Fuzzy Sliding mode control for vehicle lateral dynamics ...

POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Mechanical Engineering

Fuzzy Sliding mode control for vehicle lateral

dynamics combining Torque Vectoring with

four wheel steering for electric FSAE vehicle

Supervisor: Prof. Edoardo Sabbioni

Co-Supervisor: Ing. Michele Vignati

Master thesis of:

Andrea Giambone 921154

Fabio Lussana 920358

Academic Year 2019-2020

Abstract

Attention to the environment is significantly increasing over the last years. The

automotive world is identified as one of the biggest sources of polluting gases due

to classic internal combustion engine emissions. This has led to the implementation

of electric vehicles. Due to the different layouts and the promptness of the electric

motors, the vehicle control logic should be rediscussed. This thesis aims to design a

multi-actuated control logic for the lateral dynamics control of an electric formula

vehicle which will compete in the EFSAE championship, an engineering design com-

petition among several universities in the world. The are two available actuators:

the torque vectoring (TV), at disposal thanks to two motors, one for each wheel, on

the rear axis and the rear wheels steering (RWS). The latter has been engineered in

this thesis to be implemented on the real vehicle and to better model its dynamic

to have a more precise control model. Firstly, this dissertation carries out a review

of the state of the art of active control on road vehicles, exploring their evolution

during the years landing to the ones implemented on FEVs. A briefly review on the

steering components and the several implementation possibilities is also available.

The dissertation continues with the RWS design, focusing on the gearmotor choice

and on the FEM study of the main structural parts. Eventually, a study of the ve-

hicle control system is presented, aiming to control simultaneously the two variables

which, according to the literature, influence the most the car behaviour, i.e. the side

slip angle and the yaw rate. Thanks to the right references, the implemented con-

trol target is to obtain a high-performance vehicle capable to recover the stability in

critical conditions. The control chapter also describes how the multi-actuation is im-

plemented. A phase portrait study has been carried out in order to obtain efficiency

maps, through which the use of an actuator over the other is maximized according to

the vehicle state. Then the control logic is described. The controller is a fuzzy slid-

ing mode control, which combines a super-twisting SMC with a FLC, the first one

generates the control action, while the second one adapts the sliding surface to the

vehicle state. The final control action is weighted through the performance indexes.

Finally, different numerical simulations, run in Matlab-Simulink environment, are

reported and results are discussed to show the controller performances.

I

Sommario

L’attenzione verso l’ambiente e cresciuta sempre di piu negli ultimi anni. Il mondo

dell’automotive e riconosciuto come una delle maggiori fonti di gas inquinanti a

causa delle emissioni dei classici motori a combustione interna. Motivo che ha por-

tato allo sviluppo di veicoli elettrici. Per il differente layout di questi e la prontezza

dei motori elettrici, la logica di controllo del veicolo deve essere rivista. Questa tesi

aspira a progettare un controllo multi-attuato per la dinamica laterale di un veicolo

formula elettrico che partecipera nel campionato EFSAE, una competizione tra di-

verse universita nel mondo. Gli attuatori disponibili sono due: il torque vectoring

(TV), a disposizione grazie a due singoli motori, uno per ruota, e il sistema di sterzo

posteriore (RWS). Quest’ultimo e stato progettato in questa tesi per essere imple-

mentato sul veicolo e per avere un modello affidabile della sua dinamica da usare

nel modello di controllo. Inizialmente la dissertazione presenta lo stato dell’arte dei

controlli sui veicoli, esplorando la loro evoluzione sino a quelli attualmente usati sui

veicoli elettrici. E anche presentata una breve revisione sui componenti dello sterzo e

le differenti configurazioni. In seguito, la tesi continua con la progettazione del RWS,

concentrandosi sulla scelta del moto-riduttore e sullo studio FEM delle parti strut-

turali piu importanti. L’attenzione si sposta poi sullo studio del sistema di controllo

del veicolo, che ha lo scopo di gestire le due variabili che, secondo la letteratura, piu

influenzano il comportamento dinamico del veicolo, ossia la velocita di imbardata e

l’angolo di assetto. Grazie ai giusti riferimenti, il controllo e costruito per ottenere

un veicolo performante e che riesca a recuperare la stabilita in condizioni critiche.

Il capitolo sul controllo tratta anche dell’implementazione della multi-attuazione.

Uno studio nel piano di fase e stato fatto per produrre delle mappe di efficienza,

attraverso le quali e definito l’uso di un attuatore rispetto a un altro in base allo

stato del veicolo. Successivamente, la logica di controllo e descritta. Essa si basa

su un controllo ‘fuzzy sliding mode’, che combina un super-twisting SMC con una

FLC; il primo genera l’azione di controllo, mentre il secondo adatta la superficie di

sliding allo stato del veicolo. L’azione finale del controllo e pesata tramite gli in-

dici di performance. Infine diverse simulazioni numeriche, in Matlab-Simulink, sono

state eseguite e commentate per mostrare le capacita del controllo.

III

Acknowledgements

First and foremost, we would like to express our sincere gratitude to Prof. Edoardo

Sabbioni and Ing. Michele Vignati for guiding us, for sparing their valuable time and

for the suggestions to keep us in the right direction to achieve the desired targets.

We would also like to thank Ing. Michele Asperti for the precious help and support

in the development of all the aspect of the thesis.

We would like to express our heartiest gratitude to our parents and family for their

continuous support in all fields of our life, for motivating us and encouraging us to

give always our best.

We are also very grateful to our friends for the beautiful moments spent together in

these years, for the support and the suggestions in the difficult moments. We will

preserve these wonderful memories in our heart forever.

V

Contents

List of Figure XI

List of Tables XV

Symbols, subscript and acronyms XVII

Introduction 1

1 State of the art 7

1.1 Hybrid and electric vehicles . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Vehicle’s lateral dynamics . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Active control evolution . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Integrated control for road vehicles . . . . . . . . . . . . . . . . . . . 18

1.5 Lateral dynamic control’s actuators . . . . . . . . . . . . . . . . . . . 20

1.5.1 Brake Torque Vectoring system (BTV) . . . . . . . . . . . . . 21

1.5.2 Active differential . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.3 Indipendent electric motors . . . . . . . . . . . . . . . . . . . 23

1.5.4 Active steering . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.5 Literature review on steering systems actuation . . . . . . . . 26

1.6 Active control logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.7 Sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.8 Fuzzy logic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Vehicle model 39

2.1 Vehicle body model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Wheels model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3 Tyre model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.1 Longitudinal slip . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.2 Side-slip angle . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.3 Vertical force . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4 Electric motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

VII

2.5 Rear steering actuator . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6 Brakes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.7 Resistant forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.8 Driver model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Rear wheels steering design 59

3.1 Design concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.1 Concept 1: two linear actuators replacing tie rods . . . . . . 59

3.2 Rear wheels steering model . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Gearmotor choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Thermal model . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 Rear wheels steering system dynamic . . . . . . . . . . . . . . . . . . 71

3.5 CAD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5.1 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.4 FEM analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 Control strategy 87

4.1 Control architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 Control references . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Fuzzy sliding mode control and coordination . . . . . . . . . . . . . . 93

4.3.1 Efficency maps . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4 Torques allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Simulations and results 111

5.1 Compared control strategies . . . . . . . . . . . . . . . . . . . . . . . 112

5.2 Steering pad constant radius . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Steering pad constant speed . . . . . . . . . . . . . . . . . . . . . . . 116

5.4 Step steer manoeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Double lane change manoeuvre . . . . . . . . . . . . . . . . . . . . . 125

5.6 Braking in turn manoeuvre . . . . . . . . . . . . . . . . . . . . . . . 127

5.7 Simulations resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Conclusion 131

Bibliography 135

VIII

A MF-Tyre model 141

A.1 Longitudinal force (pure longitudinal slip) . . . . . . . . . . . . . . . 142

A.2 Lateral Force (pure side slip) . . . . . . . . . . . . . . . . . . . . . . 143

A.3 Longitudinal Force (combined slip) . . . . . . . . . . . . . . . . . . . 143

A.4 Lateral Force (combined slip) . . . . . . . . . . . . . . . . . . . . . . 144

A.5 Relaxation length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B Dimensioning of RWS components 145

B.1 Dimensioning of rack and pinion . . . . . . . . . . . . . . . . . . . . 145

B.2 Dimensioning of pinion shaft . . . . . . . . . . . . . . . . . . . . . . 146

B.2.1 Keyway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.2.2 Bushings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.2.3 Linear bearings . . . . . . . . . . . . . . . . . . . . . . . . . . 148

B.3 Datasheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

IX

List of Figures

1 Torque Vectoring explained by Bosch . . . . . . . . . . . . . . . . . . 4

2 4control system by Renault . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 series architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 parallel architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Integrated contol and g-g diagram [56] . . . . . . . . . . . . . . . . . 20

1.4 Front steering chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5 EPS mounted on the steering rack [17] . . . . . . . . . . . . . . . . . 28

1.6 EPS mounted on the steering column [17] . . . . . . . . . . . . . . . 29

1.7 sliding mode principle [18] . . . . . . . . . . . . . . . . . . . . . . . . 31

1.8 Mamdani fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.9 Sugeno fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 Vehicle 3D reference system. . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Vehicle geometrical schema. . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Vehicle 2D model schema. . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Wheel rotational equilibrium. . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Wheel rolling resistance. . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Tyre combined force ellipse. . . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Load transfer due to Ax and Ay. . . . . . . . . . . . . . . . . . . . . 50

2.8 Electric motor torque vs rotational speed. . . . . . . . . . . . . . . . 51

2.9 Front-rear braking force partition. . . . . . . . . . . . . . . . . . . . 53

2.10 Driver model used to perform close loop manoeuvres. . . . . . . . . . 57

3.1 Rear wheels steering model . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 KAG SN worm gear . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 KAG DC motor M80x40 12V . . . . . . . . . . . . . . . . . . . . . . 65

3.4 RWS motor operating points during a double lane change at 120 km/h 66

3.5 RWS motor operating points during a 15° step steer at 90 km/h . . 67

3.6 Temperature profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Power losses profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

XI

3.8 RWS motor step response . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9 View of RWS system, view from below . . . . . . . . . . . . . . . . . 75

3.10 View of RWS system, left side . . . . . . . . . . . . . . . . . . . . . . 75

3.11 View of RWS system, right side . . . . . . . . . . . . . . . . . . . . . 75

3.12 Actual system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.13 View from below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.14 Overall system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.15 Frame connection elements on transverse elements . . . . . . . . . . 79

3.16 Rack-tierod connection element . . . . . . . . . . . . . . . . . . . . . 80

3.17 Steering housing-gearmotor connection element . . . . . . . . . . . . 80

3.18 Steering housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.19 Main rod assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.20 Housing and pinion shaft assembly . . . . . . . . . . . . . . . . . . . 82

3.21 FEM analysis of the steering housing . . . . . . . . . . . . . . . . . . 83

3.22 FEM analysis of the housing-gearmotor connection structure . . . . 84

3.23 FEM analysis main rod in worst load condition . . . . . . . . . . . . 85

3.24 FEM analysis of the housing-gearmotor connection structure . . . . 86

4.1 Control system architecture. . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Yaw rate reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3 Beta reference for optimal control. . . . . . . . . . . . . . . . . . . . 92

4.4 FLC scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5 Side slip angle error membership function, as explained in the text

only a small range around zero is considered . . . . . . . . . . . . . . 94

4.6 Yaw-rate error membership function, it has an input range equivalent

to that of the side slip angle error . . . . . . . . . . . . . . . . . . . . 95

4.7 Side slip angle membership function . . . . . . . . . . . . . . . . . . 95

4.8 Performance weighting factor membership functions . . . . . . . . . 96

4.9 Stability weighting factor membership functions . . . . . . . . . . . . 96

4.10 Control variables derivatives schema . . . . . . . . . . . . . . . . . . 104

4.11 Efficency maps at 25km/h and 1.5° front steering angle . . . . . . . . 106



4.14 Efficency maps at 25km/h and 3° front steering angle . . . . . . . . . 108



4.17 ASC weighting function. . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1 Steering pad constant radius (50 m) manoeuvre on high friction road

µ = 1. Comparison between controlled and passive vehicles. . . . . . 115

XII

5.2 Steering pad constant speed (90km/h) manoeuvre on high friction

road µ = 1. Comparison between controlled and passive vehicles. . . 117

5.3 Simulation data of a steer step manoeuvre on high friction road sur-

face: steer step of 5deg at 90km/h. . . . . . . . . . . . . . . . . . . 120





5.6 Simulation data of a double lane change manoeuvre (80 km/h) on

high friction road surface. . . . . . . . . . . . . . . . . . . . . . . . . 126

5.7 Simulation data of a braking in a turning manoeuvre on high friction

road surface: half braking capacity used. . . . . . . . . . . . . . . . . 128

A.1 Tyre reference quantities . . . . . . . . . . . . . . . . . . . . . . . . . 142

B.1 Forces on pinion shaft schema . . . . . . . . . . . . . . . . . . . . . . 146

B.2 Pinion datasheet - code 30120012 . . . . . . . . . . . . . . . . . . . . 149

B.3 Keyway datasheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.4 Linear bearing datasheet - code LBPR20 . . . . . . . . . . . . . . . . 151

B.5 Bushing datasheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.6 Worm gearbox + DC motor datasheet . . . . . . . . . . . . . . . . . 153

B.7 DC motor M80x40/I 12V datasheet . . . . . . . . . . . . . . . . . . 154

XIII

List of Tables

2.1 Vehicle physical data. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Load transfer data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3 Electric motors data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4 Braking system data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5 Resistance forces tables. . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.6 Driver model coefficients. . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 Motor thermal data . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 RWS motor coupled with mechanical system . . . . . . . . . . . . . 73

4.1 Fuzzy logic rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Steering pad constant radius on high friction. Maximum lateral ac-

celeration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2 Steering pad constant speed on high friction. Maximum lateral accel-

eration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 Step-steer manoeuvre on high friction road surface. . . . . . . . . . . 119

5.4 Step-steer manoeuvre on high friction road surface. . . . . . . . . . . 121

5.5 Double lane change manoeuvre on high friction road surface. . . . . 125

XV

Symbols, subscript and

acronyms

Symbols:

Υ Time delay s

Ω Wheel angular velocity rad/s

α Wheel slip angle rad

β Vehicle side-slip angle rad

γ Sliding mode control gain ∼δ Wheel steer angle rad

θ Chassis pitch angle rad

λ1 Sliding mode control gain ∼λ1 Sliding mode control gain ∼µ Road friction coefficient ∼ξ Fuzzy logic performance weighting factor ∼φ Chassis roll angle rad

ρ Air density kg/m3

σ Fuzzy logic stability weighting factor ∼τ Transmission ratio ∼χ Performance index ∼ψ Chassis yaw angle rad

ω Natural frequency Hz

A Vehicle acceleration m/s2

Cα Cornering stiffness N/rad

Cd Drag coefficient ∼F Force N

J Moment of inertia kgm2

L Wheelbase m

Lf C.o.g. to front axle distance m

Lr C.o.g. to rear axle distance m

XVII

M Momentum Nm

R Radius m

S Vehicle front area m2

T Torque Nm

V Vehicle speed m/s

c Vehicle semi-track m

fv Rolling resistance coefficient ∼g Gravitational acceleration m/s2

hG C.o.g. height from the ground m

i x axis versor ∼j y axis versor ∼k z axis versor ∼k Longitudinal slip ∼kd Derivative coefficient ∼ki Integral coefficient ∼kp Proportional coefficient ∼kroll Roll stiffness ratio ∼kz Tyre vertical stiffness N/m

m Mass kg

s Curvilinear abscissa m

t Time s

u Tyre rolling resistance trail m

K Steering arm m

J Inertia kgm2

α Pinion pressure angle

η Efficency −

Subscript:

b Brake

f Front

fl Front left

fr Front right

i i-th element

lin Linear

r Rear

req Required

rl Rear left

XVIII

rr Rear right

sw Steering wheel

th Threshold

w Wheel

x Along x direction

y Along y direction

z Along z direction

Acronyms:

4WD Four Wheel Drive

4WS Four Wheel Steering

ABS Anti-lock Braking System

AFS Active Front Steering

ASC Anti-Slip Control

BTV Brake Torque Vectoring

c.o.g. Center of gravity

d.o.f. Degrees of freedom

DV C Dynamic Vehicle Control

DY C Direct Yaw Control

ESC Electric Stability Control

ESP Electric Stability Program

FLC Fuzzy Logic Control

FSMC Fuzzy Sliding Mode Control

FEV Full Electric Cehicle

ICE Internal combustion engine

IWM In wheel motorized

LQR Linear quadratic regulator

MPC Model Predictive Control

RAD Rear Active Differential

RWD Rear wheel drive

RWS Rear wheel steering

SMC Sliding Mode Control

STA Super Twisting Algorithm

V SC Vehicle Stability Control

V TD Variable Torque Distribution

XIX

Introduction

In recent years car manufacturers focused on researching alternatives energy sources

for automotive propulsion. Nowadays, almost all of these have realized that the

only achievable solution is the development of hybrid electric vehicles (HEV) and

fully electric vehicles (FEV). Some brands explored different solutions like hydrogen

engines, but they also propose electrified car models, which represent the only real

alternative to fossil fuel that can be implemented in relatively short time. This en-

vironmental awareness started to be relevant in the second part of the last century,

but until the ’70s there were not rules. The first time it appeared was in 1973 in

Europe, when Environment and Consumer Protection Service was set up and the

first Environmental Action Programme (EAP) was adopted [40]. From that time,

the awareness of the necessity of rules in order to protect the environment and to

improve the life’s level increases, coming out continuously with new rules, updates,

commissions and organizations to set increasingly demanding objectives. Currently,

the European Union has a long-term objective of reducing greenhouse gas (GHG)

emissions by 2050 from 80 up to 95% compared to 1990 for the transport sector,

which is accountable for more than a fifth of the EU’s GHG emissions, the target is

a 60% reduction [11]. In order to achieve these goals the European Commission has

drawn up a set of rules that limits the average CO2 emissions of all new cars and

vans registered in the EU from 2020, penalty payment of fines quantified according

to the level of emission above the target and the number of vehicles. The attention

to environmental impact and the increasingly restrictive limits on the emission of

CO2 are the two main reasons that drove manufacturers to include electrified alter-

natives within their car proposals. Some brands are so confident in the potential

and development of this kind of propulsion that expects to completely remove clas-

sical fossil fuel engine from their cars from now to 10-15 years. The main problems

of this technology are the storing of the energy, which needs big battery packages,

the charging network, which should guarantee a good diffusion to let the owners to

travel without concerns, and the recharging time, that is a lot longer than a classical

gasoline refill. Anyway, in the very last few years, understood the importance of the

electric vehicles in the future market, these problems are becoming smaller thanks

1

to commitment, research and investment of the car makers and in general of many

automotive and electrical companies. Electric vehicles, apart from the obvious zero

emission advantage when the car is running, have some relevant pros with respect

the ICE vehicles. First of all, the relocation of the space inside the car, the presence

of the electric motors and of the battery package leads to a different occupation of

the space with a layout which usually gives more space for passengers, thus more

comfort. Another point, the most important, which is the one of this study, is the

security and performance improvement thanks to the great possibility of being more

precise and effective in the active control acting directly on the electric motors, as

it will be described later.

Focusing on fully electric vehicles, which this study deals with, two main ways of

layout design exist: one motor for each axle (even on only one axle), which leads to a

more classic solution, similar to the conventional drivetrain scheme of a ICE vehicle,

with the need of a differential, or one motor for each wheel, which permits to act

on the single wheel with the torque and speed needed in each moment (limited by

the motor characteristics), offering a big opportunity in terms of control due to the

direct control on the electric motor, without dealing with mechanical parts such as

the differential, more complicated and less fast in the reactions. In our study we use

this latter solution, due to the main advantages just described.

Talking about safety on the vehicle, this has always been one of the most important

topics when dealing with cars. The first motor cars began running in the 1880s, one

of the earliest crashes resulting in fatal injury was recorded in a London newspa-

per already in 1889 [54]. The necessity of improving the safety of the vehicles was

immediately clear, the first barrier test was run by General Motors at the Milford

Proving Ground in Michigan already in 1934 [54]. At that time the causes of injury

were unknown, and improvements in design were probably related more to reducing

damage to vehicles than to reducing the risk of injury, but anyway it is evident that

safety is a problem intrinsically related with the vehicles from their birth. During

the years safety has always increased thanks to new technology and to the growing

awareness of its importance. Firstly with passive safety features, which are systems

that do not do any work until they are called to action. They become active during

an accident, and work to minimize damage and reduce the risk of injury during the

time of impact. These systems are seat belts, air bags and the structure of the vehi-

cle. These devices automatically deploy when the car gets into a crash. In parallel

active safety system are introduced and under continuous research. They work to

prevent an accident, these systems always stay active while you drive and continu-

ously work to keep you from getting into an accident. Most active safety features

are electronic and controlled by a computer. The most known active safety systems

are ABS, the first introduced, ESP and TCS (respectively Antilock Brake System,

2

INTRODUCTION

Electronic Stability Control and Traction Control System). In the modern car are

also diffused features such as forward collision warning, lane departure warning and

adaptive cruise control which approach always more the autonomous drive. Merging

passive and active controls nowadays a very high level of safety is reached.

On the other hand, active controls can be very useful to improve the performance

and the stability of a vehicle, especially in extreme situations. This approach is in-

teresting in a racing environment and it is less developed than the normal use. This

study’s aim is to develop this aspect in terms of active control on lateral dynamic

of a race electric vehicle that will compete in Electric FSAE championship. A simi-

lar work has been developed in [8] using typical proportional integral control logic.

This thesis is intended to explore and implement some innovative controls in the

literature, the sliding mode and the fuzzy logic ones, to manage several parameters

simultaneously. In a race car the safety is entrusted to the passive systems, the ac-

tive control is instead designed to help the driver to enhance the car’s performance

at maximum lateral acceleration and maximum speed working on the stability limit

of the vehicle.

To study the lateral dynamic of a vehicle two main variables must be considered:

the side slip angle, which is the angle between the transversal speed of the car’s

barycentre and the absolute speed, and the yaw rate, which is the rotational speed

around the vertical axis. The basic theory of an active theory is to fix a reference

of these quantities and manage to follow it. In order to do it there are two possible

methods. The first is the so called ‘torque vectoring’ which consists in creating a

yaw moment. Our car has two electric motors, one for each wheel, on the rear axle,

thus in order to create the yaw moment we have two possibilities: to use the brakes

or the motors, as displayed by Figure 1.

The first solution is to act on the brakes of the rear (or generally of the front, or

both) axle to break one wheel to create a difference of longitudinal force between

the two wheels and thus a yaw moment. This method, while being effective, has two

negative sides: it is only possible to dissipate energy, so to slow down one of the two

wheels, it is not possible to accelerate one of the two to have a higher difference of

torque. Besides, as just said, this system dissipate energy, so the motor is running

very inefficiently, wasting battery’s charge and reducing the maximum performance.

The cleverest method to create the yaw moment, having one electric motor per wheel,

is to use them to have the exact torque the control needs in each wheel, creating

the necessary yaw moment to follow the desired reference. Moreover, in this way

the system has no power dissipation, useful to reach the maximum performance and

to save energy, and it can introduce both positive and negative torque on the wheel

reaching every difference of moment needed (always in the motor limit). The other

instrument to manage the lateral dynamic of a car is the four-wheel steer (4WS).

3

Figure 1: Torque Vectoring explained by Bosch

The active control on the steering angle of the wheels allows to control the lateral

forces generated by the axis and influence the lateral dynamics of the vehicle. It is

very useful to influence the behaviour of a car in the curves, both in the faster ones,

where it can guarantee more stability, and in the slower ones, where it improves the

agility. For this reason, it is often used on big and very powerful SUV (sport utility

vehicle) which, due to the dimensions and to the weight, find great benefit in this

system. The main cons of this system is that it adds cost because it implies new

mechanical components and a complication in the design of the rear axle. For this

reason, it is usually used only in very expensive cars like the Bentley Flying Spur

or the Ferrari SF90 Stradale. It is possible to think that, as every technology, with

the passage of time it will be available also on cheaper cars. An example is Renault,

which already offers the 4Control system (the Renault 4WS system, shown in Figure

2) from 2007 on the best versions of Megan, Talisman and Espace. They are not

cheap cars, but for sure they are not so expensive like a Ferrari or a Bentley can be.

This thesis starts with a deep literature review (chapter 1) in which many details

on the contents are provided. This chapter firstly describes the electrification of

the vehicles, its advantages in the control world and the importance of a greener

mobility. Then it deals with vehicle’s lateral dynamic and the several control logics

which could be used in order to manage it, focusing on the two types of logic used

in this thesis: fuzzy and sliding mode control. Chapter 2 describes in every partic-

4

INTRODUCTION

Figure 2: 4control system by Renault

ular how the vehicle has been numerical modelized. The considered model is a nine

degrees of freedom one in which also brakes, electric motors, tyres and suspensions

dynamics are considered. At the end it is also described the driver model used to

perform close loop manoeuvre. This thesis deals also with the rear wheel steering

system design, as it is explained by chapter 3. In particular, the modelling of the

system, the gearmotor design and the system dynamic are carried out. In chapter 3

is also shown the CAD design in order to have a better understanding of the system

on the car and to report the FEM simulations. Chapter 4 covers the control system

design, focusing on how it is structured, explaining the control references and the

fuzzy sliding mode designed in this thesis. In chapter 5 there is the report of the

simulation. Both open-loop and closed-loop manoeuvres are performed, in partic-

ular, steering pad constant radius, steering pad constant speed, step steer, double

lane change and braking in curve manoeuvres are reported. Finally chapter 6 sums

up the thesis results, concluding the study with the considerations that came up

along the entire thesis work.

5

Chapter 1

State of the art

This chapter covers the literature review to have an overview of the arguments this

thesis deals with. The first section (1.1) discusses about the importance of the

electrification in the road vehicles world, besides it briefly describes how hybrid and

electric vehicles are designed. The section 1.2 shortly gives an explanation of the

lateral dynamic concept and how it is managed. Moreover it identifies the crucial

points to implement a lateral control system. Then in 1.3 there is an overview

of the active control evolution until nowadays, where integrated control systems

are always more present and important. The 1.4 section deals precisely with the

latter. Section 1.5 shows the actuators needed to implement the controls previously

explained, highlighting the pros and cons of the different solutions. Finally, the last

three sections cope with the active control logic topic. 1.6 gives a general overview

of the existing ones, while 1.7 and 1.8 deepen, respectively, sliding mode control and

fuzzy control logic, which are the one used in this thesis.

1.1 Hybrid and electric vehicles

In the past few years, the research and development of eco-vehicles such as elec-

tric vehicles (EVs), hybrid cars, and fuel-cell cars, which have drive-systems that

generate the driving torque with motors, has been actively undertaken due to the

increasing interest in energy and environmental problems. Several types of drive

systems have been proposed for eco-vehicles, for instance, a front- or rear-wheel-

drive system with two in-wheel motors and a four-wheel-drive (4WD) system with

independently driven front and rear motors or four in-wheel motors. In these eco-

vehicles, the driving motors can be used as generators to recuperate the kinetic or

potential energy during braking. The recuperated energy, which is stored in an en-

ergy storage device, can improve the fuel economy. Moreover, regenerative braking

from the motors can be employed to improve vehicle stability since the response

7

1.1. Hybrid and electric vehicles

time of the motor torques is faster than under conventional hydraulic braking, and

motor torque control can be accurately carried out, even in the nonlinear region of

the tire [35].

The first vehicles on the market were the hybrid ones, due to the similarity with

the classic ICE vehicle. Nowadays they are quite diffused, in many countries there

are state incentives both for hybrid and full electric cars to promote the passage

to a greener mobility. Some full electric cars, especially city-cars, are starting to

appear in the market. The main problems of a EV are related to its autonomy, to

its recharging time and to the recharging points network which nowadays are not

spread enough to guarantee the tranquillity needed by a possible buyer who wants

to afford a purchase of a EV, which, usually, costs even more than a classic ICE car.

Mainly due to these problems many people are oriented towards hybrid vehicles,

which can guarantee both the tranquillity of an ICE car and the electric vehicles ad-

vantages (although obviously having some others cons like a higher weight). These

vehicles are usually proposed with different possible drivetrain architectures. Let’s

take in consideration the two main ones: parallel and series. Other solutions are

a mix of the two and they are more complicated. The proposed architectures are

thought to solve the problem of the union between engine and motor. The series one

is based on the decoupling between the two power sources, as showed in (Fig.1.1).

Usually in this type of vehicle the fuel is the primary energy source, it is used by

the engine, which works as a generator, to power the electric motor and to recharge

the batteries, which are usually used in transient and whenever there is a peak of

power where the motor needs an extra-charge. The main pros of this system are the

Figure 1.1: series architecture

complete decoupling between engine and motors which allow to not have additional

8

Chapter 1. State of the art

mechanical pieces as clutches or gearboxes. Moreover, the control algorithm is quite

simple, the engine is usually used at its maximum efficiency point to have the best

fuel consumption. The main cons of this system is that it works like an electric vehi-

cle but with extra components. The alternative is the parallel architecture (Fig.1.2),

which is a lot more diffused in car’s field. Here the motor and the engine collaborate

Figure 1.2: parallel architecture

in order to obtain the best compromise between fuel consumption (which is usually

the main objective of a hybrid vehicle), needed power and comfort. The control

logic is very difficult because it must match all these features. Usually these types

of vehicles have different driving maps, from the eco one that tries to use the system

in the most efficient way, to the sport one which release the full power of the system.

Hybrid cars can be classified by the degree of hybridisation, which is the ratio

between the power of the motor and the engine. Typically, four categories are iden-

tified: micro, mild, full, plug-in. From the one equipped with the smallest motor

(and consequently lighter battery package and electrical equipment), to the plug-in

one which has the most powerful motor and the biggest electric autonomy thanks

to a big battery package.

Electric motor vehicles deserve a lot of attention too, being probably the long-term

future in the automotive field. In fact, in recent years, the focus of attention has

moved into the development of fully electric vehicles (FEVs), which promise to pro-

vide a personal mobility solution with zero emissions. Moreover, owing to significant

advancements in energy storage units and electric motors in terms of power density,

9

1.2. Vehicle’s lateral dynamics

this promise of modern FEVs may become a viable option for the mass market.

With these prospects, novel concepts of electric vehicle layouts are gaining more

and more importance. The first generation of fully electric vehicles was based on

the conversion of internal combustion engine driven vehicles into electric vehicles,

by replacing the drivetrains, while keeping the same driveline structure; that is, one

electric motor drive, which is located centrally between the driven wheels, and a

single-speed mechanical transmission including a differential. Such a design solution

is going to be gradually substituted by a novel vehicle architecture, based on the

adoption of individually controlled electric powertrains, with the unique possibility

to improve the vehicle dynamics control because of their intrinsic high and indepen-

dent controllability. The active control of electric powertrains allows the regulation

of the distribution of the driving torques in order to achieve desired steady-state

and transient vehicle dynamics characteristics. At the same time, if implemented

through in-wheel motors, these architectural solutions allow an improvement of the

overall vehicle packaging as less space is required by the powertrain. [12] About this

last technology also LiQuiang e Al [29] affirm that Electric Vehicles driven by in-

wheel motors present to us a practical way of designing an EV with its cost reduced

and that the performance of in-wheel motor EV is likely to perform better compared

with that of classical vehicles once a good control system is invented.

In [62] they describe with more precision that if an electric vehicle is equipped with

four in-wheel motors, it is easy to control the four tyre longitudinal forces indepen-

dently for more sophisticated vehicle motion control. In addition, the four wheels

with in-wheel motors of the vehicle are easily steered independently by some addi-

tional electro-magnetic actuators to control four tyre lateral forces for the motion

control. Thus, the electric vehicle easily becomes a full drive by-wire vehicle which

has the eight independently controllable tyre forces – four longitudinal forces and

four lateral ones. It is thus clear that in-wheel-motors are a very reliable and inno-

vative solution in order to better control the vehicle’s dynamic.

1.2 Vehicle’s lateral dynamics

Vehicle’s lateral dynamic refer to the vehicle dynamic behaviour during a curve. It

has been studied firstly by Pacejka, who in 1973 presented the steady-state turn-

ing study which includes the handling diagram and the over/neutral/understeer

concepts, which resume the car lateral behaviour and are nowadays accepted as fun-

damental cornerstones in vehicle dynamics. He also introduced the ‘magic formula’

to implement the forces exchanged by the tyre with the ground, another base con-

cept to study the car’s dynamic.

10


Let’s explain quickly the concept of Ackerman steering angle, which it is needed

in the following paragraph. It is the ideal steering angle to perform a curve, the

kinematic one which the car would follow in an ideal condition.

The over/neutral/understeer concept defines the steady state car’s behaviour in a

curve. It is a difficult and very large concept but let’s try to resume to have a little

overview to better understand the concepts that will follow. This definition of the

car’s characteristic depends from the first derivative of the difference between the

actual steering angle and the Ackerman’s one over the lateral acceleration. If it is

greater than zero the car will have an understeering behaviour, if less oversteering,

if equal neutral. In other words, fixed a curve, increasing the speed during the curve

and thus the lateral acceleration, if the requested steering angle increases with re-

spect the ideal one the car is understeering, the tendency of the vehicle will be to

enlarge the curve and to remain stable. If the requested steering angle is the same

of the ideal one than the vehicle is neutral. Last one is the oversteering behaviour,

which happens when the derivative is negative, thus the actual steering angle is less

than the ideal one and decreases increasing the speed along the curve. This means

that the car tries to close the curve independently by the steer. It can be considered

a good effect for the car’s performance, but it can lead to instability very easily,

up to an uncontrolled tailspin. This is the reason why all the car manufacturers

produce understeering vehicles. It is possible to think that the best solution would

be a neutral car, and theoretically it’s true, but there are too many variable factors

which influence the car’s characteristic in curve, as the tyres pressure and the load

distribution, which can lead the car to an unstable behaviour too easily. The only

field where the oversteering behaviour is searched in certain condition is the racing

one because the drivers are skilled and trained to control it.

It is possible to define a so-called understeering coefficient which has the same mean-

ing of the derivative, giving a measure of the change between the steering angle and

the trajectory curvature as function of the lateral acceleration [9]. Many studies on

active controls use this coefficient to find a reference to define the desired behaviour.

The handling diagram is another instrument to quickly understand the car’s steady

state behaviour in curve. It actually gives the same information of the understeering

coefficient but in a more visible way. It is a graph, thus for a trained eye it’s possible

to understand in a moment the car’s dynamic. It is possible to plot it in different

ways. A first one is to represent the lateral force over the side slip angle for each

axle, so that, having the two axle’s characteristics, it is possible to understand the

car’s behaviour. It’s enough to compare the front and the rear side slip angle at the

same lateral force value. If the front one is higher, then the vehicle is understeering,

if the rear one is higher instead the car will be oversteering and if the two values

are the same the vehicle will be neutral. From the handling diagram it is possible

11


to derive another representation, which gives the same information: the steering

angle over the lateral acceleration. From this it’s evident how usually a car has a

linear behaviour up to a certain lateral acceleration, after which it starts to behave

non-linearly until the maximum lateral acceleration, i.e. its steady-state cornering

limit. As said previously the understeering coefficient is strongly related with the

steering angle along the lateral acceleration, thus this plot is strictly correlated with

it and gives the same information.

As previously said, up to now we have only discussed the steady-state motion of

the vehicle in curve, which is the first one historically studied. However, the re-

sponse of the vehicle to the steering imposed by the driver is the superimposition

of this motion and of the transient one. While to study the steady-state motion we

can study the local stability linearizing the equations around a steady-state point,

obtaining a quite easy and understandable system, the transient motion is more

difficult to study, because it’s very difficult to implement easy equations to describe

it. To study it we need to resolve instants for instants a non-linear set of equations.

We are now studying the global stability of the vehicle which depends on the initial

disturbance. A good method to study it is to use the phase plot implemented by

the two side slip angles (front and rear) to find the so-called domain of attraction,

which is the area representing all the possible initial condition which bring to an

equilibrium point, and thus to global stability. If the initial condition was out of this

area, the vehicle couldn’t reach an equilibria point becoming unstable. The negative

side of this method is that for each combination of speed and steering angle a plot is

needed to have the complete description of the vehicle in all the possible conditions.

After a short technical introduction to better understand the concept of lateral dy-

namic study, it is clear that to implement an active control good results, both in

the transient motion and in the steady state one, are needed. In order to obtain

the maximum performance in curve the transient has to be fast but at the same

time overshoot and setting time need to be minimized. A performance index for

the steady-state motion is instead the maximum lateral acceleration reached in sta-

ble condition, the higher the better. Moreover, a more extended linear zone in the

handling diagram is appreciable. As we will see in the following chapters, there are

some tests to understand the vehicle’s behaviour in both the conditions.

From literature we know that the lateral dynamics are governed by the yaw rate,

strictly related with the lateral acceleration, and the vehicle side slip angle. Early

papers on active rear steering focus on reducing the vehicle side slip angle, more re-

cent papers focus on the controlling the yaw motion. The overshoot in yaw velocity

(“fishtailing”) is undesirable and leads to an increased workload for the driver [7].

From [29] we know that vehicles are likely to become unstable when side-slip angle

is over 12 deg if the road is dry, but if the road is wet, the vehicle will definitely be

12


out of control when the side-slip angle is over 5 deg. Forster noted that vehicles,

including tires and suspensions, must be made to fit for human’s behaviours. Ve-

hicles must be in control even when it has got severely deviated from normal track

[16, 42]. The control themes, which hold practical values, are known to be divided

into two kinds. One considers yaw rate only, such as DSC (Dynamic Stability Con-

trol) system of BMW. The other one takes both yaw rate and side-slip angle into

consideration, such as the VDC (Vehicle Dynamic Control) system of BOSH and the

VSC (Vehicle Stability Control) system of Toyota [61, 1, 2, 21]. From the discussion

above, it is evident that the major point in stabilization control is to identify both

yaw rate and side-slip angle in real-time. In order to design a reasonable controller

which can keep vehicle body always stable, those two parameters must be taken into

consideration simultaneously. Besides, the sensor that used to monitor yaw rate is

very common; however, the sensor which is used to monitor the side-slip angle is

always expensive [28], thus it is usually not present on a normal car and the side

slip angle is normally estimated.

In order to obtain the desired target many strategies and actuation systems have

been developed in literature. In [12] the authors compare three different torque vec-

toring strategies for steady state conditions: constant torque distribution (referred to

as baseline vehicle); torque proportional to the wheel vertical load; torque distribu-

tion which allows achieving the same longitudinal slip ratio on each wheel. However,

they reach the conclusion that a feedforward control in the frequency domain and a

feedback control is necessary, thus they propose another study which implements a

novel algorithm, based on optimal control, for an automated design of TV strategy in

steady-state condition. Always in [12] also the transient motion has been analysed,

comparing three different possible actuations: a differentiation of the wheel torques

within the rear axle (left-to-right torque vectoring technique), an active roll control

system capable of varying the lateral load transfer distribution between the two axles

and a four-wheel-steering (4WS) system. The conclusions of the analysis are that

the in-axle torque vectoring methodology (for the specific case study vehicle) is able

to fully compensate the load transfer and the tyre longitudinal/lateral interaction

effects due to vehicle acceleration/deceleration (a range of +/-2 is considered in the

reference). Also, this method proves to be much more effective in the compensation

than the Active Roll Control system and the 4WS system considered. For the case

study presented here, Active Roll Control is effective only for sideslip angle values

of more than 5° in deceleration and 3° in acceleration. Below this threshold, the

system is unable to compensate the effect of vehicle acceleration/deceleration. In

contrast, the 4WS system can generate the required compensation effect only for

low values of β. The conclusions of this paper are certainly interesting, but this

was only an example of one of the several study on this field. Another interesting

13


example can be the Mitsubishi four-wheel drive system described in [37]. The au-

thors describe the principles of the Mitsubishi Super-All-Wheel-Control, which is

a direct yaw moment control (DYC) strategy obtained through the distribution of

longitudinal forces and lateral forces among the four tyres. This torque-vectoring

strategy is implemented through the employment of torque-vectoring differentials,

comprising planetary gears and two clutches or brakes, in order to transfer torque

from the left wheel to the right wheel and vice-versa, independently from the loca-

tion of the faster wheel (within limits relating to the differential layout). According

to the Mitsubishi algorithm, depending on the variation of the traction coefficient,

a more balanced distribution of longitudinal and lateral forces between the left and

right wheels can be achieved during cornering.

As previously said, these are only some examples of the possible control implemen-

tations in order to act on the vehicle’s lateral dynamic. During the last years the

best car manufacturers are focusing on developing integrated control systems, imple-

menting on the vehicle more than one actuation system. For instance, one possibility

is to combinate the torque vectoring system with the four-wheel steering one, as we

will discuss later in this thesis. This is really challenging for car manufactures be-

cause it opens to many possibilities, giving the opportunity to control more than one

degree of freedom. Today, premium vehicle manufacturers are taking active systems

a step further as they continuously seek ways to deliver the most enjoyable and

pleasant driving experience. Technology currently under development allows vehicle

handling to be customized around driver’s desires, whether the preference is for a

‘fun-to-drive’ characteristic or a stable predictability. A good opportunity nowadays

is given by the electric vehicles (EVs), especially the all-wheel drive one which have

one motor for each wheel, giving the opportunity to take the precision and the speed

of response of TV at an all-new level. Thanks’ to this, in the future the hope is that

cars will be always more tailored and customized under the driver preference. Al-

ready in the present there are a lot of cars with the possibility of tuning actively

many components in order to obtain different behaviours when whished. For ex-

ample, just to report an ‘extreme’ example, the last version of Ford Focus RS had

available as optional the drift mode, where the torque was mostly delivered at the

rear axle (thanks’ to an active 4WD system based on clutches which could distribute

the torque between all the wheels as desired) and the active control where tuned in

order to let the rear part of the vehicle sliding in a controlled way. However, electric

vehicles seem to be the trend for the future, thus it’s now important to try to develop

new strategies and actuations to implement the control on such vehicles. In the liter-

ature not so much material is present, sign that it’s a quite new and interesting field.

14


1.3 Active control evolution

Safety has always been one of the most important themes in automotive world. To

ensure it many passive systems has been developed during the years. Seatbelts,

airbags, a cleverer chassis design are only some examples, each component of the

car is studied in order to be safe for the passengers. However, from about twenty

years, a huge step has been made introducing the active control on the car.

There are a lot of studies which describe how a normal driver’s experience is limited

largely to drive well within the physical limit of adhesion. In other words, a normal

driver will drive in normal adhesion situation for the 99% of his life, rarely he will

meet a limit situation so he would not be ready and difficulty he could manage

the situation because it’s anti-intuitive. From the article [59] from Bosch, one of

the most involved company in active control, it is possible to read “ Forster [23]

has analysed this situation and set up some important rules. First, the driver can

never recognize the coefficient of friction between the tires and the road and he has

no idea of the vehicle’s lateral stability margin. Second, if the limit of adhesion

is reached the driver is caught by surprise and very often reacts in a wrong way

and usually steers too much. This, he notes, is the real weak point in the system

vehicle driver-environment. Third, in traffic situations the need for the driver to

act thoughtfully has to be minimized. Forster therefore comes to the conclusion

that the concept of the vehicle including the tires and the suspension should very

strongly account for the normal human behaviour. Deviations from normal vehicle

behaviour that are inherent to the vehicle design must be controlled and reduced to

negligible differences”. This explains the area of action of the active safety system,

the limit one, where a normal driver probably wouldn’t be able to control the vehicle.

Simultaneously, the control must act in an intuitive way for the driver in order to

not create panic and thus to not get worst the manoeuvre’s recovery. The first active

control to be introduced was ABS (anti-locking braking system) due to the evident

loss of yaw response of the vehicle to steering inputs during full braking. This can be

explained easily by the combined friction theory: the friction between tyre and road

is limited, if the car is braking a certain quantity of friction is used to generate a

longitudinal force, thus the available one to generate lateral force to curve is less than

expected. If the braking force is too high and the wheel locks, the available lateral

friction is zero, so the vehicle cannot curve even if the driver is steering. This is a

very not intuitive behaviour of the vehicle which can be very dangerous for a normal

driver who doesn’t expect it. If the brake pressure induced by the driver is such

that the wheels lock, then the brake pressure must be reduced to regain steerability.

This is the work of ABS, which is done thanks to electromagnetic valves in the

hydraulic braking system which are able to keep the pressure in the wheel brakes

15

1.3. Active control evolution

below the level induced by the driver. The main goal of this system is to find the

perfect balance between the requested braking force and a good level of handling

performance by generating the right lateral force by steering. The biggest problem

for the ABS algorithm is that some information is not available, so it has to rely

on assumptions on the shape of the friction curve and on the wheel characteristics

during braking and cornering. To address these problem ABS intervention is based

on a measurable quantity, the angular acceleration of each wheel. If one wheel

decelerates too fast the braking pressure is reduced until it accelerates again. The

increase of the pressure is done stepwise in order to reduce the influence of the

transients on the wheel behaviour. Thanks to this it’s possible to maintain the

average slip value around the value which guarantees the best friction coefficient

(typically in the curve friction coefficient-slip there is a maximum for a slip value of

about 20%). The effect of the introduction of this first active control system has been

as effective as the European Union has decided that all new manufactured cars had to

be compulsorily equipped with ABS since July 2006 [34]. A first attempt to develop

a system to manage driving force (TCS, traction control system) was made using the

ABS logic, but it was a failure because of the too high total rotating inertia (with

respect the braking phase, in the driving one the engine and transmission inertia

has to be added), which prevents a significant change in the wheel acceleration, and

because of the non linear engine torque development with respect the wheel speed.

Likely in two wheels driven vehicle (2WD) it is possible to compute the free rolling

speed putting a sensor on the non-driven axle. Therefore it’s possible to calculate

the slip value, the only unknown is the friction curve which depends by the road

surface and the tyre characteristics, but it is usually taken a medium value or in the

more advance controllers it is estimated in order to be more reliable. For a 4WD

(four wheels drive) it is a bit more complicated because there is not the possibility

to compute the free-rolling speed, instead it is estimated by the motor rotation and

the torque distribution. To reduce the slip of a wheel there are several methods,

like reducing the fuel injection or the spark advance, which vary from constructor to

constructor, but we will not see them in detail because it’s not the aim of this thesis.

In order to improve the stability of the vehicle the studies have then moved on the

lateral dynamics behaviour. Driver assistance devices for vehicle dynamics primarily

produce a compensating torque for yaw disturbances. Such control systems can react

faster and more accurately than the driver when an unexpected deviation from the

desired yaw rate occurs. These studies bring quickly to the ESP (electronic stability

program), which is a feedback control to guarantee stability. Even if, especially in

the first versions, it usually takes as reference the yaw rate, its main task is to limit

the side slip angle of the vehicle in order to prevent unwanted spin. The first aim

is the stability and the safety, but also the handling performance of the car can

16


be improved if the dependence between the steering and the yaw moment can be

controlled. In order to implement it starting from the ABS and TCS equipment,

four additional sensors are needed: steering wheel angle, brake pressure, yaw rate

and lateral acceleration [59]. The controller gets as inputs the steering angle, the

requested drive and braking torque, which are the driver inputs. ESP can control

the yaw moment on the car by controlling the value of the slip at each wheel. This

can be done in different ways depending on the technology of each car, it is possible

to be done only with the braking system but there are also other systems like active

differential which are more advanced and more expensive. We will talk about them

later. The first car with ESP system was the Mercedes class S in 1995, and from

that moment experts estimate that the system has avoided about 200000 incidents

saving more than 6000 people. Due to the effectiveness of the ESP and limited

implementation cost (as long as for a base system only little additional hardware

components are needed) it is compulsory by law in Australia and United States from

2011 [34] and in Europe from 2014 [EU]. During the last twenty years technology,

especially in electronic and mechatronic areas, has made big steps introducing new

active systems which act on the yaw rate control as well as ESP. Some examples are

the active front steering (AFS), the four wheels steering (4WS), the variable torque

distribution (VTD) or the different braking (DB) [19]. Generally, all these systems

work on the stability of the vehicle, thus it is possible to refer to them as vehicle

stability control (VSC). Actually the adopted controls can be divided in two main

categories, the torque vectoring (TV), which can be made by the different usage

of the brakes, by active or self-locking differentials or by electric motors in special

electric vehicles, and the category which concerns the steering, with the control that

acts or on the front steer (AFS) modifying the steer imposed by the driver to obtain

a better response, or on the rear steer (4WS) to virtually modify the wheelbase

of the vehicle to obtain the desired behaviour in curve. We can refer to all these

controls as direct yaw-moment control (DYC) because they all produce an action on

the yaw motion of the vehicle, which is the fundamental one during curve. DYC is

proved to be one of the most promising means of chassis controls, which is able to

enhance the vehicle handling performance and, in the meantime, to improve a lot the

active safety. In literature it is possible to find many types of controls, as classical

proportional-integral-derivative (PID), fuzzy logic, optimal control and others under

investigation [14]. During the last years many studies are discussing the possibility of

integrating the yaw rate control with the ‘Beta-method’, which controls also the side

slip angle of the vehicle. This seems to be effective to enhance both the performance,

dominated more by the yaw rate control, and the safety at limit condition, which is

guaranteed by the side slip angle control. [12, 29]. In terms of performance, active

controls have proved to increase it, further to improve a lot the safety. However, a

17

1.4. Integrated control for road vehicles

control needs a reference to follow, and it is not simple to produce a reliable yaw

rate reference because the control couldn’t go over the vehicle’s handling limits, thus

sometimes the intent of the driver is physically not reachable. The second challenge,

as previously mentioned, is to create a control which is not invasive for the driver

and which makes driving easier. Nowadays the main challenge is to coordinate and

to integrate the several active controls which are present on the vehicle (usually

both TV and steering controls) so that they improve the overall control. A bad

coordination can produce a negative effect, with more controls which work worst

than only one, making the technology useless. Many studies are investigating this

field, with some of them focused on the use of electric motors due to the increasing

diffusion of the electric vehicles on the market, which seems the trend of the next

years. This type of vehicle presents interesting opportunities to re-design the active

control based on their technology.

1.4 Integrated control for road vehicles

The integration between different actuations and control strategies is one of the

most important field of study of the last years in the automotive world due to the

interesting improvements that these more complex systems aim to reach. In fact,

two or more systems which work together in the right way should guarantee a better

result than only one system. A Very good resume of this argument has been done

by the authors of [58], from which the concepts of this paragraph are taken.

Let’s start defining two important terms. The hardware of a control system is

the physical system, which covers things like sensors, actuators, power electronics,

switches and micro-processor. The hardware of modern cars is becoming very com-

plex with a lot of components, it’s easy to have 30-50 microprocessors on a single

vehicle [5], rather than many others bigger components. We are not interested in

the physical location of them because it tells relatively little about the design of the

functional control structure. We are interested in the architecture, which is defined

in [58] as “ an abstraction of the pattern (or topology) of how sensor information and

control commands interact between various control sub-systems and components ”, in

other words it’s intended as a comprehensive representation of the global Integrated

Vehicle Control System (IVCS) structure, both in its operation and its design.

The most intuitive architecture is the one called parallel, which has been used in the

past to control independently different vehicle’s functions. This means that control

hardware can be grouped into discrete subsets, with sensor information and control

demands operating in parallel processes and with no possible ambiguity or conflict

over the responses demanded of the actuators. This type of architecture has natu-

rally arisen, as different controlled sub-systems are developed and manufactured by

18


different supplier companies, or by different groups within a vehicle manufacturer.

The main problem is that the subsystems don’t communicate one each other and,

being the number of relevant degrees of freedom in a car very limited, it’s inevitable

that the subsystems interact in a wrong way, and performance conflicts arise. For ex-

ample, the use of single-wheel braking to reduce oversteer or understeer will certainly

conflict with the requirement for traction under forward acceleration [58]. Another

negative issue of this architecture is that probably many systems are overnumbered,

maybe there are two systems which could use the same hardware but, being manu-

factured and mounted distinctly, they are not. The foremost concerns are to do with

reducing complexity, improving performance and removing unnecessary and costly

duplication of hardware. Gordon et al. suggest that in the development of an IVCS

the aim is to combine and supervise all controllable subsystems affecting vehicle dy-

namic response to improve multiple-objective performance from available actuators,

reduce complexity, improve safety and comfort, reduce system costs by avoiding

unnecessary duplication and improve flexibility. The latter is another key-point of

integrated control, highlighted by Wills et al. [60] who talk about the importance of

openness, which refers to the need for the overall system to be sufficiently open to

allow an integration with other systems. It is fundamental for a modular approach,

which is always more needed in the modern automotive world. In fact, a car man-

ufacturer could build more than one type of vehicle, and for each type could have

different versions with different controls, from the cheapest one with only the basic

control, to the most advanced with many types of controls. It is fundamental, to

manage this big variety of possibilities, to design modular subsystems which can be

added and controlled always by the same software.

Tanaka et al. in [56] present a schematic figure (Fig.1.3) to indicate the domain of

operation of some typical vehicle control system, and the areas where system inte-

gration are likely to be beneficial.

This diagram underlines the possibility, thanks’ to integrated control, to enlarge and

smooth out the dynamic response of the vehicle. One of the first car manufacturer to

study this problem was Toyota, already in 1993 Hirano et al. [22] implement a four

wheel steer/four wheel drive (4WS/4WD) controller via feedforward and feedback

compensators designed using multivariable H∞ methods. Authors report improved

vehicle stability on slippery surfaces and an improved steering response. Thought

only 4WS/4WD were explicitly integrated within the control algorithm, in this paper

it’s interesting to notice that the experimental vehicle employs Local Area Network

(LAN) communications to link a central control unit to a distributed set of five con-

trol units for, further of 4WS and 4WD, ABS, engine control and electronic throttle.

Thus, it was a first attempt of fully integrated control, which guarantees also a basic

level of safety. In fact, in case of a failure of the LAN or of a local control, the

19

1.5. Lateral dynamic control’s actuators

Figure 1.3: Integrated contol and g-g diagram [56]

remaining local control units could guarantee a good level of functionality. Gordon

et al. note, however, that this architecture is not explicitly reflected in the controller

design methodology. They report that, given the consistent performance improve-

ment possibility given by integrated control, several authors have approached the

design of it via full-vehicle reference model. From [58] “ Examples include model

matched-control using a model, robust H2 and H1 design methods, and nonlinear

predictive control. [. . . ] Other authors have employed standard techniques such as

direct output feedback methods (single-loop analogue or discrete-time compensators,

sliding mode control, model reference, fuzzy logic and Artificial Neural Networks.

Two particularly common formal control methods, both essentially based on linear

systems are those of robust H2 and H1 control and optimal control.”

It is now very clear the importance to develop an integrated control system, es-

pecially in the new vehicles, where always more control systems are applied and

coexist.

1.5 Lateral dynamic control’s actuators

In the previous chapters we have discussed of the several ways to control the lateral

dynamic of the vehicle, in this one the physical mechanisms and solutions which allow

the control are presented. There are two main ways to act on the yaw motion: torque

vectoring and steering control (both front and rear). The TV systems generates a

yaw moment thanks to the uneven distribution of torque between the right and

the left side of the vehicle. There are three main actuation systems to reach this:

20


independent brakes [55, 43, 47], active differential [20, 57], independent electric

motors [12, 29, 49, 10]. The active steering control acts directly on the lateral force

generated by the tyres, changing actively the steering angle of the wheels, in fact the

lateral force depends directly by the side slip angle which is highly influenced by the

steering angle. It is possible to act actively on the front axle (front active steering

– FAS), on the rear one (rear wheel steering – RWS) or on both axles obtaining an

active four-wheels steer (4WS) vehicle [3, 39, 52, 24, 13].

1.5.1 Brake Torque Vectoring system (BTV)

The simplest and cheapest method to implement TV is the usage of independently

brakes. Thanks to them it’s possible to create a yaw moment differentiating the

braking. Compared to a passive or semi-active differential, this mechanism can be

created independently from loading and adherence on the ground. This implies that,

during a turn, the system has the ability not only to transfer all the driving torque to

the external wheel and maintain the internal one in free rolling condition but also to

further amplify the yaw moment by creating a negative traction force on the internal

wheel and increasing the driving torque on the external one [43]. It is simple and

cheap because independent brakes are already present in all the cars thanks to ABS

system. It is not a case that the first company to study and introduce the TV system

was Bosch, which is the one which first implements also ABS, to implement the ESP

system. The latter, as previously written, has been recognized as one of the most

important step-forward in active safety on vehicles. The active differential braking,

further to improve stability, can be designed also to correct even small variations in

the yaw rate which might not cause loss of stability but compromise lane keeping

[47]. Differential braking has been proved to be very effective in stability recovery

at the price of perturbing the longitudinal vehicle dynamics, and possibly causing

undesired longitudinal decelerations [55, 43]. This last concept is explained by the

dissipation of energy caused by the braking, in fact to create the requested yaw

moment at least one wheel needs to be brake causing a dissipation of the energy

supplied by the motor or by the engine. Usually the main focus of BTV is the

global enhancement of the vehicle performance, so it acts also on the throttle valve

to avoid the speed reduction associated with brake actuation [43]. This is the main

weakness of this system because in order to reach the top performance the vehicle

need to dissipate as little power as possible, in fact, at engine limit condition, it

couldn’t be able to compensate the loss of power generate by BTV.

21


1.5.2 Active differential

Differential mechanism is present on road vehicle almost since the beginning in order

to turn without an evident slip due to the different curve radius of the inner and

outer wheel. This mechanism works in a good way in normal condition but not in

particular ones, as friction split between the right and left wheel or limit condition

of adherence in a curve at high speed. In order to resolve these issues, during the

years more evolved types of differential were introduced, under the name of self-

locking differential, which could transfer the torque to the slower wheel thanks to

power dissipation through clutches. These mechanisms work in a good way but

are not suitable to fully control the yaw motion, in fact in order to obtain the

desired yaw moment the system must be capable to split the torque between the two

wheels no matter which is faster. Here comes in the active differential, introduced

to reach this aim. The latter is able, further to dissipate power, to introduce it

on the axle giving the opportunity to split the power of the motor in the desired

wheel, also in the faster one. Active differentials are found to be able to offer very

similar performance to both their ideal counterpart and to the brake-based system.

They can also deliver this performance with a fraction of the energy loss that is

observed in the brakes, thus making active differentials a viable proposition for

applying continuous yaw control below the limits of adhesion [20]. An example of

car manufacturer which has implemented this system is Audi, which describes its

sport differential as the state-of-the-art of the rear differentials. Its system relies

on a superposition gear which comprises two sun gears and an internal gear was

mounted on the left and the right of a conventional rear differential. Audi states

that it turns 10% faster than the drive shaft. A multi-plate clutch in an oil bath and

operated by an electrohydraulic actuator provides the power connection between

the shaft and the superposition gear. When the clutch closes, it steplessly forces

the higher speed of the superposition stage on the gear. Being forced to turn faster

results in the additional torque required being drawn off from the opposing wheel

on the inside of the curve via the differential. In this way nearly all of the torque

can be directed to one wheel. The maximum difference between the wheels is 1,800

Nm. The sport differential is just as effective while coasting as it is under load. It

is electronically controlled and reacts within a few hundredths of a second. Audi

developed the software itself. The controller quickly and constantly recalculates

the ideal distribution of the forces for each driving situation as a function of the

steering angle, yaw angle, lateral acceleration, speed and other information. Vehicles

with conventional axle drives tend to understeer in fast corners. With the sport

differential, it is like riding on rails. When turning into or accelerating in a curve,

the majority of the torque is directed to the outside wheel, pushing the car into the

curve. The system thus nips any tendency toward oversteer or understeer in the bud

22


[57]. It is now clear that active differential could do a big difference with respect a

normal differential system. It is the smartest and useful instrument to implement

torque vectoring on a classic vehicle with internal combustion engine or with only

one electric motor for axle (as will be described in the next chapter the situation

changes if there is one motor for wheel). The counter side of this system is that it’s

complex, it requires a lot of new mechanical components, and this means two main

problems: cost and weight. The latter is fundamental to consider because a heavier

vehicle means more consumption, more emission and less performance (mainly for

the lower power/weight ratio). The question is if the car needs this system because,

for example, it’s quite useless in a car with low performance and designed for comfort.

On the other hand, there is the cost of the system which preclude this technology

only to premium cars. Usually premium cars are the most performing ones on the

market, thus it’s normal to see active differential only on them. Examples are the M

series (performance line cars) of Bmw, Ferrari F430, Mitsubishi Lancer Evolution,

Lexus RC F and GS F.

1.5.3 Indipendent electric motors

Electrification of vehicles is now a widespread topic and it’s quite clear that in the

future they will be always more diffused, probably up to replace ICE ones. Electric

vehicles first were born with a traditional drivetrain layout, with the electric motor

placed like in classical vehicle upstream of the drive axle. Nowadays another layout

is under studies thanks to always more compact and lightweight motors: one motor

for each wheel of the drive shaft. This allows to not use the differential. Current

electric vehicle research is investigating different powertrain configurations, consti-

tuted by one, two, three or four electric motors with different performance in terms

of vehicle dynamic behaviour and energy saving targets [12]. The motor can be

directly into the wheel or on the half shaft. The first layout is more compact and

leave more space for others function, as more space on board for passengers, but

introduces a lot of weight and inertia in each wheel, worsening the comfort and the

handling due to the higher non-sprunged mass. Putting the motors on the half-shaft

gives the opposite effect. From a controller point of view, they are almost the same

things. This layout introduces a significant step forward in the yaw rate control,

in fact with one motor for wheel it is possible to give different torque creating the

torque vectoring effect. According to Bosch [49] road holding is improved by the

way these motors are connected, which lets them brake and accelerate the wheels

individually. This function could eventually enhance sports cars’ cornering, even in

borderline skid situations, as well as making it easier to handle SUVs on challeng-

ing terrain. The performance of in-wheel motor electric vehicle is likely to perform

better compared with that of classical vehicles once a good control system is in-

23


vented. Energy from kinetic of electric vehicle can be recovered by motors during

braking process, which is a remarkable feature of great use and of great value in

the future. The torque of the motor can be unleashed immediately and accurately

[29]. LiQuiang Jin et al. underlines also how electric vehicles driven by in-wheel

motors present to us a practical way of designing an EV with its cost reduced. In

addition, this system is considered to be the universal driving platform of vehicles

including EV, HEV, and FCEV. It is the evolution trend of the new generation

electric vehicle driving systems. Many novel models have been promoted into the

market by famous automotive manufacturers such as Mitsubishi, Honda, Ford, GM,

and Volvo [29]. Also, Malcolm Burgess of Lotus engineering, after a study on this

[10], affirms that ‘torque vectoring using this approach has the potential to greatly

improve response and stability, with the tuning of the control model enable vehicle

behaviour to meet driver expectations. Not only can future electric vehicles have

clear environmental advantages, but with the torque vectoring their drive systems

allow, they can potentially be both safer and fun to drive.

To summarize the two main advantages of this system with respect a classical so-

lution are: lower cost due to less component and a better dynamic thanks to the

motors precision in realising toque.

1.5.4 Active steering

Active steering is an alternative way to influence the yaw dynamic. The first studies

on the argument where only on the front axle (active front steering AFS) for obvious

reason, then also the rear wheel steer was introduced (RWS) which needs additional

components on the rear axle to allow the steer.

Active steering system has been studied for a very long time, already 50 years ago

Kasselmann and Keranen [39] designed an active control system using a gyroscope

and a proportional feedback control system to generate an additive steering input,

but it was too early for the time so we have never seen this system applied on a real

car.

One of the pioneers of front active steering was Professor Ackermann, who identifies

two main field of work of this system. One of them focuses on the attenuation of yaw

disturbances on the vehicle and the other aims at rollover avoidance. Ackermann

et al in [3] compare the system with the independent brake system to understand

which is better to create a yaw moment. They conclude that steering requires only

one fourth of the front wheel tire force compared to asymmetric braking of the front

wheels. A further advantage of steering for generating a corrective torque is that

it allows for a compensation of torques caused by asymmetric braking. The con-

tinuous operation of the active steering system yields additional advantages over

an emergency braking system, regarding comfort (e.g. under conditions of gust y

24


wind, trailer pulling, and road irregularities) and safety (no discontinuity of vehicle

dynamics in critical driving situations). Besides, keeping the active brake system on,

the combination of braking and steering allows a torque balance [3]. Ackermann has

studied also rollover protection system, explaining that the most intuitive way to

solve it is to use active suspension to apply active counter roll inclination of the roll

body towards the inner side of the curve. However, under a strict energy limitation

a combined steering and decelerating action is much more efficient. Steering has

an immediate effect on the roll dynamics, while deceleration involves more delay.

A steering/braking control system allows larger obstacle avoidance manoeuvres and

supports the driver in case of emergency, i.e. when the vehicle is close to rollover.

Many control laws are proposed and implemented on cars such as BMW 5 Series

models, or on steer by wire prototypes in which the conventional steering elements

are replaced by two electrical actuators which are positioned in the front corners of

the vehicle and turn the front wheels [52]. The main disadvantage of AFS is the

driver feeling on the steer. With a system like this the steering command by driver

is not directly correlated with the effective steering angle and this can cause pain

during the driving and a negative feeling which makes the car less comfortable. The

solution is to implement a control system which follows the driver command and, if a

correction is needed, acts softly not moving too far from the driver request. In such

a way the driver feeling will be to drive better than he actually is doing. Obviously,

if it is a system taught for safety and stability, in limit condition it should act in

anyway possible to recover the instability.

To avoid the discomfort feeling to the driver another solution has been implemented:

the active steer at the rear axle (RWS/4WS). Since the 1980’s active rear wheel

steering has caught the attention of the vehicle industry and research institutions.

Controlling the steering angle of the rear wheels can improve a vehicle’s handling

characteristics and ultimately increase vehicle safety. By steering both the front and

rear wheels at the same time the lateral acceleration can be built up more quickly

and the side slip angle of the vehicle body is reduced. Another possible advantage

is a reduction of the yaw oscillations during transient manoeuvres and the stability

of the vehicle is improved. Also, during low speed driving the turning radius of a

vehicle can be reduced by steering the front and rear wheels in the opposite direction

[24]. Several cars on the market use four-wheel steer steering (4WS) technologies.

In the first generation (Honda) of rear steering vehicles, the front wheels steering

angle is transmitted to the rear wheels mechanically by a shaft; in this case the

control law is given by a proportional law with respect the front steering angle. In

active rear steering systems, a proportional feedback control with respect the yaw

rate measurement has been proposed too, with the law depending on the vehicle

speed. The rear wheels are steered, at low speed, in the opposite direction to the

25


front wheels (out of phase) to improve manoeuvrability for instance during parking.

At high speed, the rear wheels are steered in the same direction as the front wheels

to improve stability. Also Mazda uses a feed-forward system that steers the rear

wheels like the Honda 4WS but the steering ratio depends on vehicle speed while

Nissan uses a feedback control depending on the front wheel aligning torque and the

vehicle speed to set the maximum rear steering angle [52]. Mainly due to increased

mechanical complexity, these solutions are not spread in commercial light vehicles,

however the advantages of the rear steering action have been emphasized in the

yaw control case with avoidance manoeuvre. Moving from a nominal condition (i.e.

high grip road surface) to a more critical one (i.e. icy road) the rear steering action

becomes more relevant. It could be further emphasized in harder driving situations

like braking and avoidance manoeuvre in a curve at high speed [13]. When using a

rear steering the control bandwidth of the actuation needs to be sufficient to control

the vehicle and it needs to be a safe-fault device. This means that, in case of failure

of the system, it must be deactivated bringing the vehicle again to a classic front

wheel steering car.

Marino et al. [52] assert that active front and rear steering control provides higher

controllability, enlarged bandwidth for the yaw rate dynamics, suppressed reso-

nances, more stable cornering manoeuvres and enlarged stability regions. The con-

trol law can be tuned in order to prevent the uncontrolled vehicle oscillations for

the speed range of interest. The bandwidth of the closed loop car is increased with

respect to the open loop system. A combination of AFS and RWS systems provides

a four-wheel steering system that permit to further improve stability and perfor-

mances.

1.5.5 Literature review on steering systems actuation

In this thesis a rear wheel steering system is designed in cap3, thus it is interesting

to have an overview of the components of a steering system and of the possible

actuation. Let’s start by describing the components, as represented in 1.4 (reported

the front steering to also show the steering wheel):

Steering wheel: it’s simply the crown that the driver uses as input for the

steering system. It is used to control the front steering, in fact the rotation of

the front wheels is directly connected with it by the kinematic of the steering

chain. It can be also connected with the rear steering, whether physically

connected or wired.

Steering column: it transmits the rotation of the steering wheel to the steering

gearbox. It is usually composed by two or three parts connected by U-joints

for two reasons: firstly, to adapt to the available space and to the wanted

26


steering wheel inclination, secondly, and most important, for safety reason. It

is in fact less probable that the steering column, in case of frontal impact, will

hit the driver.

Steering gearbox: it has two scopes. It converts the rotational movement of

the steering columns in the linear one of the tie rods. Then it deals with the

demultiplication of the torque required at the steering wheel. The steering

gearbox is usually configured as a rack and pinion gearbox.

Tie rods: they transfer the motion of the rack to the wheels’ hubs. In order

to design them it is important to consider that they must resist to the forces

generated by the wheels.

Figure 1.4: Front steering chain

Other components could be added to the classic steering chain in order to imple-

ment the power steering. The latter it’s nowadays present in all the road vehicles,

its basic scope is to reduce the torque needed by the driver to rotate the steering

wheel. I is worth noting that the same system could be used also to implement

active steering.

This system was firstly implemented by using a hydraulic actuator (Hydraulic power

steering – HPS), typically directly mounted on the rack housing. In this type of sys-

tem the fluid pressure normally comes by a rotary pump driven by the engine. The

27


actuator is controlled by the steering wheel: when the driver applies a torque on

the command wheel, a valve is opened that allows the flow of the pressurized fluid

to the cylinder. The higher the driver torque, the more the valve opening and then

the larger the force supplied by the power assist system [17]. The main problem of

this system is the possible leaks and the many parts needed. To avoid them during

the years companies gradually switch to the newer electric power steering (EPS),

which is more compact, reactive and manageable. This system adopts an electric

motor to introduce assist torque in the steering chain. It could be connected on the

rack housing (Fig. 1.5) through a worm gear and a pinion acting on the rack, or

directly on the column (Fig. 1.6), again by a worm gear. To implement this system

some sensors are needed to read the steering position and the torque applied by the

driver. With this information a control unit could give the right power input to the

electric motor.

The EPS system is very useful also to implement the active wheel steering because

Figure 1.5: EPS mounted on the steering rack [17]

the system is the same, the only thing to change is the ECU program which, instead

of giving an assist torque, communicates to the motor the information of the control

system (obviously in this case the steering wheel must be not physically connected,

it will be a steer by wire). Since the birth of active rear steering an electric motor

is used, thanks to the ease of control it, and the possibility to adapt the already

existent EPS systems.

28


Figure 1.6: EPS mounted on the steering column [17]

1.6 Active control logics

Linking with the quote at the end of the 1.4, several control techniques are available

in literature and almost all methods, classical and modern, have been applied for

controlling vehicle handling characteristics. In [53], [52], [12] and [43] some of the

simplest approach are presented. PID controls are based on the idea of generating

a yaw moment proportional to the error between the actual state and a reference.

Khalid El Rifai [53] uses an adaptive PID control based on the theory of adap-

tive interaction and an approximation of the Frechet tuning algorithm. The three

gains of the control terms (Ki, Kd and Kp) are updated as function of the yaw rate

errors, being the control focused on that. This structure allows for coupled adap-

tation of the PID gains and further design flexibility. Other authors [29],[55],[41]

and[63] use sliding mode techniques. In [63] Liang at al. propose a combination

of torque vectoring and rear-wheel steering in order to stabilize an off-road vehicle.

Composite controllers were designed to approach an ideal reference model taking in

consideration the longitudinal slips and gravity component effects, two parameters

which really affect the dynamics of a vehicle on soft and slope terrain as in off-road.

Etienne et al. [41] use a super-twisting algorithm, particularly adapt to deal with

the non-linearities of the system and robust against parameter variations. Particu-

larly challenging is the transient response where some difficulties could arise, mainly

due to the impossibility to estimate correctly and/or instantaneously the cornering

stiffness and the friction coefficient between tyre and road. A correct choice of the

gains and of the parameters used in the filter to smooth the signal is fundamental.

29

1.6. Active control logics

Simplicity and ability in controlling non-linearities of the system are the ground-

work of fuzzy-logic controllers [4], [31], [6] and [33]. In [6] the authors base their

strategy on the generation of a suitable yaw moment to make the vehicle follows the

target values of yaw rate and sideslip angle, resulting in a controller able to adapt

to different driving conditions such as driving manoeuvres, initial speeds and road

surfaces. Park at al. [33] use a turning stability index to examine the stability of the

vehicle while turning, minimizing the intervention and so the deceleration in order

to increase also the efficiency. Fuzzy logic has been proposed to solve the problems

of various logic which judged only the existing true and false, and it can output var-

ious values between 0 and 1. It is employed to handle the concept of partial truth,

where the truth value may range between completely true and completely false. The

main limitation of this kind of control is that it is a knowledge-based control, which

required enough data to be set up. Hierarchical controls are control with a struc-

ture organized in different layers. Generally, they are divided in three separated

levels, the so-called high-level controller, the mid-level controller and the low-level

controller. Typically used in DYC, the first one usually generates the reference yaw

moment for the mid-level controller, which distributes the torque demands among

the available actuators. The activation of each individual component is entrusted

to the low-level controller. Ono et al. [15] describes a four-wheels steering and four-

wheels traction/braking systems based on friction circle of each wheel. The first

layer calculates target force and the moment of the vehicle to achieve a desirable

vehicle motion corresponding to the driver pedal input and steering wheel angle. In

the second layer the desired force and yaw moment of the vehicle are distributed to

the target tyre forces of each wheel. The last layer actuated each wheel to achieve

the target tyre forces. Li et al. [45] use a Linear Quadratic Regulator with feedback

and feedforward to guarantee the stability of the vehicle and decrease the control

delay. Thanks to the increasingly growing computational power available on board

the vehicles, multi-objectives real time optimal control is under investigation. Model

Predictive Control (MPC) is an interesting option for controlling constrained multi-

actuated system. Every time it recomputes the new required trajectory under a

finite time horizon according to the new available information, predicting the fu-

ture system evolution to be able to select the best control action with respect to

a specified performance criterion. The main advantage of MPC is the capability

to coordinate several constrained actuators to achieve multiple goals defined by the

performance criterion. The challenging problem is achieving the optimal balance

between computational cost and complexity of the model. The definition of the

prevision length and the time discretization is fundamental in fast system dynamics

as in vehicles, to operate in real-time.

After all these examples it is clear that over the years many different proposals have

30


been adopted and tested, even if only few of them were actually applied in the real

word. The challenge, in this large possibility of choice, is to choose wisely the most

suitable control for the required application.

1.7 Sliding mode control

The sliding mode control (SMC) has been developed since 1950s and is recognized

as one of the most promising techniques for robust control. The principle of SMC is

to constrain the system trajectories to reach in finite time and remain on a sliding

surface (see (Fig.1.7)) [18]. During the last two decades since the publication of

Figure 1.7: sliding mode principle [18]

the survey paper in the IEEE transactions ons automatic control in 1977, signifi-

cant interest on variable structure systems (VSS) and sliding mode control (SMC)

has been generated in the control research community worldwide. One of the most

intriguing aspects of sliding mode is the discontinuous nature of the control ac-

tion whose primary function of each of the feedback channels is to switch between

two distinctively different system structures (or components) such that a new type

of system motion, called sliding mode, exists in a manifold. This peculiar system

characteristic is claimed to result in superb system performance which includes in-

sensitivity to parameter variations, and complete rejection of disturbances [36]. The

main problem of sliding mode is the chattering one, which is a phenomenon of finite

frequency, finite-amplitude oscillations appearing in many implementations. These

oscillations are caused by the high frequency switching of a sliding mode controller

exciting unmodeled dynamics in the closed loop. Unmodeled dynamics may be those

of sensors and actuators neglected in the principal modelling process since they are

generally significantly faster than the main system dynamics. However, since ideal

sliding mode systems are infinitely fast, all system dynamics should be considered

in the control design [26]. Guldner and Utkin in their work affirm that, however,

chattering problem can be solved with proper treatments. In particular, they pro-

pose four solutions: the boundary layer solution, the observer-based one, the regular

31

1.7. Sliding mode control

form one and the disturbance rejection. The first one is the most known and ap-

plied, typically in the sliding mode output formulation there is the sign function in

order to change the control action according to the control variable sign. This can

cause the chattering problem due to its intrinsic discontinuity. The proposed solu-

tion is to use the saturation function to smooth the sign change. It’s very simple,

it really consists in putting sat instead of sign in the control law and it is quite

useful to solve the problem, especially in simple systems. However, this solution has

a limit, it solves the discontinuity problem of the basic sliding mode, but it doesn’t

take into account the discontinuities which are inherent to the system. One possible

solution is the second one proposed, where an asymptotic observer in the control

loop can eliminate chattering despite discontinuous control laws. The key idea, as

proposed by [Bondarev et al. 1985], is to generate ideal sliding mode in an auxiliary

observer loop rather than in the main control loop. Ideal sliding mode is possible

in the observer loop since it is entirely generated in the control software and thus

does not contain any unmodeled dynamics. The main loop follows the observer loop

according to the observer dynamics. Despite applying a discontinuous control signal

with switching action to the plant, no chattering occurs and the system behaves as

if an equivalent continuous u(t)eq control was applied [26]. Both the previous solu-

tions assume that the unmodeled dynamics are completely unknown but in practical

applications sometimes it’s possible to have some information about them, even if

partially, and it worth trying to include them into the controller. Since the actuator

dynamics and the plant dynamics are block separated, i.e. the output(s) of the ac-

tuator(s) are the input(s) of the plant, a cascaded control structure can be designed

following the regular form approach or the block control principle. The last solution

proposed by Guldner et Al can be included in another recognized proposal to avoid

chattering. Integral and higher order sliding mode. In [46] Canale et Al propose

a second order sliding mode to control the yaw motion of a vehicle. Conventional

sliding mode already guarantees the robustness features suitable to deal with the

uncertainty sources and disturbances typical of automotive applications. Yet, con-

ventional sliding mode control laws produce discontinuous control inputs which can

generate high-frequency chattering, with the consequent excessive mechanical wear

and passengers’ discomfort. In contrast, SOSM controllers generate continuous con-

trol actions, since the discontinuity necessary to enforce a sliding mode is confined

to the derivative of the control signal, while the control signal itself is continuous.

Apart from the robustness features against possible disturbances and parameter

variations affecting the vehicle model, the sliding mode control methodology has

the advantage of producing low complexity control laws compared to other robust

control approaches [46].

32


1.8 Fuzzy logic control

L.A. Zadeh realized first the Fuzzy theory in 1965, as reported by Krishna et al. [50].

The idea was born in order to achieve easy and efficient controlling mechanism rather

than precision, the most important aspect according to classical control. Fuzzy logic

control (FLC) has been designed to be able to manage value of information which is

neither definitely true or false, a common area that has to be stored, analyzed and

used to classify the data and obtain a solution through an optimum management

of the them. The most intelligent fit of the data lets to operate even with external

disturbances, relying on the common sense of competent operator. Controller oper-

ative part is base on linguistic rules, series of “If -then” rules which allow the Fuzzy

logic to be applied on a variety of system, not necessarily fully determined. On the

contrary, if it is fine-tuned through optimization tools as genetic algorithm, even

poorly defined system can be managed by the Fuzzy logic. FLC are more robust

rather than classical control, because it can accept and manage a larger variety of

inputs and also for its already mentioned disturbance insensibility property, all of

this avoids having to idealized or linearized systems to be accepted by the fuzzy

logic. An input phase, an analysis phase and an output phase compose a fuzzy

controller, a structure which similarly represent its management of the information,

which are acquired, classified and “best guess” according to them. Membership

functions, or “fuzzy sets”, map the input or sensors signals at the first stage, “fuzzi-

fication” is the name of this translation process from data to fuzzy value. “The

membership function is the graphical representation of the degree of belonging of

an element to the fuzzy set.” The most important parameter of the membership

function is the number and the distribution of curves used to analyze input and

quantify the output, less important is the shape, which triangular in most of the

cases, but it could be algo trapezoidal, gaussian or custom in particular application.

Complexity and computational effort increase with the number of the curves and

according to arrangement, time delay is the natural consequence. The quantity of

membership function present in the control has to defined in order to achieve the

required level of reactivity and efficiency of the system. Fuzzy rules definition rep-

resents the most important aspect of the second stage of the Fuzzy logic. Inputs,

knows as “antecedents”, are combined using fuzzy operators, such as AND, OR and

NOT, to obtain the fuzzy sets (summarily AND simply uses the minimum weight of

all antecedents, while OR uses the maximum value, NOT give the complementary

function). The fuzzy inference represents the processing stage, which distinguished

in two methods , the Mamdani and Sugeno. As reported by Izquierdo at al. [25], the

ability to imitate the human decision capacity to control certain industrial systems

is the idea to the base of the Mamdani systems. The numerical output computation

33

1.8. Fuzzy logic control

follows these phases: the degree of consistency between observations (input, ex. “x”)

and antecedents (it is the association of a input variable to is membership function,

ex.“x is A” which correspond to “pressure is high”) of each rule is calculated. In

this passage it’s simply calculated the degree µAk(x) of the membership function

associated to each antecedent, typically with a value between zero and one. The

result of this step is a number µAk(x) for each rule “IF x is Ak THEN y is Bk”.

Then the fuzzy sets of the consequent are defined for each rule (it’s the same of the

antecedent but for the output, in this example “y is B”).

µoutputk|x(y) = min(µBk(y), µAk(x)) (1.1)

Subsequently all the fuzzy sets are aggregated to provide one single fuzzy set:

µMamdani|x(y) = maxk[µoutputk|x(y)] = maxk[min(µBk(y), µAk(x))] (1.2)

The output and final stage correspond to the defuzzification of the aggregated fuzzy

set. Centre of gravity (COG) defuzzification method is the standard for Mamdani

system to obtain one single number, the output, from the aggregated fuzzy set. It

returns the projection of the centre of gravity of the area under the membership

function. In Figure 1.8 the concept is reasumed. Alternatives are the First of

Maxima, which returns the minimum value of y for which the membership function

reaches its maximum value , and the Last of Maxima, which does the opposite. As

reported by Kalogirou [38]. The main characteristic is the linear or constant output

membership function. “IF x is A AND y is B, THEN z = px+qy+r” where p,q and

r are all constants, while A and B are fuzzy sets in the antecedent, is the typical first-

order Sugeno fuzzy rule (Fig. 1.9). To increase the fidelity to the real system, higher-

order fuzzy models. The linear relationship between each rule of the system’s input

variables makes Sugeno method more suited as interpolating supervisor of multiple

linear controllers that have to manage different dynamic operating conditions of a

nonlinear system. Moreover is very efficient as gain scheduler and appropriate for

interpolating to represent nonlinear systems. Summing up the advantages of the

two methods, for Mamdani are:

Human experience can be used to optimize the system;

Acknowledged;

More suited in case of presence of a human interface;

While Sugeno model’s are:

It is very effective in calculations and controlling;

Enhance the linear techniques;

34


Figure 1.8: Mamdani fuzzy logic

35


Figure 1.9: Sugeno fuzzy logic

36


It is used to optimize the parameters and works adaptively.

In automotive application Fuzzy Logic Control systems has been applied, generally,

as actuation control method to reach a certain refence state. Regardless of the sub-

system or the aspect of vehicle under analysis, the input for the FLC is defined as

the error between the actual state of the vehicle and a previously calculated refer-

ence value. Zhang et al. [27] and Li et al. [44] have proposed different application

of the fuzzy logic, both directed of controlling vehicle lateral stability through the

yaw rate error and the side slip angle error or the yaw error. Having these values

as input, an actuation response had been calculated using the fuzzy logic method,

which showed a satisfactory capacity to increase the stability of the vehicle, even

in different environmental conditions. Actual state of the system and errors could

be used as input, Park et al. [32] proposed to consider the actual vehicle velocity

with the yaw rate error in order to improve the stability through a torque vector-

ing distribution, but in a more efficient way. The FLC compute an index used by

the torque distribution algorithm to optimally weight up the actuation intervention.

Similarly, Guo et al. [30] used a sliding mode control to improve the lateral stabil-

ity of a vehicle, fuzzy controller is adopted to optimize the sliding mode controller

parameters, which are affected by their natural uncertainties and external distur-

bances. The sliding surface and its derivative are chosen as input and the output is

the switching coefficient. Fuzzy rules are set up to ensure the existence of the sliding

mode function and satisfying the convergence condition. To understand when Fuzzy

Logic is the most suitable control its advantages are summed up:

Conceptually simple, it is very easy to comprehend;

Its human way of reasoning makes it very appropriate to manage difficulties

in the control;

It can work well even external disturbances;

Its linguistic rule base lets it able to work multiple inputs and provide multiple

outputs. If a good comprehension of the system behaviour is present, fuzzy

rules are relatively easier to create if compared to other control method’s

parameters optimization;

Safe failures management can be integrated;

It is not based on complex mathematical analysis, hence it can be very easily

designed;

It is very economical because it works thanks to the programmer’s knowledge

of the system, not requiring high precision sensors;

37


Classical controllers’ adaptation in presence of non-linear characteristics is very

complicated, on the contrary fuzzy controllers works very well even with highly

non-linear systems;

Very simple user interface and easy end-user comprehension even if the final

user is not confident;

Dedicated integrated circuits and toolboxes are on the market;

Unknown or not completely defined information management is the discrim-

inating characteristic of fuzzy logic. Its tools helps in computing different

actions according to the system status.

38

Chapter 2

Vehicle model

In the past the process of designing a car was very different from the present one.

Thanks to the new technologies, nowadays the main design process is made by

computer simulation and models. Obviously real tests are necessary in order to

ensure that the car behaves as expected and to reach the best set-up of the system.

The reason under this approach is very simple and clever: to save time and money.

Real tests are in fact very expensive because real models need physical and human

resources to be made. However, the main aspect is the time saving, thanks to the

computer model any change to the system can be done in some seconds so in a

very short time thousands of configurations can be tested to optimize the system.

On the other hand, each modification to the real system could require new pieces,

manpower and a lot of time.

To follow this trend, the study of the new control logic on the vehicle is made in

a Matlab-Simulink environment. In this chapter is explained how all the vehicle’s

system is modelized. To simplify the comprehension the mathematical model is

divided in some subsystems, each used to describe one particular aspect of the car:

vehicle body, wheels, tyres, motors, rear steering system, brakes and driver.

2.1 Vehicle body model

To describe a complex system as a car a reference must be defined. In this case an

inertial global reference system is used to describe the vehicle trajectory, while to

write the vehicle equations of motion a local reference was used. Historically in the

automotive world the origin of the local reference is placed in the centre of gravity

(c.o.g.) of the vehicle oriented so that: x axis pointing forward, y pointing to the

left side direction and z axis upward. Classically a rotation around x axis is defined

as roll(φ), around y axis as pitch (θ) and around z as yaw (ψ) (Fig.2.1).

39

2.1. Vehicle body model

x

y

z

ψ

θ

φ

Figure 2.1: Vehicle 3D reference system.

The vertical motion of the vehicle is not correlated with lateral dynamics in flat

surface and with null displacement between wheels and body. Thus, it is possible

to neglect it in order to reduce computational complexity and time. The vehicle is

thus studied with a bi-dimensional model, but the load transfer effect is preserved

to guarantee a precise contact force estimation. Considering the suspensions charac-

teristics of the vehicle, pitch and roll motions are limited so the relative gyroscopic

effects on the vehicle dynamics are neglected as they are of lower order of magni-

tude. With these simplifications the resulting model has nine degrees of freedom

(d.o.f.), whose five are related to the vehicle body motion while the other four to

the wheels. The d.o.f. are: Vx, Vy, ψ, φ, θ, Ωfr,Ωfl,Ωrr and Ωrl. Vx is the speed

in x axis direction (longitudinal speed), Vy is the speed in y axis direction (lateral

speed), ψ is the yaw rate, φ is the roll, θ is the pitch, Ωi,j the rotational speed of the

four wheels with first subscript f for front ones, r for rear ones and second subscript

r for right and l for left.

Solving longitudinal, lateral and rotational equilibria it is possible to solve the ve-

hicle body lateral dynamics. Referring to Figure 2.3 solution of the system provides

the following system of three equations:

40

Chapter 2. Vehicle model

x

y

Lr Lf

c fc f

c rc r

G

Figure 2.2: Vehicle geometrical schema.

X

Y

xy

δfαfl

Fx,fl

Fy,fl

δrαrl

Fx,rl

Fy,rl

δfαfr

Fx,fr

Fy,fr

δrαrr

Fx,rr

Fy,rr

ψ

V β

Ax

Ay

Figure 2.3: Vehicle 2D model schema.

41

2.1. Vehicle body model

mAx =(Fx,fr + Fx,fl) cos δf − (Fy,fr + Fy,fl) sin δf+

+(Fx,rr + Fx,rl) cos δr − (Fy,rr + Fy,rl) sin δr − FresmAy =(Fx,fr + Fx,fl) sin δf + (Fy,fr + Fy,fl) cos δf+

+(Fx,rr + Fx,rl) sin δr + (Fy,rr + Fy,rl) cos δr

Jzcψ =Lf (Fx,fr + Fx,fl) sin δf + cf (Fx,fr − Fx,fl) cos δf+

+Lf (Fy,fr + Fy,fl) cos δf − cf (Fy,fr − Fy,fl) sin δf+

−Lr(Fx,rr + Fx,rl) sin δr + cr(Fx,rr − Fx,rl) cos δr+

−Lr(Fy,rr + Fy,rl) cos δr − cr(Fy,rr − Fy,rl) sin δr

(2.1)

in which m and Jzc are respectively the mass and the moment of inertia of the

vehicle evaluated in the c.o.g. with respect to the z axis. Fx,jk and Fy,jk are the

forces generated by the tyres in longitudinal and lateral directions expressed in

the tyre reference systems (see Fig.2.3) in which the first subscript expresses the

considered vehicle axis and the second one the vehicle side. δf and δr are the front

and rear steering angles, the former controlled by the driver and the latter by the

active system. Lf and Lr are the front and rear axes to c.o.g. lengths, cf and

cr are front and rear semi-tracks (Fig.2.2). Fres is the resistance force in which

the contribution of aerodynamics effect, road slope and in general dissipative forces

can be collected. Ax, Ay and ψ are the unknowns of the system (2.1) and are the

longitudinal acceleration, the lateral acceleration and the yaw rate rate evaluated in

correspondence to the c.o.g..

In eq. (2.2) and (2.3) the relationships between the model degrees of freedom and

the unknowns of the system (2.1) are reported.

~A =d~V

dt=d(Vxi + Vy j

)dt

= Axi +Ay j (2.2)

Ax =Vx − ψVyAy =Vy + ψVx

(2.3)

42


Substituting (2.3) in (2.1), system (2.4) can be obtained.

mVx =(Fx,fr + Fx,fl) cos δf − (Fy,fr + Fy,fl) sin δf+

+(Fx,rr + Fx,rl) cos δr − (Fy,rr + Fy,rl) sin δr+

−Fres +mψVy

mVy =(Fx,fr + Fx,fl) sin δf + (Fy,fr + Fy,fl) cos δf+

+(Fx,rr + Fx,rl) sin δr + (Fy,rr + Fy,rl) cos δr+

−mψVxJzcψ =Lf (Fx,fr + Fx,fl) sin δf + cf (Fx,fr − Fx,fl) cos δf+


−Lr(Fx,rr + Fx,rl) sin δf + cr(Fx,rr − Fx,rl) cos δr+


(2.4)

Once physical characteristics (m, Jzc, Lf , Lr, cf , cr) (Tab.2.1), tyre-ground forces

(Fx, jk, Fy, jk), resistant forces (Fres), steering angles (δf , δr) and the vehicle state

(V x, V y, ψ) have been determined, the system can be solved and the equations of

motion of the vehicle can be numerically integrated in time.

Finally the side slip angle (β), a crucial variable in the vehicle stability study, can

be defined as function of the longitudinal and lateral speeds (2.5).

β = arctanVyVx

(2.5)

The side slip angle is defined positive counter-clockwise.

Table 2.1: Vehicle physical data.

description symbol value units

mass m 346 [Kg]

yaw moment of inertia Jzc 116.57 [Kg m2]

front wheel base Lf 0.756 [m]

rear wheel base Lr 0.920 [m]

front semi-track cf 0.625 [m]

rear semi-track cr 0.600 [m]

c.o.g. height hG 0.246 [m]

43

2.2. Wheels model

Fx

TJΩ

FrRw

VΩ

Figure 2.4: Wheel rotational equilibrium.

2.2 Wheels model

In (Fig.2.4) the rotational wheel equilibrium is represented, the vertical one is not

considered due to the rigidity of the suspension system. Also, camber effect and gy-

roscopic ones are neglected, the first due to the low order of magnitude, the seconds

due to the small wheel system inertia and the limited maximum rotational speed.

From the forces represented in (Fig.2.4) it is possible to implement the rotational

equilibrium equation ((2.6)):

JiΩi = Ti − FxiRwi − FriRwi (2.6)

In which Ji is the inertia of the wheel, Ωi is the rotational speed of the wheel, Tiis the traction-braking torque provided from the vehicle to the wheel, Fxi is the

longitudinal tyre-ground force in the wheel reference system, Fri is the wheel rolling

resistance force and Rwi is the tyre equivalent rolling radius. To consider the load

transfer, the equivalent radius of the wheel changes in first approximation with a

linear elastic behaviour (2.7).

Rwi = Rw0 −FziKzi

(2.7)

44


Where Rwi is the tyre rolling radius, Rw0 is the nominal unloaded radius, Fzi is the

vertical load supported by the tyre and Kzi is the first order approximation of the

tyre radial stiffness.

The dynamic wheel radius variation is influenced by the effect of the longitudinal

and lateral load transfer, caused by the respective accelerations. However, this ef-

fect is of low order of magnitude and therefore neglected in this simplified model.

The last force to analyse in the wheel system is the one generated by the rolling

resistance. It is caused mainly by the hysteresis of the tyre compound, which has

a different deformation behaviour during the phases of compression and relaxation.

This asymmetry provides an uneven distribution of normal stresses (Fig.2.5) that

provides an equivalent torque against the wheel motion. In first approximation it

can be considered as proportional to the vertical load (Fzi) and to a rolling resis-

tance coefficient, as shown in eq. (2.8).

Fri = fvFzi (2.8)

Following the rolling resistance schema reported in Fig.2.5, it is possible to define

the rolling resistance as the ratio between the arm of the rolling coefficient (u) and

the tyre equivalent rolling radius.

Fv =u

Rwi(2.9)

Fv is considered constant with the speed presenting a plateau in the mean range of

speed and varying only to really high and low speeds.

It is now fundamental to present the tyre dynamic model since the contact patches

are the only way to transmit forces between vehicle and asphalt.

45

2.3. Tyre model

V

Ω

Fz

Fzu

Hysteresis

No hysteresis

Figure 2.5: Wheel rolling resistance.

2.3 Tyre model

The modelling of the forces generated by the tyres is fundamental to obtain a correct

system. These forces are the ones which determine the vehicle state. Unfortunately

they depend by a lot of factors, two among many the tyre’s non linearity and the

friction estimation, thus an exact model couldn’t exist. The most used and rec-

ognized model is the professor Pacejka one [48]. Longitudinal and lateral forces

are obtained from equation (2.10) where parameters Bi,Ci,Di and Ei are obtained

fitting the curves on the experimental data.

Fi = Di sin

(Ci arctan

(Bixi − Ei

(Bixi − arctan (Bixi)

)))(2.10)

Where Bi is the so called stiffness factor, Di is the peak factor, Ci andEi are the

shape factors. The forces are function of xi that is intended to be the longitudinal

slip ki for longitudinal forces and αi slip angle for the lateral forces. The two forces

are combined to consider the superposition of longitudinal and lateral slip effects

(Fig.2.6). More details about the model are reported in Appendix A. The MF-

tyre model (2.10) represents the specific forces generated by the tyre in a steady

state condition. In order to consider the tyre’s delay in the response, it can be

46


approximated, both for longitudinal and lateral one, to a fist order system with a

period dependent on the tyre longitudinal speed:Lx,iVx,i

Fx,i + Fx,i =Fx,i

Ly,iVy,i

Fy,i + Fy,i =Fy,i

(2.11)

where Lx,i and Ly,i are the longitudinal and lateral relaxation lengths, Vx,i is the

longitudinal speed, Fx,i and Fy,i are the steady state forces and Fx,i and Fy,i are the

dynamic forces. The dynamic forces generated by the tyre are strictly dependent

on the vertical supported load. In the next paragraphs the longitudinal slip and the

Figure 2.6: Tyre combined force ellipse.

side slip angle are explained.

2.3.1 Longitudinal slip

When a driving or a braking force is applied on the wheel a difference between

peripheral wheel speed and the wheel centre speed is generated. This phenomena

is called slip, it is an intrinsic result of the generation of a tangential force on the

contact patch. The longitudinal slip is defined as the ratio between the slip velocity

and a reference longitudinal velocity:

ki =Vx,i − ωiRwi

Vx,i(2.12)

47

2.3. Tyre model

where ki is the longitudinal slip, Vx,i is the longitudinal tyre speed in the tyre refer-

ence frame, ωiRwi the peripheral tyre speed given by the rotational speed multiplied

by the tyre equivalent rolling radius. Referring to Figure 2.3 the relation between

each tyre longitudinal slip and the model d.o.f. chosen as reference can be evaluated:

kfr =

(Vx + ψcf

)cos δf −

(Vy + ψLf

)sin δf − ωfrRwfr(

Vx + ψcf

)cos δf −

(Vy + ψLf

)sin δf

kfl =

(Vx − ψcf

)cos δf −

(Vy + ψLf

)sin δf − ωfrRwfl(

Vx − ψcf)

cos δf −(Vy + ψLf

)sin δf

krr =

(Vx + ψcr

)cos δr −

(Vy − ψLr

)sin δr − ωfrRwrr(

Vx + ψcr

)cos δr −

(Vy − ψLr

)sin δr

krl =

(Vx − ψcr

)cos δr −

(Vy − ψLr

)sin δr − ωfrRwrl(

Vx − ψcr)

cos δr −(Vy − ψLr

)sin δr

(2.13)

For small steering angles the formulas could be linearized, thus the cosine function

can be approximated to one and the sine one to the angle itself, which can be

considered zero, neglecting the contribution related to tyres lateral speed. This

results in:

kfr =Vx + ψcf − ωfrRwfr

Vx + ψcf

kfl =Vx − ψcf − ωfrRwfl

Vx − ψcf

krr =Vx + ψcr − ωfrRwrr

Vx + ψcr

krl =Vx − ψcr − ωfrRwrl

Vx − ψcr

(2.14)

The slip turns out to be negative in traction and positive during breaking.

2.3.2 Side-slip angle

The side slip angle α is the angle between the longitudinal velocity and the velocity

itself of the tyre. It is caused by the tyre lateral speed which is generated by the

misalignment between the longitudinal direction of the tyre and the vehicle’s velocity.

αi = arctanVyw,iVxw,i

(2.15)

48


where αi is the slip angle, Vxw,i and Vyw,i are the tyre longitudinal and lateral speed

respectively. Referring to the previously selected d.o.f. each tyre slip angle can be

derived in the following form:

αfr = arctan

(Vx + ψcf

)sin δf +

(Vy + ψLf

)cos δf(

Vx + ψcf

)cos δf −

(Vy + ψLf

)sin δf

αfl = arctan

(Vx − ψcf

)sin δf +

(Vy + ψLf

)cos δf(

Vx − ψcf)


)sin δf

αrr = arctan

(Vx + ψcr

)sin δr +

(Vy − ψLr

)cos δr(

Vx + ψcr

)cos δr −

(Vy − ψLr

)sin δr

αrl = arctan

(Vx − ψcr

)sin δr +

(Vy − ψLr

)cos δr(

Vx − ψcr)


)sin δr

(2.16)

The side slip angles are assumed to be positive if they are counterclockwise. Unlike

the longitudinal slip, the side slip angles are not furthermore simplified being one of

the most important non-linearities of the study of a vehicle lateral dynamics.

2.3.3 Vertical force

The vertical force on each tyre is composed by three terms. Firstly the static one,

caused by the static weight of the car on each tyre, which depends by the position of

the center of gravity with respect each wheel. The latter is a constant term. Then

there is the load transfer effect, that could be divided in the longitudinal and lateral

one. Due to the driving, braking and lateral acceleration the inertia forces weigh or

lighten the wheels. Referring to schemas Figure 2.2 and Figure 2.7 the steady state

vertical forces can be derived as:

Fz,fr =1

2mg

LrL− 1

2mAx

hGL

+mAyhG2cf

Kroll

Fz,fl =1

2mg

LrL− 1

2mAx

hGL−mAy

hG2cf

Kroll

Fz,rr =1

2mg

LfL

+1

2mAx

hGL

+mAyhG2cr

(1−Kroll)

Fz,rl =1

2mg

LfL

+1

2mAx

hGL−mAy

hG2cr

(1−Kroll)

(2.17)

where m is the vehicle mass, g the gravitational constant, Lf and Lr are the distances

between the c.o.g. and the front and rear axis respectively. hG is the height from

49

2.3. Tyre model

the ground to the c.o.g., cf and cr are front and rear semi-track and Kroll is the roll

stiffness ratio. Ax and Ay are the longitudinal and lateral accelerations that provide

dynamic load transfer. Looking to the complete kinematics chain the load transfer

is generated by the lateral and longitudinal accelerations through the pitch and roll

motions that were neglected in previous assumptions. To consider the dynamics of

the load transfer due to the suspension system dynamics Ax and Ay are modelled

as a second order system:

AxAx

=ω2

0,p

s2 + 2ξpω0,p + ω20,p

AyAy

=ω2

0,r

s2 + 2ξrω0,r + ω20,r

(2.18)

in which Ax and Ay are the steady state longitudinal and lateral accelerations, ω0,p

and ω0,r are the pitch and roll natural frequencies, ξp and ξr are the pitch and roll

damping coefficients (Tab.2.2).

y

z

G

mg

mAy

Fz,fr(rr) Fz,fl(rl)

hGx

z

G

mg

mAx

Fz,rr(rl) Fz,fr(fl)

Figure 2.7: Load transfer due to Ax and Ay.

Table 2.2: Load transfer data.


pitch natural frequency ω0,p 2π1.5 [rad/s]

pitch damping ratio εp 1 [∼]

roll natural frequency ω0,r 2π3 [rad/s]

roll damping ratio εr 1 [∼]

front-rear roll stiffness ratio kroll 0.5 [∼]

50


2.4 Electric motors

The vehicle is powered by two electric motors (EM) on the rear axis. The motors

provided by Ashwoods Electronics are used in a on-board configuration with trans-

mission semi-axis and a 1:4 transmission ratio.

Main motors data are reported in Table 2.3 and characteristics in Figure 2.8. The

motors dynamics are represented through a first order system reported in (2.19):

TmTm

=1

Υes+ 1(2.19)

in which Tm is the steady state torque generated by the motors and Υe is the

characteristic period of the motors.

Figure 2.8: Electric motor torque vs rotational speed.

2.5 Rear steering actuator

Rear steer system can be implemented in several ways. In order to guarantee the

safety of the vehicle we have set one requirement: self-locking capability in case of

failure. In most of the gear drives, when the driving torque is suddenly reduced as

a result of power off, torsional vibration, power outage or any mechanical failure at

the transmission input side, then gears will be rotating either in the same direction

driven by the system inertia, or in the opposite direction driven by the resistant

output load due to gravity, spring load, etc. The latter condition is known as back-

driving. During inertial motion or back-driving, the driven output shaft becomes the

51

2.6. Brakes model

Table 2.3: Electric motors data.


peak power Pw 15 [kW ]

peak output torque Tm 70 [Nm]

nominal input voltage V 72 [V ]

width Total motor mass m 15 [Kg]

natural period Υe 0.01 [s]

maximum rotational speed ωmAx 6000 [rpm]

transition gear ratio τG 4 : 1 [∼]

driving one and the driving input shaft becomes the driven one. There are many

gear drive applications where the output shaft driving is less desirable. In order

to prevent it, different types of brake or clutch devices are used. However, there

are also solutions in gear transmission that prevent inertial motion or back-driving

using self-locking gears without any additional devices. The most common one is a

worm gear with a low lead angle. In self-locking worm gears, torque applied from

the load side (worm gear) is blocked, i.e. cannot drive the worm [15]. The latter

is the reason why we have chosen to use a worm gear reductor in the rear steering

transmission; besides it is useful also to increase the gear ratio to let the usage of a

smaller motor. In our system we have thought to use an existent drive-box, adapted

for space reason, made by a classic rack and pinion system, and connect it to the

reductor output shaft. The rear wheel steering system has been approximated as a

first order system.δrδr

=1

Υδrs+ 1(2.20)

In which δr is the steady state rear steering and Υδr is the characteristic period of

the rear steering system (0.1s). The maximum rear steer angle is fixed following the

state of the art of actual vehicles and it ranges from −3[deg] to +3[deg].

2.6 Brakes model

Electric vehicles present the opportunity of regenerative braking by using the motors

itself, which could provide almost the same torque than in driving mode, obviously

with the opposite sign. The vehicle, for seak of simplicity, is modelized with classic

mechanical brakes on the front axle and only regenerative ones in the rear axle. For

safety reason, in the real vehicle classic brakes are presents also in the rear axle. Due

to the over-actuation the control logic of braking could be very complex to manage,

52


especially if it is taken under consideration the possibility of recharging batteries by

regenerative braking. But it’s not the aim of this thesis to investigate this topic, thus

the rear brake torque are calculated taking into account the front force requested

by the driver, maintaining the same achievable level of adherence saturation due to

longitudinal load transfer:

Fbr = FbfFz,rFz,f

= Fbfmg

LfL +mAx

hGL

mgLrL −mAxhGL

(2.21)

in which Fbr is the rear braking force, Fbf is the front braking force required by

the driver. It must be noted that during braking Ax is negative; this provides less

braking force on the rear axis with respect to the front one to balance the load

transfer. With this modulation it is possible to maximize the braking capability of

the vehicle. A sample of the ratio between front and rear braking forces is reported

in Figure 2.9.

The mechanical braking system is modelled by a second order system transfer

Figure 2.9: Front-rear braking force partition.

function (equation (2.22)). Besides, to consider the time-lag related to pressurization

of the oil system a delay of 5ms is introduced.

Tb,iTb,ireq

=kb,i

mb,is2 + db,is+ kb,i(2.22)

53

2.7. Resistant forces

Where Tb,i is the braking torque generated by each front braking disk, Tb,ireq is the

required braking torque, mb,i,db,i and kb,i are the characteristic parameters of the

braking system reported in Table 2.4.

Table 2.4: Braking system data.


brakes disk inner radius mb 0.086 [m]

brakes disk external radius mb 0.155 [m]

paddles adherence coefficient µp 0.41 [∼]

paddles contact area Ap 2.3e− 3 [m2]

brakes equivalent mass mb 0.15 [Kg]

brakes equivalent damping db 4.15 [Ns/m]

brakes equivalent stiffness kb 40 [N/m]

braking system delay Υb 50e− 3 [s]

2.7 Resistant forces

The resistant forces collect all the dissipative ones generically related to aerodynamic

resistance and track slope. However, in this simplified model, only the drag compo-

nent of the aerodynamic resistances is considered, which is related to the square of

the speed through the drag coefficient (2.23):

Fres =1

2CxSV

2x (2.23)

In which Cx is the drag coefficient, S is frontal area of the vehicle and Vx the

longitudinal speed.

Table 2.5: Resistance forces tables.


air density ρ 1 [Kg/m3]

vehicle rolling resistance fv 0.01 [∼]

vehicle drag coefficient Cd 0.3 [∼]

reference front surface S 1 [m2]

54


2.8 Driver model

In order to have a complete overview of the vehicle behaviour, closed loop simulations

are needed. In the open loop simulations the front steering wheel, brake and throttle

time-histories are imposed to the model, but in the race is the driver to give the

input to the car. Thus it’s very important to have a reliable model of the pilot, and

testing its relationship with the vehicle by open-loop manoeuvres. The model used

in this thesis has two main objectives: to follow the reference trajectory and to have

the possibility to easily switch from a amateur driver to an expert one. To match the

first aim the model is implemented considering the position error and evaluating the

yaw moment. The driver capacity is expressed mainly by his response time and the

preview lengths, in addition to all the parameters related to the control. The driver

model is a path follower based on a PD controller with position and yaw feedback

and a PI cruise controller. Referring to Figure 2.10 the total error is composed by

four parts: position and yaw moment each one evaluated at two preview length.

The steering wheel angle is evaluated through:

δsw =2∑i=1

(kp,d,iεd,i + kd,d,i ˙εd,i

)+

2∑i=1

(kp,ψ,iεψ,i + kd,ψ,i ˙ε ¨

,iψ

)(2.24)

where kp,d,i and kd,d,i are respectively the proportional and derivative coefficients

with respect to the position, εd,i and εd,i are the positional error and its time deriva-

tive, kp,ψ,i and kd,ψ,i are respectively the proportional and derivative coefficients

with respect to the yaw and εψ,i and εψ,i are the yaw error and its time derivative

using the i-th preview length.

The error related to the position (2.25) is defined as the distance between trajectory

at the curvilinear abscissa s + li and vehicle c.o.g., while the error about the yaw

(2.26) is needed to reduce the misalignment between the vehicle longitudinal axis

and the trajectory tangent.

εd,i =

√(XG,ref (s+ li)

)2 −X2G +

(YG,ref (s+ li)

)2 − Y 2G (2.25)

Where XG,ref , XG, YG,ref and YG represent respectively reference trajectory and

c.o.g. position in the inertial reference system, li is the preview length and s is the

abscissa along the trajectory.

εψ,i = ψref (li + s)− ψ (2.26)

The trajectory abscissa i is evaluated through the integration of the speed along the

time (2.27).

s =

t∫−t0

V dt (2.27)

55

2.8. Driver model

Both errors are evaluated at two different preview lengths that are dependent on

the speed and the longitudinal acceleration felt by the driver (2.28).

Li = V ti +1

2Ax t

2i (2.28)

Where ti are related to driver response time. Data used in this study are reported

in table 2.6.

For what concerns speed control a PI controlled based cruise control is implemented

to define the driving and braking torques required by the driver to respect as much

as possible the reference speed.

Table 2.6: Driver model coefficients.

description symbol value

distance proportional coefficient kp,d 0.05

distance derivative coefficient kd,d 0.001

yaw proportional coefficient kp,ψ 0.05

yaw derivative coefficient kd,ψ 0.001

distance weighting factor ηd 0.7

yaw weighting factor ηψ 0.3

56


X

Y

x

y

εd1

εd2εψ1

εψ2

Pref1

Pref2

Figure 2.10: Driver model used to perform close loop manoeuvres.

57

Chapter 3

Rear wheels steering design

The section 2.5 proposes a preview of how the rear steering system is designed,

it highlights the choice of the warm drive gearbox to have self-locking capabilities

in case of failure. Besides it introduces the dynamic model, a first order system

described in equation (2.20). The aim of this chapter is to explain how the time

constant has been estimated, through the gearmotor selection and the model of the

system. In appendix B the dimensioning and the datasheets of the mechanical pieces

present in this section are reported.

As previously written, the rack and pinion system was choosen for the rear wheel

steering, but several possibilities were taken under consideration before this choice.

The following section offers an overview of the possible designs, reporting the pros

and the cons of each system and justifying the rack and pinion choice.

3.1 Design concepts

3.1.1 Concept 1: two linear actuators replacing tie rods

Electrically driven linear actuators are used to replace all of the car’s tie rods. The

actuators will be individually controlled, allowing the control system to indepen-

dently modify steering angle on each of the rear wheels.

Advantages: This design has a high chance of optimizing the car’s dynamic

nature, such as allowing the car to change its rear toe angle while moving. The

implementation is simple because the current rear spindles can be reused, and

it offers a lot of versatility because the RWS geometry is entirely managed by

software.

Disadvantages: The disadvantage of this idea is that with two actuators in-

59

3.1. Design concepts

stead of one, it will be very heavy. Both actuators must be dimensioned for

the highest load because the car must be able to turn both ways. Adding two

actuators to a control system increases the complexity of the design process as

well as the amount of wiring necessary. Having two actuators can also increase

the likelihood of device failure and maintenance costs.

Concept 2: rack and pinion with rotary actuator

The tie rods are attached to a conventional rack and pinion system, which is actuated

by an electric motor connected to the pinion gear. This is similar to the FWS system,

which is a rack and pinion steering system, although between the electric motor and

the pinion, a gearbox would be necessary.

Advantages: Since it only uses one actuator, the device will be relatively light,

and its construction will be very close to that of the front steering rack, which

is a tested design. The regulator’s layout would be simple and straightforward,

making it simple to maintain. The rack and pinion mechanism can easily in-

tegrate the stops specified by the rules.

Disadvantages: The key drawback is that, in contrast to the two linear actua-

tor design, you do not have the same level of control over individual tire angles

for optimization.

Concept 3: Ackermann mechanism with linear actuator

A linear actuator will act directly on the steering link, rather than a rotary actuator

acting on a pinion gear as in Concept 2.

Advantages: The mechanism has many of the same benefits as concept 2, with

the extra advantage of being simpler due to the use of fewer components.

Disadvantages: The ability to fine-tune the steering angles is reduced, as it

is for concept 2, and the mechanism is not as standard as a rack and pinion

design. The speed is similar to Concept 1, which may be too slow.

60

Chapter 3. Rear wheels steering design

Concept 4: proportional hydraulic system

The front steering mechanism will be connected to a hydraulic cylinder that drives

the rear steering mechanism, which is driven by another cylinder at the back of the

car in this concept. This means the rear steering angle is equal to the front steering

angle, and the ratio between the two is determined by the system’s mechanical

advantage.

Advantages: The simplicity of the proportional hydraulic system is one of its

key advantages. The input lag is practically zero since the mechanism is di-

rectly actuated by the driver, so there is no need to design a control loop.

Since all of the actuation force comes directly from the driver, there is no need

for actuators, the device could be made very light.

Disadvantages: The system’s main drawback is that it lacks a lot of flexibility

for vehicle dynamics optimization due to the lack of a control loop; it does not

take into account the effects of the car’s speed; the ratio between the front and

rear steering angles is the same at 5 and 100 km/h; and there is no way to

switch from positive to negative 4WS or turn the system off. Another weakness

is that the system is difficult to adjust; if the steering angle ratio were to be

modified, the cylinders or mechanism will need to be replaced.

Concept 5: non proportional hydraulic system

This concept is a development of the proportional hydraulic system, which employs

a mechanism that makes the relationship between the front and rear steering angles

non proportional in order to enhance the car’s vehicle dynamics. Depending on the

desired relationship between driver feedback and rear steering angle, this type of

mechanism may be made more or less complex.

Advantages: The key advantage is that it may preserve the proportional hy-

draulic system’s simplicity of implementation while also improving vehicle dy-

namical behaviour.

Disadvantages: However, the mechanism lacks the tweaking and optimiza-

tion versatility of an electronically actuated RWS system, particularly when

it comes to integrating testing data, and the non-proportional mechanism’s

complexity will raise as vehicle dynamical behaviour demands increase. Hav-

ing a mechanical controller rather than an electrical controller, such as the

governor used on steam engines, is exactly the same thing. This also adds to

the system’s total weight.

61

3.2. Rear wheels steering model

Evaluation of design concepts

In order to evaluate which design concept will be developed further, the five options

just presented will be compared according to the following requirements: respon-

siveness, packaging, controllability, weight, safety and cost. Since concept 1 is the

only one analysed that allows an accurate control of both wheels, all other concepts

have earned a lower level. The hydraulic systems have superior responsiveness char-

acteristics, as previously stated, and all systems are thought to have lower mass than

concept 1 because they only use one actuator. All the first three options presents

possibility of self-locking device which allows to not lose the control of the rear

wheels angle. Finally, being composed by two actuators the first concept turns out

to the be the most expensive, when the cost of the others is comparable. On the

basis of the above considerations the second design concept will be developed.

3.2 Rear wheels steering model

To estimate the time constant the mechanical system has been modelled as repre-

sented in Figure 3.1. The scheme represents the steering bar (attached to the rack)

Figure 3.1: Rear wheels steering model

and the tie rods, rigidly linked for the sake of simplicity, connected to the wheels in

a point ‘K’ far from the wheel centre. K is thus the steering arm. In this model

the steering force generated by the wheel is considered as the sum of two forces, one

caused by the self-alignment moment (eq (3.1)) and another (eq (3.2)) caused by

the polar inertia of the wheel around its vertical axis (JROT ).

FR/L =MZ

K= Fy

t+ tCASTERK

(3.1)

62


Where Fy and t, calculated by Pacejka, are respectively the lateral force generated

by the tyre and the trail arm of the wheel. tCASTER is the caster trail arm, which

is estimated considering the static one, needed to guarantee the re-allignin moment

of the wheel, and the natural one due to the suspension kinematic during the drive.

FROT =MROT

K(3.2)

where:

MROT = JROT δR (3.3)

In the point of application of FP there is the rack-pinion connection, FP is thus the

force which comes from the pinion, the trigonometric term is due to the transmission

of the force between the gear teeth. Obviously FP is strictly related with the actuator

force, being the same thing except for the transmission ratio. In this model the

inertia of the steering bar is considered too (mx), where m is the sum of the mass

of the two steering arms and the bar. This is a one degree of freedom system, it is

simple to get the equation of motion (remembering that δR = xk ):

(mk + 2JROTK

)δR = FP cos(αI)− (FR + FL) (3.4)

In order to calculate the time constant of the system the dynamic of the motor is

needed too, in the next section is shown how the motor has been chosen.

3.3 Gearmotor choice

The actuator of the rear wheel steering is a DC motor. In order to choose it the

reduction ratio must be chosen too. Once the power needed from the system is

known, the choice needs to be a right compromise between the torque of the motor,

better if low because it means smaller space and lower weight, and the reduction

ratio. The idea is to have the latter bigger with a smaller motor. The reduction ratio

(equation (3.5)) is the constant which relates the rotation of the wheel around their

vertical axis (steering angle) and the rotation of the motor, further to the torque

generated by the wheel and the one requested to the actuator.

τTOT =αMδR

=MWHEEL

TM(3.5)

where MWHEEL is the torque to the wheel and TM is the motor torque.

It is useful to divide τTOT in two terms to better understand it. The first one is

the reduction added by the worm gear, τWORM . This term gives the reduction ratio

between the torque at the pinion (the one in contact with the rack) and the motor.

The other term needed to implement τTOT is the one which relates the torque and

63

3.3. Gearmotor choice

the motion between the pinion and the wheels, determined by the geometry of the

system. Finally:

τTOT =kcos(αI)

RPτWORM (3.6)

Once the total reduction ratio is obtained, it is possible to express the torque re-

quested to the motor:

TM = MWHEEL1

τTOT(3.7)

where, from the system equation of motion:

MWHEEL = MZR +MZL + (JROT +mk)δR (3.8)

As the equation highlights, the motor must counteract two torques, one coming from

the self-alignment moment of the two wheels MZR +MZL and one from the system’s

inertia. In addition, it is possible to also express the relationship between the speed

of the motor and the angular speed of the rear steering:

αMOT = δR1

τTOT(3.9)

Once the torque and speed relationships are obtained it is possible to size the motor

in terms of power. To understand the size of the forces which come to the motor,

some simulations have been run on Matlab-Simulink environment. In particular,

the rear lateral forces generated by the tyres and the inertia forces of the rear wheel

steering system have been analysed during different step steer, steering pad and

double lane change tests at different speeds. In order to ensure the safety of the

vehicle, the forces considered in the actuator design were multiplied for a security

coefficient of 2. The magnitude of the forces which the motor must counteract

depends also by the reduction ratio. Considering that the power is defined, from

this point on the idea is to find a perfect balance between the torque available and

the rotational speed of the motor by the ideal reduction ratio.

In a real mechanical system there is always a ‘choke point’, the weakest element

which defines the maximum performance of the system. In our system it is the

worm gear which, due to the low efficiency and high friction inside the mechanism,

can not support very high input speed. This limits the maximum rotational speed

of the motor, which usually could work at higher rpm. The choice of the gearmotor

system has been done comparing different companies’ products, with the aim of

taking the motor and the reductor from the same company, in order to not have

coupling problems. After this study the choice was KAG company, a German one. In

particular, the M80 motor with SN40 worm gear, a combination provided directly

by the manufacturer. More specifically, the motor is the M80x40/I Nr. 222784

(12V) DC-brushed-motor, with a nominal input power of 150W. The SN40 reductor

64


provides different possibilities of reduction ratio, our choice was the 50 one. Due to

the high ratio, the efficiency of the gear is only 35%, which is actually good because

it guarantees the self-locking capability we need. The total reduction ratio is thus

τTOT=262 (considering a pinion with 24mm of primitive diameter and a standard

20° pressure angle). The choice was not oriented on a brushless motor because a

brushed one guarantees the same performance for a lower price and it is also easier

to control. The counterpart is the wear of the brushes but in our application, which

is a race one, the actuator has not to work every day for years, so this problem is not

so relevant. Simulations show that the most demanding tests for the actuator are the

Figure 3.2: KAG SN worm gear Figure 3.3: KAG DC motor M80x40 12V

transient ones, i.e. the step steer and the double lane change. It is also interesting

to notice that the speed is an important variable in the force calculation, which,

as previously written, is made by the sum of the lateral force and the inertia ones.

Increasing the speed, the actuator must be more reactive and faster, this leads to an

increment of the acceleration and thus of the inertia forces. On the other hand, the

lateral forces grow up to their saturation limit. There is a point, at a certain speed

and front wheel steering, in which the lateral forces are saturated but the vehicle,

thanks to the control, remain stable also increasing the speed. At this point inertia

forces still increase up to the stability limit. Thus the most critical situations will

be at high speed. In particular the double lane change test has proven to be one of

the most challenging tests to control at very high speed, requiring an high level of

reactiveness to maintain stability.

Let’s report the chart with the operating points during a double lane change at

120km/h (Figure 3.4) performed by the vehicle with only the RWS system active.

This is a very limit situation, considering that usually this test is performed at

65


Figure 3.4: RWS motor operating points during a double lane change at 120 km/h

80km/h, in fact the vehicle is not able to remain into the limit trajectory provided

by the ISO 3888 for this manoeuvre at this speed, however it remains still stable so

the actuator must work also in this situation. In the plot the motor characteristic

curve is overlapped to have a visual proof that the actuator can work. Besides, the

dashed vertical line shows the maximum continuous input speed of the worm gear

reductor, highlighting how it limits the motor potential. There are not operating

points outside the overload curve during all the test, though a lot of points are over

the max continuous torque of the motor. Considering that every red point is taken

every hundredth of a second, those which are over the black line are only some picks

in the overall simulation, thus the motor can guarantee the power (see section 3.3.1).

However, during this test the maximum rotational speed reached is not so high, thus,

to guarantee the proper functioning of the actuator, it can be interesting to plot

(Figure 3.5) the 15° step steer 15° at 90km/h, which it is a demanding test due to

the quite high speed and front steering angle, besides 15° of front steer at that speed

represents the maximum allowable in order to maintain stability during this test.

Moreover, in a step steer test both transient and equilibria condition are presents,

so it is possible to visualize how the motor works in both conditions. The density

of the red points is lower in this test because the simulation time is five seconds,

66


Figure 3.5: RWS motor operating points during a 15° step steer at 90 km/h

67


compared to ten before. There is a dense cloud of operating points just under 1Nm

of torque, they represent the steady-state part of the test, where the equilibrium is

reached and the requested torque is very high due to the demanding condition of the

test. It is evident that they belong to the steady-state part of the test because the

torque is requested at very low rpm, in fact the rear wheel steering angle is almost

fixed, thus the actuator work is mainly to hold it and not to vary it. On the other

hand, there are some points at higher rpm. They represent the transient condition,

where the primarily work of the actuator is to vary as fast as possible, according

to the control demand, the rear wheel steering angle to maintain the stability after

the sudden steering stroke. This test represents an extreme one for the vehicle, in

fact during the transient phase the car reaches a very high side slip angle value,

around eight degree, which is rapidly recovered by the control system. However,

a so high level of side slip angle could be considered too much for considering the

vehicle stable, making this test unusual and extreme condition, just to verify the

actuator. Besides, because of the severity of the test, the requested torque in most

of the test is above the black line. This is not a real problem thanks to the very

short time in which the actuator works (see section 3.3.1). It is also important to

remember that the working points are calculated with a security coefficient, thus the

actuator in a normal and real situation would not have any problem to also manage

this limit case. Therefore, it is interesting to notice that the maximum input speed

allowable by the reductor is never crossed.

3.3.1 Thermal model

As seen in previous section the motor happens to work a lot in overload, the thermal

model is implemented to be sure that this is not a problem. In the worst situation

in fact the motor could overheat damaging itself. The model used in this thesis is

the following:

Ploss(t) = Cdθ

dt+θ

R(3.10)

Equation (3.10) is the first order differential equation which describes the thermal

model. Let’s now explain how to estimate the thermal constant of the motor. Solving

(3.10):

θ(t) = RPloss(t)(1− e−tτ ) (3.11)

where:

Ploss(t) = P (t)1− ηη

(3.12)

where η is the rated efficency of the motor. In steady-state condition equation (3.11)

becomes:

θSS = PlossR (3.13)

68


From (3.13) it is possible to estimate the termal resistance R:

R =θmaxPcloss

(3.14)

where Pcloss is the nominal lost power, calculated as:

Pcloss = Pc1− ηη

(3.15)

Let’s now estimate the thermal constant τ . After toverload the motor reaches its

maximum temperature, 140°, thus it is possible to write (from equation (3.11)):

θmax = RPovloss(1− e− toverload

τ ) (3.16)

As for the nominal lost power, Povloss is the overload lost power, calculated in the

same way:

Povloss = Pov1− ηη

(3.17)

It is possible to write:

Pcloss = Povloss(1− e− toverload

τ ) (3.18)

Knowing all the equations components τ can be calculated reversing equation (3.18):

τ = − toverload

ln(1− PclossPovloss

)(3.19)

Once the thermal constant is calculated, the constant C can be implemented:

C =τ

R(3.20)

Finally equation (3.11) can be resolved by using ODE45 function by Matlab®,

obtaining the temperature curve in time. The resulting constants are reported in

table 3.1. The temperature of the motor increases due to the power losses in the

Table 3.1: Motor thermal data


max temperature θmax 140 [°C]

thermal resistance R 3.83 [°C/W ]

thermal constant τ 217.70 [s]

motor constant C 4.63 [J/°C]

overload time toverload 10 [s]

69


electrical circuit, let’s imagine having a constant workload for a long time, thus a

constant power loss. In that case, for how the model is built, the motor would

have a transient in which the temperature rises until a steady state temperature.

If the motor works into its nominal range, then this temperature will be under the

maximum allowable. Therefore the motor could work in overload for limited time

periods. In the application of this thesis the motor is almost never subjected to

constant stress because the rear steering continuously changes its position under

the control request. Moreover, the tests carried out to test the vehicle are all very

short, especially the transient one which are the most demanding. During a step

steer, for example, the motor is used for only some seconds, which are not enough to

produce a significant change of temperature. However, the FSAE championship also

involves a race which lasts some minutes. To test the motor the most demanding

test (in term of motor usage), the 120 km/h double lane change, has been repeated

45 times one after the other. This is obviously an impossible situation in the reality,

but it is useful to understand the temperature behaviour under the most stressful

situation. In figure 3.6 the result is reported. After a growth in the first seconds,

Figure 3.6: Temperature profile

70


the temperature stabilizes around 42.5°C, way far to the maximum temperature

allowable. The pick of power loss (figure 3.7) in the double lane change test is quite

high, if this power was constant the motor would overheat, but, as the load is very

variable (the power loss profile (figure 3.7) is a collection of peaks) the temperature

could not raise a lot. In a real race situation, the motor could surely guarantee

Figure 3.7: Power losses profile

the maximum performance during all the race. Regarding the part of single tests

provided by the rules, the longest proof is the steering pad, but, being a steady-state

test, the motor rotates very slowly to accommodate the slow increase of speed or of

steering angle (depending by the test), thus the power loss generated are very low,

creating neglectable raise of temperature.

3.4 Rear wheels steering system dynamic

Once the motor is chosen, all the instruments to implement and to study the dy-

namic of the rear wheels steering system are present. The aim of this chapter is to

describe how to implement the transfer function of the system and to analyse the

71

3.4. Rear wheels steering system dynamic

step response, which will be useful to calculate the system time constant of equation

(2.20). Let’s start by the system equation of motion, remembering τRWS = τWORM :

J∗αM = TM − (FR + FL)RP cos(αI)

τRWS(3.21)

where J∗ is the total inertia of the system, it is composed by the sum of the iner-

tia of the motor and the one of the mechanical system (properly reported by the

transmission ratio).

J∗ = Jmec + Jmot (3.22)

where:

Jmec = (mk + 2JROTk

)1

τTOT(3.23)

For simplicity let’s consider the right and the left force equal and let’s call them:

TL = FR/LRP cos(αI)

τRWS(3.24)

Let’s express the force acting on the steering bar:

FR/L =MZ

k=Fy(t+ tCASTER)

k(3.25)

with MZ which is the tyre self-alligning moment and Fy the lateral force generated

by the tyre. It is now possible to linearize the lateral force as the product between

the tyre stiffness and the rear side slip angle:

Fy = kααR (3.26)

Remembering that the rear side slip angle can be expressed as:

αR = −β +lRVψ + δR (3.27)

Then the lateral force can be written as:

Fy = −kαβ + kαlRVψ + kαδR (3.28)

It is thus possible to express equation (3.24) as:

TL = GkαδR +G(−kαβ + kαlRVψ) (3.29)

with the term G which resumes some terms:

G =RP cosa(αI)

τRWS

(t+ tCASTER)

k(3.30)

72


The second term of equation (3.29) can be considered a disturbance:

D = G(−kαβ + kαlRVψ) (3.31)

Finally it is possible to rewrite the equation (3.21):

J∗αM +2GkαkτRWS

RPαM = TM − 2D (3.32)

The term which multiplies the angular position of the motor is the equivalent cor-

nering stiffness:

kEQ =2GkαkτRWS

RP(3.33)

Let’s now coupling the mechanical system equation and the DC motor one in the

Laplace domain: (Las+Ra)Ia = Va −KφsαM

(J∗s2 + 2cLΩ0s+ kEQ)αM = KφIa − 2D(3.34)

All terms of equation (3.34) are described in table 3.2. From this system it is

Table 3.2: RWS motor coupled with mechanical system


terminal inductance La 2.1e− 4 [H]

terminal resistance Ra 0.065 [Ω]

nominal voltage Va 212 [V ]

nominal current Ia 12.5 [A]

torque costant Kφ 3.3e− 2 [NmA ]

motor angular position αM − [rad]

total inertia J∗ 0.0191 [kgm2]

speed/torque gradient cL 0.0091 [ Nmrad/s ]

steady-state angular speed Ω0 24.98 [ rads ]

equivalent cornering stiffness kEQ 8.89e2 [ Nrad ]

possible to get the transfer function (equation (3.35)), with the voltage as input

and the angular position, which is the controlled variable, as output. In order to

introduce the tyre dynamic the relaxation length effect has been considered in the

equivalent stiffness computation.

GV α =Kφ

La(Jmec+Jmot)s3+(2LacLΩ0+Ra(Jmot+Jmec))s2+(2RacLΩ0+K2φ+LakEQ)s+RakEQ

(3.35)

73

3.5. CAD model

Giving as input the nominal voltage step to the function we are able to compute

the step response (figure 3.8) and from this the time constant. The latter is one

third of the steady-state time, which is the time needed to the system to stabilize

itself around ±5% the steady-state value. The steady-state value is 6.85 ∗ 10−3rad,

Figure 3.8: RWS motor step response

at time t = 0.18s the response enters into the steady-state band, thus the computed

time constant of equation (2.20) is: Υδr = 0.06s.

3.5 CAD model

In this section the final design of the RWS will be presented, starting from the space

management to the verification of each components against the requirements. The

whole implemented system on the FSAE vehicle can be seen in the figures 3.9, 3.10,

3.11.

74


Figure 3.9: View of RWS system, view from below

Figure 3.10: View of RWS system, left

side

Figure 3.11: View of RWS system, right

side

3.5.1 Packaging

There is no good way to measure how good a mechanism is at packaging, but there

are a few positive and negative characteristics that can be defined. A system with

mechanical components mostly located on the car’s back would be easier to reach and

tweak, while a system with mechanical components distributed across the vehicle

would be less suitable for the opposite reasons. Interfering with other mechanical

parts in such a way as to limit the functionality of any devices is not acceptable.

Moreover, a system that stays within the vehicle rear limit, represented by the

jacking bar, is preferable, avoiding any kind of problem in case of contact with other

car. Nowadays, the rear steering tie rods are directly connected to the body frame

of the vehicle, as it is shown in figure(3.12), not being structural that part will be

75

3.5. CAD model

Figure 3.12: Actual system

eliminated in order to make room for the steering rack housing. Considerations

similar to those above can be made for the positioning of the rear steering actuator,

composed by an electrical motor and a worm gearbox, farther the presence of heavy

components from the vehicle centre of gravity reduces the performance and stability.

As it is possible to see in figure (3.13), the only space available is between the motors

dedicated to propulsion and the rear of the frame; however, the two drive chains,

that connect the motors to the rear axle, make the central part of that area off

limits, thus, the whole system will be placed in a corner of the frame rear.

Overall mechanism in shown in figure(3.14), each components and the assembly will

be presented in the next sections.

3.5.2 Components

The steering rack housing is bonded to the body frame through two connection

elements welded to the latter, it is fixed by means of locking screws. As previously

reported, the supports are placed laterally with respect to the vehicle centreline due

to interference reasons and in addition they make that the steering pinion axes goes

across the two transverse elements of the frame, as shown in figure (3.15). It can be

noted notches are made to eliminate excess weight.

In order to not modify the position of the rear tie rods, changing accordingly the

wheel toe angle, two connections elements are realized between the steering rack and

the tie rods, shown by figure (3.16).

If the two subsystems steering rack housing and gearmotor were connected to the

body frame by means two different supports, it would be necessary an elastic joint

76


Figure 3.13: View from below

because it would be impossible to ensure both radial and axial alignment between

the two shafts. Furthermore, this kind of coupling does not have infinite stiffness,

thus a delay and a phase displacement could occur, specially if the actuator is quick

and subject to sudden reversals of motion. In order to avoid the use of a connector,

we opted to plug the pinion shaft directly into the worm gearboxe. Accordingly, as

it is possible to see in figure (3.17), we designed a part that connects the steering

rack housing and the gearmotor, it ensures the alignment and withstand part of the

pinion radial load through a plain bushings. Due to the relative big ratio between

the transversal section and the length of the part and the small forces and moments

acting, in fact it has to sustain only itself weight and the gearmotor’s weight plus the

reaction moment, the stresses present are very small and so the thickness is recued

as much as possible in order to lighten the connector. Its transverse section has been

dimensioned in order to make it to pass between the two transversal elements of the

body frame.

The selected worm gearbox is able to withstand a radial load of 500N, but that value

refers to loads that act on the centreline of the output shaft (producer refers to this

value also for gearboxes without output shaft). As a result, the numbers should be

compared under the same conditions. The equivalent maximum radial force allowed

by the worm gearboxe is:

Rc =R2 ∗ ad

= 74.6N (3.36)

77

3.5. CAD model

Figure 3.14: Overall system

78


Figure 3.15: Frame connection elements on transverse elements

where R2 is the maximum radial forced allowed by catalogue by the worm gearbox,

s is the position of the radial force assumed by the manufacturer and d is the actual

position of the force. The pinion creates a maximum radial load of 256.8N which

is higher than Rc, thus the shaft is held in place by two bronze flange bushings of

the type PCMF 101207 E and PCMF 121407 E, depending on the dimensions of

the shaft. Bushings are cheap, simple to build and need no maintenance, but their

friction is higher than those of comparable roller bearings. Due to the small value

their frictional moment has not been considered in the motor dimensioning. The

shaft diameter was selected according to the pinion bore. In Appendix B are reported

the calculations for the sizing of all the steering system elements: pinion, bushings,

keyway and plain linear bearings. Since the travel speed and distance would be very

low, although the loads will be reasonably heavy, plain bearings were preferred over

linear ball bearings. Another benefit is that they do not need any maintenance. The

outside of the bearings is also fitted with a seal to keep contaminants and dust out

of the mechanism.

As it is possible to see from figure(3.18), starting from the front steering housing, we

designed the rear’s one according to the other parts and the room available. It fully

encases the rack and pinion, keeping dirt out and lubricant in. The linear bearings

are press-fitted in each tube end and one of the plain bushing is placed in its position

along the pinion shaft axis.

79

3.5. CAD model

Figure 3.16: Rack-tierod connection element

Figure 3.17: Steering housing-gearmotor connection element

80


Figure 3.18: Steering housing

3.5.3 Assembly

The assembly and installing procedure are as follow. Firstly, the rack is installed in

the main rod, then the latter is inserted in the housing and the two linear bearings

with their sealings are press-fitted inside the housing, as shown in figure(3.19). The

two connection elements are linked to the main rod with M6 screws.

The bushings are mounted in the housing and in the connection structure. The

pinion is constrained to the shaft through the keyway, blocked in position with the

shaft shoulder and the Seeger ring and then assembled with the housing, as shown

in figure (3.20).The latter is screwed to the frame attachments. The connection

structure is linked to the housing with M6 screws, as it is done for the gearmotor

connecting it to the whole system in the end.

81

3.5. CAD model

Figure 3.19: Main rod assembly

Figure 3.20: Housing and pinion shaft assembly

82


3.5.4 FEM analysis

The different parts in the mechanism were analyzed with Inventor FEM tool. Mesh

sizing and mesh control were simple, and the analysis’ precision and convergence

were not extensively investigated; instead, the FEM analysis was used to identify

technical problems or to check the reasonableness of various design options.

Housing: The bushing force of 105N was added, the torque of 29.5Nm, the

moment of 10.2Nm and the force of 54N from the gearmotor were added in.

The FEM analysis of the housing revealed a stress concentration in the edge

between the tube part of the housing and the parts that are to connect the

system to the body frame. Adding a radius on the inside edge the stresses are

alleviated, the maximum Von Mises stress is around 14.91Mpa when the yield

strength of the AA350.6-F is 131Mpa.

Figure 3.21: FEM analysis of the steering housing

83

3.5. CAD model

Connection structure: The torque of 29.5Nm, the moment of 1.38Nm and the

force of 49.05N from the gearmotor were added in. The part is linked to the

housing through the three tread holes, this constraint is represented in the

simulation by fixing the surface in contact with the housing, because fixing

the holes it would not have represented realistically the joint developing a

wrong stress concentration in correspondence of them. The FEM analysis of

the connection structure revealed a stress concentration in the radius inside

the part, but it is under the yield strength of the AA350.6-F and the maximum

displacement is negligible.

Figure 3.22: FEM analysis of the housing-gearmotor connection structure

84


Main rod: The rod was extended to its end point, simulating a full steering

angle lock out so that the stresses would be maximized. The housing was fixed

and both the transversal, the longitudinal component of the rod end forces and

their carryover moment were added in. The maximum stress occurring allowed

a minimum safety factor of 1.71.

Figure 3.23: FEM analysis main rod in worst load condition

85

3.5. CAD model

Tie rod-steering rack connection: The connection of the shorter side of the

steering rack is the most strained because on the other side part of the bend-

ing moment is absorbed by the main rod itself. Not having the possibility to

implement friction constraint between the main rod and the connection ele-

ment, longitudinal force has been neglected to not have uncorrect evaluation

of the stresses. As possible to seen from figure(3.24), the stress is concentrated

around the profile of the main rod, below the yield strength of the material

AISI304.

Figure 3.24: FEM analysis of the housing-gearmotor connection structure

86

Chapter 4

Control strategy

The increasing diffusion and introduction of more and more electronic controls in

a vehicle led to have a coordination and an integration of the controls in order to

eliminate, or at least reduce, the risk to get some bad interactions among different

control strategies. The availability of two or more controls could lead to higher per-

formance of the system, if a correct coordination is realized, and moreover from the

driver point of view a smoother behaviour could be reached thanks to the interaction

among different devices. In this chapter a new combined control strategy to improve

vehicle lateral dynamics will be presented. Unlike the past, where the commonness

was the use of parallel independent control in multi-objective multi-actuated system,

the controller aims at improving both vehicle’s turning performance and stability.

This is accomplished by tracking the yaw rate reference and the side slip angle ref-

erence, the first one made to change the vehicle under-steering behaviour, while the

second to guarantee the vehicle stability. In this thesis work a hierarchical control

strategy has been implemented. It presents a layer structure usually composed by

three steps. In the first the references are defined, in the second the coordination

process and the evaluation of the control action is carried out, while in the third one

the control action is generated through the different actuators available. This kind

of approach has less computational cost and so it results more prone to real-time

implementation, even if it is usually non optimal because some simplifications are

usually carried out, despite in the second level optimal control strategies are imple-

mented (e.g. LQR). An optimal control strategy has not been implemented because

increasing the model complexity usually means to rise the computational cost and

so the capability to well manage model non-linearities and constraints on the state

are not completely exploited. As usually in the first level the values of the refer-

ences are defined and in the second one the control action evaluation is performed.

Aiming at defining coordination between TV and RWS, a control action is referred

to the rear yawing moment Mzr and the rear steering angle δr. At the third level,

87

4.1. Control architecture

the implementation of an anti-slip control (ASC) and control action allocation are

performed. The different control levels are explained in this chapter, starting from

the definition of the references of the yaw-rate ψref and of the side slip angle βref .

Then the control actions evaluation and the coordination between the TV and the

RWS are explained. The chapter ends presenting the allocation procedure and ASC.

4.1 Control architecture

The proposed control strategy has a hierarchical structure. According to the road

adhesion coefficint and the vehicle speed and front steering angle, the references

are computed and compared to the actual states, providing the tracking errors. (as

reported in figure 4.1). The FLC uses these errors and the actual value of side slip

angle to discretize which state variable to give more weight in the control system.

Two SMCs are configured in parallel and quantify the desired control action for:

Rear yaw moment needed to track the reference yaw rate and side slip angle;

Rear steering angle needed to track the reference yaw rate and side slip angle.

Control actions obtained by the two SM controllers are combined on the basis of

the performance indexes. In particular, higher priority is given to the most effective

control action. The rear steering angle is calcualted for the RWS actuator in the

final stage. Instead, the torques on the rear axis produce the rear yaw moment. The

yaw moment available torques are added to the driving/braking torque required by

the driver in the torque allocation. Finally, to avoid excessive tyre slippage, an an-

tiskid block controls torques. In real application, since side slip angle is not directly

observable with low-cost sensors, state estimation is performed. For example, on

this vehicle, as implemented in [8], it’s possible to use an extended Kalman filter to

allow control implementation.

4.2 Control references

The reference yaw-rate and side slip angle are the control targets. The yaw-rate

is determined by the steering angle imposed by the driver as well as the vehicle

under-steering behavior.

As reported by Rajamani [51],following Gillespie’s (1992) analysis of steady-state

vehicle cornering characteristics, the under-steering gradient Kv is introduced:

δ =L

ρ+KvAy (4.1)

88

Chapter 4. Control strategy

Fig

ure

4.1

:C

on

trol

syst

emarc

hit

ectu

re.

89

4.2. Control references

where δ is the front steering angle, L the vehicle wheelbase and Ay the vehicle

lateral acceleration. WIth the definition of kinematic or Ackerman steering angle δ0

equation (4.1) can be rewritten in the form of equation 4.2.

δ = (1 +KusV2)δ0 (4.2)

where the longitudinal speed Vx is approximated as the total vehicle speed V and

the under-steering coefficient is represented by Kus.

Ackerman steering angle is function of the curve radius ρ and the vehicle wheelbase

L:

δ0 =L

ρ=Lψ

V(4.3)

where ψ is the vehicle yaw-rate. Substituting equation (4.3) in equation (4.2), the

single track vehicle’s yaw rate can be expressed as a function of speed, front steering

angle, and under-steering coefficient (4.4).

ψlin =V

L(1 +KusV 2

)δ = Ψδ (4.4)

So, once selected a desired level of Kus, it is possible to set a yaw-rate reference. The

above description has a flaw in that it is based on a linearized model that ignores

tyre force curves and saturation. As a result, the reference yaw rate has been limited

to the adherence coefficient’s maximum permissible value:

ψmax =µg

V(4.5)

The switch between the linear reference ψlin and ψmax is handled via an exponential

function that asymptotically tends to the highest admissible value (4.6) in order to

prevent discontinuity in the reference yaw-rate.

ψref =

ψlin |δ| 6 δth

ψth + (ψmax − ψth)

(1− e−

Ψ(|δ|−δth)

ψmax−ψth

)|δ| > δth

(4.6)

in which δth and ψth are the threshold steering angle and its associated yaw-rate

threshold. The ψth is defined as a percentage of the maximum one (4.7) and the

threshold steering angle is evaluated through equation (4.4).

ψth = Kthψmax Kth ∈ (0; 1) (4.7)

in which Kth is the threshold percentage. The reference yaw rate is schematically

represented in figure 4.2.

90


Figure 4.2: Yaw rate reference.

In particular, in this thesis work, the under-steering coefficient is set to zero in

order to achieve a neutral vehicle behavior.

The side slip angle is described in such a way that it works in harmony with the yaw-

rate reference to enhance vehicle performance. Normally, β must be limited since a

low value ensures stability; however, it is not possible to increase yaw rate without

increasing side slip angle since the two references are in contrast, thus diminishing

the vehicle’s performance. A proper side slip angle reference is defined in such a way

that it aids the reference yaw-rate when the stability is not in trouble, but limits β

when it becomes too high and with the opposite sign with respect to the ψ.

The linearized side slip angle is defined similarly to the yaw-rate reference definition,

starting from the steady-state vehicle cornering characteristics:

β =Fy,rr + Fy,rl

Ky+ψLrVx

(4.8)

where Fy,rr and Fy,rl are respectively the lateral forces acting on the rear right and

rear left tyres, Ky,r is the equivalent cornering stiffness acting on the rear axle, Lris the rear wheelbase and Vx is the logitudinal velocity. The balance of the forces

acting on the vehicle along the transversal axis in steady state cornering, equation

2.1 can be linearized as:Fy,r + Fy,l

Ky=LfmV ψ

LKy(4.9)

where the product between V and ψ is the lateral accelaration. Substituting the

equation (4.9) in the equation (4.8), we obtain the definition of the reference side

91

4.2. Control references

slip angle:

βlin =Lrδ

L(1 +KusV 2)(1−

LfmV2

LLrKyKβ) (4.10)

The coefficient Kβ, which virtually modifies the tyre stiffness, is added to obtain a

more oversteering behaviour from the side slip angle at low lateral acceleration to

enhance the vehicle performance. The definition in equarion (4.10) does not consider

that over a certain value of side slip angle the vehicle in no more under driver control.

Several simulations have been performed to quantify the maximum admissible angle

of the vehicle, which has been used to limit the reference maximum value. In order

to avoid discontinuity in the reference side slip angle the transition between the

linear reference βlin and βmax is managed through an exponential function that

asymptotically tends to the maximum admissible value, like what is done for the

yaw-rate refence:

βref =

βlin |δ| 6 δth

βth + (βmax − βth)

(1− e−

β(|δ|−δth)

βmax−βth

)|δ| > δth

(4.11)

in which δth and βth are the threshold steering angle and its associated side slip angle

threshold, as for the yaw-rate. The βth is defined as a percentage of the maximum

one and the threshold steering angle is evaluated through equation (4.10):

βth = Kthβmax Kth ∈ (0; 1) (4.12)

in which Kth is the threshold percentage. The reference beta is schematically rep-

resented in figure 4.3.

Figure 4.3: Beta reference for optimal control.

92


Figure 4.4: FLC scheme.

4.3 Fuzzy sliding mode control and coordination

The sliding mode controllers generate the control action as function of the state

errors and their derivatives. As discussed in the previous paragraph, the conflicting

nature of the two references requires a proper matching which is performed by means

of a Fuzzy Logic Control (FLC) strategy, figure 4.4. The aim is to obtain an input

for the Sliding Mode Control, i.e. sliding surface, which exploits the benefits of

each references mitigating the conflicting aspect and taking advantages when their

targets are congruent, not necessarily coincident. The sliding surface is composed

by a properly weighted combination of the yaw-rate error and side slip angle error

and the derivative of it:

s = γe+ e (4.13)

where γ is a constant and e is the error defined as:

e = ξ(ψref − ψ

)+ σ

ψmaxβmax

(βref − β

)(4.14)

the ratio between ψmax and βmax (these two variables are function of the vehicle

speed and steering angle, so for this reason the ratio has been scheduled for every

vehicle condition) is necessary to have comparable errors; ξ and σ are two weighting

variables coming from the FLC.

The FLC analyses the relative state errors and the actual vehicle side slip angle to

compute the correct weights ξ and σ for the sliding surface (equation (4.13)). The

first two inputs are the errors of the side slip angle and yaw rate (made comparable to

the first with the same method explained previously) with respect to the references,

they are necessary to understand if the two parts of the control are working to

achieve the same target, which could be performance, stability or both, or if they are

in conflict. The actual vehicle side slip angle is the third input and it is fundamental

93

4.3. Fuzzy sliding mode control and coordination

to verify if the vehicle is at risk of instability or not. The yaw-rate error and the side

slip angle error are fuzzified into three fuzzy sets (figure 4.6 and figure 4.5) because

what we need is mainly the sign of these errors (represented by the fuzzy sets P

and N); ZE is the third fuzzy set and it is necessary to have smooth transition from

one sign to another and to consider the case where the vehicle is on the references

(e ' 0). In contrast, the third input β is fuzzified into six fuzzy sets in order to

have an higher level of discretization in the control range of interest and gradual

transition between each level of control action (figure 4.7): PB (positive big), PS

(positive small), ZP (zero positive), ZN (zero negative), NS (negative small) and NB

(negative big). Side slip angle reference is limited to ±5.5, thus the input range of

latter fuzzy input is set between −6 and 6 in order to have full weight of the control

when the vehicle exceeds for more than an half of degree the maximum β. The latter

aspects drove also the definition of the output fuzzysets (weghting factors ξ and σ)

(Figure 4.8 and Figure 4.9): PB (positive big), PM (positive medium),PS (positive

small), ZE (zero), NS (negative small) NM (negative medium) and NB (negative

big); one more membership function with respect to side slip angle in order to have

a smooth response.

Figure 4.5: Side slip angle error membership function, as explained in the text only a small

range around zero is considered

94


Figure 4.6: Yaw-rate error membership function, it has an input range equivalent to that of

the side slip angle error

Figure 4.7: Side slip angle membership function

95


Figure 4.8: Performance weighting factor membership functions

Figure 4.9: Stability weighting factor membership functions

96


The idea at base of the definition of the fuzzy logic rules (table 4.1) is to use

the actual side slip angle value to discretize if the vehicle is at risk on instability

and, consequently, modify the sliding surface with a more stabilizing effect, which

usually it is expressed in increasing the magnitude of the β − error and decreasing

the ψ − error weight. Otherwise, if the vehicle is in a safe region of side slip angle

and, thus, the main target is to maximize the performance, the two weights are

usually adjusted in order that the major contribution is given by the yaw-rate error.

As explained in the previous section, the two references are usually in contrast,

but there are some conditions where they are congruent, e.g. in a brake in turn

manoeuvre where both contributions are used to achieve the same target because

both references decrease due to the speed reduction, pointing towards a more stable

condition.

The general control action u is given by Super Twisting Algorithm, a second

order algorithm, composed by two contribution:

u(t) = λ1

√s(t)tanh(khs(t)) +

∫λ2tanh(khs(t)) (4.15)

where λ1, λ2 and kh are the SM control gains as γ in the sliding surface equation

s (4.13). The quantification of the yaw moment Mzr and of the rear steer δr is

made through SM controllers, a specific u for each actuation system, in which gains

are determined to maintain sufficient bandwidth and avoid chattering problems for

every speed and steering angle. Analyzing the effect of each gain and once found the

maximum admissible value of each one that does not induce chattering problems, an

optimization procedure has been used to find the best combination. As cost function

the sum of the relative square errors has been used, the procedure is performed in

the MATLAB environment thanks to fmincon function. A specific combination of

λ1, λ2 and kh is calculated according to the actuation system, while γ is computed

keeping it equal in both TV and RWS in order to have a unique sliding surface s

(4.13) with related benefits, which will be shown in chapter 5. Since the rear steer-

ing system works in a smaller bandwidth the state error bandwidth associated to

the rear steering angle has been limited by filtering the error with a low pass filter

with a stopping band equal to the actuator bandwidth, which has been designed in

chapter 3. This permits to limit high level of overshoots in fast transient response

having a reduction of the steady-state error.

In this thesis two actuators are available: torque vectoring and rear steering. As

better explained in chapter 5, a comparison between the single actuation and the

double one has been made. When the two actuators are used together, two possibil-

ities come out: the parallel configuration and the combined one. Having two control

variables, the concept of the parallel architecture is to assign each variable to one ac-

tuator, in the present case the side slip angle is associated to RWS and the yaw-rate

97


eψ eβ β ξ σ

P P PB ZE PM

P P PS PS PS

P P ZP PS PS

P P ZN PM ZE

P P NS ZE PS

P P NB ZM PB

P ZE PB ZE PM

P ZE PS PS PS

P ZE ZP PM PS

P ZE ZN PS ZE

P ZE NS PS PS

P ZE NB ZE PB

P N PB NS PB

P N PS PS PM

P N ZP PS ZE

P N ZN PS ZE

P N NS PS PM

P N NB PS PB

N P PB PS PB

N P PS PS PM

N P ZP PS ZE

N P ZN PS ZE

N P NS PS PM

N P NB NS PB

N ZE PB ZE PB

N ZE PS PS PS

N ZE ZP PS ZE

N ZE ZN PM PS

N ZE NS PS PS

N ZE NB ZE PM

eψ eβ β ξ σ

N N PB NM PB

N N PS ZE PS

N N ZP PM ZE

N N ZN PS PS

N N NS PS PS

N N NB ZE PM

ZE P PB ZE PM

ZE P PS PS PS

ZE P ZP PM ZE

ZE P ZN PS ZE

ZE P NS ZE PM

ZE P NB NM PB

ZE ZE PB PS PM

ZE ZE PS PM PS

ZE ZE ZP PM ZE

ZE ZE ZN PM ZE

ZE ZE NS PM PS

ZE ZE NB PS PM

ZE N PB NM PB

ZE N PS ZE PM

ZE N ZP PS ZE

ZE N ZN PM ZE

ZE N NS PS PS

ZE N NB ZE PM

Table 4.1: Fuzzy logic rules

98


to TV. The problem of this idea is that the two systems do not communicate each

other, each one has its own sliding surface s and control action u, this could lead to

conflict between the actuations decreasing the overall performance. Thus, it is very

interesting to implement a combined actuation, where both actuators controls both

the variables. The main challenge is to find a way to deal with their cooperation.

A unique sliding surface for both RWS and TV is defined in order to make both

controllers to achieve the same target, while performance index maps (see section

4.3.1) are used to coordinate the control actions created by the combined controllers.

The normalized performance index is multiplied by the rear yaw moment and the

rear steering angle control action. For simplicity, defining χMzr as the performance

index associated to the rear torque vectoring with respect to the combined control

on the yaw-rate and side slip angle at a given level of control action on the two state

variables, speed and front steering angle. χδr is the performance index associated to

the rear steering evaluated in the same conditions, the synchronized control action

can be defined as:

Mzr,comb = MzrχMzr

δr,comb = δrχδr(4.16)

where the obtained combined control actions are on the left and the control actions

given by the sliding mode controllers are present on the right.

4.3.1 Efficency maps

The combined control system needs efficiency maps to guarantee the best usage of

the actuators in following the unique sliding surface. This section explains how these

maps are carried out. The first instrument needed is the β−ψ phase plot plane, which

provides a quick way to determine a safe operating region for the vehicles states. A

phase portrait is a geometric representation of a dynamic system’s trajectories in

phase space. Curves are used to represent the system’s various paths. In the study

of complex nonlinear systems, phase portraits are very useful tools. They reveal

details about the existence of attractors, periodic orbits, and points of equilibrium

by displaying the usual trajectories of the system in state space graphically. The

phase portrait is important to better understand a system’s stability and can provide

valuable information about an actuator’s ability to affect it.

In order to build the phase portraits the vehicle’s model has to be resolved for every

possible combination of speed and steering angle to obtain the state derivatives

β, ψ. In this thesis the four contacts model is used. On the basis of the same

simplification hypothesis assumed in section 2.1, the d.o.f. of the model are reduced

to three: longitudinal speed, lateral speed and yaw rate. On this basis the vehicle

body dynamics is represented by system of equations (2.1), here after reported for

99


convenience:

mAx =(Fx,fr + Fx,fl) cos δf − (Fy,fr + Fy,fl) sin δf+

+(Fx,rr + Fx,rl) cos δr − (Fy,rr + Fy,rl) sin δr − FresmAy =(Fx,fr + Fx,fl) sin δf + (Fy,fr + Fy,fl) cos δf+

+(Fx,rr + Fx,rl) sin δr + (Fy,rr + Fy,rl) cos δr

Jzcψ =Lf (Fx,fr + Fx,fl) sin δf + cf (Fx,fr − Fx,fl) cos δf+


−Lr(Fx,rr + Fx,rl) sin δr + cr(Fx,rr − Fx,rl) cos δr+


(4.17)

According to equation (4.17), fixing Vx, δ, TV and 4WS, the state derivatives β, ψ

are computed in order to implement the phase portrait. The only way to describe

the effect of a control input at each point of the phase plane domain is to analyse

the derivatives in a grid of points wide enough to maintain a fair computational cost

while still being thick enough to adequately reflect the vehicle dynamics.In order

to determine β and ψ, the relationship among them and accelerations (longitudinal

and lateral) is needed. ψ is actually an explicit variable in equation (4.17), while β

must be made explicit. The side slip angle is used to split the speed in longitudinal

and transversal directions, thus writing the equation of Vx and Vy:

Vx = V cosβ

Vy = V sinβ(4.18)

Deriving equation (4.18) in time, it results in equation (4.19).[VxVy

]=

[cosβ −V sinβ

sinβ V cosβ

][V

β

](4.19)

Recalling the relation between longitudinal and lateral velocities to longitudinal and

lateral accelerations and yaw-rate rate, all the relationships have been made explicit:

Vx = Ax − ψV cosβ

Vy = Ay + ψV sinβ(4.20)

Longitudinal speed imposed steering angle and friction coefficient are the three main

parameters which affect the vehicle behaviour. To reduce the computational cost in

this thesis the phase plot is studied varying only the first two. The friction coeffi-

cient acts on the maximum force providable by the tyres, reducing also the gradient

of forces generation. Keeping fixed the speed, the steering angle, the side slip angle

100


and the yaw rate, a reduction of friction causes a reduction of equation (4.17) right

side member, generating a reduction of β and ψ. As it would be possible to see

in the phase portraits, this means a lower speed to converge to a stability point,

lowering the stability limit. In this thesis, fixed the friction on a high value, the

phase portraits are used to analyse the capability of the yawing moment provided

by a rear axis TV and the rear steer angle to influence the vehicle behaviours at

different speeds and front steer angles. For each combination of side slip angle, yaw

rate, longitudinal speed and wheel steering angle, equations (4.17), (4.20) and (4.19)

need to be solved. Bearing in mind the right-left turning symmetry of the vehicle

behaviour, the solution can be reduced to only null and positive front steer angles.

Moreover, imposing four parameters, the computational complexity is reduced be-

cause it’s no more necessary to introduce the responses of the tyres and the load

transfer. Anyway, longitudinal forces, needed to maintain a certain speed, must be

made explicit.

The equation (4.17), to be solved, needs the tyre-road contact forces evaluation.

The pure lateral forces are calculated on the basis of the Pacejka MF-tyre model

(A) through equation (4.21).

Fy0i= Di sin

(Ci arctan

(Biαi − Ei

(Biαi − arctan (Biαi)

)))(4.21)

in which αi is the tyre slip angle defined as (4.22) and directly dependent on the

imposed vehicle speed (Vx), side slip angle (β) and yaw-rate (ψ).

αfr = arctan

(Vx + ψcf

)sin δf +

(Vy + ψLf

)cos δf(

Vx + ψcf

)cos δf −

(Vy + ψLf

)sin δf

αfl = arctan

(Vx − ψcf

)sin δf +

(Vy + ψLf

)cos δf(

Vx − ψcf)


)sin δf

αrr = arctan

(Vx + ψcr

)sin δr +

(Vy − ψLr

)cos δr(

Vx + ψcr

)cos δr −

(Vy − ψLr

)sin δr

αrl = arctan

(Vx − ψcr

)sin δr +

(Vy − ψLr

)cos δr(

Vx − ψcr)


)sin δr

(4.22)

However, tyre lateral forces are still dependent on the vertical forces reported in

equation (4.23) due to the steady state component of the load transfer which is

101


dependent on longitudinal and lateral accelerations.Fz, fr

Fz, fl

Fz, rr

Fz, rl

=1

2m

gLrLLrLLfLLfL

+Ax

−hG

L

−hGL

hGLhGL

+Ay

hGcfKroll

−hGcfKroll

hGcr

(1−Kroll)

−hGcr

(1−Kroll)

(4.23)

In this model, since the longitudinal speed is constrained, it is possible to calcu-

late the combined lateral forces starting from the pure ones and from the residual

adherence coefficient not exploited by the longitudinal forces (4.25).

µyi =

√µ2 −

(FxiFzi

)2

(4.24)

Fyi = µyiFy0i (4.25)

in which µyi (4.24) is the part of adherence coefficient µ at disposal of the lateral

forces.

As previously written, the longitudinal forces must provide the cruise effect on the

imposed speed, thus they have to counteract the resistance forces. Neglecting the

rolling resistance on the front axle, the longitudinal equilibria is:

2Fx,r cos δr − µ(Fy0,fr + Fy0,f l) sin δf −(Fy0,rr

√µ2 −

(Fx,r/Fz,rr

)2+

+Fy0,rl

√µ2 −

(Fx,r/Fz,rl

)2)sin δr − Fres = 0

(4.26)

in which left and right rear driving forces are considered equal and defined as Fx,r.

Equation 4.26 is clearly a non-linear algebraic equation numerically solvable. It is

solved by means of MATLAB function fsolve based on the Trust-Region Dogleg

Method. The rear forces are then limited by the maximum motor allowable torque

at the defined speed.

Up to now the phase portrait analysis is performed for the passive vehicle, in fact

no stabilizing moment coming from rear steering or TV has been applied. The im-

plementation of the yawing moment caused by unequal rear force distribution must

be addressed when dealing with torque vectoring. To back-verify the admissibility

of the yawing moment imposed, the maximum permissible yawing moment under

these conditions must be assessed first. In fact the maximum friction force of the

tyre is defined as Fx = µFz while the maximum tyre force given by the electrical

motor is Fx =Tm(Ωr)

Rw. In particular maximum admissible moment is chosen as

102


the minimum between the two limiting factors, which are the maximum improve-

ment of torque providable by the motors and the force limitation due to adherence

coefficient:Mz+2cr = min

Tm(Ωr)Rw

− Fx,r; µFzrr − Fx,r; µFzrl + Fx,r

Mz−2cr = min

Tm(Ωr)Rw

− Fx,r; µFzrr + Fx,r; µFzrl − Fx,r (4.27)

whereMz+ andMz− are the maximum and the minimum allowable yawing moments

respectively in favour and in opposition to positive yaw definition. Tm(Ωr) is the

maximum allowable motor torque at the tyre rotating speed Ωr considered equal on

the left and right side and defined as the longitudinal speed over the rolling radius

Rw. In parallel, the total longitudinal tyre forces are defined as the superimposition

of the cruising forces and the ones needed to generate the required yawing moment

(4.28). Fx,frFx,flFx,rrFx,rl

=

0

0

Fx,rFx,r

+

0

0Mzr2cr

−Mzr2cr

(4.28)

Where Mzr is the imposed yawing moment.The longitudinal forces are saturated at

this point, equal to the adherence coefficient multiplied by the tyre assisted load,

and the combined lateral forces are re-evaluated using the equation to stay in the

feasibility field. Having completely defined the longitudinal and lateral forces equa-

tions (4.17) a non-linear algebraic equations system numerically solvable comes out.

As previously done, it is performed in the MATLAB environment thanks to fsolve

function based on the Trust-Region Dogleg Method.

Once the previous procedure to solve the vehicle system is implemented, it is possi-

ble to evaluate it varying the two control parameters δr and Mzr. In this way, fixing

the vehicle speed V , driver steering angle δf , rear steering angle δr and yawing mo-

ment Mzr, it is possible to understand how the controller affects the system in each

reasonable situation. This procedure is needed to carry out the efficiency maps for

the combined control, in order to optimize the usage of actuators in each moment.

Let’s start to implement the torque vectoring maps. There will be one map for each

combination of V and δf , containing one performance index for each β-ψ combina-

tion at that state. In order to obtain one map, the vehicle system is resolved for a

fixed combination of state variables (V , δf ) and varying the yaw moment produced

by TV. At this point different side slip angle and yaw rate derivatives (β and ψ

respectively) are available, one for each Mzr tested. For each of this combination

103


the following quantity is calculated:

ETV = σ∆βTV−pass + ξβpassmax

ψpassmax∆ψTV−pass (4.29)

where ∆βTV−pass indicates the difference between the side slip angle derivative of

the controlled system and the passive one, ∆ψTV−pass the same but for the yaw rate.

The ratioβpassmaxψpassmax

, where βpassmax and ψpassmax are the maximum derivatives of the

passive vehicle controlled varibales, is needed to sum comparable quantities. As it

is possible to notice in equation (4.29), also the fuzzy factors (σ and ξ) implemented

in section 4.3 are considered in this quantity. Depending by the vehicle’s state, the

control will act in a different way according to the fuzzy logic. Being a non-linear

system, this could mean to have different effects of the control on the vehicle. This

explains why it is important to include every possible fuzzy combination in the

optimization process based on equation (4.29). The biggest value of ETV , for each

combination of σ and ξ is taken. This is the maximum variation of the interested

variables that the control with torque vectoring can do for a determinate speed

and front steering angle. ETV considers both the variation of β and ψ because the

control logic implemented in this work controls both in each moment. It is possible

to notice that ETV ’s shape is very similar to the input error in the sliding mode,

it is built in fact to be coherent with the system. To help the comprehension let’s

analyse figure 4.10. It represents in the ψ − β plane in green colour the vector of

Figure 4.10: Control variables derivatives schema

the side slip angle and yaw rate derivatives of the passive vehicle in a generic side

104


slip angle-yaw-rate condition. The black arrows represent vectors of the side slip

angle and yaw rate derivatives (respectively multiplied by σ and ξ) associated to the

different level of control action (δr and Mzr) with the solid ones which represent

the only in feasible range. By selecting a generic solid one it is so possible to define

the variation in side slip angle derivative (σ∆β) and in yaw-rate one (ξ∆ψ) between

the controlled vehicle, with a certain level of control action, and the passive one.

Then the quantity of equation (4.29) is carried out and maximized. Let’s repeat the

procedure point by point, this time with the RWS system:

For each combination of V, δf , β, ψ the vehicle system is resolved, trying several

δr, getting ψ and β.

For each combination of σ and ξ the following combination is optimized:

ERWS = σ∆βRWS−pass + ξβpassmax

ψpassmax∆ψRWS−pass (4.30)

The next step is to normalize the obtained quantities. The obtained maps are

divided for the maximum values found among all the ERWS and ETV .

ETVnorm = ETVEmax

ERWSnorm = ERWSEmax

(4.31)

where:

Emax = max(ETV , ERWS) (4.32)

At this point there are two efficiency maps, one for the TV system, one for the RWS

one. They represent, with a number from 0 to 1, the capacity of the control to

manage the two control variables for a determined combination of V, δF , β, ψ, σ, ξ.

Finally, as the aim is the comparison between the two kinds of actuation systems,

the maps are then managed to have the sum of the two efficiencies equal to one,

splitting the control between the two actuators by an intuitive and simple percentage

division.χMzr =

ETVnormETVnorm+ERWSnorm

χδr =ERWSnorm

ETVnorm+ERWSnorm

(4.33)

with

0 ≤ χMzr , χδr ≤ 1

The two final maps are thus two complementary maps.

It is now interesting to show some maps at different speeds and steering angles

(figure from 4.11 to 4.16).

105


Figure 4.11: Efficency maps at 25km/h and 1.5° front steering angle



106


To represent it the map is plotted associating the colour to the performance

index, blue for zero and one for yellow. In all of them the right chart represents

the rear wheels steering map and the left one the torque vectoring one. Obviously,

for the way in which they have been created, they are complementary maps. All of

them are superimposed to the passive vehicle phase portrait so that it is easier to

carry out some conclusions on the actuation systems features.

Let’s start analysing the efficency maps at 1.5 degrees of front steering. The passive

phase portrait is shown by black arrowed lines in the performance indexes figures.

Here are some considerations on the dynamic behaviour of the passive vehicle. At low

speed in the phase portrait it is notable an exponential decay, while increasing the

speed it turns in an oscillating one. This is an indication of a slower dynamics of the

passive vehicle with the speed increase. Here it is not evident, but the relationship

between β and ψ sign from the point of view of the equilibrium point locus changes

with speed. At low speed the behaviour of the vehicle is close to its kinematic

behaviour, thus β and ψ have the same direction, or both positive or both negative,

thus it is interesting to analyse the first and the third quadrant of the phase plane.

Conversely, for higher speed, the two variables present opposite signs, so it will make

more sense to study the second and fourth quadrant. This is due to the increase

of driving forces that increase the rear axle slip angle. Also, having limited the

maximum lateral acceleration to the product between gravitational constant and

tyres adherence coefficient, an increase of speed directly causes a limitation in the

maximum yaw rate. Regarding the efficency maps, at low speed in the first and third

quadrant, where the vehicle will work, the rear wheel actuator is very efficient, while

the torque vectoring shows better performance when the variables are in counter-

phase. At 25km/h the vehicle is already fast to reach the equilibrium point, making

the presence of the actuators quite useless. Increasing the speed the trend remains

almost the same, with the torque vectoring which is very efficient at high side slip

angle, thus in critical situations, while the RWS system is more effective around the

origin of the plot. This means that the RWS will help the vehicle to perform better

when starting a curve, while the TV, due to the high reactivity of the electric motors

will help at high side slip angle to maintain stability. The RWS is not so useful at

limit situation due to its limit steering angle, in fact when the saturation of the rear

steer is reached it cannot help anymore. The TV system instead can count on a

big quantity of torque which, however, is not infinite. The latter is reflected in the

maximum efficiency of the TV at medium speed, increasing it the torque available

is less because it is used to hold the speed and so its capacity to affect the vehicle

decreases.

It is interesting to discuss also the efficiency plot increasing the front steering angle

to a value of 3 degrees (fig. 4.14, 4.15 and 4.16). Looking at the passive phase

107


Figure 4.14: Efficency maps at 25km/h and 3° front steering angle



108


portraits it is evident how the passive vehicle dynamic becomes slower increasing

the steering angle. At 75 km/h the passive vehicle could not reach an equilibria

point, as is shown in figure 4.16. The efficiency of TV and RWS in affecting the

system seems to not change, with the RWS system which is more effective along the

bisector of the first and third quadrant, while the TV which is more useful in the

higher side-slip angle zone.

4.4 Torques allocation

The allocation procedure superimposes the longitudinal force required by the driver,

which is equally partitioned between the rear wheels, to the one needed to generate

the yaw moment Mzr (4.34).

Tm,rr =Treq

2 + Mzr2cr

Tm,rl =Treq

2 −Mzr2cr

(4.34)

where Treq is the overall motors torques required by the driver.

Since the yaw moment required by the controls can be infeasible due to the motor

torque limitation, the fulfilment of throttle position demand has been selected as

soft constraint. Mz is limited in the region between Mz+ (maximum clockwise yaw

moment) and Mz− (maximum counter clockwise yaw moment).Mz+2cr = max

min

TmRw− Treq

2Rw;Treq2Rw

+ TmRw

; Mzlim

Mz−2cr = max

min

TmRw− Treq

2Rw;Treq2Rw

+ TmRw

; Mzlim

(4.35)

where cr is the rear semi-track, Tm is the maximum module of motor torque made

available by the ASC, Rw is the rear tyres rolling radius and Mzlim the minimum

level of yaw moment accepted to satisfy the throttle position requirements. This

means that in case the total required torque at the wheels exceeds the maximum

deliverable torque by the motors, the yaw moment starts to be reduced. This hap-

pens until the yaw moment decreases until Mzlim. When this threshold is overcome

the control keeps the yaw moment constant, and reduces the throttle position. It is

important to notice that if a minimum level of yaw moment is not guaranteed when-

ever the driver requires the maximum available torque at disposal, the TV control

deactivation provides a less controllable vehicle.

This condition typically arises during curves track-out when the driver requires full

throttle.

The anti slip control (ASC) limits the available motor torques as function of tyre

109

4.4. Torques allocation

longitudinal slip defined as:

ki =ΩiRw − Vxi

Vxi(4.36)

where Ωi is the wheel rotating speed, Rw the tyre rolling radius and Vxi the wheel

longitudinal speed. To do so, it limits the available motor torques multiplying them

by the weighting function reported in figure 4.17 that depends on ki.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0

0.2

0.4

0.6

0.8

1

ki

ASC(k

i)

Figure 4.17: ASC weighting function.

110

Chapter 5

Simulations and results

This chapter provides the results of the simulations used to test the control perfor-

mance. Simulations could be divided in two categories: open-loop and close-loop.

To ‘close’ the loop a driver is needed, thus in the close-loop ones a trajectory is im-

posed and the vehicle, driven by the driver, should perform it. On the other hand,

in the open-loop simulations the steering angle is imposed. Moreover, another way

to divide the simulations is by the type of manoeuvre performed, that could be a

transient one or a steady-state one. In particular the reported manoeuvres are:

Steady-state manoeuvres, in which the vehicle response under extremely low

state transitions is analysed:

– Steering pad constant radius: close-loop manoeuvre, steer is actuated by

the driver model to follow a fixed radius curve progressively increasing

speed.

– Steering pad constant speed: open-loop manoeuvre, the steering angle is

increased from null to a maximum value and the speed is maintained.

Transient manoeuvres, in which the vehicle response is analysed under fast

state variation:

– Step steer manoeuvre: open-loop manoeuvre in which speed is maintained

and a fast variation of the front steering angle is applied.

– Braking in a turn: open-loop manoeuvre where braking forces are re-

quired by the driver during a turn;

– Double lane change: closed-loop manoeuvre with a reference trajectory

defined by ISO 3888 rule.

111

5.1. Compared control strategies

5.1 Compared control strategies

The aim of this work is to analyse the capacity of two systems, the torque vectoring

and the rear wheels steering, to affect the vehicle dynamic behaviour using a fuzzy

sliding mode control, implemented as explained in Chapter 4. In order to evaluate

the systems the previously cited simulations have been run. To compare the dif-

ferent systems and to understand their potential four control strategies have been

implemented and simulated:

Single actuation: in this strategy the single actuator, whether it is TV or

RWS, controls both the control variables (β and ψ). The control logic is the

one showed in Chapter 4, the fuzzy logic is used to combine the errors (of β

and ψ), creating the sliding surface used by the SMC to produce the control

action. Obviously the performance indexes are not used, being there only one

actuator. The control gains are setted by an optimization process, again the

same expalined in the control chapter. Studying the single actuation is useful

to understand which actuator is better to influence the vehicle and finally to

verify if the use of both the actuators simultaneously (as described later) brings

advantages.

Double actuation: in this strategy both actuators work at the same time. Two

logics have been implemented:

– Parallel logic: in this logic each actuator controls one variable: the torque

vectoring manages the yaw rate while the rear steering the side slip angle

because it is recognized in literature [8] that the TV works more efficiently

on ψ and the RWS on β. This is the easiest multi-input logic, used a lot

also by car manufactures in the past, mainly because the control systems

were provided by different companies and not integrated between them-

selves. Although it is a system that works quite well, the two actuators

don’t communicate and this could lead to contrasts, as the simulations

will show. In order to avoid this problem the combined strategy needs to

be implemented.

– Combined logic: to avoid the parallel problem, the strategy developed in

the previous chapter is adopted. TV and RWS systems work simultane-

ously on both ψ and β. Through efficiency maps (described in section

4.3.1) the usage of the single actuation system is weighted taking into

account the state of the vehicle in every moment. Thanks’ to this, the

control actuation, even if it comes from two different actuations, has al-

ways only one single objective, i.e. to maximize the performance or to

recover stability depending by the vehicle’s state.

112

Chapter 5. Simulations and results

Comparing the results of all these control strategies some interesting comments can

be carried out. In addition, all the strategies are compared with the passive vehicle’s

behaviour in order to have a reference. All the simulations were carried out in a

Matlab-Simulink environment.

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5.2. Steering pad constant radius

5.2 Steering pad constant radius

The steering pad is a steady-state manoeuvre, this means that the variable which

influences the state of the vehicle changes very slow, so that it is possible to verify

the steady state vehicle behaviour at different states with only one simulation. The

steering pad constant radius is a close-loop simulation, so the trajectory is imposed,

in this case a circumference with 50m of radius. The driver model has to follow this

trajectory, operating on the front steering angle. The speed is incremented from

20km/h very slowly so that different states can be observed. The simulations are

stopped when at least one of these conditions is met: the vehicle stability limit is

reached or the vehicle trajectory deviates more than 1m with respect to the refer-

ence one. The stability limit is fixed at |β| = 5.5deg because after that value the

side slip angle increases too fast to be controlled. Simulation results are reported in

Figure 5.1. The results are reported as function of the lateral acceleration. More-

over, in the two plots in the lower side of the figure the front steering angle on the

controlled variables is reported. The latter is useful to show how the control follows

the reference in every instant, in fact the reference line, which is the dot one, is re-

ported. In the top left-hand graph the steering angle on lateral acceleration diagram

is shown, in particular normalized with respect the kinematic front steering angle

δ0. From this it is possible to understand the vehicle behaviour in curve, i.e. if it

understeering, oversteering or neutral. The passive vehicle shows a nearly constant

understeering behaviour, until it reaches the instability. This makes more difficult

for the driver to understand the vehicle’s limit. The purpose of the control system of

this thesis is to obtain a linear oversteering behaviour to highlight the vehicle’s per-

formance until the side slip angle increases too much according to the control logic.

The more oversteering vehicle behaviour with respect the passive one is due both to

the understeering gradient set to zero and the side slip angle reference which forces

a more oversteering behaviour at low lateral accelerations. Over a certain limit, the

control aims to gradually improve the stability, in order to reach the highest lateral

acceleration considering the physical limitations of the vehicle. The passage from a

performance-oriented control to a stability-oriented one wants to be very gradual,

in order to not create uncomfortable and unnatural sensations for the pilot. This

is made thanks to the implemented reference, which change very smoothly, as it is

possible to notice in the two plots in the lower side of the figure. All the control sys-

tems reaches the scope, looking at the δsw − ay chart it is evident how up to a value

of lateral accelation of about 8m/s2 the controlled vehicle presents an oversteering

behaviour, then the requested steering increases giving a pronounced understeering

behaviour which leads gradually to instability. The instability is defined mainly by

the side slip angle β. Looking at its plot it is again evident how up to a value of ay of

114


Figure 5.1: Steering pad constant radius (50 m) manoeuvre on high friction road µ = 1.

Comparison between controlled and passive vehicles.

115

5.3. Steering pad constant speed

about 9m/s2 the controlled vehicle has a better performance with respect the passive

one, having a higher (absolute value) side slip angle. After that value, the controlled

vehicle could recover the side slip angle maintaining stability up to higher lateral

acceleration. It is important to highlight how β variation is always very smooth

and intuitive for the driver, accompanying the vehicle to instability. Comparing the

different control systems, it is notable that the torque vectoring presents a slightly

more marked oversteering behaviour for lower lateral acceleration, the other ones

instead behave practically in the same way. The RWS and the parallel system have

the peculiarity of being the slowest to increment the front steering angle, making

more natural for the driver the change of behaviour, reaching very high lateral ac-

celeration for quite low steering angle. The combined system however is the best

in following both the reference, especially at high lateral acceleration the others

tend to detach from the β-reference, while the combined one could recover better

the side slip angle reaching the highest lateral acceleration, obtaining the maximum

performance, as reported in Table 5.1. The weakest control is the torque vectoring,

Table 5.1: Steering pad constant radius on high friction. Maximum lateral acceleration.

- passive TV RWS parallel combined

aymax [m/s2] 10.55 10.71 11.05 11.06 11.22

with an increment of +1.52% with respect the passive vehicle. The RWS and the

parallel are very comparable, they both have a consistent increment of maximum

lateral acceleration, respectively with an increment with respect the passive vehicle

of: 4.74% and 4.83%. The parallel logic, probably due to the not perfect cohesion

between the two actuators, seems to not have any advantage with respect the RWS

logic, even if it could count also on the torque vectoring system. The combined one

is the most performant, thanks to the collaboration of the two actuators in limit

situations it could improve of 6.35% with respect the passive vehicle.

5.3 Steering pad constant speed

This is an open-loop steady state manoeuvre. The speed is kept constant at 90km/h,

and the front steering angle is increased slowly, always to have the possibility to study

different vehicle’s state in one simulation. In Figure 5.2 the results are reported, in

the same way as before, except for the δsw − ay chart which is not normalized with

respect the Ackermann angle.

116


88

Figure 5.2: Steering pad constant speed (90km/h) manoeuvre on high friction road µ = 1.

Comparison between controlled and passive vehicles.

The behaviour reflects the previous simulation, even if here the maximum lat-

eral acceleration reached is higher than before. Let’s focus on the δsw − ay chart,

the behaviour of the controlled vehicle is more oversteering than the passive one,

requiring less front steer at equal ay (up to 9m/s2), then the requested steer gradu-

ally increases highlighting an understeering behaviour which improves the stability,

bringing the vehicle to high lateral acceleration. It is interesting to notice that the

controlled vehicle’s behaviour is linear until it has to recover the stability and the

change in the control logic (from a performance-oriented to a stability-oriented) is

very smooth. This should lead to natural feelings for the driver. Again, looking at

the β chart it is evident how the control leads the vehicle to instability linearly for

117

5.4. Step steer manoeuvre

the benefit of the driver. Focusing on the controls with the rear wheel steering, it

is evident how they reach instability when the actuator is saturated (3°). To eval-

uate the efficency of the different control systems, as before, the maximum lateral

acceleration is analysed (Table 5.2). The weakest control is, again, the TV with

Table 5.2: Steering pad constant speed on high friction. Maximum lateral acceleration.

- passive TV RWS parallel combined

aymax [m/s2] 10.63 10.85 11.24 11.64 11.67

an improvement of ay of 2.07% with respect the passive vehicle. The fact that it

is the weakest is evident also looking at the lowest part of the figure, where it is

notable that, when high lateral acceleration is reached, the vehicle with TV is the

furthest from the references. Then, in term of stability performance, there is the

RWS system, which is 5.74% better than the passive. The parallel and combined

one are very similar in the behaviour, they both do a huge step respect the other

control systems and the passive, with the combined which reaches a slightly higher

lateral acceleration (respectively the improvement with respect the passive vehicle

is 9.50% and 9.78%).

5.4 Step steer manoeuvre

In this manoeuvre a sudden step steer is imposed. The vehicle speed is kept constant

by a cruise control. This is an open-loop manoeuvre with the aim to understand

the behaviour of the vehicle during transient. It is very useful to understand the

promptness of the control action. In this section different step steer simulations are

reported, always at the same speed but with an increasing front steering angle to

make the manoeuvre more challenging. The speed is 90km/h, to be coherent with

the previous steering pad, so that the results could be comparable. In this case,

results are reported in the time domain being it the domain for evaluating transient

response.

Step steer of 5 degrees at 90km/h

Simulation results are reported in Table 5.3 and Figure 5.3.

The 5 degrees step is the only one where the passive vehicle is stable, it means the

vehicle reaches an equilibrium point. In other step steer maneouvres β increases

without control, then the vehicle is declared unstable. Looking at the β and ψ is

118


evident how all the controlled simulations report an higher promptness with respect

the passive one, with the parallel which seems to be the faster. This has also side

effects, creating an overshoot on the yaw rate. Anyway, all the controls are able to

follow the reference, overlapping it once the steady state is reached (in some tenths of

a second). Besides, the controlled vehicle reaches an higher lateral acceleration than

then passive one, highlighting how the control enhances the vehicle’s performance.

Comparing the different control methods there are not evident differences because

in this ‘soft’ situation, i.e. the five degrees step, is easy for all the control systems to

follow the reference. A very important thing to notice is that during the transient

in the parallel control action the two actuators are working one against the other

because the references are, for some moments, not coherent. The two actuators

actions, for the reference system of this model, should have opposite signs to produce

the same effect on the vehicle. In this step steer, during almost all the transient,

both MZ and δR have negative sign. On the other hand in the combined system the

two actuators work always correctly one with the other.

Table 5.3: Step-steer manoeuvre on high friction road surface.

step steer at: 90km/h, δws,ss 5[deg]

passive TV RWS parallel combined

Ay,ss [m/s2] 7.76 8.27 8.20 8.30 8.24

Ay,max [m/s2] 8.02 8.27 8.20 8.30 8.24

tAy [s] 0.62 0.44 0.37 0.36 0.31

ψss [deg/s] 17.92 19.11 18.96 19.17 19.03

ψmax [deg/s] 19.30 24.40 23.08 24.36 23.86

tψ [s] 0.34 0.55 0.48 0.47 0.34

βss [deg] -2.93 -3.30 -3.29 -3.30 -3.31

βmax [deg] -3.11 -3.30 -3.29 -3.30 -3.31

tβ [s] 0.7 0.48 0.43 0.33 0.40

Note: ?ss refer to value assumed by the ? quantity at the steady state

condition. ?max refer to the maximum value (in module) assumed by ?.

t? indicate the times between the steer step starting time (t0 = 0.4s) and

the time at which ? assume a value equal to the 90% of the steady state

one. Value − is reported if the vehicle is not stable.

119


Figure 5.3: Simulation data of a steer step manoeuvre on high friction road surface: steer

step of 5deg at 90km/h.

120



Simulation results are reported in Table 5.4 and Figure 5.4.

Increasing the steering angle, the passive vehicle could not remain stable, reaching

too high level of side slip angle too fastly. The torque vectoring shows once again to

be the weakest control, it is the slower one and the one with the highest overshoot

both in the β and ψ plots. All the control which use the RWS systems, instead,

have similar performance, having more or less the same readiness. However, con-

sidering the overall performance, thus considering both the controlled variables, the

integrated control is the one which does the best work, slightly faster, with less over-

shoot and the nearest to the references. Anyway, all the controlled systems reach the

references, the differences is notable only with a big zoom and thus it is negligible.

Moreover, this is valid also looking at the lateral acceleration plot, the controlled

systems shows the same performance, with a little advantage of the parallel system

Again, it is interesting to notice how in the transient the actuators in the parallel

logic do not work helping each other but in conflict. δr, having negative sign, is

giving a destabilizing moment to the vehicle, while MZ , being negative too, a sta-

bilizing one. This lead to the big overshoot on the ψ graph, which is not present

on the other systems equipped with RWS. Anyway, in this step steer manoeuvre

the limit is not reached yet, the actuators are in fact not saturated and there is no

problem to reach the reference.

Table 5.4: Step-steer manoeuvre on high friction road surface.

step steer at: 90km/h, δws,ss 10[deg]


ay,ss [m/s2] - 10.47 10.70 10.74 10.61

ay,max [m/s2] - 10.92 11.07 10.89 10.89

tay [s] - 0.37 0.29 0.27 0.28

ψss [deg/s] - 24.31 24.90 24.90 24.64

ψmax [deg/s] - 38.14 32.68 38.78 33.69

tψ [s] - 0.65 0.62 0.35 0.53

βss [deg] - -5.08 -5.13 -5.08 -4.99

βmax [deg] - -5.96 -5.35 -5.28 -5.12

tβ [s] - 0.88 0.43 0.32 0.37

Note: see Table 5.3

121




122



Simulation results are reported in Figure 5.3.

The 15 degrees step is also reported, here the differences could be spotted easily. The

first thing that comes up is that the TV system could not maintain stability. This

could be guessed also before since it is the slower of the group. Increasing the steering

angle it tries to recover the stability but it could not, thus the informations after 1.1s

are not reliable because, since the vehicle is unstable, the numerical computation are

no more trustworthy. As before, the RWS-featured systems behave in very similar

way, however, being a more challenging situation, the difference between the single

actuated system and the multi-actuated is evident. In particular the RWS system

needs more time to reach the reference and it is the furthest from it. For this

control system is a very limit situation, in fact for approximately one second the

rear steering is saturated. The latter coincides with a slow overshoot both on β and

ψ. The multi-actuated systems instead have no problem in reaching the stability,

even if the reaction time is longer than before. The combined system, anyway, is the

fastest to reach the equilibrium, also with the lowest overshoot, because the parallel

one has a moment in which the actuators work one against the other. The combined

is the nearest to the reference too. Once the equilibrium is reached all the three

stable controlled vehicles have similar performance.

Increasing the front steering angle or the speed more critical situations could be

reached, but the trend is now clear. The RWS would be the first to reach instability,

followed by the parallel system and the combined one. From these simulations it is

interesting to notice that the single-input control system, i.e. the TV and the RWS,

work very well, especially the rear wheel steering one, which seems to be very effective

to control both β and ψ. Obviously, the multi-actuated control logics behave better

having the possibility to split the control action on more than one actuator. The

parallel, even if sometimes during transient is not coherent between the β-control

and ψ control, does a great job, getting very close to the combined results. The

latter, thanks to its control logic which let the actuators work in the most efficient

way, have a little advantage, being always the best to follow the reference. The

results of the step steer manoeuvre are coherent with the one of the steady-state

one.

123




124


5.5 Double lane change manoeuvre

Originally known as the ”moose test” the lane change manoeuvre was transferred

to the International Standard ISO 3888-2 after a revision by the Association of the

German Automotive Industry (VDA). The ISO and VDA lane-change test is used

to evaluate the handling performance. It is based on 3 cone lanes with a total length

of 61 meters, which must be completed with maximum speed. The ISO double lane

change test consists of an entry and an exit lane and with a length of 12m and a

side lane with a length of 11m. The width of the entry and side lane are dependent

on the width of the vehicle, the width of the exit lane is constantly 3m wide. The

lateral offset between entry and side lane is 1 m and the longitudinal offset is 13.5

m. For the same lateral offset the side and exit lane has a slightly shorter longi-

tudinal displacement of 12.5m. During the test, the maximum entrance speed that

guarantees to remain inside the track boundaries is searched. The simulations are

repeated increasing slowly the entrance speed to find the maximum one, the results

are reported in Table 5.5. In order to compare the different control systems, the

simulation is also run and reported at 80km/h (Figure 5.6). Beyond the maximum

entrance speed, an index of the goodness of the control system is, at equal speed, the

front steering angle requested by the driver. The lower the better because the work-

load for the driver is less, offering a safer and an easier sensation to the drive. From

Table 5.5: Double lane change manoeuvre on high friction road surface.


Ventrance [km/h] 113.10 114.40 116.30 113.90 116.50

δws,max [deg] 4.64 5.01 3.61 5.21 3.61

βmax [deg] 3.26 3.57 3.33 3.38 3.31

ψmax [deg/s] 19.63 24.49 19.16 25.91 20.49

Aymax [m/s2] 7.53 7.94 8.59 8.83 8.49

Table data 5.5 is notable that the controlled vehicles have always a higher entrance

speed with respect the passive one. Surprisingly, the worst control system is the

parallel one, with only a little improvement with respect the passive one. Probably,

the reason of a so limited increase of preformance is the same presented in step steer

simulations paragraph 5.4, where the incoherence between the two actuators control

action is evident, unfortunately, here, the continous state change does not allow to

appreciate it. The best work is done by the RWS system and the combined one,

which ensure high entrance speed with a very low workload for the driver. The best

in class is anyway the combined one. Analysing the same speed simulation in Figure

5.6, the most important chart is the δws on time. The combined control is the one

125

5.5. Double lane change manoeuvre

which requires less front steering angle to perform the test. The difference between

the four control logics is not so evident, all the four are consistent in this test.

Figure 5.6: Simulation data of a double lane change manoeuvre (80 km/h) on high friction

road surface.

126


5.6 Braking in turn manoeuvre

The scope of this test is to analyse the response of the controlled vehicle if the

braking system is activated during a curve. The step starts at constant speed with

a step steer, when the vehicle has reached the steady state behaviour the braking

is activated. It is a very interesting test because here the friction force available

to perform the curve suddenly decreases. In this simulation the 50% of the total

braking force is applied and the vehicle responses in time are reported (Figure 5.7).

Looking at the side slip angle plot it is evident how the passive and the RWS

vehicles can not maintain stability during braking. The rear steering angle is quickly

saturated, but it is not enough to stabilize the vehicle. This is mainly due to two

phenomena: firstly, the combined friction, once the brakes are pushed a lot of friction

force is spent in the longitudinal direction, secondly, due to load transfer when

braking, the rear axle lightens up diminishing the available contact forces. Thus,

the RWS system is no more able to stabilize the vehicle if the required lateral force

is higher than that available. On the other hand, the torque vectoring system is

based on the differentiation of the longitudinal forces between the right wheel and

the left one, so also during braking it is possible to split the braking force to obtain

the same results. Thanks to this, even if the available contact force is less, the TV

system could recover the stability, even if with a marked overshoot both on ψ and β.

The multi-actuated systems have the possibility to exploit the advantages of both

systems, using all the contact force available and thus they could recover stability a

lot faster. The combined control logic is the best one, thanks to the collaboration

between TV and RWS system, it is very near to the reference during breaking and it

doesn’t suffer any overshoot. The parallel system anyway, being the two references

coherent, have a very small overshoot and, compared to the TV system, is incredibly

fast and does its job very well.

127

5.6. Braking in turn manoeuvre

Figure 5.7: Simulation data of a braking in a turning manoeuvre on high friction road

surface: half braking capacity used.

128


5.7 Simulations resume

The first aim of running the simulations was to verify that the fuzzy sliding mode

control logic implemented is good. The results were very satisfying, the intention

was to create a control system which, controlling the two variables which mainly

drive stability and performance, i.e. β and ψ, would guarantee the maximum curve

performance where possible and recovers stability in critical situation. Another

challenge was to have a very smooth passage between the performance-oriented

phase and the stability-oriented one to give to the driver very intuitive sensations.

Besides, a comparation between the different control strategies has been carried

out. After having analysed several simulations it’s evident that the combined logic

control is the best one, being the more balanced in the performance and the most

capable to follow the control references in every test. This is possible thanks to the

performance indexes which split the usage of the control in the best way possible.

The parallel control logic, considering that it is a lot simpler to implement, reaches

very good results, making a huge step with respect the single-actuation systems.

The RWS and the TV do a great job in almost every situation even if they have

their limits. In particular the torque vectoring system shows to be the slowest and

the less capable to maintain stability, however during braking it is essential. The

RWS system, on the other hand, has demonstrated to be very fast and effective until

brakes were pushed.

In the simulations the friction coefficient was always imposed. In this thesis the

control is not verified under friction changes. In order to do it, it would have been

necessary to implement an estimator to let the system know and adapt to a new

friction coefficient. It is not what this work deals with, but it is worth to do some

quickly comments. As previously written, an estimator is needed also to estimate the

side slip angle to implement the control system in a real vehicle. The side slip angle

sensors are in fact too much expensive to be present on our vehicle, and in general on

every commercial one. In [8] an extended Kalman filter for this application has been

presented and gives very good results. The main challenge with an estimator is that

it must be very reactive to identify and communicate to the control the change of

friction, otherwise the response is too slow to hold stability. The control system has

to produce new references because with low friction the stability limit is reached for

lower value of side slip angle and yaw rate (remembering that the maximum value

of their references depends on the available friction). Supposing that the estimator

works in a good way, then the readiness of the control is very important in order to

adapt quickly to the changes. Our control system has shown in the simulations to

be very ready, especially if the rear wheel steering system is present. This lead us

to think that our control would behave in a good way also in that situation.

129

Chapter 6

Conclusion

This thesis proposes a new control logic for the lateral dynamics of a FSAE electric

vehicle. The vehicle is equipped with two different control actuation systems: the

torque vectoring generated by two single electric motors on the rear axle, one for

each wheel, and the rear wheel steering system. The latter was engineered in this

study in order to have a complete system model, which permits to have a correct

dynamic model to set the control system. All the components have been designed

to counteract the forces which come from the wheels, generated by the self-aligning

moment of the tyre.

The aim of this work is to implement a control which acts simultaneously on the side

slip angle and the yaw rate, which are recognized by literature as the driven variables

of stability and performance. Usually the control logic systems control the yaw rate,

which is associated to the performance, up to a certain side slip angle, then the side

slip angle control is activated to recover the stability. This can lead to a sudden

change in the vehicle’s behaviour, which is not predictable by the driver. The target

of the control of this dissertation is to have an intuitive behaviour during all the

possible driving situations. The implemented control logic is a sliding mode control

based on a variable sliding surface which changes according to the vehicle’s state

through a fuzzy logic. The sliding mode has been chosen because it is recognized for

being a robust control, very adaptable and not very sensitive to the disturbances. A

variable sliding surface is needed because, controlling both variables simultaneously,

there must be something that weights the control in order to give more importance

to performance, i.e. to the yaw rate control, or to stability, i.e. to side slip angle. To

reach this target a fuzzy logic has been chosen, its membership functions quantify

two factors which directly influence the sliding surface. The fuzzy logic is used for

its quite intuitive implementation and for its reliability in different systems.

Another scope of this thesis is to find a new way to implement a multi-input/multi-

output control system, thus to find a way to make the torque vectoring and the

131

rear wheel steering collaborate to achieve the best result. To manage this chal-

lenge, efficiency maps have been developed thanks to phase portrait analysis. The

performance indexes weight the control action of the two actuators. Synthetically,

the performance index has been defined as the maximum variation of a quantity

that the two actuation systems are able to independently generate for a given β-ψ

point of the phase plot. This quantity is defined by the side slip angle and yaw-rate

derivatives with respect to the passive vehicle, weighted using the fuzzy logic to be

coherent with the control logic. A new coordinated control is defined based on the

performance index maps. It is a hierarchical control that presents three different

layers. In the first layer the fuzzy sliding mode control, based on a common sliding

surface for both actuators to ensure the cooperation between them, evaluate the

control action needed to follow the reference yaw-rate and side slip angle. In the

second layer, the coordination procedure is carried out based on the performance

index maps weighting the control action obtained at the previous layer and parti-

tioning the control actions between the two actuators. In the last layer, the electric

motors torque allocation is performed to fulfil the yaw moment evaluated as TV con-

trol action and torque requested by the driver. Moreover, in this layer, an anti-slip

control (ASC) has been introduced to limit the rear tyres slips. In order to under-

stand the capacity of the combined control system, which combined the use of the

two actuators through the efficiency maps, a parallel control system has been imple-

mented. The latter was very common in the past to manage a multi-input/output

system, its logic consists consists in assigning one variable to each actuator, without

exchange of information between the different actuations. For its simplicity it was

the first multi-input control logic to be used, also because car manufacturer found

themselves with different control systems provided by different companies, which

were not thought to collaborate one with the other. The main weakness of this con-

trol logic, referring to our system, is that the two actuators could work one against

the other mining the performance. Otherwise, they could also do an extra-work,

dispending energy where it is not required.

To test the control logic and the different kinds of actuation, a comparison between

the single actuation systems (TV and RWS) and multi-actuated ones (parallel and

combined), always keeping the passive vehicle as reference, has been made run-

ning several simulations in Matlab-Simulink environment. Both steady-state and

transient manoeuvres are considered and for each, both open-loop and closed-loop

manoeuvres are tested. The control has proven to be capable to well manage both

variables in every test done with every actuation system, showing high reactivity and

good adaptation. At first, the single actuation systems were tested. Both the torque

vectoring and the rear wheel steering are effective in controlling simultaneously both

the variables, always demonstrating big advantages with respect the passive vehi-

132

Chapter 6. Conclusion

cle. In particular, the RWS shows more promptness and thus it is more effective,

managing to be the best in almost every run simulation. The TV system, thanks

to the way in which it acts to create a stabilizing moment, has a clear advantage

when braking is needed during a curve. In this situation the available contact forces

on the rear axis diminish due to the load transfer, besides also the lateral force at

disposal is less due to the combined friction. This leads the RWS system to not be

effective, it is not able to generate a stabilizing force. The TV instead, acting only

on the differentiation of the longitudinal forces, could reach a good result even in

this situation. The multi-actuated systems overcome the single-actuation problems,

already the parallel system shows the potential of the collaboration between two ac-

tuators. The results evidence how the usage of both actuators simultaneously bring

advantages in almost every situation. In contrast, the simulations also highlight the

weakness of the parallel system, it is in fact evident how during transient sometimes

the two actuators fight one against the other due to different control input. This

problem disappears with the new combined control system, thanks to a coopera-

tion of the control through a single sliding surface and a high efficiency obtained

by the performance index maps, has demonstrated to improve the performance of

the vehicle in every condition with respect not only to the passive vehicle but also

to the other control methodologies. This control logic, compared to the passive

vehicle, guarantees a more oversteering vehicle up to a certain lateral acceleration,

which it is translated in less workload for the driver. Then, when critical conditions

are reached, the control manages to modify smoothly the vehicle behaviour to an

understeering one, giving to the driver a better feeling of the limit. The control

guarantees also a higher maximum lateral acceleration of the vehicle with limited

side slip angle, a better stability both at low and high speed and a better control-

lability during severe braking conditions. Moreover, the controlled vehicle shows a

higher promptness in transient situation, requiring less pilot effort during sudden

manoeuvre. A fundamental characteristic of this approach is the low computational

effort needed to implement this control in real time. The efficiency maps are in fact

evaluated off-line, letting the possibility of using a highly non-linear vehicle model,

avoiding the inaccuracies of a simple model and thus gaining in robustness.

In conclusion, the fuzzy sliding mode seems to be very consistent in the vehicle lateral

dynamic control. Moreover, the usage of a single sliding surface and the implemen-

tation of efficiency maps based on the state variables phase portraits to guarantee

the best actuators’ cooperation seem to be very promising in the multi-actuated

vehicle control system world.

133

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140

Appendix A

MF-Tyre model

MF-tyre model by Pacejka used to modelled tyres behaviour is reported [?]. The

Magic Formula model is a semi-empirical model based on the following equations.

The model parameters: p, q, r and s are non-dimensional and in addition, a set of

scaling factors λ are present. All other parameters and variable used in the equa-

tions are:

g gravity,

Vc magnitude of the velocity of the wheel contact centre C,

Vcx,y components of the velocity of the wheel contact centre C,

Vsx,y components of slip velocity Vs (of point S) with Vsy ≈ Vcy,

Vr(= ReΩ = Vcx − Vsx) forward speed of rolling,

R0 unloaded tyre radius,

Re effective rolling radius,

Ω wheel speed of revolution,

Fz0 nominal load,

dfz the normalised change in vertical load dfz =Fz − Fz0Fz0

,

κ the longitudinal slip κ = −VsxVcx

.

141

A.1. Longitudinal force (pure longitudinal slip)

Figure A.1 shows the meaning of the reported kinematic quantities.

Figure A.1: Tyre reference quantities

A.1 Longitudinal force (pure longitudinal slip)

Fx0 = Dx sin[Cx arctanBxκx − Ex(Bxκx − arctan(Bxκx))] + SV x (A.1)

κx = κ+ SHx (A.2)

Cx = pCx1λCx(> 0) (A.3)

Dx = µxFzζ1(> 0) (A.4)

µx = (pDx1+pDx2dfz)λ∗µx(> 0) (A.5)

Ex = (pEx1 + pEx2dfz + pEx3df2z )1− pEx4sign(κx)λEx(≤ 1) (A.6)

Kxκ = Fz(pKx1 + pKx2dfz) exp(pKx3dfz)λKxκ (A.7)

Bx = Kxκ/(CxDx + εx) (A.8)

SHx = (pHx1+pHx2dfz)λHx (A.9)

SV x = (pV x1 + pV x2dfz)|Vcx|/(εV x + |Vcx|)λV xλ′µxζ1 (A.10)

142

Chapter A. MF-Tyre model

A.2 Lateral Force (pure side slip)

Fy0 = Dy sin[Cy arctanByαy − Ey(Byαy − arctan(Byαy))] + SV y (A.11)

αy = α+ SHy (A.12)

Cy = pCy1λCy(> 0) (A.13)

Dy = µyFzζ2(> 0) (A.14)

µy = (pDyl + pDy2dfz)(1− pDy3γ∗2)λ∗µy(> 0) (A.15)

Ey = (pEy1 + pEy2dfz)1− (pEy3 + pEy4γ∗)sign(αy)λEy (A.16)

Kyα0 = pKy1F′z0 sin[2 arctanpKy2F

′z0)]λKyα (A.17)

Kyα = Kyα0(1− pKy3γ∗2)ζ3 (A.18)

By = Kyα/(CyDy + εy) (A.19)

SHy = (pHy1 + pHy2dfz)λHy + pHy3γ∗λKyγζ0 + ζ4 − 1 (A.20)

SV y = Fz(pV y1 + pV y2dfz)λV y + (pV y3 + pV y4dfz)γ∗λKyγλ′µyζ2 (A.21)

Kyγ0 = pHy3Kyα0 + Fz(pV y3 + pV y4dfz)λKyγ (A.22)

A.3 Longitudinal Force (combined slip)

Fxs = GxαFx0 (A.23)

Gxα = cos[Cxα arctanBxααS − Exα(BxααS − arctan(BxααS))]/Gxα0 (A.24)

Gxα0 = cos[Cxα arctanBxαSHxα − Exα(BxαSHxα − arctan(BxαSHxα))] (A.25)

αS = α∗ + SHxα (A.26)

Bxα = rBx1 cos[arctan(rBx2κ)]λxα (A.27)

Cxα = rCx1 (A.28)

Exα = rEx1 + rEx2dfz (A.29)

SHxα = rHx1 (A.30)

143

A.4. Lateral Force (combined slip)

A.4 Lateral Force (combined slip)

Fys = GyκFy0 + SV yκ (A.31)

Gyκ = cos[Cyκ arctanByκκS − Eyκ(ByκκS − arctan(ByκκS))]/Gyκ0 (A.32)

Gyκ0 = cos[Cyκ arctanByκSHyκ − Eyκ(ByκSHyκ − arctan(ByκSHyκ))] (A.33)

κS = κ+ SHxκ (A.34)

Byκ = rBy1 cos[arctanrBy2(α∗ − rBy3)]λyκ (A.35)

Cyκ = rCy1 (A.36)

Eyκ = rEy1 + rEy2dfz (A.37)

SHyκ = rHy1 + rHy2dfz (A.38)

SV yκ = DV yκ sin[rV y5 arctan(rV y6κ)] (A.39)

DV yκ = µyFz(rV y1 + rV y2dfz + rV y3γ∗) cos[arctan(rV y4α

∗)]ζ2 (A.40)

A.5 Relaxation length

Being tyres characterized by their own dynamics, the force generation can be mod-

elled as first order dynamics system (A.41).LxviFx,i + Fx,i = Fxs,i

LyviFy,i + Fy,i = Fys,i

(A.41)

where vi is the longitudinal velocity of wheel hub in wheel reference frame while Lxand Ly are the so called relaxation length respectively for longitudinal and lateral

forces. Lx is comparable to contact path length while Ly is comparable to tyre

circumference length.

144

Appendix B

Dimensioning of RWS

components

In this appendix the calculation made in order to choose the rear system components

are carried out.

B.1 Dimensioning of rack and pinion

The torque, coming from the motor, generates a maximum force Ft of 750N, consid-

ering a standard pressure angle α of 20°. The chosen pinion (30120012 in fig. B.2)

is made by C45 UNI 7845, a steel with a minimum yield strength of σyp=370MPa.

In order to verify the pinion its modulus (2 for the chosen one) must be compared

with the one calculated by the Lewis equation(eq. (B.1)):

m =

√Ft

σypY ψ(B.1)

Where ψ is a safety coefficient, taken equal to 10, a recommended value for this type

of application. Y is a factor which depends by the number of teeth (z) of the pinion,

in this case Y (12)=0.245. The result is

m = 0.91

which satisfies the request being less than 2.

With the modulus of the pinion is possible to implement its pinion pitch p:

p = mπ = 6.28mm (B.2)

The rack must have the same pitch.

145

B.2. Dimensioning of pinion shaft

B.2 Dimensioning of pinion shaft

Figure B.1: Forces on pinion shaft schema

To accommodate the pinion the pinion shaft must have a 10mm diameter. In this

section a structural verification of the shaft is carried out. The most critical section

of the shaft is where the pinion is clamped (fig. B.1). Here there is the maximum

bending moment and the torsional moment (constant on the overall shaft). The

shaft has been verified statically, not at fatigue, due to the low hours it will work.

The Von Mises criterion is used to verify it.

The force generated by the pinion on the shaft is vertical, and it’s generated by the

force’s decomposition on the pinion tooth.

F = Ft sin(α) = 256.5N (B.3)

Resolving the shaft forces equilibrium:

F1 = F b2

(a+b)2 (3− 2 ba+b) = 104.75N

F2 = F a2

(a+b)2 (3− 2 aa+b) = 151.76N

M1 = Fab2

l2= 1579Nmm

M2 = −Fa2bl2

= 2021Nmm

(B.4)

The bending moment acting on the critical section is:

Mfl = M1 − F1a = −1772.3Nmm (B.5)

146

Chapter B. Dimensioning of RWS components

It is now possible to calculate the normal stress and the shear one:

σ =MflRKt

I = −51.93MPa

τ = TRKtJ = 131.8MPa

(B.6)

Where R is the shaft radius, T the torque (9Nm), Kt is a coefficient to include the

notching effect (here Kt=1.3). The inertia could be calculated as:

J = πR4

2 = 443.87mm4

I = πR4

4 = 221.93mm4

(B.7)

Finally, the equivalent stress could be calculated by the Von Mises formula:

σEQVM =√σ2 + 3τ2 = 234.11MPa (B.8)

Considering that the shaft is made by a material with a yield strength of σyp=700MPa,

we have a security factor of 3.

B.2.1 Keyway

The keyway is needed to fix the pinion on the shaft. Considering the torque T of

9Nm on the shaft, the tangential force which stresses the keyway is:

Fk = T/R = 1800N (B.9)

With Fk it’s possible to calculate the normal and the shear stress acting on the

keyway:

τ = FkwL = 60MPa

σ = FkhL/2 = 120MPa

(B.10)

The admissible normal and shear stress are, respectively, σyp=430MPa and τyp=215MPa.

Thus the keyway is working with a safety coefficient higher than 3.5.

B.2.2 Bushings

The bushings had a Basic Dynamic Load rating in the radial direction which was

C = 8kN which is well above the approximate bearing load of 200N each. The

bushings were not analyzed for longevity.

147

B.3. Datasheets

B.2.3 Linear bearings

The chosen linear bearing is the one with 20mm of internal diameter (code LPBR20

in fig. B.4), in order to accommodate the rack shaft. The Basic Load Rating at

0,1m/s , which is similar to the velocity the rack will be traveling, is 2080N. The

maximal load in the radial direction for one of the linear bearings is F=256.5N. The

longevity and friction for the linear bearings was not analysed.

B.3 Datasheets

In this section all the datasheets of the standard components of the rear steering

system are collected.

148


Disegni CAD disponibili sul sito www.chiaravalli.com

Quantità, disponibilità e prezzicon B2B Chiaravalli

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Angolo di pressione: 20°

CON MOZZO LATERALE

MATERIALE C 45 UNI 7845

30110012 14 12 9 5 0,0130110013 15 13 10 5 0,0230110014 16 14 10 6 0,0230110015 17 15 12 6 0,0230110016 18 16 13 6 0,0330110017 19 17 14 8 0,0330110018 20 18 15 8 0,0330110019 21 19 15 8 0,0430110020 22 20 16 8 0,0430110021 23 21 16 8 0,0530110022 24 22 18 8 0,0530110023 25 23 18 8 0,0630110024 26 24 20 8 0,0630110025 27 25 20 8 0,0730110026 28 26 20 8 0,0730110027 29 27 20 8 0,0830110028 30 28 20 8 0,0830110029 31 29 20 8 0,0930110030 32 30 20 8 0,0930110031 33 31 25 10 0,1130110032 34 32 25 10 0,1230110033 35 33 25 10 0,1230110034 36 34 25 10 0,1330110035 37 35 25 10 0,1430110036 38 36 25 10 0,1430110037 39 37 25 10 0,1530110038 40 38 25 10 0,1630110039 41 39 25 10 0,1630110040 42 40 25 10 0,1730110041 43 41 30 10 0,1930110042 44 42 30 10 0,230110043 45 43 30 10 0,2130110044 46 44 30 10 0,2230110045 47 45 30 10 0,2330110046 48 46 30 10 0,2330110047 49 47 30 10 0,2430110048 50 48 30 10 0,3530110049 51 49 30 10 0,3630110050 52 50 30 12 0,2630110051 53 51 40 12 0,3230110052 54 52 40 12 0,3330110053 55 53 40 12 0,3330110054 56 54 40 12 0,3430110055 57 55 40 12 0,3630110056 58 56 40 12 0,3730110057 59 57 40 12 0,3830110058 60 58 40 12 0,3930110059 61 59 40 12 0,430110060 62 60 40 12 0,4130110061 63 61 50 12 0,4730110062 64 62 50 12 0,4930110063 65 63 50 12 0,530110064 66 64 50 12 0,5130110065 67 65 50 12 0,5230110066 68 66 50 12 0,5330110067 69 67 50 12 0,5530110068 70 68 50 12 0,5630110069 71 69 50 12 0,5730110070 72 70 50 12 0,58

Z CODICE de dp dm ØD1 Kg

1213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970

MODULO 1 CODICE de dp dm ØD1 Kg

MODULO 1,5 CODICE de dp dm ØD1 Kg

MODULO 2

30115012 21,0 18,0 14 8 0,0430115013 22,5 19,5 14 8 0,0530115014 24,0 21,0 18 8 0,0630115015 25,5 22,5 18 8 0,0730115016 27,0 24,0 20 8 0,0830115017 28,5 25,5 20 8 0,0930115018 30,0 27,0 20 8 0,1030115019 31,5 28,5 20 8 0,1030115020 33,0 30,0 25 8 0,1330115021 34,5 31,5 25 10 0,1330115022 36,0 33,0 25 10 0,1430115023 37,5 34,5 25 10 0,1630115024 39,0 36,0 25 10 0,1730115025 40,5 37,5 25 10 0,1830115026 42,0 39,0 30 12 0,2030115027 43,5 40,5 30 12 0,2230115028 45,0 42,0 30 12 0,2330115029 46,5 43,5 30 12 0,2430115030 48,0 45,0 30 12 0,2630115031 49,5 46,5 35 12 0,3030115032 51,0 48,0 35 12 0,3130115033 52,5 49,5 35 12 0,3330115034 54,0 51,0 35 12 0,3430115035 55,5 52,5 35 12 0,3630115036 57,0 54,0 35 12 0,3730115037 58,5 55,5 40 12 0,4230115038 60,0 57,0 40 12 0,4430115039 61,5 58,5 40 12 0,4630115040 63,0 60,0 40 12 0,4830115041 64,5 61,5 50 14 0,5030115042 66,0 63,0 50 14 0,5930115043 67,5 64,5 50 14 0,6130115044 69,0 66,0 50 14 0,6330115045 70,5 67,5 50 14 0,6530115046 72,0 69,0 50 14 0,6630115047 73,5 70,5 50 14 0,730115048 75,0 72,0 50 14 0,730115049 76,5 73,5 50 14 0,7330115050 78,0 75,0 50 14 0,7630115051 79,5 76,5 60 15 0,8630115052 81,0 78,0 60 15 0,8930115053 82,5 79,5 60 15 0,9130115054 84,0 81,0 60 15 0,9430115055 85,5 82,5 60 15 0,9630115056 87,0 84,0 60 15 0,9830115057 88,5 85,5 60 15 1,0030115058 90,0 87,0 60 15 1,0330115059 91,5 88,5 60 15 1,0630115060 93,0 90,0 60 15 1,0930115061 94,5 91,5 70 20 1,2230115062 96,0 93,0 70 20 1,2530115063 97,5 94,5 70 20 1,2830115064 99,0 96,0 70 20 1,3130115065 100,5 97,5 70 20 1,3430115066 102,0 99,0 70 20 1,3730115067 103,5 100,5 70 20 1,4030115068 105,0 102,0 70 20 1,4330115069 106,5 103,5 70 20 1,4630115070 108,0 105,0 70 20 1,50

30120012 28 24 18 10 0,0830120013 30 26 19 10 0,130120014 32 28 20 10 0,1230120015 34 30 22 10 0,1430120016 36 32 24 10 0,1630120017 38 34 25 10 0,1830120018 40 36 25 10 0,1930120019 42 38 25 10 0,2130120020 44 40 30 10 0,2630120021 46 42 30 12 0,2730120022 48 44 30 12 0,2930120023 50 46 30 12 0,3130120024 52 48 35 12 0,3630120025 54 50 35 12 0,3930120026 56 52 40 12 0,4530120027 58 54 40 12 0,4730120028 60 56 40 12 0,530120029 62 58 40 14 0,5230120030 64 60 40 14 0,5530120031 66 62 45 14 0,6130120032 68 64 45 14 0,6530120033 70 66 45 14 0,6830120034 72 68 45 14 0,7130120035 74 70 45 14 0,7430120036 76 72 45 14 0,7830120037 78 74 50 14 0,8630120038 80 76 50 14 0,930120039 82 78 50 14 0,9330120040 84 80 50 14 0,9730120041 86 82 60 16 1,0530120042 88 84 60 16 1,0930120043 90 86 60 16 1,1330120044 92 88 60 16 1,2330120045 94 90 60 16 1,2730120046 96 92 60 16 1,3130120047 98 94 60 16 1,4830120048 100 96 70 16 1,5330120049 102 98 70 16 1,5730120050 104 100 70 16 1,6230120051 106 102 70 20 1,6730120052 108 104 70 20 1,7230120053 110 106 70 20 1,7830120054 112 108 70 20 1,8330120055 114 110 70 20 1,8830120056 116 112 70 20 1,9430120057 118 114 70 20 1,9930120058 120 116 70 20 2,0530120059 122 118 70 20 2,1130120060 124 120 70 20 2,1630120061 126 122 80 20 2,3630120062 128 124 80 20 2,4230120063 130 126 80 20 2,4830120064 132 128 80 20 2,5530120065 134 130 80 20 2,6130120066 136 132 80 20 2,6730120067 138 134 80 20 2,7430120068 140 136 80 20 2,8130120069 142 138 80 20 2,8730120070 144 140 80 20 2,94

LARGHEZZA FASCIA B per:

Modulo 1 = 15 mm

Modulo 1.5 = 17 mm

Modulo 2 = 20 mm

LARGHEZZA TOTALE A per:

Modulo 1 = 25 mm

Modulo 1.5 = 30 mm

Modulo 2 = 35 mm

A

de

dpdm

B

D1

Figure B.2: Pinion datasheet - code 30120012

149

B.3. DatasheetsLIN

GUET

TE

84

Campo di Applicazione

LinguettaSezione

Cava

Larghezza Profondità

Diametro Albero Dimens.Nominali Tolleranze su Dimens.

NominaleTolleranze su b Albero Mozzo

Per albero Per mozzo t1 t2

d bxh bh9 h* b H9 N9 P9 D10 Js9 P9 Nom. Toll. Nom. Toll.

dafino a

68 2 x 2 0 0 2

+0,0250

0-0,004

-0,006-0,031

+0,060+0,020 ±0,012 -0,006

-0,031

1,2

+0,10

1

+0,10

dafino a

810 3 x 3 -0,025 -0,025 3 1,8 1,4

oltrefino a

1012 4 x 4

0-0,030

0-0,030

4

+0,0300

0-0,030

-0,012-0,042

+0,078+0,030 ±0,015 -0,012

-0,042

2,5 1,8

oltrefino a

1217 5 x 5 5 3 2,3

oltrefino a

1722

6 x 4 6 2,5 1,8

6 x 5 6 3 2,3

6 x 6 6 3,5 3,5

oltrefino a

2230

8 x 5

0-0,036

8

+0,0360

0-0,036

-0,015-0,051

+0,098+0,040 ±0,018 -0,015

-0,051

3 2,3

8 x 6 8 3,5 2,8

8 x 7 0-0,090 8 4

+0,20

3,3

+0,20

8 x 8 0-0,036 8 5 3,3

oltrefino a

3038

10 x 8 0-0,090 10 5 3,3

10 x 10 0-0,036 10 6 4,3

oltrefino a

3844

12 x 8

0-0,043

0-0,090 12

+0,0430

0-0,043

-0,018-0,061

+0,120+0,050 ±0,021 -0,018

-0,061

5 3,3

12 x 12 0-0,043 12 7,5 4,9

oltrefino a

4450

14 x 9 0-0,090 14 5,5 3,8

14 x 14 0-0,043 14 9 5,4

oltrefino a

5058 16 x 10 0

-0,090 16 6 4,3

oltrefino a

5860 18 x 11 0

-0,110 18 7 4,4

oltrefino a

6575 20 x 12

0-0,052 0

-0,110

20

+0,0520

0-0,052

-0,022-0,074

+0,149+0,065 ±0,026 -0,022

-0,074

7,5 4,9

oltrefino a

7585 22 x 14 22 9 5,4

oltrefino a

8595 25 x 14 25 9 5,4

oltrefino a

95110 28 x 16 28 10 6,4

oltrefino a

110130 32 x 18

0-0,062

32

0,0620

0-0,062

-0,026-0,088

+0,180+0,080 ±0,031 -0,026

-0,088

11 7,4

oltrefino a

130150 36 x 20

0-0,130

36 12

+0,30

8,4

+0,30

oltrefino a

150170 40 x 22 40 13 9,4

oltrefino a

170200 45 x 25 45 15 10,4

oltrefino a

200230 50 x 28 50 17 11,4

oltrefino a

230260 56 x 32

0-0,074

0-0,160

56

+0,0740

0-0,074

-0,032-0,106

+0,220+0,100 ±0,037 -0,032

-0,106

20 12,4

oltrefino a

260290 63 x 32 63 20 12,4

oltrefino a

290330 70 x 36 70 22 14,4

oltrefino a

330380 80 x 40 80 25 15,4

oltrefino a

380440 90 x 45

0-0,087

90+0,087

00

-0,080-0,037-0,124

+0,260+0,120 ±0,043 -0,037

-0,124

28 17,4

oltrefino a

440500 100 x 50 100 31 19,5

t2

t1

Ød1 d1-t

1

d1+t

2

b

h

Tolleranze dimensionali linguettee cave di alloggiamento

Dimensioni in mm

* i valori di scostamento si riferiscono alle zone di tolleranza h9 per sezione quadrata e h11 per sezione rettangolare

Figure B.3: Keyway datasheet

150


11

Linear plain bearings – LPBR- closed design

LPBR

Dimensions Basic load ratings Mass Designation

dyn. at stat. Linear plain

0,1 m/s 4 m/s bearing

Fw D C C4 C C C0

-0,07

mm N kg —

12 19,19 28 10 965 24 3 350 0,006 LPBR 12

14 21,21 28 12 1 370 34 4 750 0,007 LPBR 14

16 24,23 30 12 1 530 38 5 400 0,009 LPBR 16

20 28,24 30 13 2 080 52 7 350 0,011 LPBR 20

25 35,25 40 17 3 400 85 12 000 0,024 LPBR 25

30 40,27 50 20 4 800 120 17 000 0,033 LPBR 30

40 52,32 60 24 7 650 193 27 000 0,063 LPBR 40

50 62,35 70 27 10 800 270 38 000 0,088 LPBR 50

The outside diameter tolerance of the linear plain bearings is such that no additional axial fixation is required when the

bearings are fitted into a bore with a tolerance of J7 or J6.

Appropriate special seals

Dimensions Designations

Fw D B1

mm —

25 35 4 SP-25x35x4

30 40 4 SP-30x40x4

40 52 5 SP-40x52x5

50 62 5 SP-50x62x5

Appropriate special seals

Dimensions Designations

Fw D B1

mm —

12 19 3 SP-12x19x3

14 21 3 SP-14x21x3

16 24 3 SP-16x24x3

20 28 4 SP-20x28x4

Accessories for LPBR (shaft seals)

SP

Figure B.4: Linear bearing datasheet - code LBPR20

151

B.3. Datasheets

PCMF 101207 E

Boccole

Dati sulle boccoleTolleranze,Gioco in esercizio

Design delle disposizioni diboccoleTolleranze per alberi ealloggiamenti

Specifiche tecniche

Materiali PTFE composito

DIMENSIONI

d 10 mm

D 12 mm

B 7 mm

D1 18 mm

B1 1 mm

c1 min. 0.1 mm

c1 max. 0.6 mm

c2 min. 0.2 mm

c2 max. 1 mm

α ±8 20 °

r max. 1 mm

Generato dal sito il data

Pagina pagina di totale

Figure B.5: Bushing datasheet

152


*) Motor must be operated with current limitation

Max. Continuous torque [Ncm] 1500

Max. torque (short-term) [Ncm] 2000

Recommended input speed [1/min] < 3000

Permissible shaft load [axial / radial] 300 / 500

Backlash unloaded [°] 3

Temperature range [°C] -30 bis +80

Kählig Antriebstechnik GmbH Pappelweg 4 D-30179 Hannover/Germany

fon: +49 511 67493-0 fax: +49 511 67493-67 [email protected]

- Suitable for left - and right rotation and changes, duration, - and intermittent operation- Other ratios, output shafts, protective classes, connection leads and flanges on request

Application on request

Gear reduction table M80xXX/I+ SN40

Stand: 11. Februar 2016 – changes reserved

Motor M80with worm gear SN40

Dauerdreh moment

General gear data

Gearreduction

EfficiencySpeed at

motor-speed 3000rpm M80x40/I

L2 = 132mmM80x80/I

L2 = 172mm

[i = x:1] [%] [1/min] [Ncm] [Ncm] [Ncm] [Ncm]

8 70 375 213 392

10 84 300 319 588

15 72 200 410 756

30 40 100 456 840

50 35 60 665 1225

Continuous torque

Figure B.6: Worm gearbox + DC motor datasheet

153

B.3. Datasheets

0 75 150 225 300 375 450M

[Ncm / ozf.in(x 1,42)]

P [W]2 P (M)2

0

100

200

300

400

500η[%] η(M)

0

20

40

60

80

100I[A] I(M)

0

20

40

60

80

100n[min /rpm]-1 n(M)

0

1000

2000

3000

4000

5000

Typ/Type: M80x40/I MN= 38NcmWicklung/Winding: 12.18.2x0,63 UN= 24V

Ident.-Nr.: 222334 Bearbeiter/Editor: H. KhalilDatum/Date: 15.11.2019

KÄHLIG ANTRIEBSTECHNIK GMBH

0 66,67 133,33 200 266,67 333,33 400M

[Ncm / ozf.in(x 1,42)]

P [W]2 P (M)2

0

100

200

300

400

500η[%] η(M)

0

20

40

60

80

100I[A] I(M)

0

20

40

60

80

100n[min /rpm]-1 n(M)

0

1000

2000

3000

4000

5000

Typ/Type: M80x40/I MN= 38NcmWicklung/Winding: 12.9.2x0,9 UN= 12V

Ident.-Nr.: 222784 Bearbeiter/Editor: H. KhalilDatum/Date: 15.11.2019

KÄHLIG ANTRIEBSTECHNIK GMBH

4

www.kag-hannover.deKählig Antriebstechnik GmbH Pappelweg 4 D-30179 Hannover/Germany

fon: +49 511 67493-0 fax: +49 511 67493-67 [email protected]

- Brushed DC motor with permanent magnets- Ball earings- Lead wires- Chromatised housing with zinc-die-cast bearing flanges- Direction of rotation CW / CCW- Multiple combination possibilities with gears, encoders, brakes and control electro-nics

Application on request

Stand: 4. September 2020 – changes reserved

DC-Motor M80x40/IId.-Nr. 222784 (12V) 222334 (24V)

Figure B.7: DC motor M80x40/I 12V datasheet

154

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