahrzeugen b ezüglic h maximaler K omm unalität ausgelegt, so dass für jedes F ahrzeug alle
individuellen Zielgrenzw erte erfüllt sind und die Anzahl der k omm unalen Bauteile mög-
lic hst gering ist.
XI

Abstract
The automotiv e industry is lo oking for an efficien t pro duct dev elopmen t pro cess to offer
a large div ersit y of high qualit y pro ducts while reducing dev elopmen t costs. An efficien t
pro duct dev elopmen t pro cess can b e created b y shifting man y asp ects of dev elopmen t
in to the virtual w orld. This w ork describ es three con tributions to a systematic virtual
dev elopmen t pro cess: first, a metho d to deriv e quan titativ e goals for ob jectiv e quan tities,
second, a general metho d to create robust designs of complex systems and third, a metho d
for pro duct family design.
A basic requiremen t for a successful virtual dev elopmen t pro cess is to correctly quan-
tify the design goals. Therefore, sub jectiv e ev aluations of customers or test driv ers ha v e
to b e quan tified b y ob jectiv e c haracteristic v alues and corresp onding p erception thresh-
olds. Ho w ev er, this pro cedure is complex as sub jectiv e ev aluations differ with v ehicle class,
driv er and en vironmen t. A new systematic metho d for the determination of p erception
thresholds of ob jectiv e quan tities is presen ted, whic h quan tifies a relativ e relation b et w een
the sub jectiv e ev aluation and ob jectiv e quan tities. This metho d is based on a kno wledge
managemen t system, whic h stores sub jectiv e impacts b et w een design v ariables and design
categories. The con tained information can b e v erified b y sim ulation and ob jectification
kno wledge using design of exp erimen t metho ds and global sensitivit y analyses. In com bi-
nation with an absolute sub jectiv e and ob jectiv e ev aluation, p erception thresholds enable
the deriv ation of ob jectiv e design goals for a new v ehicle.
An efficien t dev elopmen t of high qualit y pro ducts requires the in tegration of uncertain-
ties in the dev elopmen t pro cess. Uncertain ties ma y o ccur in early design phases due to
limited kno wledge or requiremen t c hanges and naturally in the late man ufacturing pro-
cess. F urthermore, the n um b er of comp eting ob jectiv es of non-linear systems can b ecome
o v erwhelming. Solution space design is prop osed as a new metho d to handle sev eral re-
quiremen ts on ob jectiv es and to in tegrate robustness in a simple manner. Solution space
design is differen t from design for optimal p erformance, whic h results in a single design. A
solution space is a set of designs, whic h satisfy all requiremen ts of the system b eha viour.
A cen tral design within this set ma y b e a candidate for a robust design as uncertain ties
of the design parameters are included. F or high dimensions of design parameter, a b o x-
shap ed solution space is prop osed, whic h reduces the complexit y to handle high dimensions
b y decoupling design parameters while satisfying all requiremen ts. Decoupled in terv als of
design parameters ma y b e allo cated to sev eral exp ert design teams without high efforts of
co ordination.
Design for optimal commonalit y in pro duct families is differen t from design for optimal
p erformance. While optimal p erformance ma y b e ac hiev ed b y the c hoice of appropriate
design parameter v alues for all system comp onen ts, optimal commonalit y requires a par-
ticular sc heme of sharing comp onen ts among systems. The n um b er of p ossibilities to share
comp onen ts can b e quan tified b y Bell’s n umber and b ecomes large quic kly , th us making
optimization extremely exp ensiv e. A new pro duct family design metho d is presen ted to
optimise pro duct family designs with resp ect to commonalit y using solution spaces. In-
XI I

Abstract
stead of a p erformance optimisation of a com bination of systems, a solution space for eac h
system is iden tified, whic h con tains only go o d designs satisfying all relev an t design goals.
By o v erla ying solution spaces of sev eral systems, regions of common parameter v alues can
b e iden tified. As a simplification for high-dimensional problems, again b o x-shap ed solution
spaces are used. Th us, p ermissible ranges for all design parameters can b e iden tified that
are indep enden t of eac h other. This enables a particular algorithm to iden tify the con-
figuration with the smallest n umber of shared parameters in b o x-shap ed solution spaces.
This metho dology w orks without a-priori information on whic h comp onen ts should b e of
common design.
All three metho ds are applied to c hassis design of a new middle class v ehicle pro duct
family , whic h needs to b e designed for asp ects of v ehicle b eha viour suc h as self-steering
b eha viour, lateral stabilit y , roll-o v er, stationary roll b eha viour and dynamic roll b eha viour.
In the new p erception thresholds metho d relativ e ob jectiv e design goals are determined,
whic h are generated based on a kno wledge managemen t system in whic h the sub jectiv e
impact for middle class v ehicles is stored. These thresholds are used to quan tify the
design goals of the new v ehicle, resp ectiv e to its predecessor. In the no v el robust design
metho d these design goals are realized b y 10 design parameters: the stiffness of an ti-roll
bars and the stiffness and the start of compression of the bump stops and the reb ound
stops for the fron t and rear axle. F or simplification, the robust design approac h using
solution spaces is first applied only to the fron t and rear stiffness of an ti-roll bars. This
approac h is also extended to uncertain ties of uncon trollable parameters suc h as the b o dy
mass or the damp er c haracteristic and compared to a classical P areto-optimal robust design
optimization. Robust design using solution spaces simplifies the in tegration of robustness
enormously as no additional robustness term has to b e optimized. The robust design
approac h using solution spaces is extended to 10 design parameters b y using b o x-shap ed
solution spaces, whic h enable a simplified handling and visualization of p ermissible in terv als
for eac h design parameter. In the inno v ativ e pro duct family design metho d this design
problem is extended for sev eral v ehicles in a simple example. The pro duct family approac h
is applied to a v ehicle dynamics problem from industrial practice: a set of 13 v ehicles with
10 design parameters w as optimised with resp ect to commonalit y suc h that all design goals
are satisfied and the n um b er of differen t comp onen t designs is minimized. Pro duct family
design using solution spaces simplifies the sharing of comp onen ts within m ultiple systems
enormously as the systems do not ha v e to b e optimized in com bination.
XI I I

Abstract
XIV

1 Intro duction
1.1 Motivation and Aim of the W o rk
The automotiv e industry striv es to create exciting cars with outstanding driving p erfor-
mance in an effectiv e and short dev elopmen t pro cess. This ensures adv an tages in the
curren t c hallenging mark et en vironmen t [64]. The driving forces of the automotiv e indus-
tries for researc h on robust and pro duct family design are div erse. The tec hnical leadership
is desired in order to gain fans rather than customers. F or customer satisfaction and ef-
ficiency of the compan y , the qualit y of the pro ducts and pro cesses ha v e to b e increased
while offering a high div ersit y . F or a highly profitable compan y , p ermanen t costs ha v e to
b e reduced and mark et segmen ts ha v e to b e conquered.
The classical dev elopmen t pro cess has to pro vide high pro duct p erformance for an immense
v ariet y of pro ducts. The complexit y of a single o v er-determined system with non-linear
relations and in teractions is generally difficult to design. F or a high pro duct v ariet y , cross-
link ed comp onen ts or subsystems within sev eral pro ducts ha v e to b e designed. The amoun t
of cross-link ed comp onen ts or subsystems has to b e minimal for reasons of efficiency and
complexit y .
F or an efficien t dev elopmen t pro cess, uncertain ties ha v e to b e considered and included in
the dev elopmen t pro cess. In the early design phase only lo w detail kno wledge of a system
or a pro duct is a v ailable. Pro duct requiremen ts c hange during the design pro cess due to
sev eral reasons. These c hanges ha v e to b e k ept in mind and handled without iterations
of the dev elopmen t pro cess from the b eginning. Designed comp onen ts cannot b e created
in hardw are as desired, b ecause the design cannot b e realized exactly . Therefore it is also
imp ortan t to include uncertain ties from the man ufacturing pro cess.
A p ossible approac h for a more efficien t dev elopmen t pro cess is to shift the dev elopmen t
pro cess to w ards the early phase. In the virtual w orld the men tioned c hallenges can b e
handled and uncertain ties can b e in tegrated in the design pro cess.
The general aim of this w ork is to presen t new approac hes to reduce complexit y and enable
a b etter system design with less effort and fo cused on design goals. The dev elopmen t pro-
cess has to b e from rough to detail, as detailed system kno wledge is rare in the early phase.
Therefore, comp onen t or subsystem prop erties are determined b y requiremen ts on the sys-
tem b eha viour. Comp onen ts of predecessor systems should not just b e main tained and
used for a new system, b ecause then the new system b eha viour is just a result, whic h ma y
not satisfy the new requiremen ts. V ehicle design is difficult due to high complexit y caused
b y man y requiremen ts and design parameters. A cascaded dev elopmen t of requiremen ts
on sev eral lev els ma y help to reduce the complexit y . The la y out of a single optimal design
could lead to conflicts of goals, if man y requiremen ts ha v e to b e satisfied. The la y out of
a set of designs whic h satisfies all relev an t design criteria on the other hand could offer
more flexibilit y . Uncertain ties ha v e to b e included from the b eginning; therefore robust
design has to b e an in tegral part of the design strategy . Due to the required p erformance
1

1 In tro duction
of the v ariet y of pro ducts, pro duct family design has to b e compatible to the approac hes
men tioned. This corresp onds to four essen tial topics:
1. How c an the subje ctive p er c eption b e tr ansferr e d to quantitative obje ctive values to
enable a development pr o c ess in the virtual world?
2. Which c onc eptual layout str ate gies enable the hand ling of c omplex systems with many
r e quir ements?
3. Which r obust or r eliable design str ate gies include unc ertainties in a simple manner
and enable flexibility to r e quir ement changes?
4. Which pr o duct family design is c omp atible with other design str ate gies and is r e alize d
in a simple manner?
1.2 State of the Art
First, the ev aluation of v ehicle b eha viour is review ed for the transfer from sub jectiv e to
ob jectiv e design goals. Second, v arious conceptual la y out strategies, esp ecially in the con-
text of v ehicle dynamics are review ed. This includes trends of design pro cesses suc h as set-
based design or target cascading. Third, robust design based metho ds and reliabilit y-based
metho ds are review ed including trends as solution space design or parameter space design.
F ourth, the need for pro duct family design is essen tial, therefore common approac hes for
functional pro duct family design are review ed. Finally , the metho ds are ev aluated and the
further need for action is explained.
1.2.1 Evaluation of V ehicle Behaviour
Subjective V ehicle Evaluation
The sub jectiv e ev aluation of the v ehicle b eha viour b y driving exp erts is essen tial for the
design pro cess of v ehicle dynamics [135]. Esp ecially , the in teraction b et w een driv er, v ehi-
cle and en vironmen t influences the sub jectiv e ev aluation.The p erception of a customer is
ev aluated sub jectiv ely b y a BI-scale, from 1 to 10. BI is an abbreviation for the German
w ord "Bew ertungsindex". E.g., BI 8 means ready for pro duction, see, e.g., Heißing and
Brandl [41]. A ccording to Zomotor [135] the ev aluation b y single test driv ers can differ sig-
nifican tly . Therefore a group of test driv ers t ypically ev aluate the v ehicle b eha viour, whic h
results in a distribution of ev aluations. The test driv ers ha v e to fulfil high requiremen ts
at p erception memory and differen tiation capabilities. Sev eral asp ects can lead to sub-
jectiv e mis-ev aluation: preferences of exp erts, resp onsibilit y for the dev elopmen t pro jects,
one-sidedness to the pro ducts or tec hnical kno wledge of the v ehicle. Due to time pressure
and the v ariet y of v ehicles, test driv ers can only c hec k a fraction of v ehicle prop erties and
v ehicle v arian ts.
Subjective and Objective V ehicle Evaluation
A lot of ob jectification w orks exist already , whic h try to correlate sub jectiv e and ob jectiv e
ev aluations. Represen tativ e works can b e found, e.g., from Riedel and Arbinger [94],[95]
2

1.2 State of the Art
and Zomotor et al. [136], [137]. Nev ertheless, the quan titativ e relation b et w een sub jectiv e
and ob jectiv e ev aluation is not researc hed enough.
Often, ob jectiv e results can only b e in terpreted with exp erience, but almost no hard-
w are exists in the early design phase. Virtual dev elopmen t is only p ossible, if ob jectified
quan tities and sim ulation-based to ols are a v ailable. Therefore, quan titativ e ob jectification
kno wledge is needed for the virtual dev elopmen t of v ehicles, e.g., for comparing v ehicle
b eha viours or testing statutory regulations [135]. T ypically , the relation b et w een sub jec-
tiv e and ob jectiv e ev aluation is done b y correlation studies for eac h category of v ehicle
b eha viour or ph ysical effect. The goal is to find ob jectiv e quan tities and criteria, whic h
correlate with the sub jectiv e ev aluation of the driv ers.
Quantitative Relations b et w een Subjective and Objective Evaluations
Qualitativ e relations b et w een sub jectiv e and ob jectiv e v ehicle b eha viour categories are
often a v ailable, but quan titativ e relations are either complex or rare, see [41], [94], [95] or
[135]. Qualitativ e relations are often the result of correlation analysis. A quan tification
in form of an analytical relation b et w een correlated ob jectiv e quantities and sub jectiv e
ev aluation can b e done, e.g., b y m ultiple linear regression analysis. The p erception of the
test driv er is in terpreted as w eigh ted com bination of ob jectiv e quan tities. The w eigh tings
of the analytical relation are optimised, e.g., b y a mean square function for sev eral v ehicle
v arian ts and test driv ers. A ccording to Kriegel et al. [62], the generation of target v alues
of ob jectiv e quan tities from sub jectiv e ev aluations b y exp erts has uncertain ties due to the
confidence in terv al b ecause of the influence of driv er, en vironmen t and v ehicle category .
E.g., Dec k er [20] fo cused on the ob jectification of lateral v ehicle dynamics with the aim to
transfer the dev elopmen t of v ehicle dynamics from a hardw are-based in to a virtual pro cess.
Mostly , general trends are concluded, e.g., the more an ob jectiv e quan tit y rises, the lo w er
the sub jectiv e ev aluation b ecomes. The searc h for optimal ob jectiv e target v alues sta ys
unansw ered, e.g., at whic h phase angle of steering angle to lateral acceleration do es the
v ehicle b eha viour in a lane c hange mano euvre rise from BI 7 to 8. Recen t researc h results
sho w that optimal ob jectiv e design goals dep end on the v ehicle class, see, e.g., Harrer [39]
and the target group, see, e.g., Barthenheier [5]. Therefore, general ob jectiv e goals are
difficult to iden tify and ha v e to b e adopted.
P erception Thresholds
The effort to dev elop new systems or v ehicles is reasonable, if the sub jectiv e p erception of
the customers is impro v ed or tec hnical leadership is desired. Therefore, threshold v alues
of p erception are imp ortan t for the virtual dev elopmen t. Whether a dev elopmen t effort is
justified or not, has to b e decided based on quan titativ e relations of sub jectiv e ev aluation
and ob jectiv e quan tities, see, e.g., Harrer [39](BMW Group). Dettki [21] in v estigated the
ob jectification of straigh t line driving and giv es an extensiv e o v erview of thresholds v alues
based on the exp erience for go o d straigh t line driving b eha viour. Cornering b eha viour
is only describ ed briefly . F urther, three in terv als of impact: lo w, medium and high are
presen ted regarding the influence of test driv ers, roads and w eather. Gutjahr [37] w ork ed
for the BMW Group on the translation of sub jectiv e ev aluations in to qualitativ e and quan-
titativ e ob jectiv es for the purp ose of v ehicle design. The fo cus la y on the lateral dynamics
for the design of t yre prop erties. A short review of sub jectiv e p erception threshold v alues
3

1 In tro duction
for lateral dynamic: lateral acceleration, rotational acceleration and rotational v elo cit y
is giv en. Botev [9] co op erated with Daimler A G and giv es a short review of sub jectiv e
p erception threshold v alues for lateral dynamics. Sub jectiv e p erception threshold v alues
for v ertical and lateral dynamics can b e found in VDI 2057 [121].
1.2.2 Conceptual La y out Strategies
Classical Conceptual La y out Strategies in V ehicle Dynamics Design
Systematic approac hes are necessary for the conceptual design of v ehicle dynamics, b e-
cause of the high complexit y , consisting of o v er-determined systems and non-linear in- and
output relations. W edlin et al. [125] co op erated with V olv o Car Corp. and describ ed the
c hallenges that the c hassis designer is confron ted with. Requiremen ts on handling prop er-
ties of the v ehicle and other functional requiremen ts of further fields ha v e to b e handled. It
is imp ortan t to consider all in v olv ed prop erties sim ultaneously , when designing the c hassis
systems. As sev eral requiremen ts on the v ehicle b eha viour ma y b e in conflict, it is an
imp ortan t task to iden tify factors whic h act in the same or opp osing directions. A c hange
of design parameters ma y comp ensate or in tensify an effect. Simple mo dels of v ehicle b e-
ha viour are prop osed to get a general o v erview. A ccording to Wimmer [127] w orking for
Daimler A G, subsystems are dev elop ed as m uc h as p ossible indep enden tly and then com-
bined in the v ehicle, whic h is then ev aluated b y sim ulation and test driv ers. This approac h
includes iterations as the v ehicle b eha viour will differ from exp ectations. Virtual proto-
t yping is used for the design of v ehicle dynamics prop erties b y using classical n umerical
optimisation metho ds. Botev [9] presen ts a sim ulation-based approac h for v ehicle dynam-
ics prop erties in co op eration with Daimler A G, to o. A design la y out order is prop osed for
the design ob jectiv es of lateral and v ertical dynamics. In a subsequen t, rep etitiv e pro cess
handling and ride prop erties of the v ehicle are optimized to fulfil all goals. A ccording to
Kriegel et al. [62] co op erating with A udi A G, the ev aluation of the handling b eha viour is
classically done sub jectiv ely b y exp erts and supp orted b y ob jectiv e c haracteristic v alues
of measuremen t or sim ulation. High uncertain ties remain, if sub jectiv e targets are based
only on ob jectiv e kno wledge. F or a giv en catalogue of mano euvres, c haracteristic v alues
are ev aluated and w eigh ted to relate them to driving dynamics prop erties. The sim ulated
results are th us similar to the targets from the sub jectiv e ev aluation and a quasi-sub jectiv e
ev aluation metho d for driving dynamics prop erties is created. A ccording to K v asnic ka and
Dic k [64] (BMW Group), main systems suc h as the elasto-kinematics of the susp ension,
steering system or v ertical comp onen ts are designed b y virtual metho ds in the early dev el-
opmen t phase and then confirmed in the late dev elopmen t phase b y virtual test-rigs and
finally realized protot yp es. Requiremen ts on a pro duct cannot b e implemen ted directly in
the dev elopmen t pro cess as trade-offs ha v e to b e made whic h are partially solv ed b y sev-
eral exp ert teams. These exp ert teams use problem sp ecific metho ds, suc h as sim ulation,
hardw are and test rigs. F or the la y out of driving dynamics, a set of goals is defined in
relation to predecessor and comp etitor v ehicles. A robust and time effectiv e dev elopmen t
pro cess is desired. Mäder [70] and Röski [96]prop osed in co op eration with BMW Group
a m ulti-step design pro cess to decrease complexit y . In Huemer et al. [46] the impact of
sub jectiv e and ob jectified kno wledge is analysed b y a lo cal sensitivit y index. In p ortfolios
design parameters are clustered according to lo w in teraction b et w een lateral and v ertical
dynamics. Based on this analysis, a metho d for the conceptual la yout of lateral and v ertical
4

1.2 State of the Art
dynamics is describ ed. In a first step, lateral and v ertical dynamics is designed with simple
mo dels. In the next step more detailed mo dels are used. Designed ob jectiv e quan tities are
v erified and further comp onen t prop erties are designed with simple optimization metho ds.
Wimmer [126] dealt with virtual metho ds for the conceptual la y out and the realization
of steering systems in co op eration with the BMW Group. F or conceptual la y out, simple
principle mo dels of steering functions (of electromec hanical steering system) and for the
ev aluation of steering b eha viours w ere dev elop ed. In the in tegration phase complex mo dels
are presen t. Global sensitivit y analysis w as used for the analysis of relev an t application
and steering parameters.
In conclusion, a m ulti-step design pro cess decrease the complexit y , but it is imp ortan t
to consider all in v olv ed prop erties sim ultaneously , when designing the c hassis systems,
otherwise iterations are necessary . Complex design tasks are solv ed b y sev eral design
teams using their exp ertise and sp ecific design metho ds. Key factors whic h comp ensate or
in tensify effects on the systems b eha viour, can b e iden tified with simple mo dels.
Optimization Strategies in V ehicle Dynamics Design
F or the fundamen tals of optimization metho ds see P apageorgiou [86], P ark [87] or Kallrath
[52].
Classical Optimization Strategies. The researc h of Du [24] fo cused on the la y out of
driving dynamics prop erties and is gran ted b y V olksw agen A G. A m ulti-criteria optimisa-
tion approac h is used based on an extended m ulti-b o dy-system, whic h includes in teractions
of the differen t subsystems. The m ulti-criteria optimisation problem is transformed in a
single or a series of single optimisation problems. The optimisation is gradien t based,
where eac h gradien t is generated b y n umerical or sym b olic differen tiation of the ob jectiv e
function and the inequalit y constrain ts. Sc huller et al. [100] w ork ed for the BMW Group
and prop osed a genetic optimisation metho d for the optimisation of the lateral dynamics
of a detailed t w o-trac k mo del. Ob jectiv e quan tities are determined from differen t driving
mano euvres. A single p erformance function is built b y w eigh ted normalized deviations of
ob jectiv e v alues related to their target v alues. The more design ob jectiv es are used, the
more opaque is the resulting optim um of the p erformance function. Nev ertheless signifi-
can t impro v emen ts ha v e b een ac hiev ed in the design of lateral dynamics.
P areto-optimal Solution Sets. F or engineering design problems, it is common to han-
dle m ultiple comp eting ob jectiv es. Ho w ev er, only one feasible solution can b e realized and
will b e selected. F or m ultiple-ob jectiv e optimization problems the determination of a sin-
gle optimal design configuration requires m ultiple considerations, e.g., to maximize system
p erformance while minimizing costs. A P areto-optimal solution set is a set of solutions
that are all non-dominated with resp ect to eac h other. While mo ving from one P areto-
optimal p oin t to another, there is alw a ys a certain amoun t of loss in at least one ob jectiv e
in order to ac hiev e a certain amoun t of gain in another ob jectiv e. P areto-optimal solution
sets are often preferred to single solutions b ecause the final solution can b e determined
b y a decision mak er, who decides based on engineering judgemen t the trade-off b et w een
the ob jectiv es. Gobbi et al. [35] presen ted a pro cedure for the design of a c hassis system
to ac hiev e the desired dynamics b eha viour of the car in m ultiple driving situations. The
v ehicle mo del w as pro vided b y FIA T. P areto-optimal solutions for m ultiple p erformance
indexes (driving situations) ha v e b een computed b y using genetic algorithms. The metho d
enables the designer to distinguish b et w een P areto-optimal and non P areto-optimal solu-
5

1 In tro duction
tions, ev en when dealing with high-dimensional problems. A review of the optimisation
metho ds for road v ehicle design is giv en b y Gobbi et al. [34]. Exemplary applications to
the design of v ehicle system dynamics, activ e and passiv e safet y , and v ehicle system design
are rep orted. Sp ecial emphasis is addressed to optimization of complex systems and opti-
mal design of complex systems under uncertain t y . The consideration of uncertain t y in the
problem form ulation is a quite recen t dev elopmen t in the a v ailable optimisation to ols. Ro-
bustness along with optimal solutions is desired [34]. Sagi and Lulic [97] dealt with the use
of m ulti-ob jectiv e optimisation tec hniques in the early stage of v ehicle dev elopmen t. The
researc h is based on the use of adv anced ev olutionary algorithms in the conceptual phase
of v ehicle dev elopmen t for the calculation of P areto-optimal solutions. The optimisation of
susp ension system parameters under consideration of stabilit y , handling and ride comfort
pro vide information ab out the influence of these parameters on v ehicle dynamics. As the
n um b er of comp eting ob jectiv es can b ecome o v erwhelming in real-w orld applications, the
resulting n um b er of P areto-optimal solutions is difficult to handle. T ab oada et al. [114]
presen t metho ds to reduce the size of the P areto-optimal sets for m ultiple-ob jectiv e sys-
tem (reliabilit y) design problems. A single solution can b e determined using utilit y theory ,
w eigh ted sum or a pseudo-ranking sc heme that helps the decision mak er to select solutions
that reflect his ob jectiv e function priorities. Data mining clustering tec hniques can b e used
to group the data to find clusters of similar solutions, whic h are lik ely to b e more relev an t
to the decision mak er.
In conclusion, classically m ulti-ob jectiv e design tasks are transformed in to optimisation
problems with a single or a series of single optimisation problems with a p erformance func-
tion built b y w eigh tings, but the more design ob jectiv es are com bined the more opaque is
the resulting optim um. P areto-optimal solution sets are often preferred, e.g. generated b y
genetic algorithms, b ecause the final solution can b e determined b y a decision mak er, who
decides based on engineering judgemen t the trade-off b et w een the ob jectiv es. Nev ertheless,
for man y comp eting ob jectiv es the resulting n um b er of P areto-optimal solutions ma y b e
difficult to handle and needs additional metho ds.
Set-based Design
W ard et al. [122, 124] presen ted a metho d, whic h generates an optimal selection of a
comp onen t configuration using predefined sets of comp onen ts. This approach is called
Set-b ase d Design . System requiremen ts are form ulated as ranges of system outputs and
are used to get ranges of designs parameters, whic h satisfy these system requiremen ts. It
is not necessary to ev aluate eac h com bination of the predefined set. By using in terv als in
an iterativ e decision pro cess one final, optimal design can b e found more efficien tly .
The concept of the Set-Based Concurren t Engineering (SBCE) design used at T o y ota
is describ ed in W ard et al. [123] and Sob ek et al. [108, 109]. Decision alternativ es are
defined as sets of design alternativ es. With SBCE, T o y ota promotes m ultiple feasible de-
signs at the same time. The set-based approac h is efficien t, as dominated alternativ es are
eliminated and designers can fo cus on the relev an t, dominan t designs. A design is called
non-dominated, if none of the ob jectiv e functions can b e impro v ed in v alue without de-
grading some of the other ob jectiv e v alues. T o y ota relies esp ecially on exp ertise of their
senior engineers for efficien t elimination. A ccording to T o y ota, the SBCE approac h has
pro v en successfully , b ecause qualit y ratings increase, while the cost reduction is sufficien t
in spite of a high n um b er of deriv ativ es. P arunak et al. [89] and Sob ek [107] fo cused on
6

1.2 State of the Art
determining wh y the SBCE approac h is effectiv e. Rekuc et al. [93] extended the set-based
approac h with formal guidelines for alternativ e elimination based on imprecise informa-
tion. By taking in to accoun t uncertain parameters b etw een differen t design p ossibilities,
eliminations in the design space are made b y a maximalit y criterion. Instead of a Mon te
Carlo Sampling for the distribution of uncertain parameters an optimiser is used, whic h
determines the b est and w orst solution. Regions whic h are dominated b y the w orst so-
lution are eliminated. This is done sequen tially for eac h parameter un til a smaller set is
defined.
e.g. minimal
syste m
performance A
step 0

[


[

e.g. maximal
syste m
performance B
step B1
step A1
step B2
step A2
Figure 1.1: Set-based design approac h with
t w o design parameters x 1 and x 2 . Designer A
has to minimize the system p erformance A,
while designer B has to maximize the system
p erformance B, similar to P anc hal et al. [85].
In [85], Panc hal et al. extended the ap-
proac h to con tin uous design parameters.
An in terv al-based metho d for facilitating
in teractions of design v ariables is presen ted.
Figure 1.1 sho ws a design scenario. De-
signer A is resp onsible for design parameter
x 1 and has to minimize the system p erfor-
mance A. Designer B is resp onsible for de-
sign parameter x 2 and has to maximize the
system p erformance B. The c haracteristics
of the extremal system p erformances A and
B dep end on x 1 and x 2 . Within the design
parameter in terv als in step 0, the system
p erformance A and B is p ossible. Step B1
is the first step and reduces the range of
the con trolled parameter x 1 of designer A.
Then designer A reduces x 2 and so on. The
metho d is used for non-co op erativ e game
theory for conflict resolution in decen tral-
ized, m ultifunctional design scenarios with
shared con trol design v ariables. A b o x con-
sistency principle is used, whic h is dev elop ed in the area of in terv al arithmetic. The metho d
is applied to a linear and a non-linear optimisation problems of t w o target functions, whic h
is coupled b y t w o design v ariables.
In conclusion, Set-based design uses in terv als in an iterativ e decision pro cess to generate
one final, optimal design more efficien tly . The set-based approac h is efficien t, if dominated
alternativ es are eliminated, otherwise designers will create unnecessary costs b y needless
v arian ts.
T a rget Cascading
Kim et al. [54] describ e T ar get Casc ading as a systematic approac h for design la y out
to transform targets of the top system lev el to prop erties of comp onen ts on the b ottom
lev el. This approac h is esp ecially useful for pro ducts or systems whic h can b e separated in
hierarc hical structures and comp onen ts from differen t tec hnical domains, see Figure 1.2.
A ccording to [54] a dev elopmen t pro cess is particularly efficien t, if differen t tasks can b e
handled in parallel b y sev eral exp ert design teams. Hence a go o d co ordination b et w een
the differen t teams is necessary to consider all pro duct sp ecifications and prev en t conflict
of goals in the early phase, but often this is v ery time consuming. If co op eration can b e
7

1 In tro duction
reduced to a minim um, the exp ert design teams can concen trate on their area of activit y .
This separation can also lead to design iterations. A ccording to Kim et al. [55], the
k ey issue is the execution of design tasks separately while satisfying system consistency .
A ccording to T osserams et al. [118] system consistency is often realized b y adding a
quadratic p enalt y function to the optimization problem. The pro cess of T ar get Casc ading
is t ypically organized in four steps [55]:
1. Dev eloping appropriate analysis mo dels, whic h sim ulate the system b eha viour in an
adequate state with lo w time effort.
2. Decomp osing the system in a hierarc hical structure according to ph ysical ob jects,
kno wledge asp ects or information sequen tially . Iden tifying the system resp onses,
whic h act as link b et w een systems of differen t lev els.
3. Defining an optimization problem for eac h subsystem. F or a consisten t solution of the
whole system, a co ordination strategy for the linking variables , design v ariables whic h
ha v e influence in m ultiple subsystems, is needed. The main optimization problem
will b e solv ed first. Then the v alues of the linking variables are used as target v alues
for the sub-lev els.
4. Solving the sub-optimization problems of the subsystems. The v ariables of the sub-
system are then targets for the sub-subsystem. Minimize the difference b et w een the
targets of the top lev el and the system resp onse of the sub-lev el.
…
Master problem
linking variables
Subproblem
local v ariab le s
Subproblem
local variables
linking variables
feedback as
function of linkin g
variables

Figure 1.2: Principle of target cascading:
master problem is separated in sub-problems
with link ed v ariables, similar to Kim et al.
[54].
This metho d is applied to v ehicle system
design, e.g., in [54], [55] or [82]. In [82] an-
alytical target cascading is link ed to engi-
neering information system for sim ulation-
based optimal v ehicle design, supp orted b y
V olv o Car Corp oration. A v ehicle mo del is
separated in m ultiple system mo dels. The
targets for v ehicle b eha viour are transferred
to the system mo dels. The sim ulation of
the single systems with AD AMS R
 is done
b y parallel CPUs.
Uncertain t y and T arget Cascading.
K okk olaras et al. [61] extended the de-
terministic form ulation of analytical tar-
get cascading (A TC) to probabilistic design
to include uncertain parameters. The ad-
v anced mean v alue metho d is used to gen-
erate the required probabilit y distributions
of non-linear resp onses. The propagation of
uncertain t y is used throughout the m ulti-lev el hierarc h y system design. This approac h is
applied to a bi-lev el engine design problem in co op eration with the U.S. Arm y and General
Motors. Liu et al. [67] augmen ted the probabilistic form ulation b y robust design principles,
supp orted from F ord Univ ersity Researc h Program. Design v ariables, system resp onses and
also their targets are describ ed b y the first t w o momen ts of random quantities: the mean
8

1.2 State of the Art
v alue and the standard deviation. A ccording to the principle of robust design, the mean
v alue of the system p erformance has to reac h the target, while the standard deviation is
minimized.
Pro duct F amily Design and T arget Cascading. The researc h of F ellini et al. [28]
and K okk olaras et al. [60] are gran ted b y F ord Motor Compan y and prop osed pro duct
family design using target cascading. F or a giv en platform, hierarc hical pro duct targets
are cascaded do wn to systems, subsystems, and comp onen ts for the whole pro duct family .
Design sp ecifications are determined for all elemen ts, including the pro duct platform. The
tec hnique is successfully applied to a simple automotiv e example of t w o v ehicles that share
b o dy comp onen ts.
In conclusion, target cascading is a systematic approac h for design la y out to transform
targets of the top system lev el to prop erties of comp onen ts on the b ottom level. The
dev elopmen t pro cess is particularly efficien t, if differen t tasks can b e handled in parallel b y
sev eral exp ert design teams and system consistency can b e guaran teed. T arget Cascading
w as successful link ed to uncertain parameters and pro duct family design.
1.2.3 Robust and Reliable Design
When using optimization for design purp oses, most engineers assume that the design v ari-
ables in the problem statemen t are deterministic, but deterministic design optimization
do es not accoun t for the uncertain ties that exist in mo delling, sim ulation, man ufacturing
pro cesses, design v ariables or parameters [3]. A ccording to Zang et al. [129] in the last
t w en t y y ears, v arious non-deterministic metho ds ha v e b een dev elop ed to deal with design
uncertain ties, whic h can b e classified in to t w o groups: robust design based metho ds and
reliabilit y-based metho ds. Robust design impro v es the qualit y of a pro duct b y minimizing
the effect of uncon trollable v ariation. The pro duct shall b e insensitiv e to v arious causes
of v ariation. Robust design concen trates on the probabilit y distribution near the mean
v alues. Reliabilit y metho ds determine the probabilit y of failure of a system. The probabil-
it y distribution of a system resp onse is estimated b y the kno wn probabilit y distributions
of random parameters. The v ariation of a system resp onse is not minimized in reliabilit y
approac hes. The reliabilit y of constrain ts is similar to the robustness of the constrain ts in
robust design.
Robust Design
Definition and Clustering. P ark et al. [88] describ ed robust design as design metho d
whic h has b een dev elop ed to impro v e pro duct qualit y and reliability in industrial engineer-
ing. Unexp ected deviations from a function, that a designer initially in tended, are caused
b y v ariations in v arious engineering pro cesses. T aguc hi et al. [115] defines robustness as a
state where the tec hnology , pro duct or pro cess p erformance is minimally sensitiv e to fac-
tors causing v ariabilit y . Many researc hers ha v e defined robust design and all definitions of
robust design ha v e in common, that the designed pro duct should b e insensitiv e to external
noises or tolerances. Noise factors are also called unc ontr ol lable factors , whic h means that
they cannot b e con trolled b y the designer. Noise factors ma y b e external suc h as temp er-
ature, moisture and load, or in ternal suc h as c hanging system b eha viour due to aging, or
unit-to-unit, e.g., thic kness v ariations. The resulting distribution of the system resp onse
can b e describ ed b y its mean and v ariance v alue. Robust design can b e classified in to t w o
9

1 In tro duction
metho ds. The T aguchi metho d determines a design that mak es p erformance insensitiv e
to noises in the man ufacturing pro cess. R obust optimization adds a robustness concept to
con v en tional optimization. The ob jectiv e function and the constrain ts are redefined with
robustness indices.
system res p onse
probab ili ty
density
target

f

Step 1
Step 2
A B
C

Figure 1.3: Principle of T aguc hi Metho d:
in step one the v ariance of the system re-
sp onse is reduced, in step t w o the mean v alue
is shifted to target v alue, while the v ariance
is k ept lo w, similar to [88]
Six Sigma Qualit y strategy . A ccord-
ing to Neuman and Ca v anagh [80] and
Harry [40], the earliest approac h to reduce
a system resp onse v ariation w as the Six
Sigma Qualit y strategy . The mean of the
system resp onse lies b et w een +/- six stan-
dard deviations, whic h are the lo w er and
upp er sp ecification limit. Motorola in tro-
duced the Six Sigma Qualit y strategy as
standard and quoted sev eral billion Dollar
sa vings. Shewhart et al. [101] sho w ed that
a deriv ation from the three sigma in terv al
around the mean v alue is critical so that a
pro cess needs correction.
T aguc hi based Approac hes. Zang
et al. [129] describ ed, in the 1950s and
early 1960s, T aguc hi dev elop ed the basics
of robust design to meet the c hallenge of
pro ducing high-qualit y pro ducts. It w as
successfully applied to v arious industrial
fields suc h as electronics, automotiv e pro d-
ucts, photograph y , and telecomm unications
[116],[117]. T aguc hi prop osed metho ds of determining v ariables to mak e p erformance insen-
sitiv e to noises in the man ufacturing pro cess. T aguc hi dev elop ed a t w o-step optimization
strategy . In the first step the v ariation of the system resp onse is reduced b y using the small-
est orthogonal arra y of design v ariables, whic h are, e.g. parameters with high impact on
the system resp onse and less in teraction with other design parameter. In the second step,
the mean v alue is adjusted to the target v alue b y a so-called sc aling factor . The scaling
factor should only ha v e influence on the mean, while the noise ratio is main tained. Ho w-
ev er, it is difficult to find a scale factor in general design. In Figure 1.3 three distributions
A, B, and C of the system resp onse f are presen ted, based on differen t design parameters,
whic h are not visualized. The p erformance of differen t design approac hes (A,B,C) can b e
measured b y the distance of the mean v alue to the target v alue. The robustness is mea-
sured b y the magnitude of the v ariation. Note, design A has a b etter p erformance but a
lo w er robustness than design B. A ccording to Mon tgomery [76] man y companies rep orted
successful applications with the use of T aguc hi metho d. Designers, who used b eforehand
b est guess or one-factor-at-a-time metho ds, gained b etter results with T aguc hi’s metho d.
Robust Optimization Metho d. A ccording to Zang et al. [129] and P ark et al.
[88], robustness of the system p erformance can b e ac hiev ed b y reducing the c hange of
the p erformance with resp ect to v ariations, e.g., of uncon trollable parameters. The mean
v alue of the system p erformance is used as p erformance function and the v ariation of the
system p erformance as robustness function. Constrain ts can b e included. A p eak optim um
around whic h the system p erformance drastically c hange b y small v ariations of parameters
10

1.2 State of the Art
are unrobust solutions. Flat optima are b etter for an insensitiv e design. Chen et al.
[13] presen ts a hierarc hical m ultilev el Multidisciplinary Design optimisation metho d with
uncertain t y . Probabilistic Analytical T arget Cascading is enhanced b y considering the first
t w o statistical momen ts of in terrelated resp onses. The metho d is demonstrated on v ehicle
susp ension design as a t w o-lev el optimization problem.
In conclusion, unexp ected deviations from a system b eha viour, that a designer initially
in tended, are naturally presen t. They are called aleatoric uncertain ties. The mean v alue
and the v ariance of ob jectiv e quan tities are classical ob jectiv es for robustness. The mean
v alue should b e close to the target v alue while the v ariance has to b e minimal.
Reliabilit y Design
Reliabilit y based Design. In Reliabilit y-Based Design (RBD), uncertain ties of non-
deterministic v ariables are assumed to follo w certain probabilit y distributions. The proba-
bilit y of failure corresp onds to a failure mo de of a system. The basics of Reliabilit y-Based
Design can b e found in [38], [72], or [6]. Reliabilit y analysis is used to compute the reli-
abilit y index, whic h quan tifies the probabilit y of failure corresp onding to a giv en failure
mo de or for the en tire system [38].
probab ility
to fail

[


[

reliable
optimum
deterministic
optimum
fail
feasib le
Figure 1.4: Principle of reliable design: ap-
pro ximately 75 % design fail in a b o x around
the deterministic optim um. The reliable
solution is a trade-off system p erformance.
Similar to [3].
When a design strategy is p erformed
without accoun ting for uncertain ties, cer-
tain hard constrain ts that are activ e at the
deterministic solution ma y lead to system
failure. In Figure 1.4, it can b e recognized
that the probabilit y of failure of the deter-
ministic optim um is appro ximately 75 % in
case of a uniform distribution. The reliable
solution ma y b e a trade-off with the system
p erformance.
Reliabilit y based Design Optimiza-
tion. Aleatory uncertain ties are statisti-
cal uncertain ties: unkno wns that differ eac h
time an exp erimen t is run. These uncer-
tain ties com bined with design optimization
is t ypically referred to reliabilit y based de-
sign optimization (RBDO): a metho dology
for finding optimized designs with a lo w
probabilit y of failure. [38], [3] A traditional
RBDO using probabilit y theory t ypically
requires complete statistical information of the uncertain ties [3]. The system p erformance
criteria are describ ed b y the system p erformance functions, whic h are b ey ond zero if the
system fails. A fundamen tal problem in reliabilit y theory is the computation of the prob-
abilit y in tegral of failure. In the case of failure, the system p erformance has to b e less
than zero dep ending on the uncertain parameters including their densit y function. Diffi-
cult y in computing this probabilit y has led to the dev elopmen t of v arious appro ximation
metho ds [130]. The first-order reliabilit y metho d (F ORM) is an analytical appro ximation
of the p erformance function b y a linear function in standardized normal space. This is a
transformation of the design parameters with its mean v alue and the standard deviation
11

1 In tro duction
of its standard normal distribution. The reliabilit y index is then in terpreted as the mini-
m um distance from the origin to the limit state surface in standardized normal space. The
most lik ely failure design p oin t is searc hed using mathematical programming metho ds [8].
The second-order reliabilit y metho d (SORM) is obtained b y appro ximating the limit state
surface at the design p oin t b y a second-order surface [32]. Ho w ev er the computation of
the matrix of second-order deriv ativ es is time consuming. Rac kwitz [92] giv es a review
and some p ersp ectiv es of reliabilit y analysis. The review is motiv ated from structural re-
liabilit y and fo cuses on F ORM and SORM concepts. The errors using F ORM/SORM are
acceptable in view of the large uncertain t y . It is still m uc h b etter than crude Mon te Carlo
metho ds that is the b enc hmark to compare with or in some cases the only w a y to obtain
results. Du et al. [23] extended Reliabilit y-Based Design (RBD) with non-deterministic
v ariables within in terv als for their distribution. A metho d of RBD with the mixture of
random v ariables with distributions and uncertain v ariables with in terv als is prop osed.
Reliabilit y is considered under the condition of the w orst com bination of in terv al v ariables.
Then the effort is comparable to the classic pro cedure.
In con trast to robustness, reliabilit y-based design fo cuses on probabilit y distributions of
uncertain parameters to estimate the probabilit y of failure. Uncertain parameters lead to
the violation of constrain ts on system b eha viour and therefore to system failure.
Solution Space
Rather than seeking single design p oin ts with optimal p erformance, sets of go o d designs are
iden tified, i.e., designs that fulfil all required design criteria. A set of go o d designs is called
solution sp ac e [133]. Go o d designs are not further distinguished b y their p erformance and
are, therefore, considered as equal. The set of all go o d designs is called solution space.
Solution space is already prop osed in Lottaz et al. [68, 69], and Y annou et al. [128].
Ho w ev er, they w ere constructed from algebraic constrain ts and w ere only computed for a
small n um b er of dimensions, and are therefore not applicable to high dimensional problem
statemen ts. The iso-p erformance approac h of de W ec k and Jones [18] iden tifies regions in
the design space on whic h the system p erformance equals exactly the required p erformance.
It is similar to the solution space approac h, in that more than one design is considered
as a solution. It is differen t, ho w ev er, in that designs with b etter p erformance than the
required one are not included, th us the solution space is to o small.
Applications of Solution Space. Applications of the solution space approac h to
robust v ehicle crash design under uncertain t y is presen ted b y Zimmermann et al. [133,
132, 134].
In crash design a c haracteristic curv e of crash force and deformation scatters due to gen-
eral sources: limited man ufacturing capabilities, distributed design pro cess or scattering
in pro duction. This c haracteristic curv e has to fulfil requiremen ts for sev eral crash situa-
tions, e.g., absorb the critical amoun t of energy from the crash and reduce the maximal
deceleration under a certain uncritical lev el. Due to these reasons, in terv als (B) need to
b e designed instead of single c haracteristic curv es (A) in order to pro vide reliable protec-
tion, see Figure 1.5. F or arbitrary non-linear and high-dimensional systems, a set of go o d
designs is called the solution sp ac e . Go o d designs are not further distinguished b y their
p erformance and are, therefore, considered as equal.
In another application presen ted b y F ender et al. [31], solution space is used to iden tify
k ey parameters in high-dimensional non-linear systems that are sub ject to uncertain t y . In
12

1.2 State of the Art
Eic hstetter et al. [26], solution space is computed for the la y out of damp er c haracteristics
that are sub ject to conflicting design goals from differen t disciplines in driving dynamics.
Conflicts of goals b et w een differen t disciplines o ccur b ecause design ob jectiv es are con-
sidered one-b y-one and single designs are prop osed rather than sets of designs. Solution
space pro vides flexibilit y for satisfying further requiremen ts that are not under curren t
consideration. A single design ma y b e dominated b y further requiremen ts, while a set of
designs ma y offer alternativ es without design iterations. In Munster et al. [78] solution
space is applied to the design of the v ehicle steering system. Therefore sev eral coupled
subsystems need to b e considered sim ultaneously , but they are dev elop ed separately b y
differen t departmen ts or external suppliers. The use of solution space enabled the decou-
pling of dynamic subsystems. Requiremen ts on the top lev el suc h as v ehicle b eha viour
c haracteristics w ere divided in to requiremen ts for decoupled subsystems.
designed
character-
istic curv e (A)

F

s

realized
character-
istic curv e
designed
Intervals (B)

Figure 1.5: Realized c haracteristic curv e
of force and deformation scatters around de-
signed c haracteristic curv e (A). Designed in-
terv als (B) include the scattered c haracteris-
tic curv e.
Computing a Solution Space. A
n umerical algorithm to compute solu-
tion space for arbitrary non-linear high-
dimensional systems w as in tro duced in
[133]. The algorithm seeks a solution space
on whic h all designs are go o d, i.e., they
satisfy sp ecified design criteria. The solu-
tion space is b o x-shap ed and pro vides tar-
get in terv als for all design parameters. The
b o x-shap ed solution space is ev aluated b y
Mon te Carlo sampling and Ba y esian statis-
tics, see Lehar and Zimmermann [66]. The
effect of dimensionalit y on the n um b er of
function ev aluations necessary to compute
a solution space is studied in detail in [36].
The algorithm w as extended b y [31] for
the optimisation of solution space with con-
strain ts, whic h are used to seek target re-
gions for parameters that can b e reac hed
from a curren t bad design with minimized
effort. The prop osed metho d iden tifies k ey parameters in high-dimensional non-linear sys-
tems that are sub ject to uncertain t y . A bad design ma y b e turned in to a go o d design
b y mo ving its k ey parameters in to their target in terv als. The approac h w as applied to a
crash design problem from industrial practice. By con trast, the algorithm used for the
iso-p erformanc e analysis in [19] uses gradien t-based con tour follo wing pro cedure, whic h
is implemen ted as a m ulti-v ariable searc h algorithm that manipulates the n ull set of the
Jacobian matrix of the p erformance.
In conclusion, solution space is initially designed for crash design to handle uncertain ties.
Constrain ts on the system b eha viour result in a set of go o d designs. Solution space is
already successfully extended to the design of v ehicle dynamics for handling conflict of
goals and separate design v ariables for sev eral design teams.
13

1 In tro duction
P a rameter Space App roach
Robust Con trol. The theory of robust con trol dev elop ed multiple methods for dealing
with b ounded system uncertain ties, see A c k ermann [1] and Zhou and Do yle [131]. H-
infinit y lo op-shaping minimizes the sensitivit y of a system o v er its frequency sp ectrum and
enables robust system b eha viour or tra jectories under uncertain ties. Sliding Mo de Con trol
is a v ariable structure con trol metho d. The con trol la w can switc h from one con tin uous
structure to another in dep endence of the curren t state. There are man y more metho ds,
whic h will not b e men tioned here, see [1] or [131].
P arameter Space Metho d. F or the analysis of robustness or stabilit y of systems with
uncertain parameters sev eral metho ds are p ossible, as, e.g., the parameter space metho d.
This metho d can b e used for t w o uncertain parameters, whic h influence the co efficien ts of
the c haracteristic p olynomial of a system. If this holds true, the b oundary of the stable area
in the eigen v alue plane can b e transferred b y the solution of algebraic equations as curv es
in the t w o-dimensional parameter space. Details ab out the parameter space approac h can
b e found in [1], [2] or [11]. In general the parameter s pa ce metho d w as restricted to linear
systems and the consideration of eigen v alue criteria. Bün te [11] enhances the application
of the parameter space metho d to non-linear criteria as lo cus curv e criteria. A con trol
design with m ultiple criteria is done b y in tersection of stable areas. In [12] and [83] the
parameter space approac h w as successfully applied to dev elopmen t pro jects of the of BMW
Group. The approac h w as used for the dev elopmen t of the xDriv e all wheel driv e system,
whic h is the mark eting name for the BMW all-wheel driv e system. The approac h w as used
for the con trol of a m ultiple disk clutc hes, whic h can b e used for adoption of longitudinal,
lateral and y a w dynamics. F urthermore, the approac h w as applied to functional v alidation
of the stabilit y function of an activ e steering system. A reduction of p ossible con troller
parameters could b e done using stabilit y and robustness criteria, whic h is a pre-selection
for test driv ers. Hohen bic hler [44] extended the parameter space approac h to the con trol
design of robust PID con trollers. F or a linear system (dela y time can b e included) com bined
with a single design criterion as stabilit y , the quan tit y of all PID con trollers, whic h fulfil
the criterion are determined. F or m ultiple design criteria the designer has to c ho ose among
the in tersections. F or fixed gain v alues of k p , a stable area of the gains k d and k i is searc hed.
F urther, all v alues of k p are searc hed, whic h lead to stable area of k d and k i .
In conclusion, the parameter space metho d is similar to the solution space approac h:
the shap e of the solution space can b e determined analytically , e.g., b y b oundaries of the
stable area in the eigen v alue plane, but the metho d is limited to t w o parameters.
1.2.4 Pro duct F amily Design
A pr o duct family is a collection of sev eral similar tec hnical systems using common comp o-
nen ts. As an example, v ehicles that share b o dy parts ma y b e considered a pro duct family .
Dev eloping pro duct families rather than sev eral systems separately ma y reduce deve lop-
men t effort for comp onen ts, administration cost, logistics and price-p er-unit [103], [19].
Unfortunately , ho w ev er, the design of a pro duct family is t ypically more c hallenging than
designing eac h system indep enden tly . Design parameter v alues ha v e to b e c hosen more
carefully in order to fulfil all design goals as they will affect the p erformance of sev eral
systems sim ultaneously . Pro duct family design and platform-based pro duct dev elopmen t
has receiv ed m uc h atten tion in the literature. Simpson [103], Jose and T ollenaere [51] and
14

1.2 State of the Art
Jiao et al. [50] pro vide comprehensiv e state-of-the-art reviews of mo dular design, pro d-
uct family design and platform-based pro duct dev elopmen t. Three categories of pro duct
family approac hes are briefly summarized here.
Deriving a Pro duct F amily . Approac hes from the first category fo cus on deriving
of a pro duct family . A pro duct platform from whic h the pro duct family ma y b e deriv ed
either b y adding, remo ving or substituting one or more mo dules to a platform or b y scaling
the platform in one or more dimensions [103].
In Simpson’s review [103] he distinguished b et w een mo dule-base and scale-base plat-
forms. Simpson et al. [105] prop osed a scale-based pro duct family design, which generates
m ultiple designs b y scaling one or sev eral design parameters. Du et al. [22] prop osed
a configurational pro duct family design, also called mo dule-based pro duct family design.
This approac h aims at dev eloping a pro duct family design from whic h family mem b ers are
generated b y , e.g., substituting comp onen ts. Both approac hes increase the pro duct v ariet y
b y c hanging functional elemen ts individually for eac h system.
Minimize the Num b er of differen t Comp onen t Designs. Approaches from the
second category minimize the n um b er of differen t comp onen t designs. F ellini et al. [29]
prop oses a metho dology whic h uses first-order gradien t information, generated b y indi-
vidual optimisation of pro duct v arian ts. Under the assumption of small v ariations among
family pro ducts, this analysis can b e used to iden tify whic h comp onen t design ma y or ma y
not b e shared. In [17], sensitivit y analysis are p erformed in order to iden tify candidate
platform design v ariables. In [45], the degree of commonalit y is quan tified b y calculating
three differen t mo dularit y measures. One of these metrics uses a singular v alue decomp o-
sition of a binary design structure matrix. The rate of deca y of the singular v alues of the
design structure matrix is used as measure of mo dularit y . De W ec k et al. [19] determine
the optim um n um b er of pro duct platforms that maximizes the o v erall profit. The profit
function includes sales v olume of v arian ts and their p erformance. One of the k ey ingredi-
en ts in the pro duct v arian t lev el optimization is to c ho ose the v arian t design parameter,
that has high sensitivit y effect on the p erformance with minimal p erturbation for ac hieving
wide v arian t differen tiation with small in v estmen t. T urner and F erguson [120] syn thesize
adv ancemen ts from mark et-based design and heuristic optimization researc h to strategi-
cally construct targeted initial p opulations capable of reducing computational cost and
impro ving final solution qualit y . In [79], multi-criteria optimization is applied to decide
whether or not t w o systems should use comp onen ts of the same design. F or eac h pair of
systems, a P areto-fron t of their p erformances with one comp onen t design is computed and
compared with optimisation results with individual comp onen t designs. This information
is used b y the designer to decide on what comp onen t design should b e used. This ap-
proac h w as extended in [30] in whic h user-sp ecified p erformance losses where additionally
considered. A common comp onen t design is used, if critical threshold v alues for eac h sys-
tem p erformance are not exceeded. Unfortunately , the required computing effort of these
approac hes is significan t, as all p ossible com binations need to b e assessed.
Optimize Platform Selection and V arian t Design Sim ultaneously . Approac hes
from the third category optimize platform selection and v arian t design sim ultaneously .
Simpson et al. [104] presen t an approac h with restricted commonalit y , i.e., all or no
comp onen ts are shared. P artial commonalit y is excluded. Kha ja virad et al. [53] address
limitations of prior restrictiv e comp onen t sharing definitions b y in tro ducing a commonalit y
matrix to share comp onen ts among subsets of systems. This w a y , they extend the approac h
b y Simpson et al. [104] in order to relax the all-or-none comp onen t sharing restriction.
15

1 In tro duction
The complex problem statemen t is solv ed b y decomp osing it in to a t w o-lev el genetic algo-
rithm. On the upp er lev el, the optim um platform configuration is determined, while on
the lo w er lev el, the individual v arian ts are optimized. Cho wdh ury et al. [15] presen t an
approac h called Comprehensiv e Pro duct Platform Planning ( C P 3 ) framew ork to design
optimal pro duct platforms. The approac h pro duces a mixed in teger non-linear program-
ming problem, whic h is carefully reform ulated to allo w for the application of con tin uous
optimization. A no v el Platform Segregating Mapping F unction is used and sub-families of
pro ducts are computed using a commonalit y matrix, similarly to [53].
In conclusion, there are differen t streams of pro duct family design whic h try to increase
the v ariabilit y , minimise the design parameter v ariabilit y or iden tify candidate platforms.
T esting all p ossible com binations of configurations is time consuming and burdensome.
This includes the distribution of shared comp onen ts among the systems and the design
parameter v alues themselv es. The approac h of F ellini et al. [30] seems to promise and to
simplify pro duct family design as user-sp ecified p erformance losses are considered from the
b eginning.
1.2.5 Evaluation of Metho ds and Need fo r A ction
P erception Thresholds
The transfer of sub jectiv e ev aluations to ob jectiv e quan tities is essen tial for a virtual
dev elopmen t pro cess. F urthermore, thresholds of p erception, whic h are in this con text
thresholds of ob jectiv e quan tities, are imp ortan t for the virtual dev elopmen t [39]. While
qualitativ e relations b et w een sub jectiv e and ob jectiv e v ehicle b eha viour categories are of-
ten a v ailable, quan titativ e relations are either complex or rare, see [135],[94], [95] or [41].
A ccording to Kriegel et al. [62], the generation of target v alues of ob jectiv e quan tities
from sub jectiv e ev aluations b y exp erts has uncertain ties due to the confidence lev el result-
ing from the influence of the driv er, en vironmen t and the v ehicle category . Dettki [21],
Gutjahr [37], Botev [9] and VDI 2057 [121] already selectiv ely in v estigated sev eral p ercep-
tion thresholds, but these are not in v estigated throughout all relev an t fields of researc h.
Reasons for this are that the ev aluation of single test driv ers can differ enormously [135],
correlation studies with real v ehicles are time consuming and cost in tensiv e or afflicted
with other uncertain ties. The virtual w orld ma y b e an alternativ e. Sim ulation ma y b e a
simplification of the realit y , but the results can b e created rapidly and are repro ducible.
Requiremen t 1 R elations b etwe en subje ctive and obje ctive evaluations ar e essential for
a virtual development pr o c ess. The automotive industry has a ne e d for a systematic
simulation-b ase d metho d for the gener ation of r elations b etwe en subje ctive and obje ctive
evaluations.
Conceptual La y out Strategies, Robust and Reliable Design
The classical approac hes of Wimmer [127], Du [24] and Botev [9] include iterations in the
design pro cess, whic h should b e prev en ted from the b eginning to reduce effort and costs.
Mäder [70] and Röski [96](BMW Group) prop osed a m ulti-step design pro cess. In the
first step, complexit y is reduced b y simple mo dels. Therefore dep enden t and indep enden t
design parameters are iden tified and clustered according to lateral and v ertical dynamics.
16

1.2 State of the Art
But sev eral design parameters ma y influence m ultiple design categories. Classical opti-
mzation strategies as prop osed in [100] and [24] use single p erformance functions, whic h
are built b y w eigh ted normalized deviations of ob jectiv e v alues related to their target
v alues. Ho w ev er, the more design ob jectiv es are used the more v ague the resulting opti-
m um of the p erformance function. P areto-optimal approac hes presen ted, e.g., in [35],[97]
and [35] outp erforms classical optimization strategies with single ob jectiv e function as the
decision mak er can distinguish b et w een P areto-optimal designs according to her or his pri-
orities. The n um b er of P areto-optimal solutions is difficult to handle, although T ab oada
et al. [114] presen t metho ds to handle the n um b er. In real-w orld applications the n um b er
of comp eting ob jectiv es can b ecome o v erwhelming. Neither w eigh ted single p erformance
functions nor the high resulting n um b er of P areto-optimal solutions for high dimensional
design problems, are useful to reduce the complexit y .
The set-based design approac h, presen ted in [123], [108] or [109] and [85], is efficien t due
to a higher flexibilit y of the system parameters o v er the dev elopmen t pro cess. Dominated
designs are neglected and iterations ma y b e a v oided. In terv als of the design v ariables
are the enabler for more design freedom during the dev elopmen t duration. Ho w ev er, the
pro cedure aims for a selection of the optimal parameter setting according to system p er-
formance. Uncertain t y of design parameters is not considered. The T arget Cascading
approac hes of Kim et al. [54, 55] reduce the complexit y of system design through top-
do wn la y out. Through this pro cedure the complexit y is reduced as less detail input has
to b e considered. K okk olaras et al. [61] and Liu et al. [67] extend T arget Cascading with
robust and reliabilit y design to increase the qualit y of pro ducts, but the implemen tation
of probabilistic form ulation of target v alues in design teams is complex. K v asnic ka and
Dic k [64] and Wimmer [127] demand a robust dev elopmen t pro cess and robustness against
uncertain parameters in the conceptual la y out pro cess. Although Chen et al. [13] giv es
go o d results for a robust design with uncertain ties in susp ension design, in practice, it
is difficult to define target v alues for standard deviation of the p erformance function. In
robust or reliable design metho ds the n um b er of ob jectiv e quan tities is doubled due to a
p erformance and a robustness or reliable ob jectiv e function for eac h ob jectiv e quan tit y .
The meaning and the in terpretation of the robust ob jectiv e is difficult. Reliable design
metho ds require complete statistical information of all uncertain ties.
In [26] solution space design w as successfully applied to conflicting design goals. In [78]
this metho d w as successfully applied to decouple system requiremen ts of subsystems. The
parameter space approac h is a similar metho d and is used for robust con trol for dealing
with b ounded system uncertain ties. Stabilit y of systems with uncertain parameters is
desired. Unfortunately this metho d is limited to t w o design parameters.
Requiremen t 2 Ther e is a ne e d of the automotive industry for a r obust design str ate gy,
which de cr e ases the c omplexity and r e duc es the numb er of iter ations. Intervals of design
p ar ameters should b e use d to enable high flexibility during the development pr o c ess. R obust-
ness should b e an inte gr al p art of the metho d, so that no additional term for the evaluation
of r obustness is ne e de d. F urther, the metho d has to b e c omp atible with top-down str ate gies
such as tar get c asc ading to r e duc e the c omplexity.
Pro duct F amily Design
Scalable pro duct family design approac hes from Simpson et al. [105, 103] and Du et al. [22]
aim at increasing the pro duct v ariet y in an efficien t w a y b y c hanging functional elemen ts
17

1 In tro duction
individually for eac h system. They do not, ho w ev er, aim at minimizing the total n um b er of
comp onen ts with differen t design. Simpson et al. [106] tries to minimize design parameter
v ariabilit y b y a commonalit y index. This approac h is limited as it cannot pro duce a
result with partial commonalit y where, e.g., t w o comp onen t designs can b e used in more
than t w o systems. Either one comp onen t design is used for all systems, or all systems
ha v e differen t comp onen ts of differen t design. Nelson et al. [79] and F ellini et al. [30]
tried to iden tify candidate platforms b y building P areto-fron ts of the p erformances of eac h
pair of systems. F ellini et al. [30] used user-sp ecified p erformance losses to decide if the
effort is reasonable to couple t wo systems. Unfortunately , the required computing effort
of these approac hes is significan t, as all p ossible com binations need to b e assessed. The
approac hes from F ellini et al. [29], Dai and Scott [17] and H ¨
ltä-Otto and de W ec k [45]
ha v e in common that they try to iden tify candidate platforms. Ho w ev er, these approac hes
adopt tec hniques that are applicable to linear problems, and therefore not appropriate for
the non-linear problems considered here. F urther, a high n umerical effort is necessary for
first-order gradien t information, sensitivit y analysis or singular v alue decomp osition.
In real applications, the n um b er of comp onen t configurations of a pro duct family design
is imp ortan t, b ecause the lo w er the n um b er of configurations, the lo w er the complexit y .
Ho w ev er, the more configurations are a v ailable, the b etter the p erformance of eac h indi-
vidual system. In view of this conflict of ob jectiv es, the extension to use user-sp ecified
p erformance losses for eac h system of F ellini et al. [30] seems exp edien t. Pre-defined re-
quiremen ts on the system b eha viour do not answ er, whic h systems should b e coupled or
whic h configurations of comp onen ts should b e c hosen, but the framew ork of problem is
fixed from the b eginning.
Requiremen t 3 Ther e is a ne e d of the automotive industry for a pr o duct family design
metho d, which satisfies pr e-define d r e quir ements on the system b ehaviour and minimizes
the numb er of c omp onents. This metho d has to b e simpler and gener ate d with less effort
than the appr o aches pr esente d.
1.3 Concept of Resea rch
1.3.1 Aims of the W o rk
The concept of this researc h has to meet the demand to create a consisten t design concept
that can b e used in industrial practice in the early design phase. The concept has to
decrease complexit y and the metho ds need to b e consisten t. The w ork is fo cused on the
automotiv e design of longitudinal, lateral and v ertical dynamics, but can b e also transferred
to other fields of researc h. The aim of this w ork is based on the deficit b et w een the
requiremen ts of the automotiv e industry and the state of the art, describ ed in Section 1.2.
Aim 1 The main aim is the development of a simulation-b ase d r obust design metho d for
the c onc eptual layout phase to r e duc e c omplexity and incr e ase r obustness. A top down
metho d is desir e d, which determines r elevant design p ar ameters fr om system p erformanc e.
The high numb er of design obje ctives has to b e hand le d e asily. Unc ertainties have to b e
c onsider e d in the design phase. High flexibility of the design p ar ameters is desir e d.
18

1.3 Concept of Researc h
F or a virtual dev elopmen t pro cess the transfer of sub jectiv e ev aluations to ob jectiv e quan-
tities is imp ortan t to meet the customers’ needs. Therefore quan titativ e relations b et w een
sub jectiv e and ob jectiv e ev aluations are required. If the sub jectiv e p erception of the cus-
tomers is impro v ed, dev elopmen t effort is reasonable. A classical approac h is the correlation
study with hardw are and test driv ers to get information ab out the relation of sub jectiv e
ev aluation and ob jectiv e quan tities. Disadv an tages of this approac h are that it is time and
cost in tensiv e. The result is afflicted with uncertain t y , e.g., due to en vironmen t influences
or sub jectiv e preferences. An alternativ e sim ulation-based approac h ma y create results
faster and more repro ducible, if a simplification of the realit y in to mo del is acceptable. As-
suming a w ell-kno wn predecessor v ehicle as base, the kno wledge ab out relativ e p erception
thresholds is sufficien t to impro v e the customers’ p erception. The p erception thresholds
will differ for eac h v ehicle class.
Aim 2 A pr e c onditioning aim for the virtual development pr o c ess is the development of
a systematic simulation-b ase d metho d for the gener ation of p er c eption thr esholds. These
thr esholds have to b e vehicle class sp e cific.
In the automotiv e industry , comp onen ts are shared within m ultiple pro ducts. The com-
plexit y increases with the n um b er of comp onen t configurations. The more configurations
are a v ailable, the b etter the p erformance of eac h individual v ehicle.
Aim 3 A fol low-up aim to the r obust design metho d is the development of a metho d for a
systematic gener ation of a pr o duct family, which satisfies pr e-define d r e quir ements on the
system b ehaviour and minimizes the numb er of c omp onents.
1.3.2 Concept of Metho ds
Systematic simulation-based Metho d fo r the Generation of P erception Thresholds
The system b eha viour, or in this con text the v ehicle b eha viour, is p erceiv ed b y customers
sub jectiv ely . Desired v ehicle prop erties on a sub jectiv e level can only be achiev ed, if the
quan titativ e relation b et w een sub jectiv e and ob jectiv e quan tities is kno wn. The system-
atic generation of quan titativ e relations b et w een sub jectiv e and ob jectiv e ev aluation will
b e based on the comparison of a kno wledge managemen t system with ob jectiv e and ob jec-
tification kno wledge. The kno wledge managemen t system is assigned b y the exp ertise of
senior engineers. Sub jectiv e ev aluations ma y b e con tained in the kno wledge managemen t
system for a sp ecific v ehicle class. A kno wledge managemen t system can also b e created
b y sim ulation. Ob jectiv e kno wledge created b y sim ulation is com bined with qualitativ e
ob jectification kno wledge. If the sim ulation mo del is at an adequate lev el of accuracy
and the exp ertise of senior engineers is reliable, a comparison of the sub jectiv e and the
ob jectiv e kno wledge managemen t system lead to quan titativ e ob jectification kno wledge.
As the virtual dev elopmen t of v ehicles often starts based on predecessor v ehicles, relativ e
relations of sub jectiv e and ob jectiv e kno wledge is sufficien t for the new design, as it has to
b e impro v ed related to the predecessor v ehicle. The main question is: Ho w m uc h influence
do es the c hange of an ob jectiv e quan tit y ha v e on the sub jectiv e ev aluation?
New V alue 1 Development of a new metho d to gener ate obje ctive design go als fr om sub-
je ctive evaluations. The thr esholds for obje ctive quantities wil l b e derive d fr om a c omp ar-
ison of a subje ctive and obje ctifie d know le dge management system and use d for r elative
impr ovement.
19

1 In tro duction
Robust Design Strategy fo r the Conceptual La y out Phase
Consider a classical la y out pro cess: a design is sough t, whic h satisfies the design require-
men ts b est, e.g., ride comfort. During the dev elopment phases more and more requiremen ts
arise, e.g., requiremen ts of v ehicle dynamics, misuse and acoustics. In the classical la y out,
these requiremen ts are optimised indep enden tly , step b y step. If uncertain ties of design or
uncon trollable parameters o ccur, the system b eha viour differs from the desired b eha viour.
Therefore the design is mo dified un til a final design is c hosen. Hence conflicts of goals and
iterations in the conceptual la y out pro cess can easily o ccur. In order to cop e with man y
requiremen ts, whic h app ear during the dev elopmen t phase and to consider uncertain ties
within the parameters, a design metho d based on solution space is prop osed. Unnecessary
conflict of goals can b e a v oided and therefore the n um b er of iteration lo ops, dev elopmen t
time and dev elopmen t costs can b e minimized. It is not the p eak p erformance whic h is
desired, but a design whic h considers the uncertain ties in the conceptual design phase. In
the conceptual design phase, the distribution of uncertain ties of design or uncon trollable
parameters is often not exactly kno wn. Therefore, for unkno wn uncertain ties an uniform
distribution is assumed [43]. The quan tification of robustness is complex. T arget v alues
for v ariance or standard deriv ations are difficult to imagine. In terv als of design or uncon-
trollable parameters are a simple metho d for robustness quan tification in the input space
instead of in the output space. In general the quan tification of robustness for eac h design or
uncon trollable parameters will not b e used as input for the design problem, the robustness
will result in the design approac h.
Solution space design seems an adequate robust design metho d for sev eral reasons:
• Requiremen ts on the system p erformance can b e expressed b y threshold v alues for
system p erformance . The result is a set of design parameters, thereb y robustness
is considered. The solution space offers a flexibilit y for further requiremen ts without
iterations of the whole design pro cess. Threshold v alues for the system p erformance
are simpler than the use of mean v alues and standard deriv ation in r obust design .
• The large n um b er of design ob jectiv es is handled b y sup erp osition of solution
spaces . The determination of solution spaces for the systems can b e decoupled and
computed in parallel. This is an adv an tage in comparison to ev aluations of coupled
systems in multi-obje ctive optimization . Ev aluation iterations ma y b e significan tly
reduced.
• In general the design pro cess results in a solution space with a high-dimensional, ev en-
tually discon tin uous, v olume with non-linear arbitrary b oundaries. A b o x-shap ed
solution space, within this non-linear solution space, enables a decoupling of design
parameters. Design in terv als for eac h design or uncon trollable parameter enable
more flexibilit y during the design pro cess as prop osed in set based design. F urther-
more, decoupling of design parameters enables a parallel design of m ultiple design
teams as prop osed in set b ase d design and tar get c asc ading . Therefore, this is a
reduction of complexit y and an increase of efficiency .
• Solution space design is consisten t with a top-do wn design pro cess as with tar get
c asc ading , whic h reduces the complexit y b y dividing design tasks in sub-tasks. Re-
sulting in terv als of design parameters ma y b e used as b oundaries or requiremen ts of
the subsystem from the next system lev el.
20

1.4 Structure of the W ork
• Dep ending on the distribution of the design or uncon trollable parameters, the c hoice
of a cen tral design p oin t within the solution space is a reliable design p oin t as
determined in r eliable design . In con trast to r eliable design the distribution of the
parameters is not required or fixed from the b eginning.
Solution space design is used as a robust design metho d in the conceptual la y out pro cess.
Solution space implies a natural robustness according to v ariations of the design or un-
con trollable parameters, reliabilit y according to requiremen t of the system b eha viour and
flexibilit y for c hanging system requiremen ts in the early phase.
New V alue 2 Development of a new r obust design metho d using solution sp ac es, which
cr e ates r obustness against p ar ameter unc ertainties and flexibility for further r estrictions.
Pro duct F amily Design using Solution Spaces.
Pro duct family design is of sp ecial need in the automotiv e industry . A systematic gener-
ation of pro duct family design increases the efficiency and decease the complexit y . Using
solution space for the design do es not only increase robustness, reliabilit y or flexibilit y ,
solution spaces of m ultiple systems con tain also the information if design parameters can
b e shared. The sup erp osition of solution spaces of m ultiple designs is the k ey idea. In
extension to the robust design approac h with in terv als of design parameters, the sup erp o-
sition can also b e executed with in terv als for eac h design parameter. Without c hec king
all p ossibilities of com binations of systems or design v ariables, the use of solution spaces
enables an efficien t pro duct family design.
New V alue 3 Development of a new metho d for pr o duct family design using solution
sp ac es to r e duc e the c omplexity of c ombinatorics and to enable simultane ous engine ering.
Restriction of Researc h Field. These metho ds complemen t the classical design pro cess
as describ ed in [70], [96] and [126]. The application of the w ork lies on the design of
longitudinal, lateral and v ertical v ehicle dynamics. F or the design of steering systems see
[126] and [78]. The design of t yre c haracteristics is researc hed in [81] or [37].
1.4 Structure of the W o rk
This w ork is organized as follo ws, Figure 1.6 giv es an o v erview. In Chapter 2, the gener-
ation of p erception thresholds based on kno wledge managemen t system is presen ted.
Quan titativ e ob jectification kno wledge is deriv ed b y comparing sub jectiv e and ob jectified
kno wledge. The ob jectified kno wledge corresp onds to the sub jectiv e kno wledge and is
generated b y ob jectiv e and ob jectification kno wledge. The ob jectification kno wledge is
limited to qualitativ e information, quan titativ e information is not a v ailable. In Chapter 3
robust design with solution space is suggested. Requiremen ts on the system b eha viour
ha v e to b e fulfilled. The result is not only a single design, but a set of designs. The
concept of solution space is explained and used for robustness and conceptual la y out. In
Chapter 4 the problem statemen t of pro duct family design is pro vided. The concept
of o v erlapping solution spaces for m ultiple systems is explained and illustrated. F urther,
an algorithm is presen ted to compute pro duct family design based on b o x-shap ed high-
dimensional solution spaces. In Chapter 5 all metho ds are applied to an example from
21

1 In tro duction
+1
+1
+1
Product F amil y Design
Share compon en t s fo r less
c om ple xity
R obust Design
Sa tisfy r eq uir ements
on sys te m behaviour
Design G oals
T r an s fer su bjectiv e into
objective goa ls
Chapter 2 Chapter 3 Chapter 4
sys te m design
ݔ ଵ ݔ ଶ ݔ ଷ

Figure 1.6: Structure of the w ork
v ehicle design . A set of design categories from v ehicle b eha viour has to b e designed with
sev eral design parameters. A t the b eginning all relev ant relations b et w een sub jectiv e and
ob jectiv e ev aluations are in v estigated. Then the resulting p erception thresholds are used
for the determination of solution space for a single middle-class v ehicle. An example prob-
lem is considered with only t w o design parameters to get the idea of solution space. The
complexit y increases with more design parameters, b o x-shap ed solution space is needed.
In the end, a pro duct family design for m ultiple v ehicles is computed. Eac h v ehicle has to
satisfy its requiremen ts on v ehicle b eha viour.
22

2 Deriving Quantitative Goals fo r
Objective Quantities
The v ehicle b eha viour is p erceiv ed sub jectiv ely b y customers. A desired v ehicle b eha viour
on sub jectiv e lev el can only b e ac hiev ed with virtual metho ds, if quan titativ e relations
b et w een sub jectiv e and ob jectiv e quan tities are kno wn. In this c hapter a new metho d
is presen ted to generate p erception thresholds. P artially , sub-metho ds are already made
public in [27]. A kno wledge managemen t system with sub jectiv e ev aluations is the base
of the metho d. A quan titativ e relation b et w een sub jectiv e and ob jectiv e quan tities will
b e determined b y a comparison of the sub jectiv e kno wledge managemen t system and an
ob jectified kno wledge managemen t system.
Definition 2.1 A design c ate gory in this c ontext is a vehicle pr op erty describing a
p art of the vehicle b ehaviour. A design c ate gory is subje ctively evaluate d with BI. The
design c ate gory is obje ctifie d by sever al obje ctive quantities . A n obje ctive quantity is for
example a char acteristic value of a char acteristic curve. A design p ar ameter is use d as
synonym for a functional pr op erty of a c omp onent.
Definition 2.2 In this c ontext, a p er c eption thr eshold is a r elative value and c orr e-
sp onds to a single obje ctive quantity. A design c ate gory may dep end on a single or sever al
obje ctive quantities. A p er c eption thr eshold of an obje ctive quantity describ es the change
of the obje ctive quantity to c ause a subje ctive change of 0.3 or 1.0 BI of the c orr esp onding
design c ate gory.
The sub jectiv e kno wledge is stored in a kno wledge managemen t system, e.g., a design
structure matrix. The con tained information ma y b e repro duced b y sim ulation, if the rel-
ev an t ob jectification kno wledge is kno wn, and is then called ob jectified kno wledge. In this
con text the ob jectification kno wledge is not kno wn completely . Only qualitativ e ob jectifi-
cation kno wledge is a v ailable. This means, first, all information ab out whether a relation
b et w een ob jectiv e quan tities and design categories exists and second, if m ultiple ob jectiv e
quan tities are in v olv ed within a relation, what is their ratio. Quan titativ e ob jectification
kno wledge is not a v ailable but to b e in v estigated. The quan titativ e ob jectification kno wl-
edge is gained b y the comparison of the patterns of sub jectiv e and ob jectified kno wledge.
The result is only of use, if the sub jectiv e and the qualitativ e quan tification kno wledge is
correct and the qualit y of the sim ulation mo del is at adequate lev el or v alidated otherwise.
Definition 2.3 F our typ es of know le dge ar e distinguishe d: Subje ctive know le dge is in-
formation ac c or ding to r elations b etwe en design p ar ameters and design c ate gories. The
know le dge is gaine d by exp ertise of senior test drivers b ase d on the subje ctive human p er-
c eption. Obje ctive know le dge is information ac c or ding to r elations b etwe en design p a-
r ameters and obje ctive quantities. The know le dge is gaine d by simulation or me asur ement
23

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
fr om physic al b ehaviour of the vehicle. Obje ctific ation know le dge is information ac-
c or ding to r elations b etwe en obje ctive quantities and design c ate gories. The know le dge is
gaine d by c orr elation studies or exp erts. Obje ctifie d know le dge is information ac c or d-
ing to r elations b etwe en design p ar ameters and design c ate gories, c omp ar able to subje ctive
know le dge. This know le dge is gaine d by the c ombination of obje ctive and obje ctific ation
know le dge. This c ate gorisation is similar to [71].
2.1 V ehicle Kno wledge Management System
The v ehicle kno wledge managemen t system of the BMW Group con tains subje ctive know l-
e dge . The managemen t system is a sustainable approac h to store sub jectiv e kno wledge and
is, e.g., presen ted in [71]. This managemen t system is a domain mapping matrix, whic h
sho ws the mapping b et w een t w o domains: design p ar ameters (ro ws) and design c ate gories
(columns). F or exemplary design categories of v ehicle dynamics design see, e.g., [42].
T able 2.1: V ehicle kno wledge managemen t
system: sub jectiv e information according to
relations b et w een design parameters and de-
sign categories.
design category 1
design category 2
. . .
design category M − 1
design category M
design v ariable 1 # < . . . <
design v ariable 2 < . . . <
. . . . . . . . . . . . . . . . . .
design v ariable D − 1 < . . . #
design v ariable D < # . . . #
The cells of the matrix con tain the im-
pact of a design p ar ameter according to a
design c ate gory , see T able 2.1. The sub jec-
tiv e kno wledge of the matrix w as filled in
b y driving exp erts. T o exclude preferences
of driving exp erts, the sub jectiv e ev aluation
of sev eral exp erts is a v eraged. Note, the in-
fluence measuremen t is relativ e and related
to the subje ctive evaluation sc ale using the
unit BI, see, e.g., [41]. The influence is clas-
sified according to the degree of influence:
# high impact [1 . 0 , ∞ [ BI: influence
greater than or equal to 1 BI
< medium impact [0 . 3 , 1 . 0[ BI: influence
greater than or equal to 0.3 BI and
smaller than 1 BI.
∅ lo w or no impact [0 , 0 . 3[ BI: influence
greater than or equal to 0 BI and
smaller than 0.3 BI. The sym b ol ∅ rep-
resen ts an empt y set.
Classical Purp ose for Design. The dev elopmen t of v ehicle design is often based on
predecessor v ehicles, therefore a relativ e sub jectiv e influence measuremen t is sufficien t to
get an idea of whic h design parameter is useful to impro v e a design category , e.g., relativ e
ab out 1 BI. Analysing a design category (column), all listed design parameters with high
impact to this design category can b e iden tified. A design v ariable (ro w) ma y influence
sev eral design categories. Designing a single design category , a design v ariable with less
impact to other design categories should b e c hosen. The impact measuremen t is undirected.
This means, e.g., a high impact indicates an increase or a decrease of ab out 1 BI.
Assumption ab out Sub jectiv e Information. Driving exp erts consider their well-
kno wn range of design parameters, when assigning sub jectiv e information to a sub jectiv e
24

2.2 Generation of Ob jectiv e Kno wledge
kno wledge system. Dep ending on the v ehicle class, a certain underlying range of the
design v ariable is imagined. T est driv ers are t ypically exp erts for a certain v ehicle class.
Within the class sev eral design parameters scatter within its design range. This has to b e
considered in the generation of ob jectiv e kno wledge.
Similar Researc h. In [71] the systematic kno wledge of a v ehicle kno wledgemen t system
is used for the dev elopmen t of a design pro cess. In comparison to the sub jectiv e kno wledge
ob jectified kno wledge w as also generated. The ob jectified kno wledge w as generated with
lo cal impact analysis and qualitativ e ob jectification kno wledge. In [46] and [70] n umerical
approac hes for impact analysis of c hassis comp onen t prop erties are prop osed: a linear lo cal
impact analysis w as used without the consideration of non-linear relations or in teractions.
Mäder [70] already requested for a more detailed in v estigation. F or this researc h, the fo cus
lies on the generation of p erceptual threshold v alues. Ho w m uc h influence do es a c hange
of an ob jectiv e quan tit y ha v e on the c hange of a sub jectiv e ev aluation?
Relativ e Sub jectiv e Ev aluation. The prop osed v ehicle kno wledge managemen t sys-
tem con tains relativ e information ab out sub jectiv e ev aluations. Driving exp erts consider
v ariations of all design parameters within a v ehicle class, including p ossible in teraction
whic h ma y o ccur. The underlying range of the design v ariable is essen tial for the relativ e
sub jectiv e ev aluation of the test driv ers. The distribution of the design parameters is un-
kno wn and is therefore considered as equally distributed. All v alues of design parameters
are equally probable. The design limits are sp ecific for eac h v ehicle class.
2.2 Generation of Objective Kno wledge
Ob jectiv e kno wledge will b e generated for the purp ose of comparing the pro cessed results
with the vehicle know le dge management system . The sub jectiv e kno wledge of driving
exp erts implies v ariations of all design parameters within a v ehicle class. Therefore a
metho d is c hosen whic h enables the v ariation of sev eral design parameters at the same time.
Non-linear relations of design parameters and ob jectiv e quan tities and the in teraction of
design parameters with eac h other raise the complexit y . This complexit y is considered
b y driving exp erts and therefore con tained in the vehicle know le dge management system .
Hence to this complexit y , a metho d for quan tification is necessary . The single steps are
listed:
• Determine the design parameters , whic h will b e analysed. Sp ecify the design limits
according to the v ehicle class. An analysis of actual and predecessor v ehicles ma y b e
useful.
• The design parameters are scattered with a design of exp erimen t metho d. V ari-
ations of all design parameters at one time are desired. This enables the analysis of
non-linear relations and in teraction with other design parameters to correlate with
the sub jectiv e p erception. Cho ose the probabilit y distribution of design parameters
and generate the sample. As the probabilit y distribution of the design parameters
ma y b e unkno wn or equal for, e.g., a follo w-up v ehicle design equal distributions of
the design parameters are prop osed. The effort of this metho d increases disprop or-
tionately as the necessary n um b er of sample p oin ts rises disprop ortionately to the
dimension of the design parameters D .
25

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
• Determine all ob jectiv e quan tities J . Ev aluate the generated sample with an ad-
equate sim ulation mo del for a relev an t set of mano euvres. Based on the resulting
v ehicle b eha viour, ob jectiv e quan tities are determined. This ma y b e a time consum-
ing task, but the computation of sample p oin ts can b e parallelised.
• The quan titativ e relation b et w een design parameters and ob jectiv e quan tities will b e
determined b y a sensitivit y analysis . A global analysis is desired whic h includes
non-linearit y and in teractions with other design parameters. A v ariance-based sensi-
tivit y analysis is prop osed, e.g., Sob ol Indices. The sensitivit y indices S corresp ond
to the ob jectiv e kno wledge (T able 2.2), whic h quan tifies the relation b et w een design
parameters and ob jectiv e quan tities.
T able 2.2: Ob jectiv e informations according to relations b et w een design parameters and
ob jectiv e quan tities.
ob jectiv e quan tit y 1
ob jectiv e quan tit y 2
. . .
ob jectiv e quan tit y J − 1
ob jectiv e quan tit y J
design v ariable 1 S 1 , 1 S 2 , 1 . . . S J − 1 , 1 S J , 1
design v ariable 2 S 1 , 2 S 2 , 2 . . . S J − 1 , 2 S J , 2
. . . . . . . . . . . . . . . . . .
design v ariable D − 1 S 1 ,D − 1 S 2 ,D − 1 . . . S J − 1 ,D − 1 S J ,D − 1
design v ariable D S 1 ,D S 2 ,D . . . S J − 1 ,D S J ,D
2.2.1 Design of Exp eriments Metho d
The aim of using design of exp erimen ts metho ds is to find out as m uc h as p ossible ab out the
relation b et w een system in- and outputs with the lo w est n um b er of exp erimen ts. Therefore
design of exp erimen ts metho ds aim to distribute exp erimen ts in the design space efficien tly
to analyse the output with statistic metho ds and get reliable and ob jectiv e conclusions [58].
In this case, design of exp erimen t metho ds are used for global sensitivit y analysis. Sieb ertz
et al.[102], Kleijnen [57] and Cheng and Druzdzel [14] sho w an o v erview of existing metho ds.
T o select an efficien t design of exp erimen t metho d, a short review ab out essen tial metho d
is giv en in T able 2.3.
Requiremen ts. In the case of this researc h, it is imp ortan t to generate a nearly ho-
mogeneous sample distribution to co v er the design space at an adequate lev el. This is
imp ortan t to decrease the probabilit y to misin terpret essen tial relations. F urthermore, in
industrial applications, it is essen tial to sp ecify the n um b er of exp erimen ts to determine
the effort, e.g., of the s im ulation. In addition, it is useful to expand or impro v e an existing
analysis through the reuse of already computed exp erimen ts.
Measure of Space Co v erage. The quality of the sample distribution can b e mea-
sured b y the discrepancy index, a measure of non-uniformit y of a quan tit y of p oin ts in a
26

2.2 Generation of Ob jectiv e Kno wledge
h yp ercub e [77]. The lo w er the discrepancy index is, the higher the qualit y of the sample
distribution. Imagine an arbitrary subspace A in the space Ω . F urther, imagine the ratio
of the n um b er of p oin ts N(A) in the subspace A according to the n um b er of designs N( Ω )
in the total space Ω . The discrepancy index is lo w, if this ratio is close to the ratio of the
measure, e.g., v olume of the subspaces V(A) according to the total v olume V. F or a fixed
space Ω the qualit y of the sample distribution increases with the n um b er of exp erimen ts, if
the exp erimen ts are distributed with lo w discrepancy (dep ending on design of exp erimen ts
metho d).
T able 2.3: Comparison of design of exp erimen ts metho ds
Metho d Description Ev aluation Graphical example (2D)
F ull-
factorial
sampling
Eac h design v ariable is
equally spaced discretized.
The n um b er of samples
increases exp onen tially to
the dimension of design
parameters.
A dvantages: Lo w est dis-
crepancy v alue, b est space
co v erage. Disadvantages:
Num b er of samples is
fixed, thereb y high ev alu-
ation effort.
−1 0 1
−1
−0.5
0
0.5
1
x 1
x 2

Mon te-
Carlo
sampling
Sto c hastic sampling
metho d. Often used and
w ell kno wn metho d, based
on random n um b ers. [102]
A dvantages: F ree choice
of the n um b er of exp er-
imen ts. Disadvantages:
Generation is not repro-
ducible. High discrepancy
v alue.
−1 0 1
−1
−0.5
0
0.5
1
x 1
x 2

Latin Hy-
p ercub e
Sampling
A dv ancemen t of Mon te-
Carlo sampling. Sto c has-
tic sampling metho d,
whic h generates the
samples random but
distributes them system-
atically to get a higher
space co v erage. [102]
A dvantages: F ree choice
of the n um b er of exp er-
imen ts. Disadvantages:
Generation is not repro-
ducible. Medium discrep-
ancy v alue.
−1 0 1
−1
−0.5
0
0.5
1
x 1
x 2

Sob ol Se-
quence
Deterministic but quasi-
random lo w-discrepancy
sequences. The sequence
use a base of partitions of
in terv al b et w een 0 and 1
and reorders the partitions
in eac h dimension. [73]
A dvantages: F ree choice
of the n um b er of exp er-
imen ts. Repro ducibilit y .
Lo w discrepancy and high
space co v erage. Disadvan-
tages: All designed exp er-
imen ts are necessary for
high space co v erage.
−1 0 1
−1
−0.5
0
0.5
1
x 1
x 2

Con tin ued on next page
27

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
T able 2.3 – con tin ued from previous page
Metho d Description Ev aluation Graphical example (2D)
Halton Se-
quence
Deterministic but quasi-
random lo w-discrepancy
sequences, that app ear
to b e random for man y
purp oses and uses prime
n um b ers as its base. [59]
A dvantages: F ree choice
of the n um b er of exp er-
imen ts. Repro ducibilit y .
Lo w discrepancy and high
space co v erage. Disadvan-
tages: All designed exp er-
imen ts are necessary for
high space co v erage.
−1 0 1
−1
−0.5
0
0.5
1
x 1
x 2

Result of Review. A full-factorial sampling creates the b est sample distribution but
requires a high n um b er of samples, whic h is exp onen tially related to the dimension of the
design parameters. Randomly-distributed metho ds suc h as Mon te-Carlo and Latin Hy-
p ercub e Sampling pro vide a lo w space co v erage and discrepancy . Deterministic sequences,
suc h as the Halton and Sob ol-Sequence, pro vide a b etter sample distribution in the design
space and, consequen tly , a b etter discrepancy . The Halton-Sequence is the b est c hoice for
the requiremen ts for this researc h. The Halton-Sequence generates the most homogeneous
sample distribution, while the n um b er of exp erimen ts is set arbitrarily .
2.2.2 Sensitivit y Analysis Metho ds
Sensitivit y analysis is used for the iden tification of significan t or insignifican t design v ari-
ables or the impro v emen t of understanding of systems. In this con text design parameters
will b e also called input p ar ameters of a system and the design ob jectiv es will also b e
called system outputs . Sensitivit y analysis aims at quan tifying the relativ e impact of input
parameters x i according to v alues of assigned output v ariables y [4].
Classification. A ccording to Saltelli et al. [99] in practice t w o differen t streams of sen-
sitivit y analysis can b e iden tified: lo c al and glob al sensitivity analysis . In lo c al sensitivity
analysis input parameters are v aried one-at-a-time . The resp onse of output is in v estigated
b y holding the other input parameters fixed to a nominal or baseline v alue. Sensitivit y
ma y then b e measured b y monitoring the c hange in the output, e.g., b y p artial derivative
or line ar r e gr ession . Linear regression co efficien ts can, e.g., b e generated b y Pe arson c or-
r elation co efficien ts m ultiplied b y the ratio of the standard deriv ation of y according to x.
The Pe arson c orr elation itself indicates only the linearit y of an in- and output relation.
These approac hes do not fully explore the input space, since they do not tak e in to accoun t
the sim ultaneous v ariation of input v ariables. In teractions b et w een input v ariables cannot
b e detected [16]. In glob al sensitivity analysis a space of input parameters is in v estigated.
The induced system output is analysed b y sorting the system output o v er eac h input pa-
rameter. The system output will scatter around an a v erage v alue as all input parameters
are scattered. V ariance based sensitivit y analysis in tends to estimate ho w m uc h output
v ariabilit y is dep enden t on eac h input v ariable. The observ ed total v ariance of system
output is partitioned in to comp onen ts induced b y the input parameters. This decomp o-
sition of the v ariance is usually called analysis of v ariances (ANO V A). Sob ol’s Indices are
in tended to represen t the sensitivities for general non-linear mo dels.
28

2.2 Generation of Ob jectiv e Kno wledge
The link b et w een deriv ativ e based sensitivit y measures and global sensitivit y indices is
presen ted in [112] and [113]. It is sho wn that for highly non-linear functions the rank-
ing of imp ortan t factors using deriv ativ e based imp ortance measures ma y suggest false
conclusions in con trast to global sensitivit y measures [112].
Link to similar Researc h. One-factor-at-a-time in v estigations w ere, e.g., used b y
Huemer et al. [46] and Mäder [70]. T o set up an efficien t c hassis dev elopmen t pro cess,
an understanding of the system is necessary to a v oid unnecessary design iteration lo ops.
Huemer et al. [46] used this metho dology on the analysis of the crosswind b eha viour of
passenger cars. A t an early stage of the dev elopmen t pro cess, a certain n um b er of c hassis
parameters are only defined at a minor grade of accuracy . Mäder [70] used one-factor-at-a-
time in v estigations for a design strategy for the la y out of lateral dynamics. Wimmer [126]
used v ariance-based sensitivit y analysis for an aid at the realization phase. F or a v alid
prognosis of the system la y out high system kno wledge and detailed parametrisation are
necessary . Based on v alidated sim ulation mo dels a global sensitivit y analysis with Sob ol
Indices for the steering system is done. Application parameters of the steering system w ere
adapted due to the global sensitivit y analysis.
Uncertain t y and Global Sensitivit y Analysis. Uncertain ties migh t b e the result
of a parametric estimation or differen t assumptions whic h migh t reflect differen t views of
realit y . Imagine, e.g., a driving exp ert who considers only parts of p ossible design ranges.
Go o d practice in sim ulation-based analysis is to analyse a neigh b ourho o d of alternativ es
and dra w global conclusions. In global sensitivit y analysis, uncertain ties for eac h input pa-
rameters are considered. [98] V ariance-based sensitivit y analysis enables global conclusions
and considers non-linear relationships. F or this researc h Sob ol’s Indices [111] are used to
describ e in teractions b et ween sev eral input parameters.
Review of Sob ol’s Indices
A ccording to [111], [110] and [99], v ariance-based sensitivit y indices are ratios b et w een the
conditional v ariance V ar ( E ( y | x i )) and the total v ariance V ar ( y ) of the output y
S i = V ar ( E ( y | x i ))
V ar ( y ) (2.1)
These ratios are called first-order sensitivit y indices and pro vide quan titativ e information
ab out the imp ortance of the input x i . Assuming a non-correlated input,
n
X
i =1
S i = 1 . (2.2)
Equation (2.2) holds true for additiv e mo dels. This leads to an easy quan titativ e in ter-
pretation, b ecause eac h S i deliv ers a direct measure for the p ortion of x i on the output
v ariance V ar ( y ) . F or non-additiv e mo dels, the in teraction among the input parameters has
to b e tak en in to accoun t, and a complete decomp osition of the function y in to non-linear
summands of increasing order is required. Analogous to Equation (2.2), this leads to
n
X
i =1
S i +
n
X
i =1
n
X
j > 1
S i,j + · · · + S 1 , 2 ,...,n = 1 . (2.3)
29

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
Sob ol’s pro cedure presen ts the function y = f ( x ) in the form of a high dimension mo del
represen tation (HDMR), with the parameters x i in single summands of increasing order:
f ( x ) = f 0 +
n
X
i =1
f i ( x i ) +
n
X
i =1
n
X
j >i
f i,j ( x i , x j ) + · · · + f 1 , 2 ,...,n ( x 1 , x 2 , . . . , x n ) . (2.4)
The constan t term f 0 is considered a zeroth-order function, where the terms f i ( x i ) are
first-order functions and the remaining terms are higher order functions. If the means of
the terms are zero, all terms will b e orthogonal and the mean of the function y is
E ( f ( x )) = f 0 . (2.5)
With Equation (2.5), all higher terms can b e determined:
E ( f ( x ) | x i ) = f i ( x i ) + f 0 (2.6)
f i ( x i ) = E ( f ( x ) | x i ) − f 0 = E ( f ( x ) | x i ) − E ( f ( x )) . (2.7)
Because of the orthogonal terms, the v ariance of the function can b e represen ted through
a functional v ariance separation
V ar ( f ( x )) =
n
X
i =1
V i +
n
X
i =1
n
X
j >i
V i,j + · · · + V 1 , 2 ,...,n . (2.8)
With Equation (2.6) the separated functional v ariances can b e expressed as
V i = V ar ( f i ( x i )) = V ar ( E ( f ( x ) | x i )) − V ar ( E ( f ( x ))
| {z }
=0
. (2.9)
Equation (2.9) divided through the total v ariance V ar ( f ( x )) concludes the Sob ol Indices:
1 =
n
X
i =1
S i +
n
X
i =1
n
X
j >i
S i,j + · · · + S 1 , 2 ,...,n (2.10)
Sob ol Indices ma y b e classified as main effect S H und total effect S T . The main effect
S i = S H,i = V ar ( E ( y | x i ))
V ar ( y ) (2.11)
is a measure of the direct influence of the input parameters. The sum of main effect is
limited: n
X
i =1
S H,i ≤ 1 . (2.12)
The total effect includes additional in teractions among the input parameters. Therefore
all Sob ol Indices, whic h consider x i , are added
S T ,i = S i +
n
X
j =1 ,j 6 = i
S i,j + · · · + S 1 , 2 ,...,n (2.13)
S T ,i = V ar ( E ( y | x i ))
V ar ( y ) +
n
X
j =1 ,j 6 = i
V ar ( E ( y | x i , x j ))
V ar ( y ) + · · · + V ar ( E ( y | x 1 , x 2 , . . . , x n ))
V ar ( y )
30

2.2 Generation of Ob jectiv e Kno wledge
With Equation (2.3) and the notation S − i as sum of Sob ol Indices without the index i , the
total Sob ol Index of the input x i can b e expressed as:
S T ,i =1 − S − i =1 − V ar ( E ( y | x − i ))
V ar ( y ) = E ( V ar ( y | x − i ))
V ar ( y ) (2.14)
The sum of total effect is n
X
i =1
S T ,i ≥ 1 (2.15)
as the in teraction b et w een input parameters are considered sev eral times.
Sob ol Indices as Best Choice for this Researc h. Significan t and insignificant
input parameters can b e distinguished without assumptions of linearit y . The measure
is indep enden t of the complexit y of the relation b et w een input x i and output y . The
quan tification is undirected and related to the total v ariance of the output analysed. A
v alue of S H ,i =0 . 3 states that 30 % of the total output v ariance is caused b y x i . Therefore
the Sob ol indices ha v e a simple in terpretation. T otal Sob ol indices consider also in teraction
b et w een inputs. A total Sob ol index S T ,i =0 . 4 states that 40 % of the total output v ariance
is caused b y x i or its in teraction with other input v ariables.
_ 

[ \(

\


[


[

_ 

[\ S

\

\

_ 

[\ S

  \




 _ 



\
[ \(
6
+





[

 _   
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


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[ \(



[

_   

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
_  

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_    _  


[

Figure 2.1: Graphical interpretation of the main Sob ol Index
Graphical In terpretation of the Main Effect S H . A graphical in terpretation of
the main effect S H can b e seen in Figure 2.1. Imagine a system output y dep ending on
the input parameters x 1 and x 2 . By sorting the output y o v er the parameter x 1 a scatter
plot (gra y transparen t) is generated. Assume a normal distribution of the system output
y , the distribution function p ( y | x 1 ) can b e describ ed b y the mean E ( y ) and the standard
deviation σ ( y ) . Based on the scatter plot, a floating mean E ( y | x i ) can b e determined, whic h
describ es the a v erage of the v ariation of y ove r x 1 . Assume also a normal distribution for
31

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
the floating mean E ( y | x i ) , the distribution function is describ ed b y the mean E ( y ) and
the standard deviation σ ( E ( y | x 1 )) . The main Sob ol index can b e expressed as ratio of the
standard deviations, whic h is represen tativ e for the width of the distribution function:
S H, 1 = σ ( E ( y | x 1 ))
σ ( y ) ! 2
. (2.16)
If the standard deviations are equal, the one and only relev an t parameter is x 1 and S H , 1 = 1 .
2.3 Generation of Objectified Kno wledge
2.3.1 Objectification Kno wledge
Design categories are often ob jectified b y m ultiple ob jectiv e quan tities. The ob jectification
in form of an analytical relation b et w een correlated ob jectiv e quantities a nd sub jectiv e
ev aluation is done often b y m ultiple linear regression analysis [135]. The p erception of the
driving exp erts ma y b e separated b y w eigh tings. Ob jectification kno wledge con tains this
information according to relations b et ween ob jectiv e quan tities and design categories. F or
this researc h, exp ertise w eigh tings w are used. The kno wledge can b e presen ted as a matrix
with elemen ts w m,j ∈ [0 , 1] . The elemen ts of non-correlating ob jectiv e quan tities according
to a design category are 0. F or the purp ose of in terpretation the sum of w eigh tings is equal
to 1.
J
X
j =1
w m,j = 1 for m ∈ [ 1 , M ] , (2.17)
where m is the design category index and M the dimension of the design categories.
T able 2.4: Qualitativ e ob jectification information according to relations b et w een ob jectiv e
quan tities and design categories.
design category 1
design category 2
. . .
design category M − 1
design category M
ob jectiv e quan tit y 1 w 1 , 1 w 2 , 1 . . . w M − 1 , 1 w M , 1
ob jectiv e quan tit y 2 w 1 , 2 w 2 , 2 . . . w M − 1 , 2 w M , 2
. . . . . . . . . . . . . . . . . .
ob jectiv e quan tit y J − 1 w 1 ,J − 1 w 2 ,J − 1 . . . w M − 1 ,J − 1 w M ,J − 1
ob jectiv e quan tit y J w 1 ,J w 2 ,J . . . w M − 1 ,J w M ,J
2.3.2 Objectified Kno wledge
T o generate ob jectified kno wledge, the sensitivities of the ob jectiv e quan tities ha v e to
b e aggregated to the sensitivities of the design categories. A traceable determination is
32

2.3 Generation of Ob jectified Kno wledge
desired to enable a high in terpretabilit y . The ob jectified matrix is created b y the execution
of the pro duct of the ob jectiv e matrix with the ob jectification matrix, see T able 2.5. The
ob jectified kno wledge or sensitivit y ˆ
S m,i of the design parameters i to the design quan tit y
m is determined b y:
ˆ
S m,i =
J
X
j =1
w m,j · S j,i . (2.18)
T able 2.5: V ehicle kno wledge managemen t system: ob jectified information according to
relations b et w een design parameters and design categories.
design category 1
design category 2
. . .
design category M − 1
design category M
design v ariable 1 ˆ
S 1 , 1 ˆ
S 2 , 1 . . . ˆ
S M − 1 , 1 ˆ
S M , 1
design v ariable 2 ˆ
S 1 , 2 ˆ
S 2 , 2 . . . ˆ
S M − 1 , 2 ˆ
S M , 2
. . . . . . . . . . . . . . . . . .
design v ariable D − 1 ˆ
S 1 ,D − 1 ˆ
S 2 ,D − 1 . . . ˆ
S M − 1 ,D − 1 ˆ
S M ,D − 1
design v ariable D ˆ
S 1 ,D ˆ
S 2 ,D . . . ˆ
S M − 1 ,D ˆ
S M ,D
2.3.3 Compa rison of Subjective and Objectified Kno wledge
T able 2.6 sho ws sub jectiv e kno wledge and ob jectified kno wledge next to eac h other. The
qualit y of sim ulation, ob jectification and sub jectiv e kno wledge is high, if the sub jectiv e
and ob jectified kno wledge are highly correlated. A quan titativ e corresp ondence analysis is
only p ossible, if the thresholds are kno wn to transfer ob jectified in to sub jectiv e information.
Note, if the patterns of the ob jectified and sub jectiv e kno wledge are similar, the threshold
v alues can only b e in a small ranges. The c hoice of the threshold v alues cannot impro v e a
correlation, if the basing pattern do es not fit.
33

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
T able 2.6: V ehicle kno wledge managemen t system: sub jectiv e and ob jectified information
according to relations b et we en design parameters and design categories.
design category 1
design category 2
. . .
design category M − 1
design category M
sub. ob j. sub. ob j. sub. ob j. sub. ob j. sub. ob j.
design v ariable 1 # ˆ
S 1 , 1 < ˆ
S 2 , 1 . . . . . . < ˆ
S M − 1 , 1 ˆ
S M , 1
design v ariable 2 ˆ
S 1 , 2 < ˆ
S 2 , 2 . . . . . . ˆ
S M − 1 , 2 < ˆ
S M , 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
design v ariable D − 1 < ˆ
S 1 ,D − 1 ˆ
S 2 ,D − 1 . . . . . . ˆ
S M − 1 ,D − 1 # ˆ
S M ,D − 1
design v ariable D < ˆ
S 1 ,D # ˆ
S 2 ,D . . . . . . # ˆ
S M − 1 ,D ˆ
S M ,D
2.4 Generation of P erception Thresholds
If the sub jectiv e and ob jectified kno wledge are correlated, a high sub ject impact # meets
a high ob jectified sensitivit y v alue ˆ
S . There will b e t w o threshold v alues of the ob jectified
sensitivit y: ˆ
S < for passing from lo w ( ∅ ) to medium ( < ) sub ject impact and ˆ
S # for passing
from medium ( < ) to high ( # ) sub ject impact. An o v erview of the pro cedure is sho wn in
Figure 2.2.
Determine perception
thresholds with 3
appro ximation
Determine threshold
v ariance from
Sobol Index de fin ition
Identify sensiti vity
threshold v alues
fr om comparison of su b .
and obj. k n ow ledge

 , #
ˆ
S

 , #
V ar


 , #
q



𝑞
𝑁 (𝑞)

Figure 2.2: Generation of p erception thresholds
F or in terpretation, imagine a c hange of an ob jectiv e quan tit y ∆ q # , whic h leads to a
sensitivit y ˆ
S # , then a c hange of 1 BI is induced to the corresp onding design category . A
c hange of an ob jectiv e quan tit y ∆ q < leads to a sensitivit y ˆ
S < and to a c hange of 0.3 BI of
the corresp onding design category .
Threshold Sensitivit y and Correlation. Based on the comparison of sub jectiv e and
ob jectified kno wledge in Section 2.3 the threshold v alues for eac h design category m are
34

2.4 Generation of P erception Thresholds
concluded. The sub jectiv e and ob jectified kno wledge ha ve to correlate, otherwise the v alues
of thresholds are not usable. The threshold v alues ˆ
S m, # and ˆ
S m,< will b e determined, so
that the error b et w een sub jectiv e and ob jectified kno wledge is minimal. It is appropriate
to assume that linear correlation is sufficien t. Therefore the Pe arson Corr elation Index is
used as measure. The threshold v alues enable to transfer the quan titativ e in to sub jectiv e
sensitivities ( # , < , ∅ ). These and the exp ertise sensitivities are then transferred with their
minimal BI c hange (1, 0.3, 0) to determine the correlation, as it is more imp ortan t to meet
the high as the lo w impacts.
Threshold V ariance. The threshold v ariance ˆ
S m, # is an aggregated sensitivit y for the
design category m and ma y b e also in terpreted as Sob ol T-index. A ccording to Equation
(2.14) this Sob ol H or T-index, whic h can b e expressed as:
ˆ
S # := V ar ( E ( y | x i ) + P n
j =1 ,j 6 = i ) V ar ( E ( y | x i , x j )) + · · · + V ar ( E ( y | x 1 , x 2 , . . . , x n ))
V ar ( Y )
:= V ar #
V ar ( Y ) . (2.19)
The v ariance
V ar # = V ar ( Y ) · ˆ
S # (2.20)
induces for the corresp onding design category a deriv ation of 1 BI. Analogously , the v ari-
ance
V ar < = V ar ( Y ) · ˆ
S < (2.21)
ma y b e expressed. The v ariance V ar < induces for the corresp onding design category a
deriv ation of 0.3 BI.
P erception Thresholds. A relation b et w een a single ob jectiv e quan tit y and a design
category can b e quan tified b y a threshold v ariance of the ob jectiv e quan tit y . Imagine a
normal distribution of the ob jectiv e quan tit y based on the design of exp erimen ts sample. In
App endix A.1 some output distributions can b e seen, whic h justify this assumption. These
distributions corresp ond to the example in Section 5.2.2. The cen tral limit theorem is that
the sum of a h uge amoun t of indep enden t and iden tically distributed random v ariables will
tend to b e normal distributed, if the v ariance of the v ariables is unev en to zero and finite
[33]. Therefore a normal distribution is an ticipated for the ob jectiv e quan tities.
A normal distribution is c haracterised b y the mean E and the standard deriv ation σ or
v ariance V ar . The v ariance V ar is mainly built b y the v alues whic h ha v e a great distance
to the mean E , b ecause this distance is squared. 99.73 % of all sample p oin ts lie within a
region of 3 · σ . Therefore a critical deviation of the ob jectiv e quan tit y ∆ q can b e expressed
as:
∆ q # = 3 q V ar # = 3 q V ar ( Y ) · ˆ
S # (2.22)
∆ q < = 3 q V ar < = 3 q V ar ( Y ) · ˆ
S < (2.23)
Definition 2.4 The thr eshold values ∆ q # and ∆ q < wil l b e c al le d p er c eption thr esholds .
E.g., a change of an obje ctive quantity ∆ q # le ads to a sensitivity ˆ
S # which c orr elates with
a change of the subje ctive evaluation of 1 BI. A change of the subje ctive evaluation of 1
BI may c orr elate with the human p er c eption of a customer.
35

2 Deriving Quan titativ e Goals for Ob jectiv e Quan tities
Relations with Multiple Ob jectiv e Quan tities. Imagine, t w o ob jectiv e quan ti-
ties q 1 and q 2 whic h are assigned to a design category . The aggregated sensitivit y ˆ
S is
determined according to Equation (2.18):
ˆ
S = w 1 · S 1 + w 2 · S 2 (2.24)
Unfortunately , the sub-sensitivities S 1 and S 2 cannot b e in v erted unam biguously , if only ˆ
S
is kno wn. If the ob jectiv e quan tities q 1 and q 2 are not correlated, the ob jectiv e quan tities are
indep enden t. A c hange of eac h ob jectiv e quan tit y can induce individually the threshold
sensitivit y ˆ
S . Therefore the ob jectiv e quan tities q 1 and q 2 are handled separately . The
threshold v alues ∆ q are concluded from the threshold sensitivit y and the w eigh ting.
S 1 . # = ˆ
S #
w 1
, S 2 , # = ˆ
S #
w 2
(2.25)
S 1 ,< = ˆ
S <
w 1
, S 2 ,< = ˆ
S <
w 2
(2.26)
If the quan tities q 1 and q 2 are correlated, the increase of the ob jectiv e quan tities q 1 go es
together with an increase of ob jectiv e quan tities q 2 . A relation of the sensitivities S 1 and
S 2 ma y b e gained b y a correlation relation:
S 2 = a · S 1 (2.27)
Then the sensitivities can b e separated:
S 1 . # = ˆ
S #
w 1 + w 2 · a , S 2 , # = a · S 1 . # (2.28)
S 1 ,< = ˆ
S <
w 1 + w 2 · a , S 2 ,< = a · S 1 ,< . (2.29)
Ob jectification with correlated ob jectiv e quan tities should b e a voided to reduce the com-
plexit y . T able 2.7 sho ws the p erception thresholds of ob jectiv e quan tities according to their
design category . The relation b et w een ob jectiv e quan tities and design category is exem-
plary . F or simplicit y , a quasi-diagonal form of the ob jectification matrix is considered.
T able 2.7: P erception thresholds determined for eac h design category . Relation b et w een
ob jectiv e quan tities and design category is exemplary .
Design category ob jectiv e quan tit y q V ar ( q ) ˆ
S > ˆ
S # ∆ q > ∆ q #
Design category 1 q 1 V ar ( q 1 ) ˆ
S 1 ,> ˆ
S 1 , # ∆ q 1 ,> ∆ q 1 , #
Design category 2 q 2 V ar ( q 2 ) ˆ
S 2 ,> ˆ
S 2 , # ∆ q 2 ,> ∆ q 2 , #
q 3 V ar ( q 3 ) ˆ
S 3 ,> ˆ
S 3 , # ∆ q 3 ,> ∆ q 3 , #
. . . . . . . . . . . . . . . . . . . . .
Design category M-1 q J − 2 V ar ( q J − 2 ) ˆ
S J − 2 ,> ˆ
S J − 2 , # ∆ q J − 2 ,> ∆ q J − 2 , #
Design category M q J − 1 V ar ( q J − 1 ) ˆ
S J − 1 ,> ˆ
S J − 1 , # ∆ q J − 1 ,> ∆ q J − 1 , #
q J V ar ( q J ) ˆ
S J ,> ˆ
S J , # ∆ q J,> ∆ q J , #
Graphical In terpretation. Imagine, the absolute BI ev aluation of, e.g., a predecessor
v ehicle is kno wn. The results of Equations (2.22) and (2.23) deliv er 2 · 2 grid p oin ts for
36

2.4 Generation of P erception Thresholds
the relation b et w een design category and ob jectiv e quan tit y . Within the grid p oin ts the
c haracteristic curv e is fixed. Figure 2.3 sho ws t w o p ossibilities of c haracteristic curv es.
The sensitivit y and the p erception thresholds are undirected. The qualitativ e relation can
b e estimated b y a Sp e arman c orr elation analysis b et ween objectiv e quan tit y and absolute
sub jectiv e ev aluation or b y driving exp ertise. If a correlation exists, a monotone relation
as sho wn in Figure 2.3(a) ma y b e dra wn. The sign of the correlation decides the direction
of the gradien t. Otherwise, Figure 2.3(b) sho ws non-monotone relation.
0
q

q

0
BI

#
q 


 q


 q

#
q 

BI

0
BI

0
BI

+1
-1

(a) monotone relation: the ob jectiv e quan tity
and the sub jectiv e ev aluation are correlated
0
q

q

0
BI

#
q 


 q


 q

#
q 

BI

0
BI

-1

(b) non-monotone relation: the ob jectiv e quan tit y
and the sub jectiv e ev aluation are not correlated
Figure 2.3: Relation b et w een ob jectiv e quan tit y and the sub jectiv e ev aluations
The Figures 2.3(a) and (b) sho w only parts of the relation b et w een ob jective quan tit y
and the sub jectiv e ev aluation around the op eration p oin t. F or differen t op eration p oin t of
the same relation b et w een ob jectiv e quan tit y and the sub jectiv e ev aluation differen t t yp es
of relation ma y o ccur.
Data Fitting. Man y problems of data-fitting result of an insufficien t data prepro cessing
or of a systematic distortion of data. These problems are often of statistical nature and
ha v e to b e solv ed b efore data collection or generation: from non-represen tativ e data non-
represen tativ e results ma y b e gained. If the sub jectiv e and ob jectified kno wledge are
correlated, high sub ject impacts # meet high ob jectified sensitivit y v alues, medium sub ject
impacts < meet medium ob jectified sensitivit y v alues and lo w sub ject impacts meets lo w
ob jectified sensitivit y v alues. The base for this metho d is that the patterns of sub jectiv e and
ob jectified kno wledge matc h. Then useful and reliable p erception thresholds are generated.
The threshold v alues can only b e adopted in a certain range. There is no use for an
optimizer, if the sub jectiv e and ob jectified kno wledge are non-correlated. The more design
parameters are considered, the more exact the threshold of the ob jectiv e quan tit y can b e
determined. If driving exp erts ev aluate m ultiple design comp onen ts sub jectiv ely , they ha v e
p erception thresholds in mind, whic h can b e sough t out.
In Chapter 5.2 the metho d is applied to a lateral and v ertical dynamics design problem
of a middle class v ehicle.
37

3 Robust System Design using Solution
Space
Robustness and flexibilit y is desired in the conceptual la y out pro cess in the automotiv e in-
dustry . Solution space design is prop osed as metho d that in tegrates robustness in a simple
manner, without an additional robustness ob jectiv e. This pro cedure is a mind c hange in
the conceptual design phase: not the p eak p erformance of a system is desired, but a set of
designs whic h satisfies requiremen ts on system b eha viour and considers uncertain ties.
Definition 3.1 Design p ar ameters have to satisfy r e quir ements on system b ehaviour. R e-
quir ements on the system b ehaviour c an b e describ e d by thr eshold values for system p er-
formanc e. The r esult is a set of p ossible values of design p ar ameters. In the p ar ameter
sp ac e this set has a sp ac e with a volume and normal ly non-line ar b oundaries. This sp ac e
is c al le d solution sp ac e and may b e sep ar ate d.
Solution space design is used as robust design metho d in the conceptual la y out pro cess.
Solution space implies a robustness as in tegral part according to v ariations of design or
uncon trollable parameters. If requiremen ts on the system p erformance are made, a set
of design parameters ma y satisfy these requiremen ts. A cen tral design parameter config-
uration in this design parameter set deliv ers the maxim um robustness, satisfying all the
requiremen ts on the system p erformance. Dep ending on the distribution of the design or
uncon trollable parameters, the c hoice of a cen tral design p oin t within the solution space
is the a robust design p oin t. In con trast to r eliable design the distribution of the parame-
ters is not required or fixed from the b eginning. In real-w orld applications the n um b er of
comp eting ob jectiv es can b ecome o v erwhelming. Solution spaces for single requiremen ts
of ob jectiv e quan tities enable the sup erp osition and therefore the domination of outp er-
formed designs. Solution space deliv ers a high flexibilit y in the early phase as all p ossible
v alues of design parameters are determined. If no solution space can b e determined after
sup erp osition, a conflict of goals can b e detected early . Requiremen ts ha v e to b e c hanged
or further design parameters can b e included, if not all relev an t design parameters w ere
considered. Requiremen ts on the system b eha viour ma y c hange during the dev elopmen t
pro cess. Solution space design can b e used as top-do wn analogue to the theory of tar-
get c asc ading to reduce the complexit y . Resulting in terv als of design parameters are the
b oundaries or requiremen ts of the subsystem on the next system lev el. Imagine a b o x-
shap ed solution space, whose edges are in terv als of design parameters. In a b o x-shap ed
solution space these in terv als are indep enden t of eac h other and therefore decouple the
design parameters. The separated in terv als and design parameters ma y b e giv en to sev eral
exp ert design teams without high efforts of co ordination as prop osed in set b ase d design .
Uncertain ties in the Conceptual La y out Pro cess. Uncertain ties of design parame-
ters are large in the early dev elopmen t phase and need to b e included in the design concept.
During the course of time, kno w-ho w is generated and decisions are made. Characteristics
38

of the comp etitors and the o wn brand enable a step to w ards less uncertain t y , see Figure
3.1 at the marking p oin t pr o of of c onc ept . After the start of pro duction, uncertain ties of
the comp onen ts and the op erating p oin ts remain, whic h ha v e to b e within tolerances.
process
components

t

operating
points
uncertainty
concept series production
e arly phase late phase
proof of
concept
concept
start
start of
production

Figure 3.1: Characteristic curv es of uncer-
tain ties in the dev elopmen t pro cess.
The uncertain ties in the dev elopmen t
phase can b e clustered:
• unc ertainties of the pr o c ess : in the
early phase sev eral parameters or
comp onen ts are not kno wn. An es-
timation is used.
• unc ertainties of the c omp onents :
comp onen ts will b e realized dif-
feren t from their design. Realized
comp onen ts v ary with a sp ecific
distribution.
• op er ating c onditions : loads, t yre
c haracteristics and en vironmen t con-
ditions v ary .
In the early dev elopmen t phase high
uncertain t y o ccurs. Solution space pro-
vides robustness with resp ect to parameter
c hanges. Robust designs ha v e to b e determined as uncertain ties of comp onen ts and op er-
ating p oin ts remain.
T yp es of Uncertain t y . Uncertain t y can b e classified in to t w o categories according to
[74] and [56]:
Definition 3.2 A statistic al unc ertainty is also c al le d ale atory unc ertainty . This typ e
of unc ertainty is natur al ly pr esent and is also known as irr e ducible unc ertainty. The sys-
tem b ehaviour differs e ach time an exp eriment is run. This c annot b e r emove d by mor e
ac cur ate me asur ements. Examples ar e manufacturing unc ertainties or influenc es of the
envir onment such as temp er atur e fluctuations. A le atoric unc ertainties c an b e quantifie d by
its moments. A Gaussian distribution is describ e d by the me an and varianc e value. Epis-
temic unc ertainty is a systematic unc ertainty. This typ e of unc ertainty is due to a lack
of know le dge and is also c al le d r e ducible unc ertainty. The unc ertainty c ould b e known but
in pr actic e it is unknown. Imagine, the system b ehaviour is not me asur e d or simulate d with
a sufficient ac cur acy. Design variables may change during the design pr o c ess or the c onfig-
ur ations ar e not known yet. Base d on high know le dge of the system or pr o c ess, epistemic
unc ertainties may b e r e duc e d or a quantific ation c an b e assume d.
Uncertain t y In tegration in the System Beha viour. Bey er and Sendhoff [7] review
the in tegration of uncertain ties in the system b eha viour. A comp onen t or designed v ari-
able x i is realized differen tly as designed (epistemic) or fluctuates due to the man ufacturing
pro cess (aleatoric). A c hange of the design parameter δ x i has to b e considered. Uncon trol-
lable parameters p u , with the index u = 1 , ...U , s u c h as differen t loadings (epistemic) or
temp erature fluctuation (aleatoric) cannot b e designed. Uncon trollable parameters do not
39

3 Robust System Design using Solution Space
influence the design parameters x i but ha v e an effect on the system b eha viour f . Noise or
a random c hange of uncon trollable parameters δp u has to considered.
f = f ( x i + δx i ,p
u + δp u ) . (3.1)
The distribution of uncertain ties of design or uncon trollable parameters is often not exactly
kno wn. F or unkno wn uncertain ties an uniform distribution can b e assumed [43]. F or
completeness, uncertain t y of the system b eha viour is also p ossible, but neglected here.
Uncertain ties based on measuremen ts, assumptions, simplifications or appro ximations can
b e describ ed b y a transformation function ˜
f . The sim ulation mo del has to b e at an
adequate lev el for the design purp ose, therefore this t yp e of uncertain t y is neglected.
3.1 Principles of Solution Space
Definition 3.3 A set of design p ar ameter values satisfies al l r e quir ements on the system
b ehaviour for an arbitr ary non-line ar and high-dimensional system. This set of design
p ar ameter values is c al le d a set of go o d designs . A l l go o d designs ar e in a r e gion, which
is c al le d go o d r e gion. Ther e may exist multiple sep ar ate d go o d r e gions. Go o d designs ar e
not distinguishe d by their system p erformanc e. A l l satisfying, go o d designs ar e c onsider e d
as e qual. The r emaining designs, which do not fulfil the r e quir ements on system b ehaviour,
a re ca l l ed b ad designs . A l l b ad designs ar e in a r e gion, which is c al le d b ad r e gion. The
b ad designs ar e outp erforme d by the go o d designs.
good region
bad region

[

M
]


[



[

F U L WM ] 

Figure 3.2: Determination of a 2D solution space
Figure 3.2 sho ws a non-linear ob jectiv e quan tit y z j , whic h dep ends on t w o design param-
eters x 1 and x 2 . There is a requiremen t on the system b eha viour, the ob jectiv e quan tit y
z j has to b e less than the critical v alue z j,cr it :
z j ≤ z j,cr it . (3.2)
The gra y regions in Figure 3.3 represen t design p oin ts, whic h violate Equation (3.2).
Hence, the design parameters whic h create this system b eha viour are significan t. Therefore
40

3.1 Principles of Solution Space
the go o d system b eha viour is pro jected in the design parameter space. The pro jected
b oundaries of the system b eha viour distinguish the go o d from the bad region. All go o d
designs in the go o d region are equal, no single design p oin t is preferred. Figure 3.3 sho ws
the design parameter space of Figure 3.2. The b oundaries of the go o d region are non-
linear. The go o d region ma y ha v e inden tations and ma y b e separated in to sev eral regions.
Imagine a single v alue of the design parameter x 1 , than an in terv al of design parameter x 2
is p ossible to fulfil the requiremen ts on the system b eha viour. This in terv al dep ends on
the shap e of the solution space, see the gra y and blac k in terv al. The v alues or in terv als of
the design parameters can also b e dra wn next to eac h other in a separate figure.
good region
bad region

2
x

1
x

2 1 , x x

1
x

ub
x '
2

lb
x '
2

'
1
x

2
x

' '
1
x

lb
x ' '
2

ub
x ' '
2

'
1
x

' '
1
x

lb
x '
2

lb
x ' '
2

ub
x '
2

ub
x ' '
2

Figure 3.3: Solution space (2-D) and in terv al displa y of design parameters
If in terv als for design parameters x 1 and x 2 are desired, then a b o x-shap ed solution space
is required, see Figure 3.4. The edges of the b o x can b e in terpreted as the in terv als of the
design parameters. The in terv al for design parameter x 1 has a lo w er x lb
1 and an upp er limit
x up
1 . A b o x-shap ed solution space is not unam biguous. The b o x ma y b e shap ed to high
in terv als of x 1 or x 2 . The b o x ma y b e also shap ed in a w a y that the v olume is maximized
and deliv ers the greatest design space. Next to the design parameter space of Figure 3.4
the in terv als of the design parameters x 1 and x 2 are sho wn.
Definition 3.4 The p ar ameter intervals ar e dr awn next to e ach other and c ombine d for
r e asons of visualization. The p ar ameters may have differ ent units. This c ombination of
intervals is also r eferr e d to as c orridor . Each design within the c orridor is a go o d design.
Each c ombination of design p ar ameters within the intervals of a b ox-shap e d solution b ox
satisfies al l r e quir ements. A ny design within these intervals is a go o d design. A design
outside these intervals may b e a go o d or b ad design as the solution sp ac e is in gener al not
b ox-shap e d.
41

3 Robust System Design using Solution Space
good region
solution box
with largest
volu me
bad region

[


[


 [[


[

XE
[


OE
[ 

OE
[


XE
[



[

XE
[ 

OE
[


OE
[


XE
[


Figure 3.4: Solution space (2-D), b o x-shap ed solution space and in terv al displa y of design
parameters
Generation of 2-D Solution Space. If the n um b er of design parameters is lo w, e.g.,
equal to 2, the whole design space can b e sampled with design of exp erimen ts tec hniques
as, e.g., describ ed in Section 2.2.1. Designs whic h do not satisfy all system requiremen ts
are bad designs and mark ed gra y . The designs whic h satisfy all requiremen ts are go o d and
mark ed white. The shap e of the solution space is determined b y the relation of in- and
outputs and the output threshold v alues.
3.2 Example shap es of Solution Spaces (2-D)
The kno wledge of example shap es of solution space and the correlating in terv al visualiza-
tion ma y help in man y practical tasks for ph ysical in terpretation. The b ounded design
parameter in the parameter space can b e expressed explicitly or implicitly . The implicit
expression of the design parameters ma y giv e a simpler in terpretation in connection with
inequalit y constrain ts whic h limit the go o d region. T able 3.1 giv es an o v erview of simple
shap es.
42

3.2 Example shap es of Solution Spaces (2-D)
T able 3.1: Example shap es of 2-D solution spaces and their cor-
resp onding parameter corridors
P arameter Space Description
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameter x 1 is a non-relev an t design pa-
rameter as the design space is not limited. The v alue
of design parameter x 1 has no influence on the satis-
faction of the requiremen ts on the system b eha viour.
Design parameter x 2 is relev an t and limited to b oth
sides: α ≤ x 2 ≤ β .
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameters x 1 and x 2 are relev an t. The shap e
of the b oundary can b e expressed explicitly b y a lin-
ear equation: x 1 = − a · x 2 + b , where a is the gra-
dien t and b an offset. The shap e can b e expressed
implicitly b y a sum of the design parameters :
x 1 + a · x 2 = b . The solution space is limited due to
2 inequalit y constrain ts: α ≤ x 1 + a · x 2 ≤ β .
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameters x 1 and x 2 are relev an t. The shap e
of the b oundary can b e expressed explicitly b y a lin-
ear equation: x 1 = a · x 2 + b , where a is the gradient
and b an offset. The shap e can b e expressed implic-
itly b y a difference of the design parameters :
x 1 − a · x 2 = b The solution space is limited due to 2
inequalit y constrain ts: α ≤ x 1 − a · x 2 ≤ β .
Con tin ued on next page
43

3 Robust System Design using Solution Space
T able 3.1 – con tin ued from previous page
P arameter Space Description
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameters x 1 and x 2 are relev an t. The shap e
of the b oundary can b e expressed explicitly as a h y-
p erb ola: x 1 = a
x 2 + b , where a is a scaling factor
and b is an offset. The shap e of the b oundary can
b e expressed implicitly as a pro duct of the de-
sign parameters : ( x 1 − b ) · x 2 = a . The solu-
tion space is limited due to 2 inequalit y constrain ts:
α ≤ ( x 1 − b ) · x 2 ≤ β . T o simplify the implicit in ter-
pretation x 0
1 = x 1 − b ma y b e in terpreted as parameter
transformation.
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameters x 1 and x 2 are relev an t. The shap e
of the b oundary can b e expressed explicitly b y a lin-
ear equation: x 1 = a · x 2 + b , where a is a gradient and
b is an offset. The shap e can be expressed implicitly
b y a ratio of the design parameters : x 1 − b
x 2 = a .
The solution space is limited due to 2 inequalit y con-
strain ts: α ≤ x 1 − b
x 2 ≤ β . Note, x 0
1 = x 1 − b ma y b e
in terpreted as a parameter transformation.
2
x

1
x

2 1 , x x

1
x

ub
x 2

lb
x 2

lb
x 1

ub
x 1

2
x

ub
x 1

lb
x 1

lb
x 2

ub
x 2

Design parameters x 1 and x 2 are relev an t. The shap e
of the b oundary can b e expressed implicitly as an
ellipse of the design parameters : x 2
1
a 2 + x 2
2
b 2 = 1 or
x 2
1 +  a
b · x 2  2 = a 2 , where a is the radius and a
b is
the factor for scaling in direction of x 2 . The solution
space is limited due to an inequalit y constrain t: x 2
1 +
 a
b · x 2  2 ≤ β 2 .
There are more shap es of solution spaces, whic h ma y b e describ ed analytically . P arame-
ter transformation and v ariation of the sign of gradien ts, factors or offsets enable m ultiple
shap es. Eac h parameter next to the design parameters as gradien ts, factors or offsets can
b e isolated with in an implicit expression and limited b y b oundaries. Complex b oundaries
of solution space or tra jectories in parameter space are explicit solutions of differen tial
equations.
44

3.3 Robustness and Solution Space
3.3 Robustness and Solution Space
T aguc hi et al. [115] defines robustness as a state where the tec hnology , pro duct or pro cess
p erformance is minimally sensitiv e to factors causing v ariabilit y . Man y researc hers ha v e
defined robust design. All definitions of robust design ha v e in common, that the designed
pro duct should b e insensitiv e to external noises or tolerances.
Definition 3.5 In the sense of solution sp ac e, a r obust design is not a design which min-
imizes variations of the p erformanc e function. Designs have to fulfil r e quir ements on the
system b ehaviour, which c an b e expr esse d as ine quality c onstr aints. A l l designs, which sat-
isfy al l r e quir ements, ar e c onsider e d as e qual. Ther efor e, a r obust design in the sense of
solution sp ac e is the design which has the gr e atest distanc e to the r e gion of b ad designs.
This distance ma y b e defined b y the designer. If uncertain ties of parameters are known
or can b e assumed, the uncertain ties can b e quan tified as in terv als. If the uncertain ties of
parameters are unkno wn, an equal distribution is used. If the uncertain ties of parameters
ha v e a distribution as the normal distribution, a probable in terv al can b e estimated b y
plus min us 3 standard deviation σ around the mean v alue µ . As the parameters are
indep enden t, it is assumed that their uncertain ties are indep enden t, to o. The satisfaction
of the requiremen ts has to b e assured for all com bination of uncertain ties. Therefore the
cen ter of a b o x-shap ed solution space is prop osed as robust design. The in terv als of the
b o x ha v e to include the in terv als of the uncertain ties. Without kno wledge of uncertain ties
the design of a b o x-shap ed solution space suggests in terv als for uncertain ties. As the b o x
is v ariable within the solution space, the design exp ert ma y find a b o x whic h satisfies her
or his requests. Without design decisions, b o xes with the maximal v olume or a maximal
minimal in terv al can b e used.
3.3.1 Robustness against Design P a rameter Uncertaint y
The robustness against uncertain ties of design parameters is included in a simple manner
in the solution space approac h, see Figure 3.5. F or the determination of the solution space
no information of desired robustness is needed.
Definition 3.6 If the unc ertainties of design p ar ameters ar e indep endent and may b e
b ounde d by limits, a design x ∗
i in the c enter of a b ox-shap e d solution sp ac e wil l b e c al le d
r obust .
This definition of robustness is comparable to reliabilit y , but no probabilit y of failure in
dep endence of a parameter distribution is computed. The distance to requiremen t violation
can b e quan tified b y r i . In the pure in terv al visualization of the design parameters the
robust design has the mean v alue of eac h in terv al of design parameters.
3.3.2 Robustness against Uncontrollable P a rameter Uncertaint y
Imagine the extension of Figure 3.5 with an uncon trollable parameter p u with u = 1 .
In Figure 3.5 the v alue of p 1 w as, e.g., p 1 , 0 . In Figure 3.6 the in terv als of the design
parameters w ere extended b y the uncon trollable parameter p 1 with the v alue p 1 , 0 (gra y
visualization). With p 1 = p 1 , 0 the b o x and the in terv als are iden tical to Figure 3.5. Imagine
45

3 Robust System Design using Solution Space
good region
solution box
with largest
volu me
bad region

[


[


 [[


[

XE
[ 

OE
[


OE
[


XE
[



[

XE
[ 

OE
[


OE
[ 

XE
[



U


U


U



[



[



[



[


U

Figure 3.5: Robust design for uncertain ties of design parameters: solution space (2-D),
in terv als of design parameters and advice of robust design
good region
solution box
with largest
volu me
bad region

[


[


 [[


[

XE
[ 

OE
[


OE
[


XE
[



[

XE
[ 

OE
[


OE
[ 

XE
[




[



[

 
S


S

OE
S


XE
S


 
S

XE
S


OE
S


Figure 3.6: Robust design for uncertain ties of uncon trollable parameters: solution space
(2-D), in terv als of design parameters and advice of robust design
46

3.4 Concurren t Design
an uncertain t y in terv al for the uncon trollable parameter p 1 with the lo w er b ound p lb
1 and
the upp er b ound p lb
1 . Solution spaces with p 1 = p l b
1 and p 1 = p ub
1 are also sho wn in Figure
3.6. The set of design parameters, whic h satisfy the requiremen ts indep enden tly of the
v alue of p 1 ∈ [ p l b
1 , p ub
1 ] , is reduced (blac k b o x). The in terv als of the design parameters w ere
extended b y in terv als of the uncon trollable parameter.
Definition 3.7 Unc ontr ol lable p ar ameters p u c an b e extende d next to design p ar ameters
in the interval visualisation of the p ar ameters. If lower and upp er b ounds [ p lb
u , p ub
u ] ar e
known, the interval for the unc ontr ol lable p ar ameters c an b e fixe d.
In 2 dimensional plots, the visualization of more than 3 dimensions is hardly p ossible,
but the in terv al visualization is not b ound to restrictions. All design and uncon trollable
parameters are listed next to eac h other. Ho w ev er almost all information ab out the shap e
of the solution space is lost. Within the corridor requiremen t satisfaction is guaran teed,
outside the corridor requiremen t satisfaction ma y b e p ossible for single designs.
3.4 Concurrent Design
The concept of concurren t engineering is to consider all elemen ts of a pro duct life cycle
as functionalit y , pro ducibilit y , man ufacturing, supp ort and recycling. Eac h element has
requiremen ts to the pro duct. In practice the n um b er of requiremen ts is o v erwhelming.
A ccording to [63] all elemen ts or requiremen ts should b e tak en in to consideration in the
early design phase. Another concept of concurren t engineering is that all design activities
should b e done at the same time. This is wh y it is called concurren t. Concurren t design
ma y increase significan tly the pro ductivit y and the qualit y of a pro duct [91]. T arget
conflicts or pro duct errors can b e disco v ered early in the design pro cess when a pro ject
is still flexible. If design exp erts iden tify and solv e conflicts of goals in the early phase,
cost in tensiv e re-design in the late design phase can b e a v oided as the pro ject mo v es to
more detailed computational mo dels or first man ufactured hardw are [63]. Due to the high
n um b er of engineering tasks, sev eral design teams ha v e to op erate sim ultaneously . F or the
realization of concurren t engineering, it is necessary to share information in a manner that
mak e decision making simple.
Solution space enables flexibilit y as m ultiple designs are considered in the early phase.
F urther, the use of b o x-shap ed solution space enables the decoupling of design parameters,
whic h ma y b e requiremen ts of further subsystems of differen t design teams. Imagine a top-
do wn design metho dology , eac h in terv al of a design parameter on the top lev el is a require-
men t on the subsystem b eha viour on the sub-lev el. Eac h requiremen t on the subsystem
b eha viour can b e designed sim ultaneously b y a separate design team. The sup erp osition
of solution spaces enables the handling of the high n um b er of system requiremen ts at the
same time without compromising with w eigh ts or increasing the complexit y . Imagine t w o
design teams for t w o design criteria, the result of the first design team is sho wn in Figure
3.4 and the result of the second design team are restrictions according to the difference of
the design parameters, see Section 3.2. Figure 3.7 sho ws the sup erp osition of the solution
spaces of the t w o design teams. Go o d region ma y b e outp erformed b y additional require-
men ts on the system b eha viour. In this case the resulting go o d region after sup erp osition
is smaller than the single go o d region. The b o x-shap ed solution space of the single design
tasks are gra y . A b o x-shap ed solution space with maximal v olume whic h is blac k cannot
47

3 Robust System Design using Solution Space
good region
solution box
with largest
volu me
bad region

[


[


 [[


[

XE
[ 

OE
[


OE
[


XE
[



[

XE
[ 

OE
[ 

OE
[ 

XE
[


Figure 3.7: Concurren t engineering: sup erp osition of design criteria and solution spaces
(2-D)
b e concluded from the b o x-shap ed solution spaces of the single design tasks. The blac k
b o x-shap ed solution space lies partially outside of the gra y b o xes. F or a maxim um degree
of freedom, a b o x-shap ed solution space should b e determined after all requiremen ts of
parallel design tasks are considered.
F or simplification, a resulting b o x-shap ed solution space ma y b e gained b y the sup erp o-
sition of b o x-shap ed solution spaces from single design tasks. The p osition of b o x-shap ed
solution spaces is v ariable within the solution space. The b o x-shap ed solution space of sin-
gle design tasks ma y b e sub-optimal for sup er-p osition and w aste flexibilit y . Nev ertheless,
a significan t adv an tage is that the n um b er of system requiremen ts do es not increase the
complexit y of design as all requiremen ts can b e sup erp osed.
3.5 Solution Space fo r High-dimensional Design
3.5.1 Seeking Bo x-shap ed Solution Space
Real-w orld design problems ha v e t ypically more than t w o design parameters. In high
dimensions, a solution space of arbitrary shap e cannot b e visualized. A solution space
ma y b e determined analytically , if the system b eha viour is analytically describ ed. The
sampling tec hnique of the previous section is not applicable an ymore.
High Dimensions. Imagine for eac h design parameter, 50 % of a design space in terv al
lead to a go o d design. F or 10 dimensions of design parameters appro ximately 1 design
p oin t out of 10 3 is a go o d design ( 0 . 5 10 =9 . 8 · 10 − 4 ).
Ho w ev er, ev en in high-dimensions, b o x-shap ed solution space can b e computed, as sho wn
in [133]. This approac h is review ed shortly .
Review of Definitions. The design parameters are x i with the design parameter index
i =1 , ..., D , where D is the total n um b er of design parameters. The design ob jectiv es are z j
48

3.5 Solution Space for High-dimensional Design
with j = 1 , ..., J , where J is the total n um b er of ob jectiv e quan tities. Design constrain ts are
g k with k = 1 , ..., K , where K is the total n um b er of constrain ts. In classical optimisation
problems, the p erformance z is optimised, i.e., a design x ∗
i is sough t that yields an extreme
p erformance v alue of z ∗ while fulfilling some constrain ts g k ( x ∗
i ) ≤ 0 . In the approac h
presen ted in [133], the p erformance only has to b e sufficien t b y satisfying the inequalit y
constrain ts
z j ( x i ) ≤ z j,c (3.3)
with the b oundary v alues z j,c , whic h are either lo w er b ounds z l b
j or upp er b ounds z ub
j .
The ob jectiv e quan tities are condensed to a system p erformanc e :
z = max
j { z j − z ub
j
z ub
j
, z lb
,j − z j
z lb
,j
} , j = 1 , . . . , J. (3.4)
The system p erformance z has the follo wing prop erties:
• z quan tifies the o v erall p erformance as the w orst p erformance.
• z ≤ 0 indicates a go o d design, i.e., a system satisfies the design goals.
• The lo w est v alue of z indicates the design that has the largest distance to the critical
threshold v alues.
Constrain ts g k ( x i ) can b e included in this expression. Designs whic h satisfy Equation
(3.3) are called go o d designs, those whic h do not are called bad designs. This allo ws a
classification of the design space in to go o d and bad. The shap e of the region of go o d designs
dep ends on the system resp onses and the p erformance criteria giv en b y the p erformance
functions z j .
Seeking Bo x-shap ed Solution Space. Rather than resolving arbitrary shap es of the
go o d design space, only b o x-shap ed sub-spaces Ω are sough t. Bo x-shap ed solution space
is the Cartesian pro duct of p ermissible in terv als I i = [ x lb
i , x ub
i ] for eac h design parameter
x i , where the lo w er and upp er b oundaries are denoted as x lb
i and x ub
i , resp ectiv ely . The
v olume of the b o x is giv en b y
V (Ω) =
D
Y
i =1
( x ub
i − x lb
i ) . (3.5)
Figure 3.4 sho ws a b o x-shap ed solution space. Out of man y p ossible b o x-shap ed solution
b o xes, the one with largest v olume is sough t.
Problem Statemen t for the Iden tification of Bo x-shap ed Solution Space. F or
a giv en design space Ω ds , seek a b o x Ω ⊂ Ω ds suc h that
V (Ω) → max (3.6)
sub ject to z j ( x i ) ≤ z j c for all x i ∈ Ω (3.7)
The degrees of freedom for this optimisation problem are the in terv al b oundaries x l b
i and
x ub
i , i = 1 , . . . , D . All design p oin ts within the b o x ha v e to fulfil Equation (3.7). In order
to ev aluate expression 3.7 for candidate b o xes Ω for arbitrary non-linear high-dimensional
problems with p erformance functions f that are computed n umerically , a tec hnique from
Lehar and Zimmermann [66] is adopted that relies on Mon te Carlo Sampling and Ba yesian
statistics.
49

3 Robust System Design using Solution Space
Candidat e box
sampled by Monte
Carlo sampling
Statist ical
evaluat ion
Does the
box sti ll
move
no
yes
Candidate box
creat ed
around a goo d
sample point
Candidate box
is the solution
box
Are all bad
designs
removed?
yes
no
Grow in all
parameter
direc tions
PHASE 1
(explo ration)
PHASE 2
(conso lidation)
Remove bad
sample points
Candidate box
sampled by Monte
Carlo
Statist ical
evaluat ion
Remove bad
sample points

Figure 3.8: Algorithm to compute b o x-shap ed solution space (Figure is tak en from [31])
3.5.2 Review of the Underlying Algo rithm
An algorithm that computes b o x-shap ed solution space for arbitrary high-dimensional
non-linear problems w as in tro duced in [133], analysed in [36], and is briefly describ ed
here. Starting from a b o x that includes at least one go o d design, the algorithm computes
candidate b o xes b y iterativ e mo dification of the b oundaries and ev aluation of samples
within the candidate b o x, as sho wn in Figure 3.8.
In the exploration phase, a go o d lo cation of the b o x-shap ed solution space in the design
space is iden tified in order to extend the b oundary in terv als for each parameter as m uc h
as p ossible. In this phase the follo wing four steps are rep eated iterativ ely:
1. Compute a Mon te Carlo sample ([119]) within the candidate b o x.
2. Estimate the fraction of go o d designs and compute a confidence lev el for the estimate
based on the ev aluation of the samples b y Ba y esian statistics ([66]).
3. Iden tify a new candidate b o x that includes only go o d designs of the curren t sample
50

3.5 Solution Space for High-dimensional Design
with the so-called trimming algorithm that remo v es bad space b y relo cating b ound-
aries.
4. Extend the candidate b o x to allo w gro wth in to go o d space.
Extending the b oundaries in to design space with bad designs decreases the fraction of
go o d design space and will b e corrected b y the trimming algorithm that remo v es bad
design space from the candidate b o x. Extending the b oundaries in to space with go o d
designs will increase the v olume and mak e the candidate b o x ev olv e to w ards the maxim um
b o x-shap ed solution space. In the consolidation phase, only steps 1-3 are p erformed. The
candidate b o x is iterativ ely re-sampled, as some bad space ma y not ha v e b een detected in
the previous sample and bad space is remo v ed b y the trimming algorithm. This is rep eated
un til the fraction of go o d designs con v erges to the desired target v alue.
51

4 Pro duct F amily Design using Solution
Spaces
Definition 4.1 A ssume multiple systems with sever al c omp onents which ar e describ e d by
design p ar ameters. F or simplicity e ach c omp onent has one design p ar ameter. If the same
design p ar ameter values c an b e use d for two systems, the c orr esp onding c omp onent c an b e
shar e d in the sense of c ommonality .
The approac h presen ted in this section is already partially made public in [25] and optimises
the commonalit y of comp onen ts, without ev aluating all p ossibilities to share comp onen ts.
It is suitable for arbitrary non-linear and high-dimensional systems. Rather than seeking
single design p oin ts with optimal p erformance, sets of go o d designs (solution space) are
iden tified, whic h fulfil all required design criteria. Go o d designs are not further distin-
guished b y their p erformance and are, therefore, considered equal. When solution spaces
for design parameters asso ciated with one comp onen t of t w o differen t systems o v erlap, the
same parameter v alues ma y b e c hosen for the t w o systems. In other w ords, the comp onen t
design is the same in the sense of commonalit y .
4.1 Problem Statement fo r Pro duct F amily Design
Definitions. The system index is denoted as α = 1 , ..., ν , with ν b eing the total n um b er
of systems. The asso ciated design parameters are x α,i with the design parameter index
i = 1 , ..., d , where d is the total n um b er of design parameters. The n um b er of differen t
v alues of the parameter x i for all systems is denoted as n i = |{ x α,i }| . This is also called the
cardinalit y of the set of all parameter v alues of all v ehicles for the same parameter. If t w o
comp onen ts of the same t yp e are c haracterized b y the same parameter, and this parameter
assumes the same v alue, then they are of the same design. F or simplicit y , all comp onen ts
considered are c haracterized only b y one parameter, therefore, n represen ts the n um b er
of differen t comp onen t designs that are used in the pro duct family . The total n um b er of
differen t parameter v alues, which is equal to the n um b er of comp onen ts of differen t design,
is giv en b y
n =
d
X
i =1
n i . (4.1)
The p erformance of a system α is measured b y the obje ctive quantities z α,j , with j =
1 , ..., m , where m denotes the total n um b er of ob jectiv e quan tities p er v ehicle. The p er-
formance of eac h system is giv en b y p erformanc e functions as z α,j = f α,j ( x α,i ) . Critical
threshold v alues for ob jectiv e quan tities are used to define design goals and are denoted b y
z α,j,c . When z α,j > z α,j,c , a design fails to meet the asso ciated design goal j and is denoted
as b ad . When z α,j ≤ z α,j,c , a design meets the design goal. A design that meets all goals is
denoted as go o d .
52

4.2 Com binatorics of Common P arameters
Problem Statemen t. F or ν v ehicles with d design parameters x α,i , with m p erformance
functions f α,j and asso ciated threshold v alues z α,j,c eac h, seek x α,i suc h that
n → min (4.2)
sub ject to f α,j ( x α,i ) ≤ z α,j,c , ∀ α , j. (4.3)
Note that in this problem statemen t, all design parameters are treated equally , therefore,
no comp onen t is preferred to b e of common design. F urthermore, the solution ma y not b e
unique, as there ma y b e man y designs with the same (smallest) total n um b er of differen t
parameter v alues that satisfy expression (4.3). F or ev aluation purp oses, the ob jectiv e quan-
tities z α,j are further condensed to the vehicle p erformanc e z α and the total p erformanc e
of all v ehicles z ,
z α = max
j { z α,j − z ub
α,j
z ub
α,j
, z lb
α,j − z α,j
z lb
α,j
} , j = 1 , . . . , m. (4.4)
z = max
α { z α } , α = 1 , . . . , ν . (4.5)
The total p erformance of all v ehicles z has the follo wing prop erties:
• z quan tifies the o v erall p erformance as the w orst p erformance among all v ehicles.
• z ≤ 0 indicates a go o d design, i.e., all v ehicles satisfy their design goals.
• The lo w est v alue of z indicates the design that has the largest distance to the critical
threshold v alues.
4.2 Combinato rics of Common P a rameters
As a straigh tforw ard approac h to get optimal commonalit y , one ma y apply a sequence of
classical optimizations to eac h p ossible c hoice of common parameters, i.e., those parameters
represen ting a comp onen t prop ert y from t w o or more differen t systems that shall b e giv en
the same v alues. Unfortunately , the n um b er of p ossibilities to assign common parameters
quic kly b ecomes v ery large. As an example, for 3 systems with one comp onen t represen ted
b y 1 parameter x α, 1 for eac h system α , there are 5 p ossible c hoices of common parameters:
• All systems are assigned the same parameter v alue, and there is one degree of freedom,
i.e., x = x 1 , 1 = x 2 , 1 = x 3 , 1 .
• There are three p ossibilities to assign t w o systems one common design parameter,
first, x A = x 1 , 1 = x 2 , 1 and x B = x 3 , 1 , second, x A = x 1 , 1 = x 3 , 1 and x B = x 2 , 1 , third,
x A = x 2 , 1 = x 3 , 1 and x B = x 1 , 1 .
• Eac h system has its o wn parameter v alue, and th us its o wn comp onen t design. There
are three degrees of freedom with x A = x 1 , 1 , x B = x 2 , 1 and x C = x 3 , 1 .
Systematically coun ting the n um b er of p ossibilities to c ho ose common design parameters
leads to a so-called partition problem. A ccording to Brualdi [10], the total n um b er of
partitions on an α -elemen t set is defined b y Bell’s n um b er, B α , giv en by the recursiv e
definition
53

4 Pro duct F amily Design using Solution Spaces
B α +1 =
α
X
κ =0 α
κ ! B κ (4.6)
and B 0 =1 . Fo r ν systems with d comp onen ts, the n um b er of p ossible c hoices of common
parameters is
B ν,d =( B ν ) d . (4.7)
F or the example from Section 5.5.2 with 13 systems and 10 comp onen ts, expression
(4.7) ev aluates as B 10
13 = 27644437 10 =2 . 6 × 10 74 p ossibilities. Consequen tly , applying a
sequence of classical optimisation pro cedures to all p ossible configurations is prohibitiv ely
exp ensiv e.
4.3 Sha ring V alues of Design P a rameters using Solution
Spaces
A pro duct family with comp onen ts or design parameters of the same v alue can b e deriv ed
b y considering solution spaces of m ultiple systems. All design parameters can b e of the
same v alue, if the solution spaces o v erlap in at least a small region. This ma y not alw a ys
b e p ossible. An o v erview of constellation of t w o systems is giv en in T able 4.1.
T able 4.1: T w o example systems in parameter space
P arameter Space Description
syste m 1
syste m 2

[


[

The solution spaces of system 1 and 2 do not o v er-
lap. Eac h system is limited b y a b o x-shap ed solution
space. Note, if a v alue of a design parameter, e.g.,
x 1 is c hosen within the b o x, at least one v alue of
design parameter x 2 exists, that satisfies all the re-
quiremen ts on the system b eha viour. Outside this
b o x requiremen t satisfaction is excluded. In this case
eac h system needs its o wn v alues of x 1 and x 2 for
requiremen t satisfaction. A single design parameter
v alue ma y b e shared if the pro jections of the solution
spaces according to a design parameter o verlap. In
this case no o v erlap o ccurs at all.
syste m 1

[


[

syste m 2
The solution spaces of system 1 and 2 do not o v erlap,
but a part of the pro jection of the solution spaces
o v erlaps on to the x 2 -axis. The systems ha ve a single
o v erlap of pro jections of the solution spaces. The
solution spaces for b oth systems are reduced due to a
common v alue of x 2 . If a v alue of x 2 is c hosen within
the o v erlapp ed in terv al, x 1 of system 1 and 2 has to
b e c hosen within the asso ciated b o x in terv al of x 1 .
Con tin ued on next page
54

4.4 Deriving Pro duct F amilies from Bo x-shap ed Solution Spaces
T able 4.1 – con tin ued from previous page
P arameter Space Description

[


[

syste m 2
syste m 1
The solution spaces of system 1 and 2 do not o v erlap,
but parts of the pro jection of the solution spaces o v er-
lap on to the x 1 -axis (blac k) or x 2 -axis (gray). As the
solution spaces of system 1 and 2 do not o v erlap, the
designer can decide whic h design parameters should
b e shared. The systems ha ve a m ultiple o v erlap
of pro jections of the solution spaces. The remain-
ing design parameters has to b e c hosen within the
asso ciated b o x interv al of the system.

[


[

system 2
system 1
The solution spaces of system 1 and 2 o v erlap. The
pro jections of the solution spaces determine a b o x
for a common solution space for system 1 and 2.
This b o x for t w o systems (thic k, black) is similar to a
b o x for one system. If a v alue of a design parameter,
e.g., x 1 is c hosen within the b o x, at least one v alue
of the design parameter x 2 exists, that satisfies all
the requiremen ts on b oth system b eha viours. The
approac h is similar to the examples ab o v e or to set-
based design. Eac h time a system cannot satisfy the
requiremen ts o v er the whole in terv al of the considered
design v ariable, the interv als are reduced, iterativ ely .
The v alue of x 1 and x 2 can b e shared.
4.4 Deriving Pro duct F amilies from Bo x-shap ed Solution
Spaces
The analysis of b o x-shap ed solution space or in terv als of design parameters as sho wn in
Figure 4.1 is simpler than considering arbitrary solution space. By using b o x-shap ed
solution space for eac h system, the design parameters are decoupled and can b e shared
within sev eral systems, indep enden t of all other design parameter v alues. No v alue of
design parameter x 1 can b e shared, therefore a v alue for eac h s y s t em is needed. F or design
parameter x 2 there is an in terv al whic h o v erlaps for b oth systems. A v alue of this in terv al
can b e shared for b oth systems. F or robustness against design parameters uncertain ties
the mean v alue of the in terv al should b e used, see Section 3.3.1. Note, in high-dimensions,
b o x-shap ed solution space is used rather than arbitrarily shap ed solution space.
Once, all b o x-shap ed solution spaces are giv en, sp ecific parameter v alues need to b e
iden tified. This is not a straigh tforw ard task, as one can observ e from Figure 5.13. There-
fore, the algorithm sho wn in Figure 4.2 is applied to seek a configuration with smallest
total n um b er of differen t parameter v alues n for a giv en set of upp er and lo w er b ounds
x lb
α,i , x ub
α,i .
55

4 Pro duct F amily Design using Solution Spaces
good region
solution box
with largest
volu me
bad region

[


[


 [[


[

XE
[  

OE
[
 

OE
[  

XE
[  


[

XE
[ 

OE
[


OE
[  

XE
[  

syste m 1
syste m 2
XE
[ 

OE
[


XE
[  

OE
[  

OE
[  

OE
[
 

XE
[  

XE
[  

+1
+1
+1
Figure 4.1: Solution spaces, b o x-shap ed solution spaces and asso ciated in terv als of 2
systems for a general design problem.
Algorithm: Seek smallest n um b er of
separate design parameter in terv als.
for i =1 , ..., d do
n i =0 , Φ ← { 1 ,...,ν }
while |{ Φ }| > 0 do
Ψ ← { β ∈ Φ | x ub
β ,i > max
β ∗ ∈ Φ { x lb
β ∗ ,i } }
n i ← n i +1
I i,n i ← [max
β ∗ ∈ Φ { x lb
β ∗ ,i } , min
γ ∈ Ψ { x ub
γ ,i } ]
Φ ← Φ \ Ψ
end
end
Result: minim um n um b er n and corre-
sp onding in terv als I i,l
Figure 4.2: Algorithm to iden tify the smallest
set of parameter in terv als
In an outer lo op o v er the design pa-
rameter index i , the set of systems Φ
with system index β and design param-
eter x i is considered. As first step, a
subset Ψ of Φ with a system index γ is
determined, whose upp er b ounds x ub
γ ,i
are greater than the maxim um low er
b ound max
β ∗ { x lb
β ∗ ,i } . The required de-
sign criteria for the set of systems Ψ
can b e satisfied b y a single design pa-
rameter v alue, whic h lies within the
in terv al I i,l = [max
β ∗ ∈ Φ { x lb
β ∗ ,i } , min
γ ∈ Ψ { x ub
γ ,i } ] ,
where l is an index for the n um b er of
the subsystems in Ψ . The total n um-
b er n i of design parameter x i increases
b y one eac h time a sub-set Ψ is deter-
mined. As last step, the system set Φ is
reduced b y Ψ , b ecause a common com-
p onen t for them w as iden tified. In an
inner lo op this pro cedure is rep eated un til no system remains. Then the minimal n um-
b er of design parameter v alues n i and the corresp onding in terv als I i,l are iden tified. The
pro cedure is executed for all d design parameters in order to determine the total n umber
of comp onen ts n . Note, that parameters can b e treated indep enden tly , as their p ermissi-
ble ranges do not dep end on other parameter v alues. This w as accomplished b y c ho osing
b o x-shap ed solution space.
56

5 Application
The explained metho ds in the previous c hapters are applied to a complex design problem
of v ehicle system dynamics. The v ehicle b eha viour for a middle-class v ehicle is designed
for some asp ects of lateral dynamics and ride comfort. F or eac h design category , relativ e
sub jectiv e goals asso ciated to a predecessor v ehicle are used. F or the purp ose of sim ulation
a v ehicle dynamics mo del is used, whic h do es not fit exactly to the real hardw are due to
m ultiple simplifications, but it is at an adequate lev el to quan tify relativ e statemen ts. The
absolute result of the sim ulation is asso ciated to the real v ehicle b eha viour in hardw are.
The metho d of Chapter 2 is used to generate p erception thresholds to transfer the relativ e
v alues of sub jectiv e ev aluation in to relativ e v alues of ob jectiv e v alues. This allo ws the
transfer of relativ e sub jectiv e goals to v alues of ob jectiv e quan tities, whic h are used for
requiremen ts on v ehicle b eha viour. A set of v ehicle designs ma y satisfy these requiremen ts.
The system design metho d using solution space, explained in Chapter 3, can handle sets
of designs and a high n um b er of ob jectiv e quan tities. F urther, a robust system design
can b e realized, whic h is desired for man y reasons. T o reduce cost and effort a pro duct
family of v ehicles is designed. In Chapter 4 pro duct family design using solution spaces
w as presen ted. This approac h simplifies the sharing of comp onen ts within m ultiple systems
enormously as systems do not ha v e to b e executed coupled. The com binations of system
configurations do not ha v e to b e executed to design a pro duct family .
5.1 Problem Description
V ehicles need to b e designed for man y asp ects of v ehicle b eha viour. In the example consid-
ered here, the la y out of the self-steering b eha viour, the lateral stabilit y , roll-o v er, stationary
roll b eha viour and dynamic roll b eha viour are considered. There are man y more design
goals that need to b e satisfied, ho w ev er, they are considered in differen t design phases.
5.1.1 Design Catego ries
T ypically the general handling and ride c haracter is already decided b y the class of v ehicle
and the dimensions of the v ehicle. The design of the axles and t yre c haracteristics are
significan t for the v ehicle c haracter. Selected parts of the handling and ride c haracter
of a v ehicle are fine designed, here. Therefore design categories of the transien t lateral,
cornering and ride comfort b eha viour are considered, see, e.g., [41], [90] and [135].
• Self-steering b eha viour : Due to steering input of the driv er a y a w reaction of a
v ehicle o ccurs. The driv er compares the y a w reaction of the v ehicle with his exp ected
reaction. A deviation will b e p erceiv ed as a self-steering b eha viour. F or a go o d self
steering b eha viour, the y a w reaction of the v ehicle should b e predictable. The v ehicle
is exp ected to ha ve a neutral or a sligh t understeering self-steering b eha viour un til
57

5 Application
medium lateral accelerations. F or high lateral accelerations, the y a w amplification
should decrease to generate an indirect steering feeling.
• Roll b eha viour (handling) : When cornering, the driv er exp ects a roll momen t to
the outside of the curv e. This roll momen t is separated as lateral load transfer to
the fron t and rear axle. In dep endence of the ratio of the fron t and rear roll momen t
the self-steering b eha viour is influenced.
• T urn-in b eha viour : The v ehicle b eha viour during a change of direction is assessed,
e.g., steering from straigh t line driving in to cornering. The in-stationary v ehicle
b eha viour is ev aluated corresp onding to resp onse time and amplification of y a w rate
and lateral acceleration. The turn-in b eha viour has to b e prompt without high dela y
and prop ortional to the steering input. A high o v ersho ot of the v ehicle reaction is
not desired.
• Lateral stabilit y: The stabilit y and con trollabilit y of the v ehicle are assessed. In
case of a high o v er-steering reaction at high lateral acceleration the side-slip angle of
the rear axle can b e greater than the side-slip angle of the fron t axle due to the roll
momen t ratio and the non-linearit y of the t yre prop erties. This leads to a dynamic
unstable v ehicle b eha viour, whic h cannot easily b e handled. The driv er ma y ha v e to
coun tersteer to stabilize the v ehicle. Generally , a high o v er-steering reaction of the
v ehicle has to b e a v oided.
• Roll-o v er: When cornering or turn-in a corner, the c hassis rolls due to the cen trifu-
gal force acting at the cen tre of gra vit y . This momen tum leads to load differences
dep ending of the trac k of the v ehicle. The load outside the curv e increase while the
load inside the curv e is reduced. If the load inside the curv e is zero, the t yre will tip
up. A sim ultaneous tip-up of the fron t and rear t yre is not desired.
• Roll angle : When cornering, the driv er exp ects a roll angle due to the roll momen t.
The roll excitation is induced b y the driv er. The c hange of the roll angle o v er
lateral acceleration has to b e lo w and the c haracteristic curv e has to b e smo oth
and con tin uous. The lo w er the absolute roll angle is, the more comfortable is the
sub jectiv e p erception.
• Roll angle b eha viour (ride) : The roll excitation is induced b y the road surface.
The amplitude of roll angle, rate and acceleration is ev aluated. One-sided obstacles
lead to roll excitations of the v ehicle. This roll excitation should b e reduced.
5.1.2 Design Objectives
The relev an t design ob jectiv es and mano euvres to assess the v ehicle p erformance for the
design categories presen ted in Section 5.1.1 are listed in T able 5.1. In addition, t ypical
v ehicle resp onses for these driving mano euvres are sho wn. Again, only ob jectiv e quan tities
are pro vided that are relev an t for this particular design phase.
58

5.1 Problem Description
T able 5.1: Driving manouevres relev an t for design and their ob-
jectiv e quan tities
Design
category
Design ob jectiv es Ob jectiv e v alues T ypical v ehicle resp onse
Self-
steering
b eha viour
Sin us steer mano euvre: Driv-
ing mano euvre with lo w
constan t steering frequency ,
fixed lateral acceleration and
v ehicle v elo cit y . [49]
Ev aluated: quasi-stationary
y a w v elo cit y amplification
due to steering angle ˙
ψ
δ h stat
for a range of v ehicle v elo ci-
ties.
Ob jectiv e quan tit y: maximal
quasi-stationary y a w v elo c-
it y amplification ˙
ψ
δ h stat,max
in 1 /s .
0 50 100 150 200
0
0.1
0.2
0.3
v in k m/ h
˙
ψ
δ h s t a t in 1 / s

Roll b e-
ha viour
(handling)
Ramp steer mano euvre
with constan t v ehicle v e-
lo cit y . The steering angle
is increased un til the maxi-
m um lateral acceleration is
reac hed. [49]
Ev aluated: roll momen t ra-
tio RM R of the fron t axle
against lateral acceleration
a y .
Ob jectiv e quan tit y: mini-
m um of roll momen t ratio
RM R min in % and maxi-
m um of roll momen t ratio
RM R max in % .
2 4 6 8
56.5
57
57.5
a y in m / s 2
RM R in %

T urn-in
b eha viour
Con tin uous sine steering ma-
no euvre: driving mano euvre
with increasing steering fre-
quency , fixed lateral accelera-
tion and v ehicle v elo cit y . [49]
Ev aluated: normalized ya w
v elo cit y amplification due
to steering angle ˙
ψ
δ h ( f ) nor m
for a range for m ultiple
v elo cities.
Ob jectiv e quan tit y: max-
im um of normalized y a w
v elo cit y amplification
˙
ψ
δ h nor m,max .
0.5 1 1.5 2
1
1.05
1.1
1.15
f in H z
˙
ψ
δ h
in -

Lateral
stabilit y
T est track for a sev ere lane-
c hange mano euvre: critical
lane c hange mano euvre due
to an obstacle [47]. F or the
lateral stabilit y of the me-
c hanical susp ension system
no con trol systems are activ e.
Ev aluated: side-slip angle re-
action β in time domain.
Ob jectiv e quan tit y: maxi-
m um side-slip angle β max in
◦ .
0 5 10
−5
0
5
10
t in s
β in ◦

Roll-o v er T est trac k l t for a sev ere lane-
c hange mano euvre: critical
lane c hange mano euvre due
to an obstacle [47]. F or the
lateral stabilit y of the me-
c hanical susp ension system
no con trol systems are activ e.
Ev aluated: v ertical wheel
load F ty re of front and rear
axle in time domain
Ob jectiv e quan tit y: mini-
m um wheel load F ty r e,min in
N .
0 5 10
0
1
2
3
4
t in s
F t y r e in k N

F tyre,f
F tyre,r

Con tin ued on next page
59

5 Application
T able 5.1 – con tin ued from previous page
Design
category
Design ob jectiv es Ob jectiv e v alues T ypical v ehicle resp onse
Roll angle Quasi-stationary-state cor-
nering: driving mano euvre
with constan t radius. The
v ehicle v elo cit y is increased
un til the maxim um lateral
acceleration is reac hed. [48]
Ev aluated: roll angle c harac-
teristic φ against lateral ac-
celeration a y .
Ob jectiv e quan tit y: roll an-
gle φ ( a y ,hig h ) in ◦ at high lat-
eral acceleration.
0 2468
0
1
2
3
a y in m / s 2
φ in ◦

Roll b e-
ha viour
(ride)
V ertical road excitation: sig-
nifican t v ertical road excita-
tion for the comfort ev alua-
tion of the roll b eha viour.
Ev aluated: roll v elo cit y
˙
φ f ,low at a lo w frequency
in terv al.
Ob jectiv e quan tit y: ro ot
mean square v alue of the roll
v elo cit y RM S ( ˙
φ f ,low ) in ◦ /s .
0 0.5 1 1.5 2 2.5
0.0
5
10
f in H z
˙
φ in ◦ / s

V ehicle Mo delling and Sim ulation. All ob jectiv e quan tities are computed with
a non-linear t w o-trac k mo del for v ertical and lateral dynamics, see [65] and [75]. The
researc h is based on the sim ulation to ol ISAR (in tegrated sim ulation en vironmen t for
v ehicle and con trol systems) of the BMW Group, whic h is implemen ted in MA TLAB R
 . It
com bines the sim ulation of v ehicle, driv er, traffic and road harshness with con trol systems,
sensor, logic and actuator mo dels, see [65]. The mo del can b e adopted for the purp ose
of analysis, according to the design principle: as simple as p ossible and as complex as
necessary . The basic sim ulation mo del is a non-linear t w o-trac k v ehicle mo del that is
based on non-linear c haracteristics whic h are deriv ed from a m ulti-b o dy sim ulation in
AD AMS R
 . Differen tial equations are solv ed in the time domain. In the basic v ersion,
the v ehicle mo del has 20 degrees of freedom, including the fixed c hassis with its 6 degrees
of freedom. The t yre dynamics are describ ed b y the "magic form ula" mo del [84]. F or the
purp ose of v ehicle dynamics design in the early conceptual la y out, this sim ulation mo del is
a simplified ph ysical mo del, whic h uses functional parameters instead of, e.g., hard p oin t
lo cations of a m ulti-b o dy sim ulation mo del. The qualit y of the sim ulation mo del [65] is
considered to b e at an adequate lev el, b ecause the mo del has b een in pro ductiv e use in
the ev aluation of v ehicle dynamics for y ears and w as v alidated in sev eral pro jects against
measuremen t. Hence no comparison to m ulti-b o dy sim ulation or measuremen t is presen ted
here.
5.1.3 Design P a rameters
Ph ysics of Roll Beha viour. When cornering, the c hassis of a v ehicle rolls due to the
cen trifugal force acting at the cen tre of gra vit y . The resulting roll momen tum is supp orted
b y the roll stiffness consisting of susp ension comp onen ts as springs, b ounds, reb ounds,
damp ers or an ti-roll bars. The wheel at the outside of the curve deflects and reb ounds
on the inside of the curv e. The roll angle can b e reduced b y increased roll stiffness, but
the roll stiffness of the fron t and rear axle also influences the roll b eha viour and therefore
the self-steering b eha viour. In general, the higher the roll stiffness of one axle, the higher
60

5.1 Problem Description
the wheel load differences. Due to the degressiv e lateral force c haracteristic of the wheel,
a higher side-slip angle of the axle is necessary . The v ertical load at the outside of the
curv e increases, but the b enefit of lateral force is smaller than the loss on the inside wheel,
b ecause of the degressiv e c haracteristic. F or the necessary lateral force of the axle, the
side-slip angles of the wheels ha v e to b e increased. A high roll stiffness of the fron t axle
leads to a higher under-steering tendency , whic h is more easy to handle for the driv er.
A high roll stiffness at the rear axle leads to a higher o v er-steering tendency . F or lo w
lateral accelerations the roll stiffness has less effect, as the lateral force c haracteristic is
almost linear. The bump stop stiffness c b and the reb ound stop stiffness c r increase the
roll stiffness and enable a decoupling for single-wheel susp ension systems. A bump stop
ma y further b e used for ride comfort at obstacle crossing, but this is not in the fo cus of
this application.
Design P arameters. The stiffness of an an ti-roll bar c a increases the roll stiffness
without c hanging the stiffness for parallel deflection of the wheels of an axle. The decou-
pling of indep enden t single wheel susp ensions is reduced. One-sided obstacles influence
b oth single wheel susp ensions. Therefore a stiffness of the an ti-roll bar that is to o high
is not desired. An anti-roll bar is realized as a torsional spring. F or simplification, the
stiffness of the elemen t is equal to the stiffness of the comp onen t. A bump stop stiffness
c b increases the roll stiffness and enables the decoupling of single-wheel susp ensions. A
bump stop ma y further b e used for ride comfort at obstacle crossing. A bump stop is a
high non-linear spring elemen t, whic h limits the deflection. T ypically the bump stop is an
elastomer comp onen t on the piston ro d of the damp er. The c haracteristic of the bump
stop is determined b y the con tour and the densit y of the material. The c haracteristic
b egins with a constan t and ends with a progressiv e stiffness. The bump stop has a free
length s b , after whic h the spring is in compression and increases the stiffness of the sus-
p ension in the v ertical direction. A high step in the stiffness is not exp ected b y the driv er
and is p erceiv ed as uncomfortable. Next to the influence on loaded v ehicle b eha viour and
oscillatory b eha viour, the bump stop c b is imp ortan t for the roll b eha viour. F or middle
class v ehicles the comp onen ts of the fron t axle are activ e b efore the comp onen ts of the
rear axle. A reb ound stop stiffness c r increases the roll stiffness and enables a decoupling
of single-wheel susp ensions. A reb ound stop limits the reb ound of the wheel and consists
of an elastomer spring and/ or a steel spring whic h is in compression b efore the elastomer
spring. The reb ound stop can b egin with a constan t stiffness after the free length s r and
ends progressiv ely , analogously to the bump stop. The reb ound stop is lo cated in the tub e
of the damp ers. The reb ound stop reduces the lifting of the cen ter of gra vit y at rolling.
A bump and reb ound stop ma y also ha v e influence on v ertical deflection and the pitc h
b eha viour.
Relev an t design comp onen ts of the fron t and the rear axle for the design categories
presen ted in Section 5.1.1 are the bump stops (1), the r eb ound stops (2) and the anti-r ol l
b ars (3). The asso ciated design parameters are listed in T able 5.1. The fr e e lengths under
c ompr ession and extension denote the v ertical displacemen ts b et w een wheel and c hassis
that are necessary to activ ate the bump and the reb ound stop c r , resp ectiv ely . They are
geometrical prop erties of the comp onen t assem bly . A comp onen t assem bly of the fron t
axle is sho wn in Figure 5.1.
61

5 Application
Figure 5.1: F ron t axle assem bly of a middle
class v ehicle with (1) bump stop, (2) reb ound
stop and (3) an ti-roll bar.
T able 5.2: Design P arameters. The sub-
script letters f and r refer to the fron t and
rear axle, resp ectiv ely .
Sym b ol Unit Description
c a,f , c a,r N
mm stiffness of an ti-roll bar
c b,f , c b,r N
mm stiffness of bump stop
s 0 ,b,f , s 0 ,b,r mm free length under compression
c r ,f , c r,r N
mm stiffness of reb ound stop
s 0 ,r ,f , s 0 ,r,r mm free length under extension
5.2 Generation of P erception Thresholds fo r Middle-class
V ehicles
5.2.1 Subjective Kno wledge fo r Middle-class V ehicles
As describ ed in Chapter 2 the metho d is based on a kno wledge managemen t system, whic h
is sho wn in T able 5.3. This sub jectiv e kno wledge is v alid for middle class v ehicles only and
w as deriv ed b y driving exp erts. The design criteria presen ted in Section 5.1.1 are used.
The design parameters presen ted in Section 5.1.3 are extended b y the "genes" of v ehicles
and b y the susp ension spring and the damp er c haracteristic. The genes of vehicles consist
of the b o dy mass m , the inertia momen ts J and dimensions of the v ehicle l and w ere
extended as the parameters v ary in a pro duct family , whic h is discussed in Section 5.4.
The susp ension spring and the damp er c haracteristic are designed based on man y other
design criteria, whic h are not the fo cus of this application. As describ ed in Chapter 2,
reliable threshold v alues can only b e generated, if m ultiple relev an t design parameters are
listed.
In terpretation of Roll Angle. When cornering, the driv er exp ects a roll angle due
to the roll momen t. The c hange of the roll angle o v er lateral acceleration has to b e lo w
and the c haracteristic curv e has to b e smo oth and con tin uous. The lo w er the absolute roll
angle is, the sp ortier is the sub jectiv e p erception. The b o dy mass m , the cen tre of gra vit y
l z and the trac k l t ha v e a high impact ( # ). The pro duct of b o dy mass, lateral acceleration
and the distance b et w een the v ertical cen tre of gra vit y and the roll axle determine the roll
momen t, whic h is transferred b y the trac k to forces on the susp ensions. The roll momen t
is equal to the pro duct of roll stiffness and roll angle. The stiffness of an ti-roll bars c a
and susp ension springs c s con tribute the most to the roll stiffness. Therefore the stiffness
of an ti-roll bars and susp ension springs also ha v e high sub jectiv e impact ( # ) on the roll
angle. The stiffness of the bump and reb ound stop are p erceiv ed to ha v e a medium impact
( < ).
62

5.2 Generation of P erception Thresholds for Middle-class V ehicles
T able 5.3: V ehicle kno wledge managemen t system for middle class v ehicle: sub jectiv e
informations according to relations b et w een design parameters and design categories.
Self-steering b eha viour
Roll b eha viour(handling)
T urn-in b eha viour
Lateral stabilit y
Roll-o v er
Roll angle
Roll b eha viour (ride)
b o dy mass m # # < # # <
inertia momen t J xx # #
inertia momen t J y y
inertia momen t J z z # < <
wheelbase l w # < # <
cen tre of gra vit y l x # # # # #
cen tre of gra vit y l z < < < < # #
trac k l t < < # #
damp er c haracteristic d ( v ) # < #
an ti-roll bar stiffness c a < # < # # # <
susp ension spring stiffness c s < # < # # # <
bump stop c b # <<<
free length u. comp. s 0 ,b < <
reb ound stop c r <
free length u. reb ound s 0 ,r
T able 5.4: Obje ctific ation K now le dge : Relation b et w een ob jectiv e quan tit y and sub jectiv e
design category
Self-steering b eha viour
Roll b eha viour (handling)
T urn-in b eha viour
Lateral stabilit y
Roll-o v er
Roll angle
Roll b eha viour (ride)
maximal quasi-stationary y a w v elo cit y amplification ˙
ψ
δ h stat,max 1 0 0 0 0 0 0
maxim um of roll momen t ratio R M R max 0 0.5 0 0 0 0 0
minim um of roll momen t ratio R M R min 0 0.5 0 0 0 0 0
maximal y a w v elo cit y amplification ˙
ψ
δ h max 0 0 1 0 0 0 0
maxim um side-slip angle β max 0 0 0 1 0 0 0
minim um v ertical t yre force F ty r e,min 0 0 0 0 1 0 0
roll angle φ ( a hig h ) at high lat. acceleration 0 0 0 0 0 1 0
ro ot mean square v alue of roll angle velocity RM S ( ˙
φ f ,low ) 0 0 0 0 0 0 1
In case of stationary cornering and ideally plain roads the inertia momen ts and the
63

5 Application
damp er c haracteristic ha v e no influence. This sub jectiv e p erception is v alid for a middle
class v ehicle as the driving exp erts had their exp erience with middle class v ehicles in mind.
The v ariation of parameters is therefore limited to design spaces corresp onding to middle
class v ehicles. Only within this design space, a design parameter has high impact #
according to a design category . In other w ords, only within these in terv als, c hanges of the
parameters lead to a c hange of the sub jectiv e ev aluation o v er 1 BI.
Asso ciated Ob jectified Kno wledge. The assignmen t of ob jectiv e quan tities to design
categories is done b y driving exp erts. A general assignmen t is already sho wn in T able 5.1.
If m ultiple ob jectiv e quan tities are assigned to a single design category w eigh tings are
necessary . T able 5.4 sho ws an o v erview of the ob jectification kno wledge. As describ ed
in Section 2.3.1, for simplification correlating ob jectiv e quan tities are reduced. F or this
design purp ose only the design category r ol l b ehaviour (hand ling) needs w eigh tings as the
minim um and the maxim um of the roll momen t ratio can b e designed separately , e.g., b y
bump or reb ound stops. Both ob jectiv e quan tities are imp ortan t, therefore their impact is
a v eraged.
5.2.2 Generation of Objective Kno wledge fo r Middle-class V ehicles
Design P arameters. The design parameters of the sub jectiv e kno wledge in T able 5.3
are analysed. The first 8 design parameters are also called genes of the vehicle . F or this
analysis, the design parameters of the susp ension are doubled: 7 design parameters for eac h
axle. In total 22 design v ariables are v aried. In the sim ulation mo del the c haracteristics
of comp onen ts are describ ed as v ectors or matrices. The c haracteristics ma y b e scaled
or parametrised b y a mathematical mo del. T o reduce the n um b er of design parameters,
scaling parameters are preferred.
Design Space. The limits of the design space are not equally p erceptible. The limits
of the design space are iden tified b y an analysis of the v ariation of the design parameters
of middle class v ehicles. Note, that this range is imp ortan t as the driving exp erts ha v e
this range in mind when deriving the sub jectiv e kno wledge.
Design of Exp erimen t. The v ariation of all design parameters at one time is desired
to find out as m uc h as p ossible ab out a relation b et w een system inputs and outputs. The
design parameters are scattered with a design of exp erimen t metho d. The probabilit y
distribution of the design parameters is equally distributed as eac h v alue of the design
parameters is equally probable. In this researc h case, it is imp ortan t to generate a homo-
geneous sample distribution to co v er the design space at an adequate lev el. F urthermore,
in industrial applications, it is essen tial to sp ecify the n um b er of exp erimen ts to determine
the effort, e.g., of the sim ulation. The Halton-sequence is the b est c hoice for the require-
men ts of this researc h. The Halton-sequence generates the most homogeneous sample
distribution, while the n um b er of exp erimen ts is set arbitrarily . F or 22 design parameters
a sample of 2000 sample p oin ts is created.
V ariance-based Sensitivit y Analysis. F or eac h sample p oin t all ob jectiv e quan tities
listed in T able 5.4 are ev aluated using a non-linear t w o-trac k mo del. Mano euvre data
are used as input for the sim ulation mo del. The v ehicle b eha viour ma y differ for eac h
exp erimen t. The computation of all ob jectiv e quan tities for all exp erimen ts is a time
consuming task, but the computation of sample p oin ts can b e parallelized. F or a graphical
analysis all v alues of an ob jectiv e quan tit y can b e sorted o v er eac h design parameter. This
is called sc atter plot . All design parameters with exclusion of the sorting design parameter
64

5.2 Generation of P erception Thresholds for Middle-class V ehicles
1 1.1
2
2.5
3
3.5
m in k g
ϕ ( a y ,h i g h )
1 1.5
2
2.5
3
3.5
d f in N s/ m
1 1.5
2
2.5
3
3.5
d r in N s/ m
1 1.5
2
2.5
3
3.5
c a , f in N / m m
1 1.5
2
2.5
3
3.5
c a , r in N / m m
1 1.5
2
2.5
3
3.5
c b , f in N / m m
1 1.1
50
55
60
m in k g
R M R ( a y ,h i g h )
1 1.5
50
55
60
d f in N s/ m
1 1.5
50
55
60
d r in N s/ m
1 1.5
50
55
60
c a , f in N / m m
1 1.5
50
55
60
c a , r in N / m m
1 1.5
50
55
60
c b , f in N / m m
1 1.1
0.1
0.12
0.14
m in k g
R M S ( ˙
φ ( f l o w ))
1 1.5
0.1
0.12
0.14
d f in N s/ m
1 1.5
0.1
0.12
0.14
d r in N s/ m
1 1.5
0.1
0.12
0.14
c a , f in N / m m
1 1.5
0.1
0.12
0.14
c a , r in N / m m
1 1.5
0.1
0.12
0.14
c b , f in N / m m

Figure 5.2: Scatter plots: ob jectiv e quan tities sorted o v er rising design parameter.
scatter within its design space at eac h v alue of the sorting design p oin t. If a design
parameter is imp ortan t for c hange of an ob jectiv e quan tit y , a trend within the scattered
ob jectiv e quan tities can b e recognized. Figure 5.2 sho ws an extract of scatter plots for 3
ob jectiv e quan tities and 7 design parameters. Eac h line is asso ciated to the same ob jectiv e
quan tit y and eac h column to the same design parameter.
In terpretation of the Stationary Roll Angle. The stationary roll angle at high
lateral acceleration φ ( a y ,hig h ) sho ws high linear dep endence of the b o dy mass m and the
fron t stiffness of the an ti-roll bar c a,f . The absolute v alue of the rear stiffness of the an ti-roll
is a magnitude smaller as the fron t stiffness, therefore the effect is minimal. The higher
the b o dy mass or the smaller the stiffness of the an ti-roll bar, the higher the resulting roll
angle. A scaling of the fron t d f and rear d r damp er c haracteristic has no effect, as the
roll angle is stationary . Non-linearities of design parameters do rarely app ear. The free
lengths under compression or reb ound are candidates for significan t non-linearities, but
the influence of their stiffness is almost negligible here.
Sensitivit y Analysis. F or a go o d global sensitivit y analysis the neigh b ourho o d of
the designs is in v estigated. The impact of the design parameters can b e quan tified b y a
v ariance-based sensitivit y analysis. The total Sob ol indices S T are used to condense the
quan titativ e relation b et w een design parameters and ob jectiv e quan tities to a single v alue.
Non-linear relation and in teraction are considered. T able 5.5 sho ws an o v erview of the
generated ob jectiv e kno wledge consisting of Sob ol indices.
65

5 Application
T able 5.5: Obje ctive K now le dge : Relation b et w een design parameters and ob jectiv e quan-
tit y
maximal quasi-stationary y a w
v elo cit y amplification ˙
ψ
δ h stat,max
maxim um of roll momen t ratio
RM R max
minim um of roll momen t ratio
RM R min
maximal y a w v elo cit y amplifica-
tion ˙
ψ
δ h max
maxim um side-slip angle β max
minim um v ertical t yre force
F ty re,min
roll angle φ ( a y ,hig h ) at high lat-
eral acceleration
ro ot mean square v alue of the roll
v elo cit y RM S ( ˙
φ f ,low )
b o dy mass m 0.19 0.00 0.00 0.58 0.02 0.09 0.26 0.00
inertia momen t J xx 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.65
inertia momen t J y y 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
inertia momen t J z z 0.00 0.00 0.00 0.24 0.01 0.03 0.00 0.00
wheelbase l w 0.12 0.00 0.00 0.02 0.08 0.01 0.00 0.00
cen tre of gra vit y l x 0.67 0.02 0.02 0.08 0.31 0.09 0.00 0.00
cen tre of gra vit y l z 0.01 0.00 0.00 0.04 0.01 0.11 0.28 0.00
trac k l t 0.00 0.00 0.00 0.01 0.00 0.03 0.10 0.14
fron t damp er c haracteristic d f ( v ) 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.09
rear damp er c haracteristic d r ( v ) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.08
fron t an ti-roll bar stiffness c a,f 0.00 0.40 0.43 0.01 0.24 0.31 0.20 0.01
rear an ti-roll bar stiffness c a,r 0.00 0.02 0.02 0.00 0.01 0.01 0.01 0.01
fron t susp ension spring stiffness c s,f 0.00 0.09 0.09 0.00 0.08 0.08 0.04 0.00
rear susp ension spring stiffness c s,r 0.01 0.47 0.39 0.01 0.20 0.16 0.10 0.04
fron t reb ound stop c r ,f 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
rear reb ound stop c r ,r 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00
fron t bump stop c b,f 0.00 0.00 0.01 0.00 0.03 0.03 0.00 0.00
rear bump stop c b,r 0.00 0.00 0.02 0.00 0.02 0.00 0.01 0.00
fron t free length u. reb ound s 0 ,r,f 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
rear free length u. reb ound s 0 ,r,r 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
fron t free length u. comp. s 0 ,b,f 0.00 0.00 0.01 0.00 0.01 0.00 0.00 0.00
rear free length u. comp. s 0 ,b,r 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
In terpretation of Results. The sum of the Sob ol indices of the ob jectiv e quan tit y
r ol l angle at high later al ac c eler ation is appro ximately equal to 1. Therefore no in teraction
of design v ariables o ccurs for this ob jectiv e quan tit y . The b o dy mass, the v ertical cen tre
of gra vit y , the trac k, the stiffness of the fron t an ti-roll bar and the rear susp ension spring
ha v e high impacts. The stiffness of the rear an ti-roll bar, rear bump stop and the fron t
susp ension spring ha v e a medium impact. The remaining design parameters ha v e no
influence. A Sob ol index S T = 0 . 2 as for the quan tification of the fron t stiffness of the
an ti-roll bar and the roll angle ma y b e in terpreted as: The v ariation of the stiffness within
its limits causes 20 % of the total v ariation of the roll angle within all design parameters.
66

5.2 Generation of P erception Thresholds for Middle-class V ehicles
5.2.3 Compa rison of Subjective and Objectified Kno wledge of
Middle-class V ehicles
Generation of Ob jectified Kno wledge. The ob jectified kno wledge is generated b y
scalar m ultiplication of the ob jectiv e matrix, see T able 5.5 and the ob jectification matrix,
see T able 5.4. In this case almost all design categories are link ed with a single ob jec-
tiv e quan tit y . Therefore the ob jectified matrix is v ery similar to the ob jectiv e matrix or
kno wledge. Only for the design category r ol l b ehaviour (hand ling) the sensitivities of the
maxim um and minim um roll momen t ratio are aggregated. F or a comparison with the
sub jectiv e ev aluation, the sensitivities of the comp onen ts of the fron t and rear are added.
T able 5.6 sho ws the ob jectified kno wledge next to the sub jectiv e kno wledge. The qualit y
of sim ulation, ob jectification and sub jectiv e kno wledge is high, if sub jectiv e and ob jectified
kno wledge correlate highly . Note, a quan tified correlation is not p ossible without threshold
v alues to transfer the ob jectified in to sub jectiv e information. If the pattern of sub jectiv e
and ob jectified kno wledge do es not fit, the c hoice of the threshold v alues cannot impro v e
a lo w corresp ondence. The patterns of the sub jectiv e and ob jectified kno wledge matc h
in this case, for a negativ e imaginary example see App endix A.2. P arameters with high
sensitivities meet high sub jectiv e impacts.
Sensitivit y Thresholds for eac h Design Category . Imagine t w o design categories.
The first design category can easily v ary o v er sev eral BI. Man y design parameters ha v e
a high impact ( # ). The second design category hardly has design parameters with high
impacts. The design category cannot b e c hanged easily . F or b oth design categories the
sum of the sensitivities (Sob ol indices) is equal to one or ab o v e as the v ariance induced b y
certain design parameters is divided b y the total v ariance. The design categories ma y ha v e
differen t sub jectiv e p erception ranges, therefore the thresholds v alues for the sensitivities
ha v e to b e differen t for eac h design category .
Sensitivit y thresholds can b e concluded from the comparison of ob jectified and sub jectiv e
kno wledge. As the pattern or the relativ e order of the impact of the ob jectified and
sub jectiv e kno wledge matc h, thresholds can b e concluded. Note that the order of thresholds
hold true:
ˆ
S # > ˆ
S < > 0 . (5.1)
Sub jectiv e impacts can b e arranged to the aggregated sensitivities. This arrangemen t is
done in a w a y that the matc hing of the ob jectified and sub jectiv e kno wledge is maximised.
Theoretically , the true sensitivit y thresholds ˆ
S # lies within the range of the lo w est sensi-
tivit y arranged to a high impact and the greatest sensitivit y arranged to a medium impact.
The more parameters are analysed, the smaller the range gets and the more precisely the
threshold can b e iden tified. F or simplicit y , the lo w est sensitivit y v alue arranged to a high
sub jectiv e impact # is used as sensitivit y thresholds ˆ
S # . This is done similarly for ˆ
S < .
A correlation is built of the sub jectiv e kno wledge and the transferred ob jectified kno wl-
edge. Similar to the in terpretation of the sub jectiv e ev aluation, # is replaced b y 1 and <
b y 0.3. A Pe arson c orr elation index is built for eac h design category , see T able 5.6. All
design categories get a high correlation index.
Result of Comparison. There is esp ecially one mismatc h, whic h needs to b e discussed
in detail. The sub jectiv e effect of the damp er c haracteristic ma y not b e v erified b y the
ob jectified kno wledge. The damp er c haracteristics influence the dynamical roll momen t
distribution, whic h determines later al stability and r ol l-over b eha viour. F or the design
67

5 Application
T able 5.6: Ov erview: Comparison of Subje ctive K now le dge and Obje ctifie d K now le dge
Self-
steering
b eha viour
Roll b e-
ha viour
(handling)*
T urn-in b e-
ha viour
Lateral sta-
bilit y Roll-o v er Roll angle
Roll b e-
ha viour
(ride)
sub. ob j. sub. ob j. sub. ob j. sub. obj. sub. ob j. sub. ob j. sub. ob j.
b o dy mass m # # 0.19 0.00 # # 0.58 < < 0.02 # # 0.09 # # 0.26 < 0.00
inertia momen t J xx 0.00 0.00 0.00 0.00 # # 0.10 0.00 # # 0.65
inertia momen t J y y 0.00 0.00 0.00 0.00 0.00 0.00 0.00
inertia momen t J z z 0.00 0.00 # # 0.24 < < 0.01 < < 0.03 0.00 0.00
wheelbase l w # # 0.12 0.00 < < 0.02 # # 0.08 < < 0.01 0.00 0.00
cen tre of gra vity l x # # 0.67 # # 0.02 # # 0.08 # # 0.31 # # 0.09 0.00 0.00
cen tre of gra vity l z < < 0.01 < 0.00 < < 0.04 < < 0.01 # # 0.11 # # 0.28 0.00
trac k l t 0.00 0.00 < < 0.01 0.00 < < 0.03 # # 0.10 # # 0.14
damp er c haracteristic d ( v ) 0.00 0.00 0.00 # 0.00 # < 0.01 0.00 # # 0.17
an ti-roll bar stiffness c a < 0.00 # # 0.44 < < 0.01 # # 0.25 # # 0.32 # # 0.20 < < 0.01
spring stiffness c s < < 0.01 # # 0.52 < < 0.01 # # 0.28 # # 0.24 # # 0.14 < < 0.04
reb ound stop c r 0.00 0.00 0.00 0.00 < < 0.03 0.00 0.00
bump stop c b 0.00 # # 0.02 0.00 < < 0.05 < < 0.03 < < 0.01 0.00
free length u. reb ound s 0 ,r 0.00 0.00 0.00 0.00 0.00 0.00 0.00
free length u. comp. s 0 ,b 0.00 < < 0.01 0.00 < < 0.01 0.00 0.00 0.00
threshold v alue ˆ
S # 0.12 0.02 0.08 0.08 0.09 0.20 0.14
threshold v alue ˆ
S < 0.01 0.01 0.01 0.01 0.01 0.01 0.01
correlation index 0.95 0.95 1.00 0.82 0.95 1.00 0.95
*Roll b eha viour (handling) is a sub-design category of self-steering b eha viour
68

5.2 Generation of P erception Thresholds for Middle-class V ehicles
categories later al stability and r ol l-over the sub jectiv e impact cannot b e v erified. P ossible
reason ma y b e the simple v ariation b y scaling. F or lateral stabilit y or roll-o v er high relativ e
v elo cities of deflections o ccur. The damp er force for high relativ e v elo cities can b e designed
differen tly . Ho w ev er, this w as neglected for simplification. The sub jectiv e impact ma y also
b e o v er-estimated for this class of v ehicle. Nev ertheless, the matc hing of the parameters
and design categories is at a high lev el.
5.2.4 P erception Thresholds of Middle-class V ehicles
A ccording to the metho d describ ed in Section 2.4, the sensitivit y thresholds ˆ
S < and ˆ
S #
can b e transformed to p erception thresholds ∆ q > and ∆ q # . First of all, the arrangemen t
of ob jectiv e quan tities to design categories is imp ortan t. F or a single relation, the sensi-
tivit y threshold can b e directly used as sensitivit y of the ob ject quan tit y . This is the case
for self-ste ering b ehaviour, turn-in b ehaviour, later al stability, r ol l-over, r ol l angle and r ol l
b ehaviour (ride) . F or r ol l b ehaviour (hand ling) tw o ob jectiv e quan tities are used: the min-
im um and maxim um roll momen t ratio, whic h are indep enden t of eac h other. Therefore,
the sensitivit y threshold is used for b oth ob jectiv e quan tities indep enden tly . The v ariance
of an ob jectiv e quan tit y q is determined b y the total v ariance V ar ( q ) and the sensitivit y
thresholds according to Equations (2.20) and (2.21). The p erception thresholds ∆ q # and
∆ q < are determined b y Equations (2.22) and (2.23). T able 5.7 sho ws the results for the
data in T able 5.6.
T able 5.7: P erception thresholds determined for eac h design category . Relation b et w een
ob jectiv e quan tities and design category is exemplary .
Design category ob jectiv e quan tit y q V ar ( q ) ˆ
S > ˆ
S # ∆ q > ∆ q #
Self-steering b eha viour ˙
ψ
δ h stat ( v hig h ) 3.786 10 − 4 1 /s 2 0.01 0.12 0.006 1 /s 0.02 1 /s
Roll b eha viour (handling) RM R min 6.53 P P 2 0.01 0.02 0.76 PP 1.08 PP
RM R max 7.17 P P 2 0.01 0.02 0.80 PP 1.13 PP
T urn-in b eha viour ˙
ψ
δ h nor m,max ( v hig h ) 4.046 · 10 − 3 0.01 0.08 0.02 0.05
Lateral stabilit y β max 16.63 ◦ 2 0.01 0.08 1.22 ◦ 3.46 ◦
Roll-o v er F ty r e,min 79330 N 2 0.01 0.09 84.5 N 253 N
Roll angle φ ( a y ,hig h 0.061 ◦ 2 0.01 0.20 0.07 ◦ 0.33 ◦
Roll b eha viour (ride) RM S ( ˙
φ f ,low ) 0.35 ◦ 2 /s 2 0.01 0.14 0.18 ◦ /s 0.66 ◦ /s
In terpretation of P erception V alue. Within the middle-class v ehicles, a c hange of
the roll angle at high lateral acceleration φ ( a y ,hig h ) of ab out 0.33 ◦ leads to a sub jective
c hange of the ev aluation of 1 BI. F or deviations of angles, Botev [9] prop oses the gen-
eral p erception threshold of 0.5 ◦ . Due to reasons of cognitiv e influences, an individual
scattering of this limit is estimated. The com bination of angle, rotational v elo cit y and
acceleration is imp ortan t [9].
Graphical In terpretation. Imagine, a sp ort y middle class v ehicle with a roll angle
φ ( a y ,hig h ) = 3 ◦ and an absolute BI ev aluation of BI 7 for the design category R ol l angle . A
monotone relation b et w een ob jectiv e quan tit y and absolute sub jectiv e ev aluation can b e
assumed based on driving exp ertise. The smaller the roll angle, the higher the sub jectiv e
ev aluation. With the use of the p erception thresholds a c haracteristic curv e around this
op erating p oin t can b e dra wn, see Figure 5.3.
69

5 Application
7 BI

 07 . 0

 33 . 0

BI

) ( , h ig h y
a


 0 . 3

 33 . 0

6 BI

 7 . 2

 3 . 3

8 BI

 07 . 0

Figure 5.3: Characteristic curve for subjective BI ev aluation ov er roll angle at high lateral
acceleration
5.3 Robust Design Strategy of Anti-roll Ba rs
V ehicles need to b e designed for man y asp ects of v ehicle b eha viour. In this example, the
la y out of the r ol l b ehaviour (hand ling), later al stability, r ol l-over, r ol l angle and dynamic
r ol l b ehaviour (ride) are considered. There are man y more design goals that need to b e
satisfied, ho w ev er, they are considered in differen t design phases.
Design Scenario. Imagine the design of a sedan middle class v ehicle based on a
predecessor v ehicle. The new design has to b e sp ortier and reac h b est in se gment for the
design categories r ol l b ehaviour (hand ling) . Therefore an increase of 1 BI is desired. The
v ehicle b eha viour of later al stability and r ol l-over should b e main tained and r ol l angle and
dynamic r ol l b ehaviour (ride) ma y decrease ab out 1 BI.
F or simplicit y , in this scenario, only the stiffness of the fron t and rear an ti-roll bar are
design parameters. Sev eral v ehicle v arian ts are planned, all v arian ts are planed to get the
same fron t and rear an ti-roll bar. The v arian ts include sev eral motors and differen t driving
mo des. Therefore the v ehicle mass can v ary ab out +/-5 %, e.g., +/-75 kg and the damp er
c haracteristics can v ary ab out +/-10 %, e.g., +/- 100 N at a damp er v elo cit y of 0.5 m/s.
In dep endence of the c hange of mass, the stiffness of the susp ension spring is adopted
to obtain a comparable natural frequency of the oscillation and that the v ehicles ha v e the
same initial pitc h angle.
Use of P erception Thresholds. Based on the kno wledge presen ted in T able 5.7 the
requiremen ts ab o v e can b e transformed to requiremen ts on ob jectiv e quan tities. The lo w er
and upp er limit of roll momen t distribution will b e shifted up ab out 1 PP (p ercen t p oint).
The roll angle can increase ab out 0.33 ◦ and the middle roll v elo cit y rise ab out 0.66 ◦ /s .
Based on the p erformance of the predecessor v ehicle, new threshold v alues for the ob jectiv e
quan tities are set. The roll momen t ratio has to b e b et w een 55 and 60 % fron t load. The
side-slip angle of the v ehicle is limited to a critical v alue β max.cr it for lateral stabilit y . The
v ertical t yre force has to b e ab o v e 1000 N for at least one t yre in case of a critical roll-o v er
situation. The roll angle at high acceleration is limited to maximal 3 ◦ and the mean roll
70

5.3 Robust Design Strategy of An ti-roll Bars
v elo cit y is limited maximal to 5 ◦ /s . T able 5.8 giv es an o v erview of lo w er (lb) and upp er
(ub) b ounds for eac h considered ob jectiv e quan tit y .
T able 5.8: Threshold v alues of ob jectiv e quan tities for a sedan: lo w er (lb) and upp er
b ounds (ub).
φ ( a y ,hig h ) in ◦ RMR in % β max in ◦ F ty re,min in N RM S ( ˙
φ f ,low ) in ◦ /s
lb ub lb ub lb ub lb ub lb ub
sedan - 3,0 55 60 - β max,cr it 1000 - - 5.0
System P erformance. The p erformance of the v ehicle is describ ed b y Equation (3.4).
F or this example the system p erformance results in:
z ( x i ) = max  φ ( a y ,high ) − φ ub
φ ub , RM R lb − R M R min
RM R lb , R M R max − RM R ub
RM R ub , (5.2)
β max − β ub
max
β ub
max , F lb
ty re,min − F ty r e,min
F lb
ty re,min
, RM S ( ˙
φ f ,low ) − R M S ( ˙
φ f ,low ) ub
RM S ( ˙
φ f ,low ) ub  .
As a reminder, the system p erformance z has the follo wing prop erties:
• z quan tifies the o v erall p erformance as the w orst p erformance.
• z ≤ 0 indicates a go o d design, i.e., a system satisfying the design goals.
• The lo w est v alue of z indicates the design that has the largest distance to the critical
threshold v alues.
5.3.1 Shap e of Solution Space of Anti-roll Ba rs
Figure 5.4 sho ws a solution space con taining go o d designs with resp ect to differen t design
criteria. The corresp onding threshold v alues are listed in T able 5.8. The x- and y-axis
represen t the an ti-roll bar stiffnesses c a of the fron t (f ) and rear (r) axle, resp ectiv ely .
The shap es of the solution space can b e explained ph ysically . Figure 5.4 sho ws, e.g., a
restriction of the solution space with resp ect to comfort. The ro ot mean square v alue of
the roll angle v elo cit y increases for high stiffness of the fron t an ti-roll bar ( differ enc e of
design p ar ameters ). Imagine an excitation on the righ t side, whic h causes a deflection of
the righ t wheel. Through the coupling of the righ t and left side of the an ti-roll bar, the
left wheel will reb ound. Therefore the roll reaction and consequen tly the roll v elo cit y rise.
The higher the an ti-roll bar stiffness of the fron t or rear axle, the smaller is the admissible
roll angle for a fixed roll momen t ( sum of design p ar ameters ). A dditionally the b oundary
of the solution space due to roll o v er is sho wn. The required minim um remaining t yre force
has to b e ab o v e a critical threshold v alue ( differ enc e of design p ar ameters ). Generally , for
the fron t axle, a short lift-off is una v oidable in a sev ere lane-c hange mano euvre, therefore,
the stabilit y reserv e is demanded from the rear axle. Hence, the stiffness of the rear axle
is limited in order to reduce t yre load differences. The higher the rear t yre load differences
are, the higher is the required rear slip angle and, therefore, the higher is the side-slip
angle ( differ enc e of design p ar ameters ). The fundamen tal ph ysics on the fron t axle is the
same, but the higher the slip angle of the fron t axle, the smaller the maxim um side-slip
71

5 Application
angle and therefore the higher the lateral stabilit y . The higher the rear stiffness and the
lo w er the fron t stiffness, the smaller is the lev el of roll momen t distribution ( r atio of design
p ar ameters ). F or more details see the App endix A.3.
5.3.2 Concurrent Engineering of a Solution Space of Anti-roll Ba rs
good region
roll-over
optimum
stability
oversteering
understeering
roll angle
roll dy namics
c a,f in N/mm
c a,r in N/mm

Figure 5.4: Solution space for the fron t (x-
axis) and rear (y-axis) an ti-roll bar stiffness.
Designs are excluded b y requiremen ts from
roll b eha vior, roll-o v er, lateral stabilit y and
self-steering b eha vior.
Figure 5.4 sho ws the shap e of the solution
space for the requiremen ts listed in T able
5.8. In this case a second design team is
resp onsible for the design categories self-
ste ering b ehaviour and turn-in b ehaviour .
The ob jectiv e quan tities are the stationary
y a w v elo cit y amplification ˙
ψ
δ h stat and the
maximal normalised y a w v elo cit y amplifi-
cation ˙
ψ
δ h nor m,max . Both ob jectiv e quan ti-
ties ha v e a lo w er and an upp er b ound. The
b ounds are listed in T able 5.9
Shap e of Second Solution Space.
The sub jectiv e p erception of directness of a
v ehicle correlates with the stationary y a w
v elo cit y amplification ( r atio of design p a-
r ameters ). The higher the v alue of ˙
ψ
δ h stat ,
the directer b ecomes the v ehicle b eha viour
or the higher is the y a w v elo cit y due to
steering input. The lo w er b ound of ˙
ψ
δ h stat
reduces the solution space from the b ottom,
the v ehicle b eha viour is to o indirect. The upp er b ound limits v ehicle b eha viour, whic h is
to o direct. The agilit y of the v ehicle corresp onds to the maximal y a w v elo cit y amplification
( sum of design p ar ameters ). The higher the v alue of ˙
ψ
δ h max , the more agile is the v ehicle
b eha viour. The righ t diagonal b ound limits reduced agilit y , while the left diagonal b ound
limits increased agilit y . The resulting solution space is sho wn in Figure 5.5.
10 15 20 25 30 35 40 45
0
2
4
6
8
10

c a, f in N/mm
c a, r in N / m m

Design Space 2

Figure 5.5: Solution space with fo cus on
self-steering and turn-in b eha viour.
T able 5.9: Threshold v alues of ob jectiv e
quan tities for sedan of second design team.
˙
ψ
δ h stat ( v hig h ) in 1/s ˙
ψ
δ h max ( v hig h ) in -
lb ub lb ub
sedan 0.365 0.369 1.20 1.23
Figure 5.6 sho ws the o v erla ying of the solution spaces from b oth design teams. Require-
men ts on the system b eha viour can b e separated in to m ultiple design teams without design
72

5.3 Robust Design Strategy of An ti-roll Bars
iterations as the solution spaces can b e o v erlaid. The blac k lined solution space sho ws the
common solution of the fron t and rear stiffness of the an ti-roll bar for all requiremen ts.
10 15 20 25 30 35 40 45
0
2
4
6
8
10

c a, f in N / m m
c a , r in N/mm

Design Space 1
Design Space 2
Common Design Space

Figure 5.6: Ov erla ying of solution spaces
with differen t fo cus.
F or simplification, the requiremen ts of the second design team are not considered in the
follo wing applications.
5.3.3 Classical App roach: Robust Design Optimization
F or comparison to the solution space approac h, this classical robust design optimization
approac h w as executed, whic h is similar to [13]. A P areto-fron t for the system p erformance
ˆ z and the robustness σ ( ˆ z ) is generated. The designer ma y decide b et w een a p erformance
and a robustness shift. T able 5.10 sho ws the goals for the v ehicle b eha viour and the
constrain ts for lateral and v ertical stabilit y . T arget v alues for the ob jectiv e quan tities are
denoted b y *, lo w er and upp er b ounds b y lb and ub. The constrain ts ha v e to b e fulfilled
for eac h p ossible solution.
T able 5.10: Goals and constrain ts for ob jectiv e quan tities for sedan.
φ ∗ ( a y ,hig h ) in ◦ RM R ∗
min/max in % β ub
max in ◦ F lb
ty re,min in N R M S ∗ ( ˙
φ f ,low ) in ◦ /s
sedan 3.0 56/58 β max,cr it 1000 4.5
The p erformance function for robust design optimization is similar to Equation (5.2).
ˆ z ( x i ) = max
j
abs ( z j ( x i ) − z ∗
j )
z ∗
j
. (5.3)
The system p erformance ˆ z quan tifies the o v erall p erformance as the w orst p erformance
of all ob jectiv es. The p erformance function ˆ z quan tifies the largest distance to the target
v alues in p ercen t, e.g. ˆ z = 0 . 05 means: the greatest deviation of one unsp ecified ob jectiv e
quan tit y from its target v alue is 5 %. All other deviations are equal or smaller. The lo w est
v alue of ˆ z indicates the design that has the smallest maxim um distance of all target v alues.
In tegration of Robustness. The design parameters x i are again the fron t c a,f and the
rear stiffness of the an ti-roll bars c a,r . Uncontrollable parameters p u , with u as index for
73

5 Application
the uncon trollable parameters are: the b o dy mass, whic h v aries +/-5 % and the damp er
c haracteristics whic h v ary +/-10 %. F or ev aluation, the uncon trollable parameters are
scattered within their ranges. The r obustness obje ctive is giv en b y the standard devia-
tion of the p erformance function σ ( ˆ z ( x i , p u )) and quan tifies the exp ected deviation of the
p erformance function ˆ z . The p erformance function ˆ z indicates the greatest deviation of
one unsp ecified ob jectiv e quan tity in p ercen tage terms, while σ ( ˆ z ) denotes the exp ected
deviation in p ercen t p oin ts (PP) according to the system p erformance ˆ z . In con trast, in
classical RDO the standard deriv ation is asso ciated to the mean v alue.
Result. Figure 5.7 sho ws the result of a genetic algorithm with a p opulation of 400
individuals (designs x i ), 5 generations and 100 neigh b our individuals p er individual for the
ev aluation of robustness. In summary 200 thousand design p oin ts w ere ev aluated.
0.08 0.09 0.1 0.11 0.12
0
0.1
0.2
0.3
0.4

ˆ z
σ ( ˆ z )

Figure 5.7: P areto-fron t b et w een p eak p erformance ˆ z and robustness ob jectiv e σ ( ˆ z ) .
T able 5.11 sho ws the b est designs according p erformance and robustness. The b est
design according to p erformance has a maximal deviation of 7.5 % to the target v alues.
A ccording to this p erformance within the scattered uncon trollable design parameter an
a v erage deviation of 12.8 PP o ccurs. This corresp onds to an a v erage maximal deviation of
20.3 %. The design, according to robustness, has a maximal deviation of 9.2 % and only
an a v erage deviation of 2.4 PP . This corresp onds to an a v erage maximal deviation of 11.6
%
T able 5.11: Results of robust design optimization.
c a,f in N /mm c a,r in N /mm ˆ z σ ( ˆ z )
b est p erformance 19.84 2.28 0.075 0.128
b est robustness 21.06 0.86 0.092 0.024
F or b etter results, the target v alues migh t b e adopted or more design parameters migh t
b e included in further iterations. The robust design p erforms the b est according to v aria-
tions of the uncon trollable parameters.
5.3.4 New App roach: Robust Design with Solution Space
Robustness against Uncertain ties of Design P arameters. Robustness against un-
certain ties of design parameters is included in a simple manner in the solution space ap-
proac h. If the uncertain ties of design parameters are indep enden t and ma y b e b ounded b y
74

5.3 Robust Design Strategy of An ti-roll Bars
limits, a design x ∗
i in the cen ter of a b o x-shap ed solution space called here robust. Figure
5.8 sho ws the solution space of Section 5.3.1 and a b o x-shap ed solution space. The cen ter
of this b o x-shap ed solution space is regarded as a robust design. The b o x ma y arbitrarily
b e mo v ed within the solution space. In this case a robust design migh t b e c a,f = 33.0
N/mm and c a,r = 8.5 N/mm. The fron t an ti-roll bar stiffness can ha v e an uncertain t y
of +/-4.0 N/mm and the rear an ti-roll bar stiffness can ha v e an uncertain t y of +/-1.2
N/mm. Note, the uncertain ties can o ccur in com bination. Despite of these uncertain ties,
requiremen t satisfaction is guaran teed.
10 15 20 25 30 35 40 45
0
2
4
6
8
10
c a, f in N / m m
c a, r in N / m m

Figure 5.8: Robustness against uncertain ties of design parameters (2D).
Robustness against Uncertain ties of Uncon trollable P arameters. Uncon trol-
lable parameters p u as men tioned in Section 5.3.3 are the b o dy mass, whic h v aries +/-5 %
and the damp er c haracteristics, whic h v aries +/-10 % due to differen t motors and driving
c haracters of the v ehicle. Figure 5.9(a) sho ws the reduction of the solution space due to
v ariation of the b o dy mass. The solution spaces for separate b o dy masses are o v erlaid.
The blac k lined solution space indicates the minimal solution space. Within this solution
space all requiremen ts on v ehicle b eha viour are satisfied with a b o dy mass v ariation of +10
% and -5 %. If the b o dy mass rises, due to the roll angle requiremen t, more stiffness of the
fron t and rear an ti-roll bar is needed. A lo w er b o dy mass reduces the roll angle and do es
not lead to requiremen t violation. Figure 5.9(b) sho ws the reduction of the solution space
due to v ariation of the damp er c haracteristic. The fron t and rear damp er c haracteristics
are scaled equally . With decreasing damping of the v ehicle, the roll angle v elo cit y require-
men t reduces the solution space. Eac h solution space is sampled with 200 design p oin ts,
in summary 1000 design p oin ts are sampled. Within this solution space, all requiremen ts
on v ehicle b eha viour are satisfied with a damp er c haracteristic v ariation of +5 % and -10
%.
75

5 Application
10 15 20 25 30 35 40 45
0
2
4
6
8
10

c a, f in N / m m
c a, r in N / m m

0.95*m
1.00*m
1.05*m
1.10*m

(a) Robustness against mass v ariations
10 15 20 25 30 35 40 45
0
2
4
6
8
10

c a, f in N / m m
c a, r in N / m m

0.90 d
0.95 d
1.00 d
1.05 d

(b) Robustness against damp er v ariations
Figure 5.9: Ov erla ying of solution spaces with differen t v alues of uncertain parameters.
Comparison to Robust Design Optimization. Figure 5.10 sho ws a solution space
approac h for the design problem describ ed in Section 5.3.3. The solution space is reduced
as the b o dy mass and the damp er c haracteristic can v ary . Note, the target v alues z ∗
of the ob jectiv e quan tities are replaced b y b ounds for the solution space approac h. The
differences to the optim um of ob jectiv e quan tities, whic h are acceptable for the designer,
are included and used in order to find solutions. Sev eral designs satisfy all requiremen ts
on the v ehicle b eha viour, therefore a set of robust solutions is prop osed. All designs within
the thic k blac k lined solution space in Figure 5.10 satisfy all requiremen ts while the b o dy
mass and the damp er c haracteristics v ary ab out +/-5 % and +/-10 % The robust solution
space (blac k line) is reduced in comparison to the non-robust reference (gra y line) due to
the robustness against the uncon trollable parameters.
The optim um of RDO (square sym b ol) lies within the robust solution space at the
b oundary of the maxim um roll momen tum ratio RM R max . As Figure 5.9(a) and (b) sho w
this b oundary indep enden t of b o dy mass and damp er c haracteristics. The robust optim um
of RDO is the design with the lo w est standard deriv ation of the system p erformance,
but a little c hange of the fron t or rear stiffness of the an ti-roll bar leads to violation of
requiremen ts. Uncertain design parameter w ere not included in the RDO approac h but
ma y b e included.
In comparison to the result of RDO, the solution space approac h generates a set of robust
designs. Within this set, a robust design considering uncertain ties of design parameter can
b e considered without additional effort. A robust design migh t b e c a,f = 20.0 N/mm and
c a,r = 2.8 N/mm. The fron t and rear stiffness of the an ti-roll bars can v ary ab out +/-2.5
N/mm and +/-1.0 N/mm.
The handling of uncertain ties of design and uncon trollable parameters is simpler with
solution space than with robust design optimization. The robustness is naturally in tegrated
for design parameters and can b e easily extended for uncon trollable parameters. F urther,
in con trast to RDO the desired robustness can b e c hanged after the computation of the
solution space.
76

5.4 Pro duct F amily Design for V ehicle System Dynamics Design
10 15 20 25 30 35 40 45
0
2
4
6
8
10
c a , f in N / mm
c a , r in N / mm

solution space (reference)
robust solution space
(uncontrollable parameter)
robust optimum (RDO)
robust design (design and
uncontrollable parameter)

Figure 5.10: Solution spaces for reference and robustness against uncon trollable pa-
rameters. F or comparison, the robust optim um of the RDO approac h is added. Uncertain
parameters are the b o dy mass: [0 . 95 , 1 . 05] · m and the damp er c haracteristic: [ 0 . 9 , 1 . 10] · d .
In addition, a robust design, whic h considers design parameter uncertain ties is sho wn.
5.4 Pro duct F amily Design fo r V ehicle System Dynamics
Design
Problem Description. Three v ehicles with t w o design parameters eac h are to b e designed
for maxim um commonalit y . F or simplicit y , the analysis is restricted again to the t wo design
parameters c a,f and c a,r , denoting the stiffness parameters of the an ti-roll bars of the fron t
and real axle, resp ectiv ely . F or the design ob jectiv es, sp ecific threshold v alues for lo w er
(lb) and upp er b ounds (ub) are pro vided in T able 5.12. F or ev aluation purp oses, the
ob jectiv e quan tities z α,j are further condensed to the vehicle p erformanc e z α and the total
p erformanc e of all v ehicles z , The three v ehicles consist of v ehicle (1) as sedan, v ehicle (2)
as station w agon and v ehicle (3) as coup é. F urthermore, threshold v alues for a comfortable
v ehicle c haracter are presen ted, whic h are, e.g., used for the design of a sedan or station
w agon. The thresholds for a sp ort y v ehicle c haracter as for the coup é are defined with a
reduced roll angle and relaxed ride goals. Again, only ob jectiv e quan tities are pro vided
that are relev an t for this particular design phase. Therefore, for the assessmen t of self-
steering b eha viour, only the roll momen t distribution is relev an t, as t yre and other c hassis
prop erties are considered in differen t design phases.
T able 5.12: Threshold v alues of ob jectiv e quan tities for sedan, station w agon and coup é.
middle-class v ehicle φ ( a y ,high ) in ◦ R M R in % β max in ◦ F ty r e,min in N RM S ( ˙
φ f ,low ) in ◦ /s
lb ub lb ub lb ub lb ub lb ub
(1) sedan - 3.0 55 60 - β max,cr it 1000 - - 5.0
(2) station w agon - 3.0 55 60 - β max,cr it 1000 - - 5.0
(3) coup é - 2.5 55 60 - β max,cr it 1000 - - 6.0
5.4.1 Classical App roach: Optimizing V ehicles Sepa rately
Cho osing the optimal configuration of common and individual comp onen ts is difficult in
pro duct family design. Examining all p ossible configurations is computationally prohibitiv e
as w as sho wn in Section 4.2.
77

5 Application
As an alternativ e approac h, one could iden tify optimal parameter v alues for eac h v ehicle
separately and then try reassigning these parameter v alues in v arious com binations to
other v ehicles to minimize the n um b er of differen t parameter v alues n . This is similar to
the approac h of F ellini et al. [30]. T able 5.13 sho ws in part (1) the result of separate
v ehicle optimisations with resp ect to z α . F or eac h v ehicle, parameters could b e iden tified
with z ≤ 0 , i.e., all design goals w ere reac hed. The asso ciated total n um b er of differen t
parameter v alues is n = 6 .
All reassigned p ossible pro duct families are based on the separately optimized parameter
v alues of configuration (1) and are presen ted in App endix A.4. Among all reassigned
solutions, configuration (2c) is the b est configuration, whic h is sho wn in T able 5.13 in part
(2) The rear an ti-roll bar stiffness is shared among all v ehicles. Here, the total n um b er of
differen t parameter v alues is n = 4 , while the total p erformance z ≤ 0 .
T able 5.13: Results of the approac hed based on separate v ehicle optimisation.
P arameter V ehicle 1 V ehicle 2 V ehicle 3 n i
(1) Separate p erformance optimisation for eac h v ehicle
c a,f in N /mm 21.93 (A) 24.77 (B) 18.77 (C) 3
c a,r in N /mm 2.34 (A) 2.30 (B) 2.40 (C) 3
z i -0.014 -0.012 -0.019 n = 6
(2c) Best configuration obtained b y separate optimisation
c a,f in N /mm A B C 3
c a,r in N /mm B B B 1
z i -0.013 -0.012 -0.018 n = 4
5.4.2 New App roach: Deriving Common P a rameter V alues from
Solution Spaces
The new approac h b ypasses the com binatorics of distributing common parameters, b ecause
single designs are not ev aluated sequen tially . Instead, solution spaces are constructed
and then compared for differen t systems in order to iden tify appropriate configurations of
common parameter v alues. F or the three v ehicles considered, solution spaces, taking all
design criteria in to accoun t, are sho wn in Figure 5.11. Also, the optima with resp ect to
the v ehicle p erformance z α are sho wn. Note, that the design space of the fron t an ti-roll bar
stiffness is limited due to w eigh t. As the solutions spaces of v ehicles 1 and 2 o v erlap, pairs
of v alues for the an ti-roll bar stiffness of the fron t and rear axle exist, whic h satisfy the
design goals for b oth v ehicles. Similarly , the solution spaces of v ehicles 1 and 3 o v erlap.
All p ossible configurations of common parameter v alues can b e iden tified in Figure 5.11.
V ehicles 1 and 3 can share b oth parameter v alues. F or v ehicle 2, the rear an ti-roll bar
stiffness can b e the same as for v ehicle 1 and 3. The fron t an ti-roll bar stiffness, ho w ev er,
has to b e differen t. The sp ecific v alues are sho wn in T able 5.14.
78

5.5 High Dimensional Design Problems
14 16 18 20 22 24 26 28
0
0.5
1
1.5
2.0
2.5

c a ,f in N/ mm
c a ,r in N / mm

Vehicle 1
Vehicle 2
Vehicle 3
Optimum
Common design

Figure 5.11: Solution spaces (lines), common designs (big dots), and p erformance optima
from Section 5.4.1 (small dots) are sho wn in the space of an ti-roll bar stiffness c a v alues of
the fron t (x-axis) and rear (y-axis) axle.
T able 5.14: Configuration deriv ed from o v erlapping solution spaces.
P arameter V ehicle 1 V ehicle 2 V ehicle 3 n i
c a,f in N /mm 20.60 (A) 25.00 (B) 20.60 (A) 2
c a,r in N /mm 2.25 (A) 2.25 (A) 2.25 (A) 1
z i -0.005 -0.007 -0.005 n = 3
Separate v ehicle optimisation yielded a total n umber of n = 4 comp onen ts (3 fron t
and 1 rear axle an ti-roll bar stiffness v alues), whic h is necessary for the design of three
v ehicles. In comparison, the solution space approac h sho ws that a configuration with a
total n um b er of differen t comp onents of n = 3 is sufficien t (2 fron t and 1 rear axle an ti-roll
bar stiffness v alues). Figure 5.11 illustrates wh y the approac h that relies on optimising
v ehicles separately will inevitably deliv er a w orse result: The p erformance optima do not
lie in the o v erlapping region of solution spaces. Therefore, a larger n um b er of differen t
parameter v alues and, th us, comp onen ts of differen t designs is required.
5.5 High Dimensional Design Problems
Real-w orld design problems ha v e t ypically more than t w o design parameters. In high
dimensions, solution space of arbitrary shap e cannot b e visualized and a sampling tec hnique
of the previous section is not applicable an ymore. Design parameters next to the stiffness
of the fron t and rear an ti-roll bar c a,f and c a,r are the stiffness of the fron t and rear bump
stop c b,f and c b,r with their free length under compression s 0 ,b,f and s 0 ,b,r and the stiffness
of the fron t and rear reb ound stop c r ,f and c r ,r with their free length under reb ound s 0 ,r,f
and s 0 ,r ,r . In total these are 10 design parameters. Solution space cannot b e visualized
79

5 Application
and analysed at an adequate lev el. 2-D solution space ma y only b e created, if all other
parameters are fixed to a certain v alue or in terv al. The alternativ e is to w ork with b o x-
shap ed solution space. The edges of the h yp er b o x can b e illustrated next to eac h other.
5.5.1 Robust Design fo r High Dimensional Design Problems
The design problem of Section 5.3.4 is extended b y 8 further design v ariables. Note a b o x
shap ed-solution space can b e shifted in the original solution space. In Figure 5.12 the gra y
dashed lines sho ws a b o x-shap ed solution space for a sedan depicted as upp er and lo w er
limits for eac h design parameter. The connecting lines b et w een limits are only for the
purp ose of visualization.

0
1
2
3
4
5
6
7
N/mm, mm, kg , −

c a,f 10c a,r c b,f c b,r s 0,b,f s 0,b,r c r,f c r,r s 0,r,f s 0,r,r m/500 f d,f /4 f d,r /4
Design 1 (referenz)
Design 2 (robust against m,d f , d r )

Figure 5.12: Bo x-shap ed solution space for a sedan depicted as upp er and lo w er limits
for eac h design parameter. The connecting lines b et w een limits are only for the purp ose
of visualization. Num b ers in the x-lab el are scaling factors.
If the design parameter v alues are set in the middle of the corridor, robustness against
uncertain ties of design parameters is considered in a simple manner. The half in terv al giv es
the range of robustness for eac h design parameter. The gra y corridor ma y b e the corridor
with the largest v olume. Note an in terv al can b e extended, if other in terv als are allo w ed
to decrease. The gra y stripp ed corridor is not robust against uncon trollable parameters
as the b o dy mass or the damp er c haracteristics, as the last three parameters are set to a
single v alue. The blac k corridor has fixed in terv als for the uncon trollable parameters as
robustness. In consequence, the in terv als of the design parameters are reduced. In the
robust case, the stiffness of the an ti-roll bars and bump stops is reduced. The free lengths
80

5.5 High Dimensional Design Problems
are reduced, so the bump stop and the reb ound stop are in action already at lo w deflection
to handle higher masses.
Bo x-shap ed solution spaces or corridors can also b e o v erlaid if sev eral design teams
w ork sim ultaneously on the design of differen t ob jectiv e quan tities. In case of violation of
requiremen ts in terv als migh t b e adjusted.
5.5.2 Pro duct F amily Design fo r High Dimensional Design Problems
In this section, a real-w orld problem with all 10 design parameters is considered. The
pro duct family comprises 13 v arian ts of a middle class v ehicle and its deriv ativ es (including
v arian ts of a sedan, long sedan, station w agon, coup é, con v ertible and grand tourismo).
F rom no w on, eac h v arian t will b e referred to as vehicle . F or eac h v ehicle, the goals for
handling and ride are presen ted in T able 5.15. V ehicles 3, 4 and 11 ha v e differen t design
goals as they are designed to b e more sp ortiv e. Therefore, design goals for lateral dynamics
are stricter and design goals of v ertical dynamics are relaxed. All other v ehicles ha v e the
same design goals. Figure 5.13 sho ws the b o x-shap ed solution spaces for 13 v ehicles as
p ermissible in terv als for eac h design parameter with upp er and lo w er b oundaries. As long
as all design parameter v alues are within the computed in terv als, all design goals are met.
T able 5.15: Critical threshold v alues for ob jectiv e quan tities
V ehicle φ in ◦ R M R in % β max in ◦ F ty re in N R M S ( ˙
φ f ,low ) in ◦ /s
lb ub lb ub lb ub lb ub lb ub
1,2,5-10,12,13 - 3,0 55 60 - β max,cr it 1000 - - 5.0
3,4,11 - 2.5 55 60 - β max,cr it 1000 - - 6.0
In terpretation of Results. The an ti-roll bars for v ehicle 1 and 3 are of common
design, but v ehicle 2 needs an individual stiffness of the fron t an ti-roll bar, b ecause it is
a station w agon with a higher cen ter of gra vit y and mass. Therefore, the roll momen t
increases. In order to fulfil the requiremen ts of the roll momen t distribution, v ehicle 2
needs a higher fron t an ti-roll stiffness than v ehicle 1 (sedan) and 3 (coup é). The remaining
comp onen ts almost o v erlap, see, e.g., the stiffness of the reb ound and bump stop c b . V ehicle
2 also needs a higher rear reb ound stiffness and a lo w er free length under extension, whic h
are necessary b ecause of the increased rear axle load of a station w agon. Applying the
algorithm sho wn in Figure 4.2, n = 26 differen t parameter v alues can b e defined with
whic h the design goals for all 13 v ehicles can b e satisfied, 4+3 for an ti-roll bar stiffness,
2+2 for bump stop c b stiffness, 2+3 for reb ound stop c r stiffness, 3+2 free lengths under
compression and 2+3 free lengths under extension. The parameter v alues ma y b e c hosen
an ywhere within the in terv als pro vided in T able 5.16. The cen ter of the in terv als ma y b e
c hosen in order to b e robust against unin tended deviations from the target v alue. T able
5.17 sho ws what parameters v alues are assigned to what v ehicle. Note that a maxim um of
4 differen t parameter v alues is required for eac h parameter, although there are 13 differen t
v ehicles. The distribution of common parameters w as iden tified b y the prop osed metho d
without an y a-priori kno wledge.
81

5 Application
T able 5.16: In terv als for shared design parameter v alues represen ting comp onen ts of same
design. The unit of the stiffness c and the free length s is in N/mm or mm. The iden tify er
A, B, C, D denote the resp ectiv e in terv als for eac h design parameter.
c a,f c a,r c b,f c b,r s 0 ,b,f s 0 ,b,r c r ,f c r,r s 0 ,r ,f s 0 ,r ,r
A 2.4 - 2.5 2.9 - 5.2 0.5 - 3.0 1.3 - 4.1 0.5 - 1.9 0.9 - 2.3 0.6 - 3.2 0.5 - 1.8 1.5 - 3.8 1.3 - 2.3
B 2.7 - 2.9 6.4 - 7.1 5.7 - 6.1 6.0 - 9.9 2.1 - 3.0 6.8 - 10 6.6 - 9.6 3.2 - 4.4 7.0 - 10 3.1 - 7.7
C 2.9 - 3.3 8.3 - 9.1 - - 5.1 - 9.8 - - 7.0 - 9.9 - 8.0 - 10
D 3.4 - 3.4 - - - - - - - - -
T able 5.17: Configuration table. The same sym b ols A, B, C, D for differen t v ehicles
indicate comp onen ts of same design.
P arameter
V ehicle 1
V ehicle 2
V ehicle 3
V ehicle 4
V ehicle 5
V ehicle 6
V ehicle 7
V ehicle 8
V ehicle 9
V ehicle 10
V ehicle 11
V ehicle 12
V ehicle 13
n i
c a,f A B A D C A A C D D D C C 4
c a,r C B C C B A B C C C C B A 3
c b,f B B B A A B A B B B B B B 2
c b,r B B B B A B B B B B B B B 2
s 0 ,b,f B A B C C C C C C C C C C 3
s 0 ,b,r B B B B A B B B B B B B B 2
c r ,f A A B B B A B B B B B B B 2
c r ,r A B B B B C C B C B C C A 3
s 0 ,r ,f B B B B B A B B B B B B B 2
s 0 ,r ,r B A B C C C C C C C C C C 3
82

5.5 High Dimensional Design Problems

0
20
40
60
80
100
120
N/mm, mm

c a,f 10 c a,r c b,f c b,r s 0,b,f s 0,b,r c r,f c r,r s 0,r,f s 0,r,r
Vehicle 1
Vehicle 2
Vehicle 3

(a)

0
20
40
60
80
100
120
N/mm, mm

c a,f 10 c a,r c b,f c b,r s 0,b,f s 0,b,r c r,f c r,r s 0,r,f s 0,r,r
Vehicle 1
Vehicle 2
Vehicle 3
Vehicle 4
Vehicle 5
Vehicle 6
Vehicle 7
Vehicle 8
Vehicle 9
Vehicle 10
Vehicle 11
Vehicle 12
Vehicle 13

(b)
Figure 5.13: Solution spaces for (a) v ehicles 1-3 and (b) all 13 v ehicles sho wn as upp er and
lo w er limits for eac h design parameter. The units of stiffness v alues c and free lengths s are
N/mm and mm, resp ectiv ely . The connecting lines b et w een limits of differen t parameters
are only for the purp ose of visualization.
83

6 Discussion
Generation of P erception Thresholds
Handling Uncertain ties. F or the metho d presen ted in Chapter 2 and applied in Section
5.2, class-sp ecific sub jectiv e kno wledge has to b e generated b y driving exp erts. Uncertain-
ties ma y o ccur in the relativ e sub jectiv e ev aluation of the impact of design parameters
according to design categories. Uncertain ties of a single driving exp ert are reduced b y
merging or a v eraging the ev aluation of sev eral driving exp erts. F urthermore, if driving
exp erts explain or justify their ev aluation b y telling in-/output relations and the design
space of the parameters in mind, the traceabilit y is increased. Explanations are useful
for v alidation. Uncertain ties of the ob jectification kno wledge ma y b e reduced in a similar
w a y . Sev eral driving exp erts or senior engineers describ e the qualitativ e relation b et w een
ob jectiv e quan tities and design categories in mind. F or relativ e w eigh tings it is hardly
p ossible for driving exp erts to set absolute n umerical n um b ers, but the use of sym b ols
(a, b, c) ma y simplify and accelerate the w eigh ting pro cess. If m ultiple driving exp erts
ev aluate a relation b et w een a design parameter and a design category , all other design
parameter v alues scatter within a certain range in mind. T o repro duce these uncertain ties
or get the influence of the neigh b ourho o d of designs, design of exp erimen ts metho ds and
global sensitivit y analysis are used. If design parameters and their design spaces are fixed,
a sim ulation mo del is used for the ev aluation of exp erimen ts. Uncertain ties o ccur due
to mo del qualit y or accuracy . Therefore the mo del needs to b e v erified steadily against
measuremen t.
Generation of Threshold Sensitivities. Driving exp erts use p erception thresholds
in mind for the arrangemen t of sub jectiv e ev aluation. These p erception thresholds can
b e concluded from the comparison of sub jectiv e and ob jectified kno wledge. If the pat-
tern of sub jectiv e and ob jectified ev aluations matc h eac h other, threshold v alues for the
aggregated sensitivities ˆ
S or p erception thresholds can b e concluded. P attern means in
this con text the relativ e order of impacts or sensitivities of design parameters for a certain
design category . F or eac h design category individual threshold v alues are concluded as
the relativ e sub jectiv e ev aluation of the design categories are differen t. The more design
parameters are considered, the exacter the thresholds of the ob jectiv e quan tit y can b e de-
termined. With the kno wledge of the aggregated sensitivit y thresholds, the sub jectiv e and
ob jectified kno wledge can b e correlated. The matc h of the pattern is a necessary request
for high corresp ondence. The threshold v alues are only adopted for a direct comparison of
sub jectiv e and ob jectified kno wledge.
84

Solution Space Design and Robust Design using Solution
Space
Change in Mindset. In classical optimisation problems a design x ∗
i is sough t, whic h
optimizes the system p erformance. The target v alues for the system p erformance z ∗ are
giv en b y the designer. All target v alues ma y not b e compatible and the single design x ∗
i
will lead to deviations from z ∗
j . The designer has to use w eigh tings for the optimization of
a single total p erformance function or to select a design after generating a P areto-optimal
fron t for m ultiple p erformance functions in order to shift off the deviations. In con trast, the
solution space approac h uses inequalit y constrain ts for the system b eha viour. F or a go o d
design all inequalit y constrain ts ha v e to b e satisfied. All go o d designs constitute a set of
go o d designs. Critical v alues for the system p erformance z j,cr it are giv en b y the designer.
Designs, whic h violate a criterion, are not desired. Classical optimisation represen ts a
sp ecial case of solution space design, if critical v alues of system p erformance limit the
system p erformance to z ∗
j . A single design is sough t, instead of a set of go o d designs.
Handling Requiremen t Changes. If critical v alues for system p erformance z j,cr it
c hange, the shap e of the solution space ma y c hange. Note that not ev ery inequalit y con-
strain t determines the shap e of the solution space, as an inequalit y constrain t ma y b e
dominated b y other inequalit y constrain t. If the shap e of the solution space c hanges, a
b o x-shap ed solution space has to b e adopted. The sample p oin ts, whic h are ev aluated for
the solution space, do not ha v e to b e re-ev aluated as c hanges of critical v alues can b e done
in real time with meta-mo dels. F or high-dimensional b o x-shap ed solution space, the set of
designs can b e ev aluated for differen t critical v alues and extended and trimmed if necessary .
F urther, requiremen ts o ccurring suddenly in the design pro cess can b e added b y ev aluating
the relev an t ob jectiv e quan tit y for eac h design p oin t or at least eac h design p oin t in the
solution space. If some design p oin ts do not satisfy the criteria, they are excluded. The
shap e of the solution space c hanges and b o x-shap ed solution space is adopted.
Numerical Effort to Generate Solution Space. F or t w o or three dimensional
solution space the whole design space can b e sampled. Therefore 200 to 300 sample p oin ts
are sufficien t, if using a lo w-discrepancy sampling metho d. F or high dimensional design
problems it is not adequate to sample the whole design space to generate a solution space.
The design space can b e sampled with a high sample n um b er to generate meta-mo dels
with high qualit y . F or 20 parameters 5000 sample p oin ts are sufficien t, if using a lo w-
discrepancy sampling metho d. A b o x-shap ed solution space with 10 parameters as, e.g.,
in the example problem in Section 5.5.2 requires appro ximately 10 4 ev aluations of the
p erformance functions to con v erge, see [133].
Handling Conflict of Goals. T arget conflicts ma y o ccur in classical design as w ell
as in solution space design. If in the design of a solution space a conflict of goals o ccurs,
no solution space remains. A conflict of goals can b e confirmed and visualized b y solution
space to other design exp erts. All constellations of design parameters lead to violation
of at least one requiremen t. A p ossible strategy is to w eak en a relev an t requiremen t of
the conflict of goals un til a solution space is created. This can b e done in real-time, if
meta-mo dels are used. A probable solution of the conflict of goals is then quan tified b y the
p ossible shifts of critical v alues. If critical v alues cannot b e shifted or this is not desired,
further relev an t design parameters ha v e to b e included in the design pro cess. This includes
high effort: a new design of exp erimen ts plan is needed, exp erimen ts ha v e to b e executed
85

6 Discussion
and ev aluated. Therefore all relev an t design parameters should b e included in solution
space design from the b eginning.
Detrimen ts and Benefits of Bo x-shap ed Solution Space Ho w ev er, as a total
solution space that con tains all go o d designs is generally not b o x-shap ed, in b o x-shap ed
solution space go o d designs will b e neglected. The loss of solution space is discussed in
detail in [133]. A t least 2 corners of a b o x shap ed solution space lie on the b oundary of the
total solution space. Therefore designs at an in terv al limit ma y b e next to designs whic h
violate a requiremen t. The qualit y of the sim ulation and of the ob jectification kno wledge
ma y induces uncertain t y , to o. A b o x-shap ed solution space ma y op en a lot of p ossibilities.
The cen tral design in a b o x-shap ed solution space is a candidate for a robust design. If
the cen ter design is c hosen for all design parameters within the in terv als, uncertain ties
of the design parameters can o ccur indep enden tly of eac h other with a range of +/- the
half design parameter in terv al. F or high-dimensional systems, b ox-shaped solution s pa ces
are computed instead of arbitrarily shap ed solution spaces. This mak es it p ossible to
visualize the results in in terv als of design parameters. Box-shaped solution space decouple
parameters. Imagine parameters of subsystems whic h ha v e to b e designed themselv es b y
design teams, analogously to target cascading. Decoupled parameters are also necessary
for the use of the pro duct family design algorithm for high dimensional parameter design.
Due to decoupled parameters eac h parameter can b e shared within its in terv al with other
systems without influencing other parameters or requiremen t satisfaction.
Robust Design with Solution Space. In classical robust design optimization, the
range of scattered design or uncon trollable parameters is needed from the b eginning. The
robustness ob jectiv e is ev aluated for eac h design candidate b y c hec king the system p erfor-
mance with scattered parameters within their range. These ranges of uncertain parameters
ma y c hange due to c hanged costumer requiremen ts, impro v emen t of comp etitors or pro-
duced subsystems of suppliers with higher or lo w er accuracy . If the p ermissible ranges
of uncertain parameters c hange, classical robust design optimization has to b e rep eated.
F or robust design with solution space, no a-priori kno wledge ab out ranges of uncertain
parameters is required. The cen ter of a b o x-shap ed solution space suggests a robust design
for indep enden t uncertain ties of the design parameters. A b o x-shap ed solution space can
b e re-arranged within the shap e of the solution space to adopt the ranges of uncertain
parameters without a new determination of the solution space.
Pro duct F amily Design
Simplifying Pro duct F amily Design b y Ov erla ying Bo x-shap ed Solution Spaces.
A corridor for eac h system is computed and o v erlaid. The in terv als of the design param-
eters ma y not fit for eac h design parameter. Within the ov erlapping in terv als, the design
parameters can b e shared for sev eral systems, indep enden t of the other design parameters.
Note that the b o x-shap ed solution space ma y b e pulled or pushed within the parameter
space. If solution spaces with maximal v olume are o v erlaid pulling or pushing of limits
of design parameter in terv als lead to decreased v olume. Certain limits of design in terv als
can only b e increased b y decreasing others. In terv als of other design parameters ma y then
not o v erla y an y more and further comp onen t configurations are necessary . Note that the
use of the greatest solution b o xes increases the probabilit y for o v erla ys. The use of the
flexibilit y of solution b o xes will b e fo cus in future researc h.
86

Comm unal Corridors f or Sev eral Systems. Imagine c hec king comm unalit y of 2
systems with m ultiple design parameters. A comm unal corridor for b oth systems can
b e computed. If no solution space results, this means all design parameters cannot b e
coupled for these t w o systems. Or with other w ords, at least one design parameter cannot
b e coupled, but it is not kno wn whic h one. If a comm unal corridor for 2 systems exists, all
design comp onen ts can b e shared. If the decision is made to com bine these t w o systems
the minimal n um b er of comp onen ts for all systems is affected. The com bined systems ma y
also b e com bined with clusters of other systems with differen t v alues of design parameters.
All com binations of systems w ould ha v e to b e c hec k ed.
Numerical Effort of Pro duct F amily Design using Solution Spaces. Imagine
a design of pro duct family for 13 systems as presen ted in Section 5.5.2. A ccording to
Section 4.2 the n um b er of configurations for 10 parameters and 13 v ehicles is 2 . 6 × 10 74 .
As an imp ortan t adv an tage of the prop osed solution space metho d, solution spaces can
b e computed for eac h system separately . This reduces data managemen t effort and ma y
sa v e time when sim ulations run in parallel. 13 v ehicles require 10 5 ev aluations of the
p erformance functions. This n um b er is still small compared to the num b er of configurations
if all com binations w ere computed.
Uniqueness of Pro duct F amily Design with Solution Spaces. A ccording to the
problem statemen t in Section 4.1, sev eral differen t pro duct family designs ma y b e a solution
with equal total n um b er n of differen t parameter v alues. Imagine pulling and pushing the
in terv als of the fron t an ti-roll bar stiffness for all systems in Section 5.5.2 together. All
system requiremen ts ha v e to b e satisfied b y separating other design parameter v alues and
decoupling shared in terv als. Ho w ev er, a solution for the pro duct family ma y b e found,
whic h has the same total n um b er of comp onen ts. The solution for a total minimal n um b er
of comp onen ts is not unique. This do es not compromise practical applicabilit y . Either the
designer prefers a certain distribution of comp onen ts or the maximal n um b er of comp onen ts
n i can b e minimized.
Comp onen ts with Multiple Design P arameters. F or simplification m ultiple de-
sign parameters for a single comp onen t ma y b e treated indep enden tly in the con text of the
n um b er of parameter v alues. Imagine 2 design parameters whic h are coupled in a single
comp onen t but treated indep enden tly: design parameter 1 is needed 3 times and param-
eter 2 is needed 2 times. If all configurations are needed for the satisfaction of all system
requiremen ts 3 · 2 = 6 comp onen ts are needed. In Section 5.5.2 all design parameters are
separated and arranged to their o wn comp onen t. Then, the n um b er of design parameters
corresp onds to the n um b er of comp onen ts, whic h has to b e minimized. Imagine the free
damp er length is fixed b y a comm unal damp er and not v ariable due to free length of re-
b ound or free length of compression. E.g., the fron t bump stop has 2 design parameters:
the stiffness c b,f and the free length under compression s 0 ,b,f . F or this example all config-
urations can b e generated b y T able 5.17. All com binations of c b,f and s 0 ,b,f are: A C (3
v ehicles), BA (2 v ehicles), BB (2 v ehicles) and BC (7 v ehicles). In this case 4 bump stops
are needed, but a maxim um n um b er of 2 · 3=6 bump stops w ould ha v e b een p ossible.
87

7 Conclusion and Outlo ok
7.1 Conclusion
The automotiv e industry is lo oking for an efficien t pro duct dev elopmen t pro cess to offer
a large div ersit y of high qualit y pro ducts while reducing dev elopmen t costs. An efficien t
pro duct dev elopmen t pro cess can b e created b y shifting man y asp ects of dev elopmen t
in to the virtual w orld. Therefore, the automotiv e industry has the needs for a systematic
sim ulation-based metho d for the generation of relations b et ween subjectiv e and ob jectiv e
ev aluations, a robust and flexible design metho d and a pro duct family design metho d,
whic h minimizes the n um b er of comp onen ts to handle and reduce complexit y .
A new metho d to generate p erception thresholds of ob jectiv e quan tities w as presen ted
to quan tify c hanges of sub jectiv e ev aluations. A p erception threshold is corresp onding to
a single ob jectiv e quan tit y and is a relativ e v alue. It quan tifies the c hange of the ob jectiv e
quan tit y to cause a c hange of the sub jectiv e ev aluation index (BI) of the corresp onding
design category . Sub jectiv e ev aluations of design parameters according to design categories
are stored in a kno wledge managemen t system. The con tained informations ma y also
b e pro duced or v erified b y simulation and compared to the sub jectiv e kno wledge. The
results are p erception thresholds of ob jectiv e quan tities. Design parameters are scattered
with design of exp erimen ts metho ds within a range driving exp erts estimated. F or eac h
exp erimen t all ob jectiv e quan tities are ev aluated whic h are essen tial for the ev aluation
of the design categories. The qualitativ e relation b et w een ob jectiv e quan tities and design
categories are stored in the ob jectification kno wledge database. A global sensitivit y analysis
with Sob ol indices is used to get the relativ e impact of the design parameters according
to the ob jectiv e quan tities. Using the ob jectification kno wledge, these sensitivities are
transferred according to the design categories. Quan titativ e ob jectification kno wledge or
p erception thresholds are gained b y the comparison of the patterns of sub jectiv e and
ob jectified kno wledge. These thresholds are essen tial for target v alues of system design as
they are often quan tified sub jectiv ely and relativ ely to predecessor or comp etitor systems.
Solution space design is prop osed as a metho d whic h in tegrates robustness naturally ,
without a robustness ob jectiv es and generates flexibilit y to c hanges of design parameter.
This pro cedure is a mind c hange, as it is not the design whic h leads to the desired p eak p er-
formance, but a set of designs whic h satisfies requiremen ts on system b eha viour and ma y
therefore consider uncertain ties. If requiremen ts on the system p erformance are made,
a set of design parameters ma y satisfy these requiremen ts. A cen tral design parameter
configuration in this design parameter set deliv ers the maximal robustness, whic h p ossibly
satisfies all the requiremen ts on the system p erformance. In real-w orld applications the
n um b er of comp eting ob jectiv es can b ecome o v erwhelming. Solution space for single re-
quiremen ts of ob jectiv e quan tities enable the sup erp osition and therefore the domination
of outp erformed designs. Solution space deliv ers a high flexibilit y in the early phase as all
p ossible v alues of design parameters are determined. The set of solutions enables quic k
88

7.1 Conclusion
resp ond to c hanges of requiremen ts. Imagine a b o x-shap ed solution space whose edges
are in terv als of design parameters. In a b o x-shap ed solution space these in terv als are in-
dep enden t of eac h other and decouple therefore the design parameters. These decoupled
in terv als of design parameters ma y b e separated to sev eral exp ert design teams without
high efforts of co ordination, as prop osed in set b ase d design or tar get c asc ading .
A new pro duct family design metho d is presen ted to optimise pro duct family designs with
resp ect to commonalit y using solution spaces. This is a new metho d for robust pro duct
family design. Instead of a p erformance optimisation of single systems, the largest solution
space for eac h system is iden tified. Solution space con tain only go o d designs whic h meet all
relev an t design goals. By o v erla ying solution spaces of several systems, regions of common
parameter v alues can b e iden tified. As simplification for high-dimensional problems, b o x-
shap ed solution space is used. Th us, p ermissible ranges can b e expressed for all design
parameters that are indep enden t of the other parameter v alues. This enables a particular
algorithm to iden tify the configuration with the smallest n um b er of shared parameters
in b o x-shap ed solution spaces. This metho dology w orks without a-priori information on
whic h comp onen ts should b e of common design. The metho d is applicable to arbitrary
non-linear high-dimensional systems with sp ecified p erformance functions and asso ciated
threshold v alues.
The presen ted metho ds are applied to an example design problem of v ehicle system
dynamics. A new middle class v ehicle needs to b e designed for man y asp ects of v ehicle
b eha viour. In the example considered here, the la y out of the self-steering b eha viour, the
lateral stabilit y , roll-o v er, stationary roll b eha viour and dynamic roll b eha viour are con-
sidered. There are man y more design goals that need to b e satisfied, ho w ev er, they are
considered in differen t design phases. Relev an t design parameters of the fron t and the rear
axle are the stiffnesses of the bump stops, the reb ound stops and the an ti-roll bars and the
free lengths under compression and extension to activ ate the bump and the reb ound stop,
resp ectiv ely . A new design of v ehicle system dynamic for a middle class v ehicle is desired,
therefore relativ e sub jectiv e ev aluations of design categories asso ciated to the predecessor
v ehicle are used. P erception thresholds are generated to transfer relativ e sub jectiv e ev alu-
ation to relativ e c hanges of ob jectiv e quan tities. A sub jectiv e kno wledge for middle class
v ehicles is used to generate p erception thresholds for the stationary and maximal y a w am-
plification, the maximal and minimal roll momen tum ratio, the maximal lateral side-slip
angle for stabilit y , the minimal t yre force b efore tip up, the roll angle at high lateral accel-
eration and the roll v elo cit y for ride. Based on these p erception thresholds, the sub jectiv e
ev aluation is transferred to requiremen ts on the system b eha viour. All approac hes or pro-
cess steps are first applied only to the fron t and rear stiffness of an ti-roll bars to reduce the
complexit y and increase the understanding. The shap e of solution space of the front and
rear stiffness of the an ti-roll bars due to m ultiple requiremen ts on the v ehicle b eha viour
are explained in detail. The concept of concurren t engineering is also applied b y o v erla ying
solution spaces of the design parameters from differen t design teams, whic h ha v e to fulfil
sev eral requiremen ts of the v ehicle b eha viour. F or robust design a classical approac h with
P areto-optimal robust design optimization is presen ted to sho w the con trast of the new
solution space design metho d whic h includes robustness in a simple manner. The handling
of uncertain ties of uncon trollable parameters is similar to design parameters. In terv als of
uncon trollable parameters are extended next to in terv als of design parameters. T o reduce
cost and effort, a pro duct family of v ehicles is designed. In a simple example a pro duct
family for 3 v ehicles is considered. Firstly , a classical approac h of pro duct family dev elop-
89

7 Conclusion and Outlo ok
men t is applied. The pro duct family design approac h using solution spaces surpasses the
result of the classical approac h and simplifies the sharing of comp onen ts within m ultiple
systems strongly: system do not ha v e to b e executed in coupled manner with the same
design parameter setting. The com binations of system configurations do not ha v e to b e
executed. The robust design approac h is then extended to 10 design parameters. The use
of high dimensional b o x-shap ed solution space enables robustness, simplifies the handling
and the visualization as p ermissible parameter in terv als next to eac h other. The pro duct
family approac h is applied to a v ehicle dynamics problem from industrial practice: a set of
13 v ehicles with 10 design parameters represen ting the prop erties of an ti-roll bars, bump
stops, reb ound stops for the fron t and rear axle w as optimised with resp ect to commonalit y .
7.2 Outlo ok
The system design metho ds presen ted here are still in a v ery early stage. There are ev en
more fields of researc h whic h seem to b e promising.
Clustering Design Problems from Global Sensitivit y Analysis. V ehicles need to
b e designed for man y design asp ects of v ehicle b eha viour. T o reduce the complexit y , global
sensitivit y analysis can b e used to separate a total design problem in to sub-design problems,
whic h are more or less decoupled. The ob jectiv e kno wledge can b e re-organized as quasi-
triangle form. Then parameters or groups of parameters can b e designed analogously as
to ho w linear equation systems are solv ed. In other w ords, all connections b et w een design
parameters and ob jectiv e quan tities can also b e visualized as a graph with no des and edges.
This graph ma y b e separated in sub-graphs suc h that as the lo w est n um b er of edges are
cut. Then the design complexit y is reduced to sub-design problems.
T arget Cascading and Solution Space. The complexit y of a system ma y b e sepa-
rated in to sub-systems. F or eac h system, sub-system and sub-sub-system solution space
can b e generated. Imagine the design parameters of the system are the ob jectiv e quan tities
of the sub-system as prop osed in T ar get Casc ading . A c hallenge of this approac h is the
consistency of the whole system. Classical optimisation target v alues of ob jectiv e quan ti-
ties ma y not b e reac hed and as a result the v alues of the design v ariables shift and the total
p erformance of the system c hanges. The approac h of T ar get Casc ading seems totally com-
patible to solution space, but w as not in fo cus in this w ork. In terv als of design v ariables
are the b ounds of an ob jectiv e quan tit y of the next sub-lev el. Using solution space and
target cascading, the satisfaction of requiremen ts on system p erformance or consistency of
the system is guaran teed o v er m ultiple design lev els.
Com bination of Solution Space and Sensitivities. Sensitivity indices ma y b e
used for the algorithm to compute b o x-shap ed solution spaces in Section 3.5.2. In eac h
iteration the candidate b o x is extended in to the parameter space to searc h for go o d designs.
Extending the b oundaries in to space with go o d designs will increase the v olume and mak e
the candidate b o x seek to w ards the maxim um b o x-shap ed solution space. Bad designs are
remo v ed from the b o x b y a trimming algorithm. F or 10 design parameters, a bad design
can b e remo v ed in 10 differen t w a ys. G ene rally a trim step is preferred, whic h increases
the fraction of go o d design space, but this dep ends on the sample distribution. Imagine
a bad design is remo v ed b y trimming irrelev an t parameters. The trimming algorithm
ma y b e supp orted b y sensitivit y indices to prev en t the trimming of irrelev an t parameters.
P arameters should b e trimmed, if a cut remo v es as m uc h as p ossible bad designs.
90

7.2 Outlo ok
Com bination of Solution Space and Sub jectiv e Ev aluation (BI). F or the design
pro cess solution space is used to express the set of go o d designs whic h satisfy all require-
men ts on system b eha viour. If relev an t ob jectiv e quan tities are com bined, the shap e of
solution space can also b e in terpreted as con tour lines for design categories or sub jectiv e
ev aluations (BI). Maps of design parameters ma y b e designed in connection to sub jectiv e
ev aluation. Driving exp erts ma y create solution space or v erify m ultiple sub jectiv e ev alu-
ations. If driving exp erts create solution space with sev eral con tour lines for a sub jectiv e
ev aluation (BI 5, BI 6, BI 7 and BI 8) p erception thresholds can b e concluded from the
distance of the con tour lines and their asso ciated ob jectiv e quan tities.
Rotating Bo x-shap ed Solution Space. Bo x-shap ed solution space includes robust-
ness and decouple parameters and can b e n umerically computed for high dimensions. Ho w-
ev er b o x-shap ed solution space neglects go o d design p oin ts. A b o x-shap ed solution space
ma y b e rotated. Then these design parameters are not indep enden t an y more. A certain
angle of rotation ma y increase the v olume of go o d designs enormously . Therefore more
flexibilit y is gained to reduce design iterations. This ma y b e also useful for pro duct family
design as a higher probabilit y of matc hing for sev eral systems ma y o ccur. F urthermore,
the algorithm to compute b o x-shap ed solution space of Section 3.5.2 can b e used, if an
angle of rotation is kno wn, e.g., from appro ximation of a 2-dimensional solution space.
Solution Space with Maxim um Minim um In terv als. F or high dimensional prob-
lem statemen ts, b o x-shap ed solution space with maximal v olume is computed. An alter-
nativ e for robust design w ould b e this problem statemen t with maximal minimal in terv als
for design parameters. F or a giv en design space Ω ds , seek the minimal in terv al of all
I i = [ x lb
i , x ub
i ] for eac h design parameters suc h that
min
i ( I i ) → max (7.1)
sub ject to f ( x ) ≤ f c for all x ∈ Ω (7.2)
Imagine, for example, scaling factors as design parameters: this b o x enables a certain
relativ e minim um robustness for all design parameters.
Using Bo x-shap ed Solution Space around Solution Space. Bo x-shap ed solution
space is easy to handle and can b e visualized for an arbitrary n um b er of design parameters.
A b o x-shap ed solution space within the shap e of the solution space decouples all design
parameters. Outside a b o x-shap ed space around a solution space, no com bination of design
parameter v alues can b e found that satisfy all requiremen ts. Within suc h a b o x-shap ed
space around a solution space, for eac h v alue of a single design parameter at least one
com bination of all other design parameters can b e found that satisfies all requiremen ts.
This statemen t seems w eak, but ma y help in practice, e.g., these b ounds are tuning aids
for driving exp erts and for further disciplines suc h as for pro duct family design.
Extension of Pro duct F amily Design. F or the iden tification of in terv als of design
parameters whic h can b e shared within m ultiple systems, b o x-shap ed solution spaces with
maximal v olume are used for high dimensional design parameters. This b o x ma y b e shifted
(pulled and pushed) within the solution space. This ma y lead to a decrease of the v olume
but p erhaps to an increase of commonalit y . Bo x-shap ed solution space around solution
space of eac h system ma y help to iden tify whic h systems can definitely not b e coupled.
If b o x-shap ed solution space around solution space of systems do not o v erlap, no design
parameter v alue can b e shared. Corridors do not need to b e pulled or pushed to eac h
other. F urthermore, pro duct family design should also b e extended for comp onen ts with
91

7 Conclusion and Outlo ok
sev eral functional prop erties or design parameters, resp ectiv ely .
Con trol Systems and Mec hanical Systems. Solution space design ma y b e applied
to an y design problem of v ehicle dynamics or in general to an y kind of system design.
The design parameter ma y b e functional prop erties of the susp ension, the axle, steering
system, t yre c haracteristic or con trol systems. P arameters of con trol systems ma y b e
handled differen tly as their v alue will b e deterministic and exactly implemen ted. F urther,
the parameters of con trol systems can b e set individually for eac h v ehicle or system to
individualize the b eha viour. In practice, parameters of con trol systems are adjusted for
w orst case situations of critical v ehicles as the capacit y of driving exp erts is limited. The
parameters of the con trol systems ma y b e extended next to the design of the mec hanical
system. Alternativ ely , the v alue of the parameters of the con trol systems are optimized to
maximise the v olume of a b o x-shap ed solution space in order to increases the robustness
of the design parameter. Con trol systems settings of m ultiple systems ma y b e shared to
reduce v arian ts and adjustmen t effort.
92

A App endix
A.1 Histogram of Objective Quantities
0.3 0.35 0.4 0.45
0
50
100
150
200

N
maxi mal quasi- statio n a r y
y a w vel oci ty amp l ificati on
˙
ψ
δ h stat,m ax i n 1 /s
50 55 60 65
0
50
100
150
200
N
maxi m u m of ro ll
momen t ratio
RM R m a x in %
45 50 55 60 65
0
50
100
150
200
N
minim um of ro ll
momen t ratio
RM R m i n in %
1 1.2 1.4
0
50
100
150
200
250
N
maxi mal ya w v e lo ci t y
a m pli fi ca t i o n
˙
ψ
δ h m a x
0 10 20 30
0
50
100
150
200
N
maxi m u m s i d e- sli p a n g l e
β m ax i n ◦
0 1000 2000
0
50
100
150
200
250
N
minim um v ert i ca l t yre fo r ce
F tyre, m i n in N
2 2.5 3 3.5 4
0
50
100
150
200
250
N
rol l a n g le a t h i gh
l ateral a ccel eratio n
φ ( a y ,high ) in ◦
0.04 0.06 0.08 0.1
0
50
100
150
200
250
N
roo t mean squ a re v a l u e
o f t he r o l l v el o cit y
RM S ( ˙
φ f, lo w ) i n ◦ / s

Figure A.1: Histogram of scattered ob jectiv e quan tities
93

A App endix
Figure A.1 sho ws the histogram of the ob jectiv e quan tities analysed in T able 5.5 and
designed in Section 5.3 in the application. F or the generation of the p erception thresholds
in Section 5.2.4 a normal distribution of the scattered ob jectiv e quan tities is assumed.
Figure A.1 confirms the statemen t in this case to use a normal distribution for all ob jectiv e
quan tities.
A.2 P attern Matching of Subjective and Objectified
Kno wledge
In T able A.1 in the first 4 columns, random sub jectiv e and ob jectified kno wledge are
presen ted next to eac h other. The fitting is bad as the correlation index is lo w after
selecting the b est threshold v alues ˆ
S . These examples sho w that the threshold v alues
cannot b e fitted in order to correlate the sub jectiv e and ob jective kno wledge. In the 5 th
column the patterns or the relativ e order of impacts corresp ond. Therefore threshold v alues
can b e concluded, whic h lead to a high correlation.
T able A.1: V ehicle kno wledge managemen t system: sub jectiv e and ob jectified information
according to relations b et w een design parameters and design categories.
design category 1
design category 2
design category 3
design category 4
design category 5
sub. ob j. sub. ob j. sub. ob j. sub. obj. sub. ob j.
design v ariable 1 # 0.01 0.05 < 0.01 < 0.15 < 0.05
design v ariable 2 0.29 < 0.10 # 0.01 0.05 0.01
design v ariable 3 0.01 # 0.01 0.03 0.06 0.31
design v ariable 4 < 0.11 < 0.05 0.05 # 0.29 # 0.10
design v ariable 5 # 0.00 0.31 < 0.20 # 0.05 < 0.04
design v ariable 6 # 0.05 < 0.22 < 0.26 0.00 # 0.26
design v ariable 7 < 0.21 0.00 0.01 < 0.08 < 0.09
design v ariable 8 0.15 0.08 0.14 < 0.15 0.04
design v ariable 9 0.09 # 0.05 # 0.03 # 0.14 0.05
threshold v alue ˆ
S # 0.02 0.05 0.02 0.29 0.10
threshold v alue ˆ
S < 0.01 0.01 0.01 0.08 0.04
correlation index -0.45 0.07 -0.20 0.61 1.00
A.3 Details of the Solution Space Shap e of Anti-roll Ba rs
Figure A.2 sho ws sev eral solution spaces con taining all go o d designs with resp ect to differ-
en t design criteria. The corresp onding threshold v alues are listed in T able 5.1. The x- and
y-axis represen t the an ti-roll bar stiffnesses of the fron t and rear axle, resp ectiv ely . F at
94

A.3 Details of the Solution Space Shap e of An ti-roll Bars
sym b ols indicate bad designs, the asso ciated ob jectiv e quan tit y that exceeds its threshold
is sho wn in the legend.
10 20 30 40 50
0
2
4
6
8
c a, f
c a,r

go o d design

R M S ( ˙
φ f,low )

(a)
10 20 30 40 50
0
2
4
6
8
c a, f
c a , r

go o d design
R M S ( ˙
φ f,low )
φ ( a y ,h igh ) m ax
F t y r e ,m in

(b)
10 20 30 40 50
0
2
4
6
8
c a, f
c a,r

go o d design

R M S ( ˙
φ f,low )
φ ( a y ,h igh ) m ax
F t y r e ,m in
β m ax

(c)
10 20 30 40 50
2
4
6
8
c a, f
c a,r

go o d design

R M S ( ˙
φ f,low )
φ ( a y ,h igh ) m ax
F t y r e ,m in
β m ax
R M R m ax
R M R m in
O ptimum

(d)
Figure A.2: Solution space of fron t and rear an ti-roll bar stiffness c a considering (a) roll
dynamics, (b) roll-o v er,(c) maxim um swim angle and (d) roll momen t ratio.
Figure A.2(a) sho ws a restriction of the solution space with resp ect to comfort. The ro ot
mean square v alue of the roll angle v elo cit y increases for high stiffness of the fron t an ti-
roll bar ( differ enc e of design p ar ameters ). Imagine an excitation on the righ t side, whic h
causes a deflection of the righ t wheel. Through the coupling of the righ t and left side of
the an ti-roll bar, the left wheel will reb ound. Therefore the roll reaction and consequen tly
the roll v elo cit y rise. In Figure A.2(b) the roll angle criteria ( sum of design p ar ameters ) is
sho wn. The higher the an ti-roll bar stiffness c a of the fron t or rear axle, the smaller is the
admissible roll angle for a fixed roll momen t. A dditional the b oundary of the solution space
due to roll o v er is sho wn. The required minim um remaining t yre force has to b e ab o v e
a critical threshold v alue ( differ enc e of design p ar ameters ). Generally , for the fron t axle,
a short lift-off is una v oidable in a sev ere lane-c hange mano euvre, therefore, the stabilit y
reserv e is demanded from the rear axle. Hence, the stiffness of the rear axle is limited in
order to reduce t yre load differences. In Figure A.2(c) the restriction due to the maxim um
side-slip angle is added ( differ enc e of design p ar ameters ). The higher the rear t yre load
differences are, the higher is the required rear slip angle and, therefore, the higher is the
side-slip angle. The fundamen tal ph ysics of the fron t axle is the same, but the higher the
95

A App endix
slip angle of the fron t axle, the smaller is the maxim um side-slip angle and therefore the
higher is the lateral stabilit y . In Figure A.2(d), t w o more criteria are added: the maxim um
and the minimal distribution of the roll momen t related to the fron t axle ( r atio of design
p ar ameters ). The higher the rear stiffness and the lo w er the fron t stiffness, the smaller is
the lev el of roll momen t distribution.
A.4 A classical App roach constructing Platfo rm
Architectures
T able A.2 sho ws in part (1) the result of separate v ehicle optimisations with resp ect to
z α . F or eac h v ehicle, parameters could b e iden tified with z ≤ 0 , i.e., all design goals w ere
reac hed. The asso ciated total n um b er of differen t parameter v alues is n = 6 .
In T able A.2, sev eral candidate configurations are ev aluated. In configurations 1a-3a,
the parameter v alues of one v ehicle from part (1) are assigned to all three v ehicles sim ul-
taneously . Then the total n um b er of differen t parameter v alues is n = 2 , ho w ev er, not all
design goals are met, as z > 0 . In the configurations 1b-3b, the fron t an ti-roll bar stiffness
is shared and the rear an ti-roll bar stiffness remains individual from the optimisation in
part (1). Here, the total n um b er of differen t parameter v alues is n = 4 . Ho w ev er, again
z > 0 , as the fron t an ti-roll bar stiffness cannot b e shared. In the configurations 1c-3c,
the rear an ti-roll bar stiffness is shared, and again n = 4 . This time, z ≤ 0 , therefore,
the same an ti-roll bar stiffness for the rear axle can b e used for all v ehicles. Among all
reassigned solutions, based on the separately optimized parameter v alues of configuration
(1), configuration (2c) is the b est configuration. The rear an ti-roll bar stiffness is shared
among all v ehicles. Here, the total n um b er of differen t parameter v alues is n = 4 , while
the total p erformance z ≤ 0 .
T able A.2: Results of the approac hed based on separate vehicle
optimisation
P arameter V ehicle 1 V ehicle 2 V ehicle 3 n i Description
c a,f in N /mm 21.93 (A) 24.77 (B) 18.77 (C) 3 (1) Separate p erformance optimisation for
c a,r in N /mm 2.34 (A) 2.30 (B) 2.40 (C) 3 eac h veh icle
z i -0.014 -0.012 -0.019 n = 6
c a,f in N /mm A A A 1 (1a) Platform based on the optimization of
c a,r in N /mm A A A 1 v ehicle 1
z i -0.014 0.003 0.014 n = 2
c a,f in N /mm B B B 1 (2a) Platform based on the optimization of
c a,r in N /mm B B B 1 v ehicle 2
z i 0.011 -0.012 0.119 n = 2
c a,f in N /mm C C C 1 (3a) Platform based on the optimization of
c a,r in N /mm C C C 1 v ehicle 3
z i 0.002 0.019 -0.0194 n = 2
c a,f in N /mm A A A 1 (1b) Platform based on the optimization
c a,r in N /mm A B C 3 of vehicle 1
Con tin ued on next page
96

A.4 A classical Approac h constructing Platform Arc hitectures
T able A.2 – con tin ued from previous page
P arameter V ehicle 1 V ehicle 2 V ehicle 3 n i Description
z i -0.014 0.010 0.014 n = 4
c a,f in N /mm B B B 1 (2b) Platform based on the optimization
c a,r in N /mm A B C 3 of vehicle 2
z i 0.007 -0.012 0.028 n = 4
c a,f in N /mm C C C 1 (3b) Platform based on the optimization
c a,r in N /mm A B C 3 ofv ehicle 3
z i 0.014 0.112 -0.019 n = 4
c a,f in N /mm A B C 3 (1c) Platform based on the optimization of
c a,r in N /mm A A A 1 v ehicle 1
z i -0.014 -0.006 -0.018 n = 4
c a,f in N /mm A B C 3 (2c) Platform based on the optimization of
c a,r in N /mm B B B 1 v ehicle 2 Best configuration obtained
b y separate v ehicle optimisation
z i -0.013 -0.012 -0.018 n = 4
c a,f in N /mm A B C 3 (3c) Platform based on the optimization of
c a,r in N /mm C C C 1 vehicle 3
z i -0.001 -0.003 -0.019 n = 4
97

List of Figures
1.1 Set-based design approac h with t w o design parameters x 1 and x 2 . Designer
A has to minimize the system p erformance A, while designer B has to max-
imize the system p erformance B, similar to P anc hal et al. [85]. . . . . . . . 7
1.2 Principle of target cascading: master problem is separated in sub-problems
with link ed v ariables, similar to Kim et al. [54]. . . . . . . . . . . . . . . . 8
1.3 Principle of T aguc hi Metho d: in step one the v ariance of the system resp onse
is reduced, in step t w o the mean v alue is shifted to target v alue, while the
v ariance is k ept lo w, similar to [88] . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Principle of reliable design: appro ximately 75 % design fail in a b o x around
the deterministic optim um. The reliable solution is a trade-off system p er-
formance. Similar to [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Realized c haracteristic curv e of force and deformation scatters around de-
signed c haracteristic curv e (A). Designed in terv als (B) include the scattered
c h a r a c t e r i s t i c c u r v e . ............................... 1 3
1 . 6 S t r u c t u r e o f t h e w o r k .............................. 2 2
2.1 Graphical in terpretation of the main Sob ol Index . . . . . . . . . . . . . . 31
2.2 Generation of p erception thresholds . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Relation b et w een ob jectiv e quan tit y and the sub jectiv e ev aluations . . . . . 37
3.1 Characteristic curv es of uncertain ties in the dev elopmen t pro cess. . . . . . 39
3.2 Determination of a 2D solution space . . . . . . . . . . . . . . . . . . . . . 40
3.3 Solution space (2-D) and in terv al displa y of design parameters . . . . . . . 41
3.4 Solution space (2-D), b o x-shap ed solution space and in terv al displa y of de-
s i g n p a r a m e t e r s ................................. 4 2
3.5 Robust design for uncertain ties of design parameters: solution space (2-D),
in terv als of design parameters and advice of robust design . . . . . . . . . 46
3.6 Robust design for uncertain ties of uncon trollable parameters: solution space
(2-D), in terv als of design parameters and advice of robust design . . . . . . 46
3.7 Concurren t engineering: sup erp osition of design criteria and solution spaces
( 2 - D ) ....................................... 4 8
3.8 Algorithm to compute b o x-shap ed solution space (Figure is tak en from [31]) 50
4.1 Solution spaces, b o x-shap ed solution spaces and asso ciated in terv als of 2
systems for a general design problem. . . . . . . . . . . . . . . . . . . . . . 56
4.2 Algorithm to iden tify the smallest set of parameter in terv als . . . . . . . . 56
5.1 F ron t axle assem bly of a middle class v ehicle with (1) bump stop, (2) re-
b ound stop and (3) an ti-roll bar. . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Scatter plots: ob jectiv e quan tities sorted o v er rising design parameter. . . . 65
98

List of Figures
5.3 Characteristic curv e for sub jectiv e BI ev aluation o v er roll angle at high
l a t e r a l a c c e l e r a t i o n ............................... 7 0
5.4 Solution space for the fron t (x-axis) and rear (y-axis) an ti-roll bar stiffness.
Designs are excluded b y requiremen ts from roll b eha vior, roll-o v er, lateral
stabilit y and self-steering b eha vior. . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Solution space with fo cus on self-steering and turn-in b eha viour. . . . . . . 72
5.6 Ov erla ying of solution spaces with differen t fo cus. . . . . . . . . . . . . . . 73
5.7 P areto-fron t b et w een p eak p erformance ˆ z and robustness ob jectiv e σ ( ˆ z ) . . 74
5.8 Robustness against uncertain ties of design parameters (2D). . . . . . . . . 75
5.9 Ov erla ying of solution spaces with differen t v alues of uncertain parameters. 76
5.10 Solution spaces for reference and robustness against uncon trollable param-
eters. F or comparison, the robust optim um of the RDO approac h is added.
Uncertain parameters are the b o dy mass: [ 0 . 95 , 1 . 05] · m and the damp er
c haracteristic: [0 . 9 , 1 . 10] · d . In addition, a robust design, whic h considers
design parameter uncertain ties is sho wn. . . . . . . . . . . . . . . . . . . . 77
5.11 Solution spaces (lines), common designs (big dots), and p erformance optima
from Section 5.4.1 (small dots) are sho wn in the space of an ti-roll bar stiffness
c a v alues of the fron t (x-axis) and rear (y-axis) axle. . . . . . . . . . . . . . 79
5.12 Bo x-shap ed solution space for a sedan depicted as upp er and lo w er limits
for eac h design parameter. The connecting lines b et w een limits are only for
the purp ose of visualization. Num b ers in the x-lab el are scaling factors. . . 80
5.13 Solution spaces for (a) v ehicles 1-3 and (b) all 13 v ehicles sho wn as upp er
and lo w er limits for eac h design parameter. The units of stiffness v alues c
and free lengths s are N/mm and mm, resp ectiv ely . The connecting lines
b et w een limits of differen t parameters are only for the purp ose of visualization. 83
A.1 Histogram of scattered ob jectiv e quan tities . . . . . . . . . . . . . . . . . . 93
A.2 Solution space of fron t and rear an ti-roll bar stiffness c a considering (a) roll
dynamics, (b) roll-o v er,(c) maxim um swim angle and (d) roll momen t ratio. 95
99

List of T ables
2.1 V ehicle kno wledge managemen t system: sub jectiv e information according
to relations b et w een design parameters and design categories. . . . . . . . . 24
2.2 Ob jectiv e informations according to relations b et w een design parameters
a n d o b j e c t i v e q u a n t i t i e s . ............................ 2 6
2.3 Comparison of design of exp erimen ts metho ds . . . . . . . . . . . . . . . . 27
2.4 Qualitativ e ob jectification information according to relations b et ween ob-
jectiv e quan tities and design categories. . . . . . . . . . . . . . . . . . . . . 32
2.5 V ehicle kno wledge managemen t system: ob jectified information according
to relations b et w een design parameters and design categories. . . . . . . . . 33
2.6 V ehicle kno wledge managemen t system: sub jectiv e and ob jectified informa-
tion according to relations b et w een design parameters and design categories. 34
2.7 P erception thresholds determined for eac h design category . Relation b e-
t w een ob jectiv e quan tities and design category is exemplary . . . . . . . . . 36
3.1 Example shap es of 2-D solution space and their corresp onding parameter
c o r r i d o r s..................................... 4 3
4.1 T w o example systems in parameter space . . . . . . . . . . . . . . . . . . . 54
5.1 Driving manouevres relev an t for design and their ob jectiv e quan tities . . . 59
5.2 Design P arameters. The subscript letters f and r refer to the fron t and rear
a x l e , r e s p e c t i v e l y . ................................ 6 2
5.3 V ehicle kno wledge managemen t system for middle class v ehicle: sub jectiv e
informations according to relations b et w een design parameters and design
c a t e g o r i e s . .................................... 6 3
5.4 Obje ctific ation K now le dge : Relation b et w een ob jectiv e quan tit y and sub jec-
t i v e d e s i g n c a t e g o r y ............................... 6 3
5.5 Obje ctive K now le dge : Relation b et w een design parameters and ob jectiv e
q u a n t i t y ..................................... 6 6
5.6 Ov erview: Comparison of Subje ctive K now le dge and Obje ctifie d K now le dge 68
5.7 P erception thresholds determined for eac h design category . Relation b e-
t w een ob jectiv e quan tities and design category is exemplary . . . . . . . . . 69
5.8 Threshold v alues of ob jectiv e quan tities for a sedan: lo w er (lb) and upp er
b o u n d s ( u b ) . ................................... 7 1
5.9 Threshold v alues of ob jectiv e quan tities for sedan of second design team. . 72
5.10 Goals and constrain ts for ob jectiv e quan tities for sedan. . . . . . . . . . . . 73
5.11 Results of robust design optimization. . . . . . . . . . . . . . . . . . . . . . 74
5.12 Threshold v alues of ob jectiv e quan tities for sedan, station w agon and coup é. 77
5.13 Results of the approac hed based on separate v ehicle optimisation. . . . . . 78
5.14 Configuration deriv ed from o v erlapping solution spaces. . . . . . . . . . . . 79
100

List of T ables
5.15 Critical threshold v alues for ob jectiv e quan tities . . . . . . . . . . . . . . . 81
5.16 In terv als for shared design parameter v alues represen ting comp onen ts of
same design. The unit of the stiffness c and the free length s is in N/mm
or mm. The iden tify er A, B, C, D denote the resp ectiv e in terv als for eac h
d e s i g n p a r a m e t e r . ............................... 8 2
5.17 Configuration table. The same sym b ols A, B, C, D for differen t v ehicles
indicate comp onen ts of same design. . . . . . . . . . . . . . . . . . . . . . 82
A.1 V ehicle kno wledge managemen t system: sub jectiv e and ob jectified informa-
tion according to relations b et w een design parameters and design categories. 94
A.2 Results of the approac h based on separate v ehicle optimisation . . . . . . . 96
101

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