On the Design of Actuator and Control Systems in Early Development Stages

On the Design of Actuator and Control Systems in Early

Development Stages

Amir Zare

On the Design of Actuator and Control Systems in Early

Development Stages

Vom Fachbereich für Physik und Elektrotechnik

der Universität Bremen

zur Erlangung des akademischen Grades

Doktor

–Ingenieur (Dr.-Ing.)

genehmigte Dissertation

von

M.Sc. Amir ZAre

wohnhaft in München

Referent:

Prof. Dr.-Ing. K. Michels

Korreferent:

Prof. Dr. M. Zimmermann

Thisthesis isbasedon the resultsobtained during mytimeasPhD student in the years from 2014 until 2017 at BMW Group and submitted for thedegree of Do torofEngineering (Dr. -Ing)at theUniversityofBremen. Theresear h des ribed herein was ondu ted under the supervision of Prof. Dr.-Ing. Kai Mi hels inthedepartment of InformationandAutomation Engineering (IAT), University ofBremen.

This work is done by myself originally, ex ept where a knowledgment and referen es are made to previous work. I onrm that neither this nor similar dissertation has been or is being submitted for any other quali ation at any other university.

Part ofthis work hasbeen presentedinthe following publi ations.

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A.Zare, K. Mi hels, M. Zimmermann, L. Rath-Maia, On the Design of A tuator andControl Systems in Early Stages of Development, In: has-sis.te h plus 2017,Juni20-21 2017,Muni h,Germany.

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A. Zare, D. Odenthal, M. Zimmermann, M. Wahle, Generis he Vors-teuerungfuerGesamtfahrzeug,InventionDis losure,registered,Nr. 3456, Appli ation Date: 04.05.2017.

I would like to express my gratefulness to my thesis supervisorProf. Dr.-Ing. Kai Mi hels for his ontinuous guidan e and advi e. His Support, patien e, and pre ious inputs in our dis ussions are of utmost value to me. I am also very thankful to my supervisor Prof. Dr. Markus Zimmermann for initiating this resear hproje tinthe rstpla e,hisguidan ethroughout theproje tand the en ouragement and supporthe hasgiven me. Thetime at BMW AG was really intriguing and ri h in experien es. I would espe ially like to thank Dr. Martin Wahlefor his generous supportand ounseling at thebeginning of the proje t and his ontinuous valuable dis ussions about most of the topi s in this resear h. Iwould like to extend my thanksand appre iation to Dr. Vena Gianpiero forwilling to helpmeinme hani al topi s.

I owe mu h to my olleagues and friends for their kind en ouragement. A spe ial a knowledgment goes to my friends Ali Shariati, Brahim Soltani, Ali Tu hi, Pegah Pezeshkan, and Nadia Parvizi who have been always there for me. Iwouldalso like to sin erelythankall mystudents,spe iallyLu as Rath-Maia, Sebastian S hoder, Yuran Liang and Julian Ruggaber. Their ex ellent ontributions have signi antly enhan ed the quality of my resear h and its results.

Last but not least, I would like to sayI am indebted to my parents Hassan Zare and Fariba Shadaei who have supported and guided me pre iously and arefully throughout my edu ation years. They all kept me going and have alwaysbeen asour e of inspirationto me.

Vehi le development fa es new hallenges due to the in reasing requirements on vehi le performan e, e.g., on ride, omfort andsafety. Addingme hatroni systems to the hassis helps to rea h more ambitious design goals, however, it also adds to the omplexity of the design pro ess. The number of design variablesin reasessigni antly: inadditiontotheme hani al hassis,a tuators of me hatroni systems and ontrol logi parameters ae t the overall vehi le performan e. The omplex intera tion between many design variables poses a parti ular hallenge. Engineersneedtomakeimportantdesignde isionsinearly development stages, typi ally about the me hani al hassis, without knowing thenal design,inparti ular thedetailparameters of me hatroni systems.

The present dissertation illustrates the design variables of logi s as well as a tuators oflateralvehi ledynami sme hatroni systemsbyproposing appro-priate prin iples of logi fun tions and fun tional models of a tuators. The ee ts oflogi saswellasa tuatorsonobje tive drivingdynami s targetswith respe t to dierent driving maneuver will bepointed out. Besides, a so- alled dependen ygraphofme hatroni -me hani alsystemswillbeestablished,whi h helpsengineersto ontrolthe omplexityinthedesignpro edureofme hatroni systems.

Moreover, thedissertation applies thetheoryofsolution spa esfor ndinga robustsolutionspa efortheparametrizationof ontrolsystemlogi parameters andalayoutofa tuators. Inthisway,solutionintervalsofme hatroni systems' design variableswill bedeterminedbysettingquantitative requirementsonthe obje tive driving dynami s targets. For this purpose, mathemati al surrogate models will be introdu ed whi h redu e the optimization time and des ribe orrelations between designvariablesand drivingdynami stargets. These sur-rogate models will be derived by ma hine learning methods su h as arti ial neuralnetworksandsupportve torma hines. Intheend,thesolutionintervals of design variables will be onverted into thesolution veri ation variables for verifyingend-produ tssu hasa tuators.

Ame hatroni rearsteering systemis onsideredasan appli ation example. Requirementsonthedynami sandoverallperforman eofthea tuatorandthe asso iated ontrol logi parameters arederived byformulatingrequirementson the onsidered obje tive driving dynami s targets. The derived requirements

state error,maximumrear angle ofa rear steering a tuator or damping fa tor of dynami feedforward ontrol, proportionalterm of feedba k ontroller, et .

Contents

List ofFigures 12

List ofTables 15

1 Introdu tion 17

1.1 Content andmotivation . . . 17

1.2 State oftheart . . . 21

1.3 Stru ture ofthe dissertation . . . 27

2 Vehi le Dynami s Theory 29 2.1 Single-tra k model . . . 29

2.2 A tive RearSteering . . . 42

2.3 One-Sided Brake Intervention . . . 46

2.4 Driving Maneuversand the obje tive driving performan es . . . 48

2.5 Simulation environment, Two-tra k model . . . 56

3 Lateral vehi ledynami s ontrol systems 59 3.1 Stati feedforward ontrol . . . 59

3.2 Centralized dynami feedforward ontrol . . . 66

3.3 Centralized disturban efeedforward ontrol . . . 71

3.4 Referen e input generatorand drivingsituation identier . . . . 73

3.5 Centralized feedba k ontroller . . . 77

3.6 Priorization, allo ation and summation unit . . . 81

4 Fun tional a tuator modeling 87 4.1 Modeling methodology . . . 87

4.1.1 Empiri al modeling pro edure . . . 87

4.1.2 Physi al andsemi-physi al modeling pro edure . . . 89

4.2 Appli ation to rear steeringsystem . . . 90

4.2.1 Summary . . . 104

5 Robust design of a tuators and ontrol systems 106 5.1 Dependen ygraph . . . 108

5.2 Bottom-up mapping . . . 109

5.2.1 Designof experiments . . . 110

5.2.2 Responsesurfa e model . . . 112

5.2.3 Classi ation . . . 116

5.3 Top-downmapping . . . 124

5.3.1 Solutionspa es . . . 125

5.4 Computing requirementson veri ation variables . . . 129

6 Appli ation 136 6.1 Results forrobust designing oftherear steeringsystem . . . 136

7 Con lusionand future work 151

Table of Symbols

Symbol Des ription

V

X

i

(·), E

X

i

(·)

Varian e or meanofargument (.) taken over

X

i

Sgn(.)

Signof argument (.)

V ol(.)

Volumeof argument (.)

n(.)

Numberofargument (.)

max(.)

Maximumof argument (.)

Discrepancy(.)

Dis repan yof argument (.)

ψ

Yawangle

˙

ψ

Yawangle velo ity

¨

ψ

Yawa eleration

˙

ψ

ref

Referen e yaw anglevelo ity

˙

ψ

act

A tual yawangle velo ity

˙

ψ

ST M

Yaw angle velo ity al ulated from the single-tra k model

˙

ψ

ay

Laterala eleration yaw anglevelo ity

φ

Rollangle

θ

Pit h angle

δ

H

Steeringwheelangle

δ

f

Front wheelsteering angle

δ

r

Rearwheelsteering angle

δ

r,stat

Output oftheARS stati feedforward ontrol

δ

ef f

Ee tivewheel steeringangle

c

f

Front total sideslipstiness

c

f,mod

Modiedfront total sideslipstiness

c

r

Reartotal sideslipstiness

c

r,mod

Modiedrear total sideslipstiness

β

Slipangle

˙

β

Slipangle velo ity

m

Mass

v

Velo ity

F

x

For e in dire tion of the x-axis (Vehi le longitudinal axis)

Symbol Des ription

F

x,r

Longitudinal for ea ting on thetire oftherear axle

F

y

For e indire tion ofthey-axis(Vehi lelateral axis)

F

y,f

Lateralfor e a tingon thetire ofthefront axle

F

y,r

Lateralfor e a tingon thetire oftherear axle

F

x,d

External for e indire tion of thex-axis(Vehi le longi-tudinal axis)

F

y,d

Externalfor eindire tionofthey-axis(Vehi lelateral axis)

F

z

For e indire tion ofthez-axis (Vehi leverti al axis)

F

total

Total for e

J

z

Vehi leinertiaindire tionofthez-axis(Vehi leverti al axis)

g

Gravitation onstant

R

Path radius

α

f

Sideslip angle onthefront axle

α

r

Sideslip angle ontherear axle

l

Wheelbase

EG

Self-steering gradient

δ

A

A kermann angle

r

dyn

Dynami wheelradius

b

r

Tra k widthofa vehi le

r

w

Wheelradius

A

pistol

Diskarea

µ

temp

Fri tion oe ient ausedbythedis temperature

a

y

Longitudinal a eleration

a

x

Laterala eleration

h

Center ofgravity hight

l

f

Distan e ofthe enter ofgravityfrom thefront axle

l

r

Distan e ofthe enter ofgravityfrom therearaxle

δ

f r

Front steering angleof theright wheel

δ

f l

Front steering angleof theleftwheel

M

br,stat

Output oftheBRstati feedforward ontrol

i

M

z,f f

Translation fa tor of thestati yawmoment

ω

des

Desired vehi lenatural frequen y

D

des

Desired vehi ledamping ratio

Symbol Des ription

a

y,act,measured

Measured a tual laterala eleration

˙

β

ST M

Sideslip angle omputed fromthesingle-tra kmodel

˙

β

act

A tualside slip anglevelo ity ofa vehi le

φ

Roll angle

K

p

Stati gain

T

w

Time onstant

˙δ

out

Output signalvelo ity

φ

Roll angle

T

w

Time onstant

˙δ

out

Output signalvelo ity

˙δ

in

Output signalvelo ity

˙δ

cu

Upperbound ofthe modeloutputvelo ity

˙δ

lu

Lowerboundof themodeloutput velo ity

Ω

ds

Designspa e

C

Box onstraint

ǫ

Sla kvariable

x

act

Designvariableof ana tuator

x

log

Designvariableof a ontrol systemlogi

I

act

Internal fora variableofan a tuator

I

log

Intervalfor a variable ofa ontrol systemlogi

T

ris

Risingtime

T

st

Stabilization time

SE

Stati error

M

z,d

External disturban emoment indire tion ofthez-axis

M

z,F B

Yaw moment generated byfeedba k ontroller

LF D

Longitudinal for edistribution

v

ch

Chara teristi velo ity

v

crit

Criti al velo ity

ω

0

Vehi lenatural frequen y

D

Vehi ledamping ratio

T

z

Vehi letime onstant

M

z

Yaw Moment

M

z,f f

Yaw Moment generated byfeedforward ontrol

M

z,stat

Yaw Moment generated bystati feedforward ontrol

Symbol Des ription

M

z,df f

Yaw Moment generated by disturban e feedforward

ontrol

F

br

Brake for e

r

w

Wheelradius

a

y

Laterala eleration

a

y,max

Maximumlateral a eleration

SAF

Steeringwheel anglefa tor

f req

Frequen y

T

eq

Equivalent timedelay

i

ARS

Translation ratio of a tive rearsteering system

v

trans

Transient velo ity

M

br

Brake moment

a

y,ack

A kermann laterala eleration

ω

f

Naturalfrequen y fa tor

D

f

Dampingratio fa tor

T

f

Time onstant fa tor

K

os

Indi ator of oversteering

K

cs

Indi ator of ounter steering

g

K

p

Weightingfa tor ofthefeedba k ontroller propotional term

F

c

Counter for e

List of Figures

1.1 Development phases ofthe vehi le dynami s development . . . 17

1.2 Multi-Obje tive Designwithiterations . . . 19

1.3 Multi-Obje tive Designbasedon solutionspa es . . . 19

1.4 Blo kdiagramofa losed-loop ontrolsystem . . . 22

2.1 Single-tra k model . . . 30

2.2 Steeringbehaviorwith onstant radius . . . 37

2.3 Stationary yawvelo ityampli ation vsVelo ity . . . 39

2.4 Naturalfrequen y and Dampingratio vsVelo ity . . . 42

2.5 Rearsteering systemof BMW . . . 43

2.6 Fun tionality ofa tiverear steering system . . . 44

2.7 Basi fun tionalityofthe dynami brakinga tion . . . 47

2.8 Open-Loopand Closed-Loop Maneuvers . . . 48

2.9 QuasiSteady State Cornering Maneuver . . . 49

2.10 SineWith Dwell Maneuver . . . 51

2.11 Brake WhileCornering Maneuver . . . 52

2.12 WEAVE Maneuver . . . 53

2.13 ContinuousSineSteering Maneuver. . . 54

2.14 The CSST asso iated obje tivetargets . . . 55

2.15 Tyre lateral for es andslip angles [66℄ . . . 57

2.16 S hemati ofthevehi letwo-tra k model. . . 58

3.1 Stru ture oftheproposed ontrol system . . . 59

3.2 Chara teristi urve ofARS stati feedforward ontrol . . . 60

3.3 The impa tof theARS stati feedforward ontrol . . . 63

3.4 Brake systemstati feedforward ontrol . . . 65

3.5 The omparison between the self-steeringdemand . . . 66

3.6 The bodediagram ofthevehi le dynami s . . . 70

3.7 The sideslip angle response . . . 71

3.8 The s hemati of thegenerated yawmoment. . . 72

3.9 The yaw velo ityresponse duringthe BRWCmaneuver . . . . 72

3.11 Driving Situations Identier (DSI)unit. . . 77

3.12 Feedba k ontroller . . . 79

3.13 Weighting fa torof thefeedba k ontroller . . . 80

3.14 Vehi leyawangle velo ityand sideslip angle . . . 81

3.15 Distribution of the wholeyawmoment . . . 82

3.16 Distribution of the yawmoment ofea h ontrolsystemunit . . 83

3.17 Distribution of the dynami feedforward ontrol yawmoment . 84 4.1 Empiri al modeling . . . 89

4.2 Stru ture ofsemi-physi al modeling. . . 90

4.3 s hemati ofthe a tiverear steering a tuator ontherear axle . 91 4.4 S hemati ofthetest rigsetup ofARS . . . 92

4.5 S hemati ofthesinusinput . . . 93

4.6 Sinus responseof theARS a tuator regarding

F

c

= 5.5

kN . . 94

4.7 Comparison ofthestep responses . . . 94

4.8 FastFourier Transform . . . 96

4.9 Bode diagramof thesinus responses . . . 97

4.10 Bode Diagram ofthereal systemvsthemodel . . . 98

4.11 The response oftheASR reala tuator and themodel . . . 99

4.12 The response oftheASR a tuator to

F

c

= +5.5

KN . . . 99

4.13 The responseof the ASR a tuator to

F

c

= −5.5

kN . . . 100

4.14 Comparison between dierent outputs . . . 101

4.15 For eand velo ity measurements . . . 102

4.16 Power hara teristi urve of the ARSa tuator . . . 103

4.17 Semi-physi al modeloftheARS a tuator . . . 103

4.18 The response ofdierent models to the stepinput. . . 105

5.1 Designpro edure basedon theV-Model . . . 107

5.2 Three-enablers . . . 108

5.3 The most twoe ient sampling methods intermsof dis repan y 111 5.4 The pro edureof generatinginput-output data . . . 112

5.5 A typi als hemati ofa simple feedforward neural network . . 113

5.6 ANN-a tivation . . . 114

5.7 Modelquality . . . 115

5.8 AvailableOutputs. . . 117

5.9 ANNPredi tion of a onstraint onthe meta-modeloutput . . . 118

5.10 Wronginterpolationof ANN. . . 118

5.11 The s hemati of SVMfun tionality . . . 119

5.14 The ombinationof SVMand ANN. . . 123

5.15 Representation of good andbad region inthedesignspa e . . . 126

5.16 Twodimensional se tional views. . . 127

5.17 Collapsedtwo dimensional se tional views . . . 128

5.18 Solution-Box . . . 129

5.19 Solutionintervals . . . 130

5.20 For e-Velo ity hara teristi urve of theperforman e test . . . 131

5.21 Stepresponseand its hara teristi parameters . . . 132

5.22 Behaviorofea h stepresponse . . . 133

5.23 Bode diagramof atransfer fun tion. . . 134

5.24 Extended V-Model forthe design of a tuators . . . 134

6.1 The exampleof thedependen y graph . . . 136

6.2 QSSC-CV . . . 138

6.3 Regression plots for CV

ψ

˙

∆1s

and

( ˙

ψ/δ

H

)

stat,max

. . . 139

6.4 Regression plots for CV

a

y,max

and

T

eq, ˙

ψ/δ

H

. . . 140

6.5 Solutionspa e ofthe a tuator's parameters . . . 141

6.6 Solutionbox withthelargestinterval . . . 142

6.7 Proje tion ofthe solution box . . . 144

6.8 ARS Solutionintervals(normalized) . . . 145

6.9 Solutionspa e for the hara teristi urve . . . 145

6.10 Test-rigsetup of ARS . . . 146

6.11 MinimumARS a tuator for e-velo ity hara teristi urve . . . 147

List of Tables

2.1 Denition ofvehi lestatesa ordingto Olley . . . 35

2.2 Denition ofthe vehi lestates. . . 36

2.3 Chara teristi values for QSSC . . . 50

2.4 Chara teristi values for SWD. . . 52

2.5 Chara teristi values for BRWC . . . 53

2.6 Chara teristi values for WEAVE . . . 54

2.7 Chara teristi values for CSST . . . 55

4.1 The kinemati s propertiesof ARS. . . 91

4.2 Properties ofthestep test . . . 92

4.3 Properties ofthesinusoidaltest . . . 93

4.4 Predi ted parameters . . . 97

4.5 FitPer ent for all ounterfor es . . . 98

4.6 Comparison ofthemodel outputwiththesystemoutput . . . . 100

4.7 Parameters for the hara teristi urve . . . 103

6.1 Designvariables ofthe ARSme hatroni system . . . 137

6.2 Considered obje tive drivingdynami s performan emeasures . 139 6.3 Mis lassi ation of alltrained SVM for ea hCV . . . 140

Abbreviations

Abbreviation Des ription

ESP Ele troni Stability Program

DSC Dynami Stability Control

CV Chara teristi Value

ECU Ele troni Control Unit

STM Single-Tra kModel

ARS A tiveRearSteering Control System

DSI Driving StateIdenti ation

ICM Integrated Chassis Management

LTI Linear TimeInvariant System

LPV Linear Parameter VaryingSystem

FPEM Frequen y DomainPredi tonError Estimate

CAN Controller AreaNetwork

RSM ResponseServi e Ma hine

SVM Support Ve tor Ma hine

NN Neural Network

ms Millise ond

FFT FastFourier Transform

DSI Driving SituationIdentier

YVA YawVelo ityAmpli ation

RWD RearWheelDrive

FWD Front WheelDrive

AWD AllWheel Drive

CG Center ofGravity

LFD Longitudinal For eDistribution

EG Self-SteeringGradient

BS Brake System

QSSC QuasiSteady StateCornering

SWD SineWithDwell

BRWC Brake While ornering

ABS Antilo kBrakingSystem

1 Introdu tion

1.1 Content and motivation

Vehi ledynami sareofgreatimportan einvehi ledevelopment. Thedynami driving behaviorof a vehi leis important for thesafetyof the vehi le passen-gers and the environment. In addition, it is an important pur hase aspe t for potential ustomerswho value thedriving experien e. One of theaims of the vehi ledynami sdevelopmentistodesignvehi lesoptimallya ordingtosafety and ustomerrequirements. BMW,asapremiummanufa turer,isparti ularly well-known for its sporty vehi les. Therefore, a great deal of importan e is atta hed to thedynami design of thevehi le.

The design pro edure of the vehi le dynami s is done in two phases: Early phase and late phase. In ea h phase, dierent parts of the hassis is designed and tested, gure1.1.

Figure1.1: Development phases ofthevehi ledynami s development

In order to a hieve the optimal design of the hassis, me hani al variables inuen ing the driving dynami s are determined in the early phase of vehi le dynami s development. The potential of onventionalme hani al omponents ae ting the driving behavior in a onstru tive way is rea hing its limits and is often asso iated with ompromises when we are oping with many diverse ustomers requirements. However, modern hassis omposed of a wide range of me hatroni systems oer thepossibility ofhaving more sheer pleasureand safetybyinuen ingthedrivingbehaviora tivelywithoutmeetingany on es-sion. Withtheaidof hassisme hatroni systems,thevehi le anbeadaptedto meet individualsafety and ustomer requirements. Yet, there are intera tions between the dierent me hatroni systems,sothat, for example, two separate

ontrolled input. Inthis ase,thea ura yof theregulationswill be impaired. In the worst ase, thedriving behaviorbe omesworse than an unregulated or evenunstablevehi le. Therefore,itisusefulto determine,rst,the ontrol sys-tem and a tuators parameters and the intera tion between them and se ond, their impa ts on thedriving dynami s performan e measures. Asa result, the rst goalofthis workisto develop theprin iple modelof su h ontrol systems andtheirasso iateda tuatorsinthesimulation environmentandtoexplainthe mathemati al relations between them.

In the past, vehi le design was iterative. The departments submitted the system requirements one after theotheruntil thedesigned omponents nally mettherequirements,gure1.2 . Su hanapproa hmayleadto oni tsofgoals regarding overall performan es of a vehi le, sin e ea h department attempts to rea h its spe i omponent requirements instead of rea hing the overall vehi le performan e. Additionally, it annot guarantee the reprodu ibility of the development for the next vehi le generations. Another disadvantage of iterative design is that it is highly time- onsuming. Be ause, if all designed omponents, after assembling in a vehi le, do not a hieve the overall vehi le performan e, alldesigndepartmentsshouldstartoverthedesigningpro edure. Moreover, the designed omponents su h as a tuators and ontrol systems must be able to opewithdierent variants ofa vehi le withdierent weights and properties, whi h are not ompletely determined in early stages of the vehi le development. In other words, we are dealing with dierent kinds of un ertainties due to undened properties, whi h ause the omplexity in the designpro edure. Wealsoknowthatthea tuatorsandtheirasso iated ontrol systems have strong intera tions. So, it would be more pra ti al to design a tuatorsand ontrolsystemstogetherinsteadofdesigningea honeseparately.

Figure1.2: Multi-Obje tive Designwithiterations

Asa onsequen e,the se ondgoalofthe dissertationisto developa method whi h onfronts the omplexity and ensures the robustness, transparen y and reprodu ibilityofthedevelopmentof ontrolsystemsanda tuators. Besides,it isimportanttohaveamethodwhi hisableto onsiderallvehi lerequirements formulatedbydierentdesigndepartmentsatthesametime,gure1.3,inorder to nd the bestdesign possibilities for ea h omponent. In this way,the time of thedevelopment isredu eddrasti ally.

•

New stati feedforward ontrol approa hes for a rear steering and braking system: Stati feedforward ontrols are essential in the ontrol engineering inorderto a elerate the ontrol loop anddo not in-uen e the stability of the system under investigation. Moreover, they regulatethevehi leinstationaryridesbeforethefeedba k ontroller. As a onsequen e,wedevelopdierent hara teristi urvesasstati feedfor-ward ontrolfor dierenta tuatorsassembledinavehi le. Thegoalisto hange thestationary responseof thevehi leto thestationarysteering.

•

A new entralized dynami feedforward ontrol approa h for lateral vehi le dynami s ontrol: An open loop solution an fully takeadvantageofthepotentialofmoderndevi es. Adynami feedforward ontrol anmodifythedynami behaviorofthe ontrolledinputofaplant. Asaresult,weproposeanewapproa hforadynami feedforward ontrol inorderto enhan ethe dynami behaviorofavehi lein aseofdynami steering. Thegoalistomodifythetransientresponseoftheyawvelo ity of avehi lebefore the feedba k ontroller a tsoverthat.

•

A new simplied feedba k ontrol approa h for lateral vehi le dynami s ontrol: As we know, the rst and foremost taskof a on-troller is to stabilize a system. So, we introdu e a new approa h for modeling su h a ontroller based on the available ideas in the state of the art. Beside this goal, stabilizing the vehi lein riti al situation, the developed ontrollershouldenhan etheresponseofthevehi leinnormal driving situations. Subsequently, the proposed approa h onsiders also all otherrequirementsinaddition to stabilization.

•

A new approa h for fun tional modeling of a tuators, in par-ti ular a rear steering system, for the design pro edure in the early stage of the development: A new approa h is presented how a fun tional model of an a tuator an be built. This will be done based on the measurement data gathered from the test-rig. Su h a fun tional model isadequateto overthemost important properties ofan a tuator and isofhighinterestintheearlystageofthevehi ledynami s develop-mentforthedesignpro edure. Themodelisidentiedwithafewnumber ofparameters and anbeintegrated inthesimulation environmentofthe full vehi lemodel.

•

A new approa h for data fusion for deriving mathemati al sur-rogate models: Using mathemati al surrogate models instead of the fullyphysi alvehi leandthe ontrolsystemmodels a eleratesthe

opti-mizationtime. Anewapproa hforapplyingneuralnetworksandsupport ve tor ma hine will be proposedfor thedata fusion.

•

A new approa h for dening a tuator test rig tests and veri- ation variables: Designing an a tuator an be done by formulating requirementsonitsfun tionalmodelparameters. However,these require-ments annotbetransmittedinthisformtosuppliersforthe onstru tion. Theymustbe onvertedintotherequirementssetdownontheveri ation variables oftest-rigs. The a tuator onstru tedbysuppliers hasto fulll these requirements. Asa result, the veri ation testsand theasso iated variables areproposedinthis dissertation. Theway therequirementson the fun tionalmodelare onverted into therequirementson the veri a-tion variables is also laried.

•

Appli ation of system engineering theory to a robust a tuator design androbustparametrizationofa ontrolsystem: Findinga robustsolutionspa eforparameterizingthedeveloped ontrolsystemand onstru ting a tuators, whi h opeswiththevariation ofvehi le proper-ties inthe development of avehi le, isgetting of highinterestinthelast fewyears. We onsiderthetheoryofdesignoflarges ale systemssubje t to un ertainty, whi hhasbeen beingdeveloped inthelast threeyears in the system engineering, in order to nd a robust solution spa e for the developed ontrol systemand fun tional a tuator modelparameters.

1.2 State of the art

For a hieving theabove-mentioned ontributions, we rst develop ontrol sys-tem logi and fun tional a tuator models for thevehi le lateral dynami s and se ondamethodtoparametrizethe ontrolsystemlogi parameters anddesign ana tuatorrobustly. Therefore,abriefreviewofthebasi softhe ontroltheory will be given inthis se tion. Afterwards, state of theartwill be explained for lateral vehi ledynami s ontrol systems,a tuator modeling,arobust a tuator design, and ontrol systems.

Normally, a losed-loop ontrol system onsists of dierent subsystems. A feedforward ontrol,afeedba k ontroller,ana tuatorandaplantareexamples of su h subsystems, gure1.4 . Basi ally,a feedforward ontrol an be divided into threegroups,namelystati ,dynami anddisturban efeedforward ontrol. Typi ally,a stati feedforward ontrol isafa tor whi h a eleratesthe ontrol loopand anbeadjustedindependentlywithout onsiderationofthestabilityof

transfer fun tion of a plant and a feedba k ontroller [47 ℄. It an also redu e the stati deviation of the error between the referen e input and the lose-loopsystemoutput, in ase of usingjust proportionalfeedba k ontroller [67℄. Moreover, a dynami feedforward ontrol an modify the transient behavior of the ontrolled input, sent to the plant and partially relieve the feedba k ontroller from regulating the transient phase ofthe error between thesystem output and the referen e input. Mostly, the dynami feedforward ontrol is designed based onthe desireddynami behavior of an output ofa losed-loop system.

Figure1.4: Blo kdiagramof a losed-loop ontrol system

Ifthe disturban e (

D

) penetrating intheplant ismeasurable, adisturban e feedforward ontrol anbedesignedbasedonthedynami ofthedisturban ein order to ompensatefor itsee t ontheplant. Theresultof su hregulation is thatthefeedba k ontroller shouldnot ompensate fortheee t ausedbythe disturban ewhi happearsontheerrorandthe ontrolpro edurewilltherefore bea elerated. Ifthereisstill anerrorbetween referen einputandthe losed-loop system output despite the regulation of all feedforward ontrols, it must then be regulated by a feedba k ontroller. The al ulated ontrolled input (

U

d

) hasbeen then setinto theplant throughan a tuator.

After this brief review of the basi s of the ontrol theory, we turn ba k to state of the art about the development of the lateral vehi le dynami s ontrol systems. There area tually three dierent kindsof approa hes for developing su h ontrol systems:

•

Developingaspe i ontrolapproa hforea ha tuatoravailablein has-sis. Su h approa h is alledde entralized ontrol system.

•

Developing de entralized ontrol systemsfor various a tuators and a o-ordinating systemwhi h oordinatesthedierent ontrolsystemsby

pri-oritizing the requirements of ea h ontrol system. In this way, a better driving performan e will be a hieved. Su h an approa h is dened as integrated vehi le dynami s ontrol system inthe vehi le dynami s liter-atures.

•

Developing a entralized ontrol system whi h generates a global on-trolled input regarding to the driver input and independent from a tua-tors.

[76℄explains thedieren e between entralized andde entralizedvehi le dy-nami s ontrol systems. A entralized vehi le dynami s ontrol system gener-ates a generi ontrolled input (e.g. generi yaw moment) independent from implemented a tuators in a vehi le. Subsequently, this ontrolled input will be distributedin dierent a tuators (e.g. dierential lo k,brake, steering sys-tems). A de entralized ontrol system, however, in ludes ontrol systems for ea h a tuator. Inthear hite ture ofthis kind of ontroller,a oordinator may be in orporated to manage the output of all fun tions to rea h the optimal vehi ledynami s performan e.

As pointed out, the de entralized ontrol system an be with or without a oordinating system. In the last few de ades, mu h resear h has been done about developing a de entralized ontrol systemfor ea ha tuator without im-plyingany oordinatingsystem. These ontrolsystemshavebeenproposedjust for one a tuator. For instan e, [51℄ proposes anew te hnique for developing a ontrol systemonly for a rear steering a tuator. There,an adaptive ontroller regardingthe drivingsituationisintrodu edwhi hisbasedonthephaseplane method. [16℄ introdu es an approa h for designing a stationary value for the rearsteeringanglebasedontheoptimizationofthe losedloopsystem,i.e. the single-tra kmodel(STM)ofthevehi lewithfrontandrearsteeringangles. The optimization is arriedoutwithrespe ttoanobje tive riterion formulatedon theamplitudeof thefrequen yresponseofthevehi lelaterala eleration. [34℄ introdu es a new ontrol on ept for the torque ve toring based on quadrati gaussion theory to improve the lateral dynami s of the vehi le. This ontrol approa h onsiders the fa t,thatthe only a tuator implementedinthevehi le isatorqueve toringa tuator. But, oftentherearemorethanone a tuator im-plemented inavehi le. If ea ha tuator follows the ommand of itsasso iated ontroller without onsideration of what all other ontrollers are performing, the overall ontrolled performan e of a vehi le an be impaired or sometimes unstable. Consequently, there has been mu h resear h in the last de ade in nding a oordinating system for de entralized ontrol systems of vehi le

dy-by introdu ing the urgen y hara teristi urve for oordinating between rear steering and brake system. A oordinating strategy between available ontrol systemsofvariousa tuatorsinavehi leisdevelopedin[58℄to a hieve abetter vehi leperforman ebyavoiding the oni ts enariosbetween dierent ontrol systems. A platformfor fun tions,a tuators, and ele tri al devi eswhi h on-ne t these fun tions and a tuators is presented in [70 ℄. Moreover, it spe ies howto integrate dierent ontrol approa hesof dierent a tuatorsinonly one platform.

The fo us of this dissertation lies, however, on the development of entral-ized ontrol approa heswhi h onsider thedriverinputs and generate aglobal ontrolled input independent from a tuators. As a result, state of the art of su h ontrol system approa hes is explained. [8 ℄ usestwo dierent ontrol ap-proa hesfor ontrollingbrakeandsteeringsystemtoimprovethelateralvehi le dynami s. Thesteeringsystem ontrolsthedeviationbetweenthereferen eand a tual yaw velo ity. Additionally, the braking system ontrols the velo ity of the side slip angle to keep thestability of the vehi leunder ontrol. Both ap-proa hesgenerateayawmomentas ontrolledinput. Thisworkalsointrodu es anSIindexto oordinatethegeneratedyawmomentsofthesetwodierent on-trol strategies. [29℄ proposes an approa h how to enhan e the steerability of a vehi le by ontrolling the deviation between the referen e yaw velo ity and the output yaw velo ity. It also laries how to enhan e the stability of the vehi leby ontrolling theerror betweenthe referen eanda tualsideslipangle. The ontrolled input is a global yaw moment independent from implemented a tuators. This work uses dierent rule bases for distributing the generated yaw moment between front and/or rearsteering and brakingsystem. The dis-advantage is that the amount of the yaw moment for distributing should be predened whi h in reasesthe omplexityofthedesignand parametrizationof the ontroller parameters. A ordingly, if the amount of the yaw moment is small, then the front steering systems will be a tivated. If itis medium, then rearsteeringwillbea tivatedandifitislarge,thenbrakingsystemmustbethe only a tuator to implement the requirement of the ontroller. [49 ℄ a hieves a ontrolling approa hwhi h al ulatesawhole torqueandfor eforapredened traje tory. These yaw torqueand longitudinal and lateral for es are indepen-dent from implemented a tuators. Subsequently, an optimization method is developed inthis work whi h distributes the yaw torque and for es optimally between a tuators. Su han ideahasbeen alsoinvestigated by[53 ℄ and [23℄.

In this dissertation, a new approa h is proposed for fun tional modeling of an a tuator. Thus, itis ne essaryto do areview of theavailable works inthe eld of a tuator modeling. There are a tually dierent kinds of methods for modeling of an a tuator. Two of them are, for example, physi al or empiri al

modeling. [55℄ shows how a semi a tive damper an be modeled physi ally and fun tionally. Fun tionally,thea tuator ismodeled bydividing thesystem dynami s into ele tri al dynami s and me hani al ones. Theele tri al part is represented by a transfer fun tion whose parameters are tted, su h that the bestpossible modeloutput, as lose aspossible to themeasurement output, is a hieved. Theproposedsimplied modelis derived,at theend, byjust stimu-latingthea tuatorwithdierent frequen iesandgettingthebodediagramand tting a transfer fun tion whose bode diagram ts to the one obtained from themeasurement. [65℄ explainshowto ompute anidealized modelofan a tu-atorbydeningase ond-ordertransferfun tion whosedampingratio depends on themaximal bearable moment or for e ofthea tuator underinvestigation. It, however, does not onsider the nonlinearity existing in the a tuator. [26℄ des ribes how to model a variable dierential lo k for the design pro edure. It is just fo used on the performan e of the a tuator in the time domain by stimulatingthea tuator withastep signal. Based on thestepresponseof the a tuator, the a tuator is modeledbyarst-order transfer fun tion.

After developing ontrol system logi aswell as a tuator models, a method isdemandedfor developing newa tuatorsaswellasparametrization of ontrol system logi s robustly, if we ope with the development of new vehi les. So far,manyapproa heshavebeen establishedfor designingnewme hani al om-ponentsinthe vehi ledynami s development. Mostof theresear hes fo uson thedevelopmentofvirtualprototypesandvirtualmethodswhi hrstly ontrol the omplexityofthe development andse ondly,de reasethetimeof develop-ment. [60 ℄ develops a tool for the vehi le dynami s development in order to ontrol the omplexity by keeping thetool user friendly, onsistent, modular, s alable,and real-timeexe utive. [63℄also develops asimulation tool,inwhi h an engineer an hange the properties of me hani al omponents of a hassis and ompare their ee tson the drivingdynami s very qui kly. Inthis way,it an keep the omplexityunder ontrol andredu ethetimeofthedevelopment pro ess, sin e building up a virtual prototype is less time onsuming than a real prototype. [43 ℄ develops a method for the virtual development of hassis omponents whi h inuen e the vehi le lateral dynami s. The method deals with the fa t, how the properties of ea h hassis omponent ae t the lateral vehi le dynami s. It is done by sensitivity analysis. However, the intera tion of omponentsis not onsidered. In addition, the surrogate models are devel-oped in this work for dierent omponents, whi h dene the main omponent properties. These surrogate models are then designed withrespe t to driving dynami s performan emeasures. Thede ien yofofallthese methods isthat they an neither onsider omplex intera tions of omponents under

develop-ride, omfort and safety, at the same time. As a onsequen e, [78℄ proposes a new strategy based on thesystemengineering to keep the omplexity under ontrol regarding the produ ts' properties qualitatively and quantitatively in thedevelopment pro ess. Inaddition,itpresentsanewnumeri al method,the so- alled solution spa es, for developing produ ts quantitatively besides many design parameters and design goals. A ording to this method, a omponent withmanydesign variables an be developed quantitatively withrespe ttoall design goals. Thismethod hasbeen applied inmanyprodu tdevelopmentsof me hani al omponentsinthelastveyears. Forinstan e,[59℄appliesthis the-oryfor thevirtualdevelopmentoffun tionalpropertiesofaxles. Moreover,the author ompares thedevelopment ofaxles bythis theorywith theother avail-able lassi development methods and determines that a development based on this theory delivers a better a ura y and robustness with respe t to the onsidered driving targets. Besides, this theoryis applied in[75 ℄ to formulate requirementsondesignvariablesoffun tionalmodelsoftiresandaxles regard-ing all onsidered driving dynami s performan e measures. It is also used by [22 ℄ to develop requirementson thedesign variables of thefun tionalmodelof dampers. Aswehaveseen, alltheabove-mentionedresear hesaredonefor the development ofonly me hani al omponent.

There area fewresear hes intheeldof the parametrization of ontrol sys-temsora tuatordesigns. Forexample,[13 ℄appliestheparameterspa emethod with respe t to eta, beta and theta stability riteria in order to nd solution spa es for ontrol systemparameters. The de ien y of this method refersto the number ofparameters whi h an be onsidered in this method. Notmore than a dened number of parameters an be onsidered by this method. Be-sides, theonly onsideredrequirementfor theparametrization pro edureisthe stability riterionofthesystem. Italsoneedsthelineartransferfun tionofthe openor losedloopsystem. Itmeansanon-linearsystemmustbelinearizedfor this method. Thismethodisalso applied by[14℄ tond thesolution spa efor theparametrizationofthefrontsteeringsystem ontrolparameterswithrespe t to theunstru tured un ertainties inthe dynami model of thevehi le su h as mass, inertiaandet . [64℄proposesamethodbasedontheknow-how manage-ment system. The author presents a quantitative method for parametrization of ontrol systems based on driving dynami s performan e measures. In this ase, he onsiders dierent driving dynami s performan e measures and sets quantitative requirements on them. Afterwards, he varies the parameters of the ontrolsystemsafewtimesandlookswhi hsetofparameter onguration satises all driving dynami s performan e measures. However, the problemof this method is, that it nds an optimum onguration and not a robust so-lution spa e and annot take many dierent ombinations of parameters into

a ount. The appli ation ofthe method hasjustbeen done for stationaryand dynami al feedforward ontrol parameters and hasnot onsidered theee t of thefeedba k ontroller,referen einputgenerator, anddisturban e feedforward ontrolonthedrivingdynami sperforman emeasures. [20℄developsamethod basedonavirtualplatform,onwhi htheparametrizationofthe ontrolsystem parameters ofbrakeinterventionsisdetermined,su hthatthegoal oni t be-tween vehi lesafetyandagilityisdenied. Forthispurpose,theauthordevelops a newdriving maneuver,whi h in ludesthesetwo targets. There,a hara ter-isti valuewithrespe tto thismaneuverisdevelopedforassessingwhetherthe safetyandagilityaremet. Justoneoptimumdesign for theparametrization is a hieved without onsiderationof otherdriving targets.

1.3 Stru ture of the dissertation

In Chapter 2, the theory of vehi le lateral dynami s is explained, sin e we will develop ontrol system logi and a tuator models for the vehi le lateral dynami s. Also,thefundamentalbasi sofa tiverearsteeringsystemand one-sided brake intervention will be rephrased in Chapter 2. Moreover, driving dynami s performan emeasuresfor rideand safetyareexplainedand areview of the non-linear modelof a hassisis giveninChapter 2.

InChapter3, ontrolsystemlogi sforthelateralvehi ledynami sare devel-oped. Allthefun tionsaredevelopedbasedontheideaofavehi le entralized ontrol system. However, stati feedforward ontrols aredesigned for ea h a -tuator assembledina vehi leseparately. Inuen es ofea hfun tion on vehi le dynami s performan eare thoroughlyinvestigated.

InChapter4,anewapproa hisintrodu ed toexploitthefun tionalmodelof an a tuator basedonthe measurement datafrom dierent tests arriedouton thetest-rigofana tuator. Thisapproa hanalyzesthebehaviorofana tuator in the time and frequen y domain. It also observes the non-linear behavior of the a tuator and proposes a te hnique for modeling the non-linearity. The approa his thenapplied for modeling therearsteering a tuator asexample.

InChapter5,weintrodu eamethodfortherobustparametrization ofa on-trol systemand design ofan a tuator,whi h isbased ontheso- alled solution spa es existing insystem engineering. The method demonstrates thedemand ofhavingveri ationtestsandvariablesfor theveri ationpro edureofan a -tuator after formulating requirements on its design variables. These testsand variables are then explained in details and the method in system engineering is, hen e, expanded.

terizing the developed ontrol system in Chapter 2 and designing a new rear steering a tuator modeled inChapter4.

2 Vehi le Dynami s Theory

Asabasi understandingofthevehi leisne essaryforthemodeling,designand evaluationofee tsofavehi ledynami s ontrol system,thevehi ledynami s theory, in parti ular the lateral one, isdes ribed in more details in this hap-ter. In the beginning, the linear single-tra k model and its transfer fun tion aredes ribed. Thesimulationenvironment,thetwo-tra kmodelanditsspe ial featuresarethenexplained. Inaddition,itisne essaryto larifythe fundamen-tals of tire behavior in longitudinal and lateral dire tions, as they hange the vehi le performan e while orneringsigni antly. Dierent driving maneuvers with their obje tive targetsare explained. Theyare thenused for the investi-gation of the developed ontrol system logi s and the design pro edure of an a tuator and ontrol systemlogi parameters.

2.1 Single-tra k model

Not only to investigate thevehi le behavior during ornering, but also to bet-terdes ribethe drivingdynami s, mathemati alsurrogate modelsareusedfor modeling the driving dynami s. The single-tra k model is a simplied and linearized modelfor the mathemati al representation of thelateral vehi le dy-nami s. Here,varioussimpli ationsaremadein omparisontotherealvehi le. Thesingle-tra kmodelenablesafastandsimpleanalysisofthelateraldynami behaviorof vehi lesinthelinearrange. Thedegree offreedomofthesystemis signi antlyredu edbymeansoflinearization,whi hmakesthe al ulationand analysisofthedrivingbehavior onsiderablyeasier. Themoststriking simpli- ationisprobablythemergingofthewheel onta tpointsonthefrontandrear axles. The simplied vehi le onsistsof a single front and a singlerear wheel. Therefore,thevehi leshrinksintoasingle-tra kmodel(STM).Inthisway,the lateral for ebuild-up of thewheels and the inuen es of kinemati s and om-plian es are ombined toformalinearsideslipstiness. Thevehi le's enterof gravity is also set at theheight of the road surfa e. Other simpli ations are thatrolling,i. e. turningaroundthex-axisofthevehi le,andpit hing,i.e. the rotationofthevehi learoundthey-axis,areprevented. Thesimpliedand lin-earizedsingle-tra kmodelisillustrated intheabove-mentioned simpli ations

Figure 2.1:Single-tra k model

Asa result, the modelis redu ed to the two degrees of freedom for theyaw movement aroundthez-axisinthe enter ofgravityof thevehi leandtheslip angle(

β

),whi hdes ribesthedeviationofthedire tionofthe enterofgravity speed from the longitudinalaxisof thevehi le[48 ℄.

Themotionequationofthevehi le anbederivedfromthekinemati softhe single-tra k model. The for e equilibrium inthe vehi lelongitudinal dire tion is asfollows [52℄:

m ˙v = F

y,f

sin(β −δ

f

)+F

x,f

cos(β −δ

f

)+F

y,r

sin(β −δ

r

)+F

x,r

cos(β −δ

r

)+F

x,d

(2.1) The for eequilibrium inthevehi lelateral dire tionis asfollows:

mv( ˙

β+ ˙

ψ) = F

y,f

cos(β−δ

f

)−F

x,f

sin(β−δ

f

)+F

y,r

cos(β−δ

r

)−F

x,r

sin(β−δ

r

)+F

y,d

(2.2) Thetorqueequilibrium atthe enter ofgravity (CG)is asfollows [52 ℄:

J

z

ψ = F

¨

y,f

l

f

cos δ

f

+F

x,f

l

f

sin δ

f

−F

y,r

l

r

cos δ

r

−F

x,r

l

r

sin δ

r

+M

z

+M

z,d

(2.3)

m

,

l

f

and

l

r

express the vehi le mass, the distan e of the enter of gravity (CG) from the front axle and rear axle, respe tively.

F

x,d

,

F

y,d

are external for es in ea h dire tion and

M

z,d

is a disturban e torque.

M

z

stands for the yawmoment generated bylateral vehi le ontrol systems.

The longitudinal and lateral for es a ting on tires are formulated from the vehi le propertiesasfollows:

F

total

= ma

x

F

x,f

= (1 − LF D)F

total

F

x,r

= LF DF

total

F

y,f

= c

f,mod

(δ

f

− β −

l

f

v

ψ)

˙

F

y,r

= c

r,mod

(δ

r

− β +

l

r

v

ψ)

˙

(2.4) where

c

f,mod

= c

f

(1 −

a

x

g

h

l

r

)

c

r,mod

= c

r

(1 +

a

x

g

h

l

f

)

(2.5)

Here, the longitudinal for e distribution (LFD) is the proportion of the driv-ing or braking for e a ting on the rear axle. In this dissertation, vehi les are onsidered asrear wheeldrive, while The

LF D = 1

. Furthermore, we add an assumption to thisSTM modelto later developa fun tion whi h ompensates theinuen eofthelongitudinalfor eswhile ornering. Theassumptionisthat the stati pit h motion is onsidered proportional with longitudinal dynami s inthesingle-tra kmodel. Through brakingora elerating, ani k-anglearises, whi h indu es a stati hange of wheel loads at the front and the rear axle. This hanges the whole sideslip stiness at the front and rear axle, whi h is dependentonthewheelloads[54℄. Themodiedsideslipstinessesatthefront and rear axle arerepresented as

c

f,mod

and

c

r,mod

,respe tively.

c

f

and

c

r

rep-resent the total sideslip stinessof the front and rear axle.

a

x

,

h

and

g

stand for thelongitudinala eleration,the enterofgravityheight andgravityvalue, respe tively.

In the single-tra k model, the angles are assumed to be small [30 ℄. There-fore, equation 2.2 and 2.3 an be linearized at thestraight ahead drive witha onstant velo ity,longitudinal a elerationand longitudinalfor e distribution. Moreover, in order to explain mathemati al formulations of the single-tra k model more easily, we onsider no ee ts of disturban es,

M

z,d

= 0

,

F

x,d

= 0

and

F

y,d

= 0

. From the above-mentioned equations 2.1, 2.2, 2.3 and 2.4 , the state spa e modelis thendened asfollows:

 ˙β

¨

ψ



=

"

−

c

f,mod

+c

r,mod

+F

total

mv

−1 +

c

r,mod

l

r

−c

f,mod

l

f

mv

2

c

r,mod

l

r

−c

f,mod

l

f

J

z

−

c

r,mod

l

2

r

+c

f,mod

l

2

f

J

z

v

# 

β

˙

ψ



+

"

_c

f,mod

mv

c

r,mod

+F

total

mv

c

f,mod

l

f

J

z

−

(c

r,mod

+F

total

)l

r

J

z

# 

δ

f

δ

r



+

 0

₁

J

z



M

z

(2.6)

Inserting 2.5inequation2.6 resultsin:

 ˙β

¨

ψ



=



a

11

a

12

a

21

a

22

 β

˙

ψ



+



b

11

b

12

b

21

b

22

 

δ

f

δ

r



+

 0

₁

J

z



M

z

(2.7) where

a

11

= −

(c

r

+ c

f

)gl

r

l

f

+ a

x

(h(c

r

l

r

− c

f

l

f

) + mgl

r

l

f

)

mgl

r

l

f

v

a

12

= −1 +

gl

r

l

f

(c

r

l

r

− c

f

l

f

) + a

x

h(c

r

l

2

r

+ c

f

l

2

f

)

mgl

r

l

f

v

2

a

21

=

gl

f

l

r

(c

r

l

r

− c

f

l

f

) + a

x

h(c

r

l

2

r

+ c

f

l

f

2

)

J

z

gl

r

l

f

a

22

=

a

x

h(c

f

l

3

f

− c

r

l

3

r

) − gl

r

l

f

(c

r

l

2

r

+ c

f

l

f

2

)

J

z

gl

r

l

f

v

b

11

=

c

f

gl

r

− a

x

c

f

h

mgl

r

v

b

12

=

c

r

gl

f

+ a

x

(c

r

h + mgl

f

)

mgl

f

v

b

21

=

l

f

(c

f

gl

r

− a

x

c

f

h)

gl

r

J

z

b

22

= −

l

r

(c

r

gl

f

+ a

x

(c

r

h + mgl

f

))

gl

f

J

z

(2.8)

If we want to onsider only the lateral vehi le dynami s without the stati ee tofthelongitudinalfor esonthe ornering(

a

x

= 0

),thestatespa emodel be omesasfollows:

 ˙β

¨

ψ



=

"

−

c

r

+c

f

mv

−1 +

c

r

l

r

−c

f

l

f

mv

2

c

r

l

r

−c

f

l

f

J

z

−

c

r

l

2

r

+c

f

l

2

f

J

z

v

#

|

{z

}

A

β

˙

ψ



+

"

_c

_f

mv

c

r

mv

0

c

f

l

f

J

z

−

c

r

l

r

J

z

1

J

z

#

|

{z

}

B





δ

f

δ

r

M

z





(2.9)

This model onsiders only theyaw motion of a vehi le while ornering and negle ts alllifting, rollingandpit hingmotions. Theeigenvalues ofthematrix A are al ulated fromthe hara teristi polynomialasfollows:

det(A − λI) = 0

(2.10)

λ

2

_{− (a}

11

+ a

22

)λ + (a

11

a

22

− a

12

a

21

)

(2.11) where

a

11

= −

c

r

+ c

f

mv

a

12

= −1 +

c

r

l

r

− c

f

l

f

mv

2

a

21

=

c

r

l

r

− c

f

l

f

J

z

a

22

= −

c

r

l

r

2

+ c

f

l

2

f

J

z

v

(2.12)

Thesolution ofthe hara teristi polynomialis

λ

1,2

=

a

11

+ a

22

2

±

s

 a

11

+ a

22

2



2

− (a

11

a

22

− a

12

a

21

)

(2.13) Thestability riterion a ordingto Hurwitz statesthatall oe ientsofthe polynomial must be positive. Only the onstant term an be negative. From this follows:

(a

11

a

22

− a

12

a

21

) > 0

(2.14) whi hmeans

c

f

c

r

l

mJ

z

v

2



l + v

2

m(c

r

l

r

− c

f

l

f

)

c

f

c

r

l



(2.15)

Thismodelis thenalways stable,ifthe riterion below ismet:

omes:

v

2

_crit

<

c

f

c

r

l

2

m(c

f

l

f

− c

r

l

r

)

(2.17) where

l = l

f

+ l

r

(2.18) Stationary steering

On the basisof the stationarysteering behavior, various driving dynami s pa-rameters an beobtained fromthesingle-tra k modelto assessthedriving be-havior. The hara teristi sof stationarysteering behaviorare theself-steering behaviorand theresultant self-steeringgradientaswellasthe hara teristi or riti al speed ofthevehi le.

stationary steeringbehavior and gradient

Self-steering behaviorisanimportant riterionfor assessingthedriving behav-ior of vehi les. In this ase, the steering angle demand, independent of the driver's inuen e, is determined for a ir ular drive at a onstant radius and an in reasing driving speed. The entrifugal for e ae ts the turning motion of the vehi le, resulting in sideslip angle on the tire. This results in a higher steeringangleeortto rossthe urve. Duringthestationary ir ulardrive,the additional steeringangletotheA kermannangle[30 ℄isinvestigatedinrelation to the lateral a eleration

a

y

. The onditions for the stationary ornering are asfollows:

v = const.

˙

ψ = const. ⇒ ¨

ψ = 0

β = const. ⇒ ˙β = 0

(2.19) A ordingly,

ma

y

= F

y,f

+ F

y,r

= F

y,f

l

l

r

ma

y

l

r

l

= F

y,f

= c

f



δ

f

− β −

l

f

v

ψ

˙



|

{z

}

α

f

(2.20) analogously,

ma

y

l

f

l

= F

y,r

= c

r



−β +

l

r

v

ψ

˙



|

{z

}

α

r

(2.21)

Equations2.20 and2.21 deliverwith

ψ =

˙

v

l

(small angle):

α

f

− α

r

|

{z

}

∆α

=

m

l

 l

r

c

f

−

l

f

c

r



a

y

= δ

f

−

l

f

+ l

r

v

ψ = δ

˙

f

−

l

R

|{z}

δ

A

(2.22)

Where,

R

and

l

stand for thepath radius andthewheelbase, respe tively. Basedon the A kermann angle (

δ

A

) and thesimpli ations resulting bythe stationary ir ular drive, it is possible to derive the steering angle from the single-tra k model as inEquation 2.23. The equation shows that, in addition tothegeometri steeringangle,thedrivermustalsoprovidethevehi lewithan additional steering angleto ompensate forthe sideslipangledieren e,

∆α

.

δ

f

=

l

R

|{z}

δ

A

+

m

l

 l

r

c

f

−

l

f

c

r



|

{z

}

EG

a

y

|

{z

}

∆α

(2.23)

To hara terize steeringbehavior, thetermsundersteerandoversteeraswell as neutral driving behavior are introdu ed here. For the evaluation of the vehi lebehaviora ording to Olley,thesideslipangle dieren e isused, Table 2.1 .

Conditions States

∆α = α

f

− α

r

< 0

Oversteer

∆α = α

f

− α

r

= 0

Neutralsteer

∆α = α

f

− α

r

> 0

Understeer

Table2.1: Denition ofvehi le statesa ording to Olley

This denition onsiders the absolute steering angle. However, thesteering anglegradient

dδ

f

/da

y

ismoreimportantthantheabsolutesteeringanglewhile

self-steering gradient (EG) is a vehi le parameter that depends on the vehi le mass, the positionof the enterof gravityandthetotal sideslipstinessofthe front and rearaxles.

EG =

m(c

r

l

r

− c

f

l

f

)

lc

v

c

r

(2.24)

The self-steering gradient des ribes the self-steering behavior of the vehi le and indi ates how mu h the lateral a eleration ae ts the steering angle re-quirement. Therefore,the front steeringangle demandis expressedasfollows:

δ

f

= δ

A

+ EG · a

y

(2.25)

The required steering angle for a stationary ir ular drive is omposed of two parts: theA kerman's angle dened bythevehi le geometry and the self-steering angle. The driving ondition dierentiates between neutral steer, un-dersteer andoversteer, asdenedinTable2.2 , basedon theself-steering gradi-ent. Neutral steering means, that thesteering angle must remain onstant at anyspeed or lateral a eleration[66℄.

Conditions States

EG < 0

Oversteer

EG = 0

Neutralsteer

EG > 0

Understeer

Table2.2: Denition of thevehi le stateswithrespe tto theself-steering gra-dient

Figure 2.2shows thesteering angle above thelateral a eleration for a sta-tionary ir ular motioninthelinearand non-linear range.

Figure 2.2:Steering behavior in stationary ornering while the road radius is onstant

Inthisgure,thesteeringeort at a onstant radiusat dierent velo ities is investigated. Thesteeringangle demandremainsun hanged,ifthevehi lehas aneutralproperty. For

EG > 0

,asteeringeorthastobeprovidedinaddition to the A kermann angle. Conversely, in the ase of oversteering, the steering angledemandisredu edbyanegativegradient. Atalaterala elerationofless than approx.

4m/s

2

, this gradient is quasi- onstant. However, an in reasing gradiento ursathighlaterala elerationbe ausethepropertiesofthetiresare non-linear intermsofsideslipstinessdueto wheelload hangesandin reased lateral for es onthetires.

Asalreadydes ribed,theyawvelo ity

ψ

˙

isoneofthemostimportant output variables inthe driver-vehi le ontrol loop. Therefore,itis ne essaryto exam-inethedrivingbehaviorwiththehelpoftheyawvelo ityresponsetoasteering angle input under stationary onsideration. The stationary yaw velo ity am-pli ation, also known as the steering angle-related ir ular motion value, is introdu edinthis ontext. Themathemati alrelationshipbetweentheyaw ve-lo ityampli ationandfront axlesteeringangle anbederivedfromequations 2.9 and2.19:

δ

f

s[mvl

f

c

f

] + δ

f

[c

f

c

r

l] =

˙

ψs

2

[mJ

z

v] + ˙

ψs[J

z

(c

f

+ c

r

) + m(c

f

l

2

f

+ c

r

l

2

r

)] + ˙

ψ

[c

f

c

r

l

2

− (c

f

l

f

− c

r

l

r

)mv

2

]

v

(2.26)

˙

ψ(s)

δ

f

(s)

=

[mvl

f

c

f

]s + c

f

c

r

l

[mJ

z

v]s

2

+ [J

z

(c

f

+ c

r

) + m(c

f

l

2

f

+ c

r

l

2

r

)]s +

[c

f

c

r

l

2

−(c

f

l

f

−c

r

l

r

)mv

2

]

v

(2.27) byreformulating:

˙

ψ(s)

δ

f

(s)

=

v

l +

m

_l



l

r

c

f

−

l

f

c

r



v

2

|

{z

}

=



˙

ψ

δf



stat

|

{z

}

Stationary

.

1 + T

z

s

1 +

2D

_ω

0

s +

1

ω

2

0

s

2

|

{z

}

Transient (2.28)

The stationary yaw velo ity ampli ation, equation 2.29 , depends signi- antly onthedrivingspeed. Itss hemati progressionasafun tion ofspeed is shown ingure2.3

˙ψ

δ

f

!

stat

=

v

l +

m

_l



l

r

c

f

−

l

f

c

r



v

2

=

v

l + EG.v

2

(2.29)

Figure 2.3:Thebehaviorofstationary yawvelo ityampli ationregarding ve-lo ity

Avehi lewithatenden ytoundersteer,nowadaysastandardforprodu tion of vehi les, has a slowly in reasing stationary yaw velo ity ampli ation. At a ertain speed

v

ch

, the yaw gain rea hes its maximum value indi ated by



_˙

ψ

δ

f



stat,max

and goesdownagainwithin reasingspeed. But, theyawvelo ity

ampli ation of an oversteer vehi le tends to innity at the riti al velo ity

v

crit

, whi h is undesirable for a driver, as he/she annot ontrol the vehi le at this velo ity. The riti aland hara teristi velo ityare then al ulated as follows: 1. For understeer vehi les (

EG > 0

):

˙ψ

δ

f

!

stat

=

v

l + EG.v

2

(2.30)

d

dδ

f

˙ψ

δ

f

!

stat

=

l − EG.v

2

(l + EG.v

2

₎

2

= 0 → v

2

ch

=

l

EG

(2.31) 2. For oversteer vehi les (

EG < 0

):

d

dδ

f

˙ψ

δ

f

!

stat

=

l − EG.v

2

(l + EG.v

2

₎

2

= 0 → v

2

ch

= ∞ → v

2

crit

= −

l

EG

(2.32)

A ordingly,

v

_ch

2

_{= −v}

_crit

2

=

l

2

m

c

f

c

r

l

r

c

r

− l

f

c

f

(2.33)

And the stationary yawvelo ity ampli ation an be onverted asfollows:

˙ψ

δ

f

!

stat

=

1

l

v

1 +

_v

v

2

2

ch

(2.34)

The hara teristi speed is learly illustrated by the self-steering behavior. Equation 2.34 shows that the vehi le is ontrolled in a neutral manner when

v

ch

isequaltoinnity. If

v

ch

isgreater thanzero, thevehi lehasatenden yto understeer and an be ontrolled atanyspeed. However, if

v

ch

is imaginaryor

v

crit

isreal,thevehi lehasanoversteeringtenden y. It anbe ontrolledstable atspeedsbelow

v

crit

. From

v

crit

on,however,theyawampli ationgoesagainst innityandthereforethevehi lebe omesunstableandthe ontrollabilityislost. However,su hanoversteeringdesign for series-produ ed vehi les hasnotbeen put on the marketfor years dueto safetyreasons.

Transient steering

Inorder toassessthedynami driving behaviorandfor thesubsequent ontrol design, it is also ne essary to know the driving behavior regarding transient steering. Inthis ase,yawnaturalfrequen yandyawdampingare hara teristi variables and arebrieyexplained below.

Thetransferfun tionofthevehi le anbederivedfromthestatespa emodel, equation 2.9 , asfollows:

G

_{(s) = (Is − A)}

−1

B

(2.35)

β

˙

ψ



=

"

G

_β/δ

_f

(s) G

_β/δ

_r

(s) G

_β/M

_z

(s)

G

_ψ/δ

˙

_f

(s) G

_ψ/δ

˙

_r

(s) G

_ψ/M

˙

_z

(s)

# 



δ

f

δ

r

M

z





(2.36)

Amongallthese transferfun tions, whi h an be easilyderived,thetransfer fun tion

G

˙

ψ/δ

f

(s)

isour pointofinterest,sin ewe useitinthenext hapterfor developing lateral vehi le dynami s ontrol systems. As a onsequen e, some featuresof thistransferfun tion willbestudied. The transferfun tionhasthe following form: