# ABS for FSAE-Thesis

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Design and Simulation of an ABS Control Schemefor a Formula Student Prototype

Joo Pedro Carrapio Ferro

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisor: Prof. Paulo Jorge Coelho Ramalho OliveiraCo-supervisor: Prof. Joo Miguel da Costa Sousa

Examination Committee

Chairperson: Prof. Joo Rogrio Caldas PintoSupervisor: Prof. Paulo Jorge Coelho Ramalho Oliveira

Member of the Committee: Prof. Duarte Pedro Mata de Oliveira Valrio

May 2014

2

Acknowledgments

The work developed in this thesis a continuance of the knowledge and innovative spirit acquired dur-

ing the three years spent in Projecto FST. For their unflagging help, suggestions and encouragement,

I would like to specially thank my team-mates, with whom Ive designed, spent sleepless nights, dis-

cussed, learned, travelled, cheered, celebrated and shared the same passion for engineering and mo-

torsports.

To my thesis supervisor Professor Paulo Oliveira and co-supervisor Professor Joao Sousa, whose

insight, knowledge and advice were equally important on pursuing this ambitious but highly enlightening

challenge.

To my friends, for their motivation, everlasting friendship and comprehension when I just could not

be there.

To my family, for their inestimable support and incessant belief, not only during this work, but through-

out the entire degree. A special dedication to my grandfather, who provided me the inspiration and

engineering genes.

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Abstract

Anti-lock braking system (ABS) has the objective of controlling wheel slip so that maximum friction

force is attained while maintaining adequate steerability and stability during hard braking. Beyond the

inestimable contribution to road safety, ABS may also be used in racecars as a driving-aid system to

enhance braking performance.

This thesis addresses the design of an ABS seeking the implementation on a Formula Student racing

prototype. The proposed scheme utilizes a cascade control architecture: a PID-type fuzzy controller with

a Takagi-Sugeno-Kang fuzzy inference system is designed for wheel slip control on the outer loop, whilst

a brake pressure PD controller is adopted in the inner loop. Wheel slip estimation solution, resorted on

a complementary filter, is also developed.

The performance of the ABS is assessed with a full vehicle model, sustained by vehicle dynamic

principles. The model is integrated with brakeline dynamics and a tire friction model based on reliable

experimental data. Straight line brake simulations are performed and results are evaluated in terms

of braking efficiency, under different and variable conditions. A complete lap within a typical Formula

Student circuit is also simulated, for the cases with and without ABS.

Keywords: anti-lock braking system (ABS), Formula Student, fuzzy controller, PID controller,wheel slip estimation, vehicle model, tire model

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Resumo

O sistema de travagem ABS (do ingles anti-lock braking system) tem o objetivo de controlar o escor-

regamento da roda de forma a atingir a maxima forca de atrito, mantendo dirigibilidade e estabilidade

suficientes durante uma travagem brusca. Para alem da inestimavel contribuicao para a seguranca

rodoviaria, o sistema ABS pode tambem ser usado em carros de competicao como sistema de as-

sistencia a` conducao para melhorar o desempenho em travagem.

Esta tese trata do projeto de um sistema ABS para implementacao num prototipo de competicao

do tipo Formula Student. O esquema proposto utiliza uma arquitetura de controlo em cascata: um

controlador fuzzy do tipo PID com um sistema de inferencia fuzzy do genero Takagi-Sugeno-Kang e

projetado para o controlo do escorregamento da roda no anel exterior, enquanto para o anel interior e

adotado um controlador de pressao do tipo PD. Uma solucao de estimacao para o escorregamento da

roda, com recurso a uma solucao de filtro complementar, e tambem desenvolvida.

O desempenho do sistema ABS e avaliada com um modelo completo do veculo, sustentado nos

princpios da dinamica de veculos. O modelo e integrado com a dinamica da linha de travagem e

um modelo de atrito do pneu baseado em dados experimentais fidedignos. Simulacoes de travagem

em linha reta sao executadas e os resultados examinados em termos de eficiencia de travagem, sob

condicoes diversas e variaveis. E tambem simulada uma volta completa a um circuito de Formula

Student tpico, para os casos com e sem ABS.

Palavras-chave: sistema de travagem ABS, Formula Student, controlador fuzzy, controladorPID, estimacao do escorregamento da roda, modelo de veculo, modelo de pneu

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction 1

1.1 Brief history of ABS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation (Formula Student) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Work contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Vehicle Dynamics 7

2.1 Global Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Horizontal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Vertical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Lagrange Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Vehicle Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4 Tire Vertical Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Wheel Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Free-body diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Brakeline Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Steady-state equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Transient behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.3 Hydraulic Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Tire Model 25

3.1 Types of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Pacejka Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3.1.2 Burkhardt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.3 Neural Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Data Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Methodology for Pacejka and Burckhardt models . . . . . . . . . . . . . . . . . . . 31

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 ABS Overview 35

4.1 Objectives of ABS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 ABS Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Wheel Slip Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 Longitudinal Slip Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.2 Control Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.3 Main difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.1 Threshold Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.2 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.3 Sliding-mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4.4 Intelligent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Proposed Approach 45

5.1 Problem data and requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.1 FS Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1.2 Competition characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1.3 FST 05e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Proposed Control Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Inner Loop - Brake Pressure Controller . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.2 Outter Loop - Wheel Slip Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Design Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 ABS Design 51

6.1 Brake Pressure Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1.1 PID Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1.2 PWM Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1.3 Open/closed loop step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1.4 PD Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Wheel Slip Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.1 FIS Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.2 FIS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2.3 PID Gains Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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6.2.4 Reference Wheel Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3 Wheel Slip Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3.1 Complementary Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 ABS Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.4.1 Minimum brake pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.4.2 Wheel slip and wheel slip rate trigger . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.4.3 Low velocity trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Simulation Results and Analysis 67

7.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.1.1 FST 05e Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2 Straight Line Hard Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2.1 Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.2.2 Varying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2.3 Without pressure controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.3 Complete Lap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8 Summary and Conclusions 77

8.1 Future work and research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Bibliography 83

A SAE Definitions 84

A.1 Axis Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.1.1 Earth-Fixed Axis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.1.2 Vehicle Axis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.1.3 Intermediate Axis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.1.4 Tire Axis System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.2.1 Vehicle Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

A.2.2 Tire Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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List of Tables

2.1 Coordinates of tire i on the (X,Y ) axis system. . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Hydraulic modulator solenoid valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Magic Formula coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 MATLABs lsqcurvefit parameters [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 MSE fitting result for Pacejka, Burkhardt and Neural Network tire models. . . . . . . . . . 34

4.1 Closed-loop variables for wheel slip control. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1 Closed-loop variables for pressure control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1 Tuned PD gains for pressure controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Fuzzy sets and membership functions (MFs) parameters for inputs eSX and eSX . . . . . 59

6.3 Takagi-Sugeno-Kang (TSK) consequent function values Ck. . . . . . . . . . . . . . . . . . 60

6.4 Fuzzy rules for wheel slip controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Tuned PID gains for fuzzy wheel slip controller. . . . . . . . . . . . . . . . . . . . . . . . . 62

7.1 FST 05e parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Other model and simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3 Braking distance and time results with and without ABS, with constant road . . . . . . . . 70

7.4 Braking distance and time results with and without ABS, with varying road . . . . . . . . 72

7.5 Braking distance and time without PD pressure controller. . . . . . . . . . . . . . . . . . . 72

7.6 Lap time results with and without ABS for Autocross Event. . . . . . . . . . . . . . . . . . 74

8.1 Results summary with and without ABS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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List of Figures

1.1 ABS tests by Mercedes-Benz on S-Class passenger car, 1978. . . . . . . . . . . . . . . . 2

1.2 Williams FW15C featuring ABS, 1993. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Team from Projecto FST and FST 05e prototype. . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Vehicle dynamics overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Planar tire forces on vehicles corner i [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Planar vehicle dynamics [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 7 DOFs vibrational model [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Longitudinal load transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Lateral load transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Aerodynamic load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Freebody diagram of the wheel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 Schematic of basic brakeline dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10 Step response of brakeline dynamics Tb(t) for = 0.1 s. . . . . . . . . . . . . . . . . . . . 22

2.11 Schematic of brakeline dynamics with hydraulic modulator. . . . . . . . . . . . . . . . . . 23

2.12 Schematic of a hydraulic modulator [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.13 Simulink implementation of the hydraulic modulator. . . . . . . . . . . . . . . . . . . . . . 24

3.1 Classification of tire models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Pacejkas Magic Formula curve and geometric parameters. . . . . . . . . . . . . . . . . . 27

3.3 Artificial neural network (ANN) scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Tire test at Tire Research Facility (TIRF) [4]. . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Excerpt data from a complete run. In red, data for FZT = {50, 150, 250, 350} lbs, = =0 and p = 96.5 kPa (14 psi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Plot of SX vs. FXT for four sweeps of different FZT , with = = 0 and p = 96.5 kPa

(14 psi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.7 Pacejka Model vs. TTC Data for FZT = {150, 250, 350} lbs. . . . . . . . . . . . . . . . . . 333.8 Burkhardt Model vs TTC Data for different velocities. . . . . . . . . . . . . . . . . . . . . . 33

3.9 Neural Network Model vs. TTC Data for FZT = {150, 250, 350} lbs. . . . . . . . . . . . . . 34

4.1 Buildup of yaw moment induced by large differences in friction coefficients [5]. . . . . . . 36

4.2 Anti-lock braking system (ABS) components diagram. . . . . . . . . . . . . . . . . . . . . 36

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4.3 ABS action zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 ABS control loop components [5]: 1-Hydraulic Modulator, 2-Master Cylinder, 3-Wheel

caliper, 4-ECU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5 ABS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1 Formula Student Germany 2013 endurance event. . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Proposed control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Schematic of brakeline dynamics and pressure controller. . . . . . . . . . . . . . . . . . . 48

5.4 PID-type fuzzy controller scheme [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1 PID controller scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2 PWM conversion scheme [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 PWM implementation with fc = 50 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4 Open-loop schematic for pressure control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.5 Open-loop variable-step response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.6 Closed-loop variable-step response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.7 Linearisable approximation of brakeline system (a = 400, b = 1.28, h = 0.22 and v = 100. 566.8 Step response of original and approximated system. . . . . . . . . . . . . . . . . . . . . . 57

6.9 Step response with PD controller for pcal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.10 Fuzzy inference system diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.11 Membership functions of fuzzy inference system (FIS). . . . . . . . . . . . . . . . . . . . . 60

6.12 Output surfaces of FIS models for wheel slip controller. . . . . . . . . . . . . . . . . . . . . 62

6.13 Block diagram of vS estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.14 Block diagram of complementary filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.15 Slip velocity vS estimation results with complementary filter. . . . . . . . . . . . . . . . . . 65

7.1 Comparative simulation results for constant , with (solid) and without (dashed) ABS. . . 70

7.2 Comparative simulation results for varying , with (solid) and without (dashed) ABS. . . . 71

7.3 Control scheme without pressure inner loop. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.4 Comparative simulation results with (solid) and without (dashed) pressure PID controller. 73

7.5 GG diagram of an Autocoross lap with and without ABS. . . . . . . . . . . . . . . . . . . . 74

7.6 Brake pedal actuation within lap perimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.7 Velocity plot over an Autocross lap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.8 Velocity distribution within lap perimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.1 Vehicle Axis System [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2 Tire and Wheel Axis System [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.3 Tire Force and Moment Nomenclature [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xiv

List of Symbols

Symbols defined per standard SAE J670-2008 [7] are properly indentified. Italized words and phrases

indicate definitions in the same standard.

coefficient of friction; membership function parent.

Greek

tire slip angle.

vehicle sideslip/attitude angle.

SW steering-wheel angle.

steer angle (road wheel steer angle).

inclination angle.

W wheel-spin velocity .

W0 reference wheel-spin velocity .

roll angle.

yaw/heading angle.

pitch angle.

spring spring deflection/compression.

ARB ARB torsional spring rate.

brakeline lag; time integration variable.

Latin

SX tire longitudinal slip ratio.

a longitudinal distance from the vehicle CG to the front axle center line.

b longitudinal distance from the vehicle CG to the rear axle center line.

cS suspension damping coefficient.

cT tire damping coefficient.

F vehicle force.

fc modulator signal frequency.

f function.

h vehicle CG height above ground.

Ixx vehicle roll moment of inertia.

Iyy vehicle pitch moment of inertia.

Izz vehicle yaw moment of inertia.

xv

K gain.

kS suspension rate (wheel rate).

cdamper damping coefficient of suspension damper.

kspring spring rate of suspension spring.

kT tire normal stiffness (tire spring rate).

L wheelbase.

M vehicle moment .

m total vehicle total mass.

MR motion ratio.

ms sprung mass.

mu unsprung mass.

Re effective rolling radius.

RL tire loaded radius.

T track (track width, wheel track); time constant.

Ts sample time.

a vehicle acceleration.

v vehicle velocity .

xE , yE , zE earth-fixed coordinate system.

xV , yV , zV vehicle coordinate system.

X,Y, Z intermediate axis system.

z (w/ subscript) vertical displacement/travel (bounce/hop) in the direction of ZV .

Subscripts

bp brake pedal.

F front.

L left.

R rear; right.

r road/ground; DOF index.

s sprung mass.

T tire.

u unsprung mass/wheel.

X longitudinal/roll component.

Y lateral/pitch component.

Z vertical/yaw component.

xvi

Acronyms

Symbols defined per standard SAE J670-2008 [7] are properly indentified. Italized words and phrases

indicate definitions in the same standard.

ABS anti-lock braking system.

ANN artificial neural network.

ARB anti-roll bar.

CAD computer aided design.

CG center of gravity.

DOF degree of freedom.

ECU electrical control unit.

FIS fuzzy inference system.

FS Formula Student.

GA genetic algorithm.

MF membership function.

MF Magic Formula.

MSE mean squared error.

PWM pulse-width modulation.

SMC sliding-mode control.

TIRF Tire Research Facility .

TSK Takagi-Sugeno-Kang.

TTC Tire Test Consortium.

VAF variance accounted for.

xvii

xviii

Chapter 1

Introduction

The anti-lock braking system (ABS), also Antiblockier-Bremssystem in German, is an active safety sys-

tem present in most of passenger cars and trucks. Originally designed for aircrafts, it prevented the

wheels from locking under braking by regulating the brake line pressure independent of brake pedal

force. Beyond simply avoiding lock-up, modern ABSs manipulate wheel speed to achieve a desired slip

level range where braking distance is minimized while keeping steering stability.

Beyond the inestimable contribution to road safety, ABS may also be used in racecars as a driving-

aid device. By maximizing the braking force, the driver is able to brake later and while cornering, saving

precious time in every corner. Moreover, preventing the wheels from locking also considerably reduces

tire wear. Although forbidden in most high-end racing disciplines like Formula 1, driving-aids like ABS

are allowed in Formula Student (FS) and teams may benefit from its implementation.

1.1 Brief history of ABS

The first ABS dates back to 1929 and is credited to the french automobile and aircraft pioneer Gabriel

Voisin [8]. It was developed for aircraft use as threshold braking on airplanes is nearly impossible. Later

in 1945, the first set of mechanical ABS brakes were implemented on a Boeing B-47 to prevent spin outs

and tires from blowing [9]. In 1952, Dunlops Maxaret automatic and fully mechanical brake control was

one of the most important devices in the history of the aviation safety. Back then, considerably improved

braking efficiency and the elimination of the pilots fear of over-braking, which could result in skidding

and burst of tires, has resulted in a marked reduction in landing distances (up to 30%) [10]. ABS brakes

were commonly installed in airplanes thenceforth.

Though already acknowledged as a revolutionary system in the aviation scenery during the 1960s,

fully mechanical systems saw limited automobile use on high end automobiles only. In 1972, the british

Jensen lnterceptor automobile became the first production car to offer a Maxaret-based ABS. Low reli-

ability of system electronics and low public awareness, allied to the additional cost to the buyer, led to

their quiet withdrawal from the market in the middle 1970s [11].

1

The development of digital electronics changing from analog to integrated circuits and microproces-

sors resulted in a major automotive milestone with the introduction of the Bosch ABS on the Mercedes-

Benz S-Class passenger car in 1978 [11]. It was the first completely electronic four-wheel multi-channel

ABS. BMW and others succeeded shortly. Japanese brake and vehicle manufacturers introduced ABS

brakes based on the Bosch system as well as their own designs by the middle 1980s. The Bosch sys-

tem was used in 1986 Corvette and Cadillac Allante, followed by Ford in 1987. Since the late 1980s

and early 1990s, ABSs were found on nearly all top models of every manufacturer. By the late 1990s,

practically all passenger cars and light trucks were equipped with four-wheel ABSs, either as an option

or standard equipment.

Figure 1.1: ABS tests by Mercedes-Benz on S-Class passenger car, 1978.

As more complete accident statistics became available over the years, the contribution of ABS to

road safety is now unquestionable and has already saved thousands of lives over the years. In the

European Union, all passenger cars require ABS as a standard equipment since 2007, as well as other

safety devices. This measure will be extended to motorcycles in 2016.

From the moment that braking performance enhancement by the ABS was recognized, racing cars

were equipped with the state-of-the-art ABS technology. An experimental antilock braking system ABS

was used for the first time in the Porsche Bergspyder in 1968. In the Formula 1, it was used for the last

time in the notable Williams FW15C in 1993 and banned by FIA ahead of the 1994 season, along with

traction control and active suspension.

2

Figure 1.2: Williams FW15C featuring ABS, 1993.

1.2 Motivation (Formula Student)

Formula Student is a motosport and engineering competition between university teams across the globe.

Students are challenged to design and build a single-seat racing car in order to compete in several events

worldwide. Teams are evaluated by experienced judges from professional motorsport and automotive

industry, not only according to the dynamic performance of the prototype but also regarding its design

and manufacturing features, cost analysis, and marketability.

At Instituto Superior Tecnico, a FS team was founded in 2001 under the name of Projecto FST and

has built its fifth and most innovative prototype in 2013: FST 05e. Featuring a complete carbon-fiber

monocoque, rims, suspension rods and aerodynamic package, two AC electric motors (55 kW in total)

are used to drive 230 kg from 0 to 100 km/h in less than 3 s.

Besides the great number of innovations presented in the FST 05e, an ABS system has never been

used by the team in any of their prototypes. Commercial ABS kits are very expensive and a self-made

design has been yet a very complex and time-consuming task. However, the constant seek for the best

performance possible allied with the experience attained by the the team and the availability of new

development tools (sensors on the car, computer car model) has cleared the path to the implementation

of this system for the first time.

1.3 Objective

The purpose of this work is thus to design and model an effective and reliable solution for the ABS control

problem that can also be simple and inexpensive to implement on a FS prototype. In other words, the

3

Figure 1.3: Team from Projecto FST and FST 05e prototype.

more favourable the following factors are, the better and more suitable the designed controller is:

Performance of the prototype in competition, measured in terms of lap times and/or overall score.

Implementation by Projecto FST team members on the upcoming prototypes, concerning required

knowledge, necessary components, assembling and maintenance.

Price of all components, excluding the ones already available on the prototype.

Although only computational design and simulation is performed here, it is also the intent of this

thesis to pave the way for further physical implementation of ABS on an upcoming prototype. It aims to

be both a learning tool and a design guide for actual and future team members of Projecto FST.

1.4 Work contributions

It is relevant to outline the following contributions present in this thesis:

A self-developed vehicle model with 14 degrees of freedom (DOFs) was built for the design of theABS and subsequent simulation. The complete model, whose basis are described in Chapter 2,

includes the horizontal dynamics, relating tire longitudinal and lateral forces with the 3 DOFs for

vehicle position and orientation in the inertial reference plane; a 7-DOFs vibrational model of the

vehicles sprung and unsprung mass, integrated with load transfer and aerodynamic loads; a model

for the wheel dynamics, replicated for each corner, that outputs wheel angular velocity as a function

of the brake torque and tire longitudinal forces. Though the complete model is suitable for any four-

wheel vehicle, it is loaded with accurate parameters for the FST 05e prototype, measured either

experimentally or via CAD tools.

4

Brakeline dynamics, relevant for ABS design, were modelled and integrated in the complete vehiclemodel. It takes into account the hydraulic lag phenomenon and a pressure modulator model with

simple transient hydraulic equations.

A tire model was designed and coupled with the full vehicle model using experimental tire dataoutsourced from a renowned tire test facility. Three different fit models were used and compared,

including the well-known and widely used Pacejkas Magic Formula (MF), the Burkhardt velocity-

dependant equation, and an self-developed new approach using artificial neural networks (ANNs).

The proposed ABS control architecture resorts to a cascade structure with a brake pressure PDcontroller in the inner loop and a self-developed PID-type fuzzy controller for wheel slip regulation

in the outer loop. Beyond clearly reducing the braking time and distance, results evidence the high

adaptability of a FIS to the nonlinear dynamics involved, model uncertainties or external variances.

A wheel slip estimator is designed using a complementary filter, i.e., a Wiener filter equivalent tothe stationary Kalman filter solution, which merges information provided by sensors (accelerometer

and wheel speed encoder) over distinct, yet complementary frequency regions. Moreover, this

type of filter ensures stability and a single design parameter representing a compromise between

responsiveness and noise tolerance.

5

1.5 Outline

Chapter 2 presents the structure, concepts and mathematical basis that constitute the full vehicle model

and the integrated brakeline dynamics.

Chapter 3 describes the tire models tested in this work and the corresponding fitting to the experi-

mental tire data.

Chapter 4 introduces the main concepts and components of an ABS, and a problem definition from

the control theory point of view. The chapter ends with a literature review on the existing control methods.

Chapter 5 outlines the proposed control structure based on the available data and design require-

ments for the FST 05e prototype and the FS competition.

Chapter 6 describes the design of the ABS proposed in the previous chapter, including the pressure

controller, the wheel slip controller and the wheel slip estimator.

Chapter 7 presents the simulation results achieved with the ABS designed in Chapter 6. Different

and time-varying conditions are tested and the results are compared with the no-ABS situation.

Chapter 8 concludes this thesis by summarizing the main results and outlining further subjects to be

developed.

6

Chapter 2

Vehicle Dynamics

Vehicle dynamics is a very diverse and extensive topic. It refers to all the mechanical, physical or

aerodynamic principles governing the vehicle motion. For an ABS application, dynamics under braking

and tire-road interaction are the most important aspects to consider on a full vehicle model for further

controller design and simulation purposes. Hence, this chapter introduces the basic principles behind

the dynamics of the main sub-systems composing the aforementioned model.

After a global overview on the interdependencies between the subsystems (Section 2.1), the dy-

namics of each model are described in the following order: horizontal dynamics (Section 2.2); vertical

dynamics, including vibrational model, load transfer and aerodynamic (Section 2.3); wheel dynamics

(Section 2.4); brakeline dynamics with hydraulic modulator (Section 2.5).

2.1 Global Overview

The full vehicle model may be decomposed in five interconnected sub-models that are dependant on

each other, as depitected by the diagram in Fig. 2.1:

Horizontal Dynamics is a 3-DOFs model for the kinematics of the vehicles center of gravity (CG) (ac-

celeration, velocity and position/heading) on an inertial reference plane, as a mass point subjected

to tire longitudinal and lateral forces from the Tire Model.

Vertical Dynamics output the tire vertical load on each wheel, as a transient reaction force to the longi-

tudinal and lateral load transfers (due to acceleration) and the load due aerodynamics (dependent

on the velocity). Transient loads and displacements are measured by a 7-DOFs vibrational model

of the cars suspension and tires.

Brakeline Dynamics model the hydraulic braking system, which for an ABS-equipped vehicle also in-

clude an hydraulic modulator. Given an external input of pedal brake force (also from the driver), it

outputs the brake torque that is sent to the Wheel Dynamics.

7

BrakelineDyamics

Tb WheelDynamics

SX TireModel

FXT

HorizontalDynamics a

VerticalDynamics

v

FZT

Fbp

Figure 2.1: Vehicle dynamics overview.

Wheel Dynamics apply the brake torque from Brakeline Dynamics and the current tire longitudinal

force from the Tire Model and calculates the resulting wheel angular acceleration. The real-time

calculation of the longitudinal slip ratio SX (Section 4.3.1) that is returned to the Tire Model may

be considered part of the Wheel Dynamics.

Tire Model is an empirical model of the complex friction dynamics between the tire and the pavement.

It maps the tire forces and moments as a function of a set of input variables, namely the longitu-

dinal slip ratio SX and tire vertical load. Due to its importance and complexity, this is addressed

separately on Chapter 3.

The forthcoming sections describe in more detail the first four dynamics mentioned above. The actual

set of values used hereinafter is listed in the FST 05e parameterss Table 7.1.

2.2 Horizontal Dynamics

Horizontal dynamics are related to the kinematics of the car on the ground plane, as an inertial reference

plane. It is a 3-DOFs model that outputs the vehicle position in the Earth-fixed coordinate system

(XE ,YE)1 and its heading/yaw angle 2. Computation comprises three steps:

1. Resolution of individual tire longitudinal and lateral forces into a single force ~F = (FX , FY ) and

moment MZ acting on the vehicles CG. From the planar free-body diagram shown in Fig. 2.2:

1See Appendix A.12See Appendix A.2

8

,

,

, ,

,

Figure 2.2: Planar tire forces on vehicles corner i [2].

FX =i

FX,i =i

FXT,i cos i FY T,i sin i (2.1a)

FY =i

FY,i =i

FY T,i sin i + FY T,i cos i (2.1b)

MZ =i

MZT,i i

FX,i.yi +i

FY,i.xi (2.1c)

for i = FL, FR, RL, RR

where xi and yi designate the coordinates of tire i on the (X,Y ) axis system, as listed on Table 2.1.

i FL FR RL RR

xi a a b b

yiTF2 TF2 TR2 TR2

Table 2.1: Coordinates of tire i on the (X,Y ) axis system.

2. Calculation of resulting forces FX and FY and moment MZ into linear and angular acceleration

and velocity of the vehicle axis system3. As explained in [2], Newton-Euler equations of planar

motion for a rigid body in a coordinate frame attached to CG are:

FX = m(aX vY

)(2.2a)

FY = m(aY + vX

)(2.2b)

MZ = Izz (2.2c)3See Appendix A.1

9

( , )

Figure 2.3: Planar vehicle dynamics [2].

where aX = vX and aY = vY represent the scalar value of the components of vehicle acceleration

vector ~a in the direction of the X and Y axis4, respectively.

3. Vehicle velocity and acceleration conversion from the vehicle axis system to the Earth-fixed axis

system. This is achieved through the use of a rotation matrix:

xE

yE

0

=

cos sin 0

sin cos 0

0 0 1

vX

vY

0

(2.3)

4. Calculation of the vehicles path (position and yaw over time) by direct integration:

xE(t) = x0E +

t0

xE() d (2.4a)

YE(t) = y0E +

t0

yE() d (2.4b)

(t) = 0 +

t0

() d (2.4c)

where x0E , y0E and

0 refer to the respective initial positions.

4See Appendix A.1

10

2.3 Vertical Dynamics

Vertical dynamics are simulated through a 7-DOFs vibrational model of the cars suspension and tires, as

a system of masses (sprung and unsprung), springs and dampers. This allows the real-time calculation

of the tire loads when the vehicle is subjected to load transfer and aerodynamics. As depicted in Fig. 2.4,

the 7 DOFs correspond to:

Vertical displacement of the sprung mass zs (bounce);

Pitch angle5 of the sprung mass ;

Roll angle6 of the sprung mass ;

Vertical displacements of unsprung masses (hop) zu,FL, zu,FR, zu,RL and zu,RR.

Figure 2.4: 7 DOFs vibrational model [2].

For ease of calculations, the displacement, pitch and roll angles of the sprung mass can be translated

into displacements of its four corners as follow:

zs,i = zs + xi + yi (2.5)

2.3.1 Lagrange Method

The equations of motion for system can determined by applying Lagrange method [2]. For small and

linear vibrations the Lagrange equation assumes the following form:

d

dt

(K

qr

) Kqr

+D

qr+V

qr= fr r = 1, 2, , n (2.6)

5See Appendix A.26See Appendix A.2

11

where K, D and V are kinetic energy, dissipation function and potential energy, assuming the respective

forms

K =1

2mszs

2 +1

2Ixx

2 +1

2Iyy

2 +1

2

mu,F2

z2u,FL +1

2

mu,F2

z2u,FR +1

2

mu,R2

z2u,RL +1

2

mu,R2

z2u,RR (2.7)

V =1

2kS,F

(zu,FL zs + a + TF

2

)2+

1

2kS,F

(zu,FR zs + a TF

2

)2+

1

2kS,R

(zu,RL zs b + TR

2

)2+

1

2kS,R

(zu,RR zs b TR

2

)2+

1

2kT (zr,FL zu,FL)2 + 1

2kT (zr,FR zu,FR)2 +

1

2kT (zr,RL zu,RL)2 + 1

2kT (zr,RR zu,RR)2 +

1

2ARB,F

( zu,FL zu,FR

TF

)2+

1

2ARB,R

( zu,RL zu,RR

TR

)2(2.8)

D =1

2cS,F

(zu,FL zs + a + TF

2

)2+

1

2cS,F

(zu,FR zs + a TF

2

)2+

1

2cS,R

(zu,RL zs b + TR

2

)2+

1

2cS,R

(zu,RR zs b TR

2

)2+

1

2cT (zr,FL zu,FL)2 + 1

2cT (zr,FR zu,FR)2 +

1

2cT (zr,RL zu,RL)2 + 1

2cT (zr,RR zu,RR)2 (2.9)

Since the suspension spring is not directly actuated, but through a series of suspension mechanisms

(tire to upright, pullrod/pushrod and bellcrank), an equivalent spring rate called suspension rate or wheel

rate, ks is used in the equations.

kS =kspringMR2

(2.10)

where MR stands for motion ratio, defined as

MR =zu zsspring

(2.11)

The same rationale is applied to the damper, where cS is defined as

cS =cdamperMR2

(2.12)

Equation 2.6 is then evaluated for each one of the r = 1, . . . , 7 DOFs, from which a system of 7

independent differential equations is obtained. In the matrix form:

[M] {q}+ [C] {q}+ [K] {q} = {F} (2.13)

12

where [M], [C] and [K] are 7-by-7 matrices for the mass, damping and stiffness, respectively. Thevectors q, q and q represent the acceleration, velocity and position for each one of the 7 DOFs.

[M] =

ms 0 0 0 0 0 0

0 Ixx 0 0 0 0 0

0 0 Iyy 0 0 0 0

0 0 0mu,F

2 0 0 0

0 0 0 0mu,F

2 0 0

0 0 0 0 0mu,R

2 0

0 0 0 0 0 0mu,R

2

(2.14)

[C] =

2 cS,F + 2 cS,R 0 #1 cS,F cS,F cS,R cS,R

0 #2 0 cS,F TF2 cS,F TF2 cS,R TR2 cS,R TR2

#1 0 #3 a cS,F a cS,F b cS,R b cS,R

cS,F cS,F TF2 a cS,F cS,F + cT 0 0 0

cS,F cS,F TF2 a cS,F 0 cS,F + cT 0 0

cS,R cS,R TR2 b cS,R 0 0 cS,R + cT 0

cS,R cS,R TR2 b cS,R 0 0 0 cS,R + cT

(2.15)

#1 = 2 b cS,R 2 a cS,F

#2 = cS,FTF

2

2+ cS,R

TR2

2

#3 = 2 cS,F a2 + 2 cS,R b

2

13

[K] =

2 kS,F + 2 kS,R 0 #1 kS,F kS,F kS,R kS,R

0 #2 0 #4F #4F #4R #4R

#1 0 #3 a kS,F a kS,F b kS,R b kS,R

kS,F #4F a kS,F #5F ARB,FTF 2 0 0

kS,F #4F a kS,F ARB,FTF 2 #5F 0 0

kS,R #4R b kS,R 0 0 #5R ARB,RTR2

kS,R #4R b kS,R 0 0 ARB,RTR2 #5R

(2.16)

#1 = 2 b kS,R 2 a kS,F

#2 = kS,FTF

2

2+ kS,R

TR2

2+ ARB,F + ARB,R

#3 = 2 kS,F a2 + 2 kS,R b

2

#4i = kS,iTi2

+ARB,iTi

#5i = kS,i + kT +ARB,i

Ti2

The matrix of F is a 7-by-1 column matrix with the total forces on each DOF as a function of theexternal forces Fs,i and Fu,i.

{F} =

Fs,FL Fs,FR Fs,RL Fs,RR

Fs,FL TF2 + Fs,FR TF2 Fs,RL TR2 + Fs,RR TR2

Fs,FL a+ Fs,FR a Fs,RL b Fs,RR bM

cT zr,FL Fu,FL + kT zr,FL

cT zr,FR Fu,FR + kT zr,FR

cT zr,RL Fu,RL + kT zr,RL

cT zr,RR Fu,RR + kT zr,RR

(2.17)

14

2.3.2 Vehicle Loads

At any point in time, the vehicles sprung and unsprung masses are subjected to the following types of

load:

Static load

Dynamic load

Load Transfer

Aerodynamics

Static Load

Static load corresponds to the load when the car is stationary and subjected to its own weight. Simple

free-body diagram calculations for the sprung and unsprung mass result in:

F sts,FL = Fsts,FR =

ms g

2

b

L(2.18a)

F sts,RL = Fsts,RR =

ms g

2

a

L(2.18b)

F stu,FL = Fstu,FR =

mu,F g

2(2.18c)

F stu,RL = Fstu,RR =

mu,R g

2(2.18d)

Load Transfer

When the car accelerates, inertial forces are generated on the CG of the sprung and unsprung masses.

For a positive longitudinal acceleration (drive), this has the effect of transferring the load from the front

to the rear axle, whilst a positive lateral acceleration (left curve) transfers the load from the inner to the

outer tires.

With the help of the free-body diagram for longitudinal load transfer, shown in Fig. 2.5, the sum of the

longitudinal forces and moments results in:

Fwtxs = ms aXhsL

(2.19a)

F xwtu = (mu,F +mu,R) aXhuL

(2.19b)

Lateral load transfer is similarly calculated from the free-body diagram in Fig. 2.6.

Fwtys,F = ms aYb

L

hsTF

(2.20a)

Fwtys,R = ms aYa

L

hsTR

(2.20b)

15

Lba

FX

2FF wtx -2FRwtx

h

Figure 2.5: Longitudinal load transfer.

Fwtyu,F = mu,F aYhuTF

(2.20c)

Fwtyu,R = mu,R aYhuTR

(2.20d)

The total load change due to load transfer is, respectively, for each corner of the sprung and unsprung

masses:

Fwts,FL = Fwtxs

2Fwtys,F (2.21a)

Fwts,FR = Fwtxs

2+ Fwtys,F (2.21b)

Fwts,RL =Fwtxs

2Fwtys,R (2.21c)

Fwts,RR =Fwtxs

2+ Fwtys,R (2.21d)

Fwtu,FL = Fwtxu

2Fwtyu,F (2.21e)

Fwtu,FR = Fwtxu

2+ Fwtyu,F (2.21f)

Fwtu,RL =Fwtxu

2Fwtyu,R (2.21g)

Fwtu,RR =Fwtxu

2+ Fwtyu,R (2.21h)

16

TF ,TR

FX

-Fiwty Fiwty

h

Figure 2.6: Lateral load transfer.

Aerodynamic Load

Aerodynamic forces are, along with tire forces and gravity, the only external source of forces acting

on the car. The total aerodynamic force is commonly expressed into three components acting on the

vehicles CG:

Drag is the force component acting parallel and opposite to velocity of the vehicle.

FD =1

2ACD v

2 = cD v2 (2.22)

where is the air density, A is a reference frontal area and CL is the lift coefficient.

Lift is the force component acting perpendicular velocity of the vehicle. Since this force points down-

wards in a racecar, pushing the sprung mass against the ground, it is frequently called Downforce.

FL =1

2ACL v

2 = cL v2 (2.23)

Pitching Moment is the moment that results when the above forces are translated from the center of

pressure cp to the vehicles CG.

M = cM v2 (2.24)

Since the aerodynamic package (front wing, rear wing and undertray) are part of the sprung mass,

aerodynamic loads are only applied on sprung mass DOFs. From the free-body shown in Fig. 2.7, the

17

load change due to aerodynamic forces is given by:

L

ba

FD

2FF aero -2FRaero

h

FL

CP

MA

Figure 2.7: Aerodynamic load.

F aeros,FL = Faeros,FR =

FD hs + FL bM2L

(2.25a)

F aeros,RL = Faeros,RR =

FD hs + FL a+M2L

(2.25b)

2.3.3 Inverse Kinematics

Since the available parameter values are taken from the static deflection state, i.e., measured when the

car is stationary and loaded only with its own weight (static load), the vibrational model is excited solely

with dynamic loads, i.e., the external forces on vector {F} (Eq. 2.17) are given by:

Fs,i = Fwts,i + F

aeros,i (2.26a)

Fu,i = Fwtu,i (2.26b)

Equation 2.13 is numerically solved in order of q, using previous time-step values for q and q.

{q} = [M]1(

[F ] [C] {q} [K] {q})

(2.27)

18

or, in the discrete difference form

qi((k + 1)Ts) = [M]1(

[F(kTs)] [C] qi(kTs) [K] qi(kTs))

(2.28)

Velocity and position can then be calculated through integration over time.

q(t) =

q()d (2.29a)

q(t) =

q()d (2.29b)

2.3.4 Tire Vertical Load

Total tire vertical load is the sum of the static loads and the transient load retrieved from the vibrational

model, as a reaction force on the road for zr,i = zr = 0:

FZT,i = Fsts,i + F

sts,i + cT .zu,i + kT .zu,i (2.30)

Therefore, vertical dynamics yield the tire vertical loads as a function of the vehicle acceleration and

velocity vectors.

FZT,i = f(Fs,i, Fu,i) = f(aX , aY , vX , vY ) (2.31)

2.4 Wheel Dynamics

The dynamics of the wheel under braking are one of the most important areas in Vehicle Dynamics when

developing an ABS controller. As described in the beginning of this chapter, Wheel Dynamics deal with

the sum of forces and moments acting on the wheel to calculate the rotational kinematics of the wheel,

namely wheel angular acceleration W and velocity W . With the knowledge of the vehicle velocity,

Wheel Dynamics ultimately calculates and outputs the actual longitudinal slip ratio SX , necessary for

the Tire Model.

2.4.1 Free-body diagram

The free-body diagram shown in Fig. 2.8 yields the following equations:

m

4vX = FXT (2.32)

IW W = TW RLFXT (2.33)

19

Figure 2.8: Freebody diagram of the wheel.

where the wheel torque TW can be positive or negative, either this comes from the motor or the brakes.TW = Td, if TW > 0 (Driving Torque)TW = Tb, if TW < 0 (Braking Torque) (2.34)

Longitudinal tire forces FXT , as described in the next chapter, depend on the longitudinal slip ratio

SX . Its calculation, though part of the Wheel Dynamics, is addressed later in Section 4.3.1.

2.5 Brakeline Dynamics

Brakeline refers to the mechanical and hydraulic systems that transform the force applied on the brake

pedal Fbp into a brake torque Tb exerted by the calipers on each wheel. These include the brake pedal,

master cylinders, brake lines, wheel calipers and rotors. In the absence of an ABS controller, brakeline

components can be divided as depicted in Fig. 2.9.

Figure 2.9: Schematic of basic brakeline dynamics.

2.5.1 Steady-state equations

Brake pedal acts as a lever that multiplies Fbp by a geometric constant designated as Pedal Ratio, PR.

The resultant force is distributed to the master cylinders responsible for the front and rear brake lines,

20

according to the brake bias BB (equations 2.35).

Fmc = Fbp PR (2.35a)

Fmc,F = Fmc BBF (2.35b)

Fmc,R = Fmc BBR (2.35c)

Assuming an incompressible a brake fluid and ideal efficiency for brakeline components, pressure is

kept constant:

pmc = pcal = p =FmcAmc

=FcalAcal

(2.36)

Thus, the relation between the force on the master cylinder piston and the force exerted on caliper

pistons is given by the squared inverse of the ratio between its diameters:

Fcal = FmcAcalAmc

= Fmc

(dcaldmc

)2(2.37)

The hydraulic pressure translated into mechanical force by the caliper is then applied on a couple of

brake pads, one on each side of the rotor, so a clamping force is generated. With the knowledge of the

coefficient of friction between the brake pads and the rotor, pads, brake torque is finally calculated:

Fpads = 2Fcal (2.38a)

Tb = FbRb = pads FpadsRb (2.38b)

where Rb designates the effective brake radius, the distance between the rotor center and the center of

pressure of the caliper pistons.

Introducing equations 2.37 and 2.38 on Eq. 2.35a, brake torque becomes a linear function of the

brake pedal force:

Tb = f(Fbp) = 2pads

[PR

(dcaldmc

)2] Fbp Rb

= K Fbp = 1.02 Fbp(2.39)

2.5.2 Transient behaviour

For most applications, the response characteristics of hydraulic brake systems are such that the time

lags between input and output variables are very small, typically less than 0.5 s [12]. However, when

it comes to ABS design, the dynamic response of an hydraulic brake system and its individual brake

components becomes important [11].

21

The main dynamic elements in a typical brake system are the brake lines. The flow of brake fluid

from the master cylinder to the wheel cylinder is a function of fluid viscosity, cross-sectional flow area,

and brake line length [11]. As fluid viscosity increases, the time interval between the application of force

to the brake pedal and operation of the wheel brake increases and may affect the performance of an

ABS. To account for this hydraulic lag, a dynamic behaviour is introduced in the Simulink model through

a first-order transfer function with unitary gain and time constant :

G =1

s+ 1(2.40)

The dynamic response of applying a pedal force of 1 N is depicted in Fig. 2.10. As expected, the

steady-state output (static gain) is 1.02 Nm.

Figure 2.10: Step response of brakeline dynamics Tb(t) for = 0.1 s.

2.5.3 Hydraulic Modulator

A hydraulic pressure modulator is an electro-hydraulic device for reducing, holding, and restoring the

pressure within a hydraulic circuit. It forms the hydraulic link between the brake master cylinder and the

wheel-brake cylinders (see Figures 2.11 and 4.2) and is essential for any ABS application.

The pressure is regulated by an ECU that sends a control signal for energizing one or both inlet and

outlet solenoid valves, according to Table 2.2. The maximum pressure is determined by the pressure

on the master cylinder. Depending on the design, this device may also include a pump/motor assembly,

accumulator and reservoir, as depicted in Fig. 2.12.

22

Figure 2.11: Schematic of brakeline dynamics with hydraulic modulator.

Figure 2.12: Schematic of a hydraulic modulator [3].

Mode Inlet Valve (State) Outlet Valve (State)

Increase/Restore Open (0) Closed (0)

Hold Closed (1) Closed (0)

Decrease Closed (1) Open (1)

Table 2.2: Hydraulic modulator solenoid valves.

Hyraulic modulator valve positions work as described here:

Open When the valve is open, pressure from the master cylinder is free to pass through the brake

circuit. In this condition, the amount of brake pressure is directly controlled by the driver.

Closed/Hold When the valve is closed, the pressure in the circuit is retained and cannot increase,

regardless of how hard the driver pushes the brake pedal.

Release In this position, not only the brake circuit is isolated from the brake pedal, but the amount of

braking pressure on the wheel is actively reduced by pumping the brake fluid back to the master

cylinder reservoir. This is felt by the driver as a reaction force on the brake pedal.

23

The hydraulic modulator behaviour is simply modelled as follow:

u = (0, 0) dpdt = pmcppmc Rinc Pressure increase

u = (1, 0) dpdt = 0 Pressure hold

u = (1, 1) dpdt = Rdec Pressure decrease

(2.41)

where u is the binary control signal for inlet and outlet valves, Rinc and Rdec are the rates of pressure

increase/decrease. Fig. 2.13 shows the pressure over time for an example input control signal u.

Figure 2.13: Simulink implementation of the hydraulic modulator.

24

Chapter 3

Tire Model

Tires are the primary source of the tractive, braking, and cornering forces/torques that provide the han-

dling and control of a car. For that reason, they are one of the most important components and its

understanding in respect of magnitude, direction and limit of those forces/torques is essential for any

application in vehicle dynamics and/or vehicle control system. However, the complexity of tire behaviour

is not yet completely understood even in professional motorsports theyre often known as black magic

and the development of a model for all driving conditions in real-time is still a very challenging task.

In this chapter, three tire models are described: two semi-empirical models, Pacejka (Section 3.1.1)

and Burkhardt (Section 3.1.2); and ANN-based model (Section 3.1.3). The models are then used to

fit the experimental data (illustrated in Section 3.2), whose methodology is described and results are

compared in Section 3.3.

3.1 Types of models

Several tire models can be found in the literature. Depending on how one approaches the problem,

models can be classified between theoretical and empirical (Fig. 3.1).

From experimental

data only

Using similarity

methods

Through simple

physical model

Through complex

physical model

Empirical Theoretical

Figure 3.1: Classification of tire models.

25

Nevertheless, the objective of every model is to output the tire horizontal forces and/or moments as

a function of one or more rolling conditions: tire vertical load, longitudinal slip ratio, slip angle, inclination

angle, temperature, pressure and so on.

[FXT , FY T ,MXT ,MY T ,MZT ] = f(FZT , SX , , , T, p, . . .) (3.1)

3.1.1 Pacejka Model

Pacejka1 tire model [13] is the most widely used tire model to calculate steady-state tire force and

moment characteristics for use in vehicle dynamics. As a semi-empirical model, it is particulary useful to

represent the tire as a vehicle component in a vehicle simulation environment, as in this work. This model

is classified as semi-empirical because, despite being based on measured data, it contains structures

that find their origin in physical models like the brush model [14].

Pacejka tire model is given by the parametric equation, known as MF:

y = D sin [C arctan {Bx E (Bx arctanBx)}]Y (X) = y(x) + SV

x = X + SH

(3.2)

where Y is the output variable FXT , FY T or MZT and x is the main x-axis input variable, namely tan or

SX . The remaining parameters listed in Table 3.1 represent geometric characteristics of the MF curve,

as shown in Fig. 3.2.

B stiffness factor

C shape factor

D peak value

E peak value

SH horizontal shift

SV vertical shift

Table 3.1: Magic Formula coefficients

When multiple inputs are needed for the tire model (e.g., varying tire vertical load and road coefficient

of friction is important for this application), the aforementioned parameters are itself given by parametric

functions (of parameters pi, qi, ri or si) whose inputs are the supplementary input variables FZT , and

p. The actual expressions for these functions differ depending on the type of tire data to be fitted:

Longitudinal Force for pure longitudinal slip, FXT = f(SX , = 0)

Lateral Force for pure side slip, FY T = f(SX = 0 , )1Hans Bastiaan Pacejka (Rotterdam, 1934), Professor emeritus at Delft University of Technology in Delft, Netherlands, is an

expert in vehicle system dynamics and particularly in tire dynamics, fields in which his works are now standard references.

26

Figure 3.2: Pacejkas Magic Formula curve and geometric parameters.

Aligning Torque for pure side slip, MZT = f(SX = 0 , ).

Longitudinal Force for combined slip, FXT = f(SX , )

Lateral Force for combined slip, FY T = f(SX , )

Aligning Torque for combined slip, MZT = f(SX , )

In this thesis, only the first case is considered for the development of the tire model. Assuming

constant pressure p and inclination angle , the simplified equations in the Pacejkas 2002 version are

displayed next (scale factors i are ignored).

FXT = Dx sin (Cx arctan (Bxkx Ex (Bxkx arctan (Bxkx)))) + SV x (3.3a)

kx = SX + SHx; (3.3b)

dfz =FZT F 0ZT

F 0ZT(3.3c)

Cx = pCx1 (> 0) (3.3d)

Dx = xFZT (> 0) (3.3e)

x = (pDx1 + pDx2dfz) (3.3f)

Ex = (pEx1 + pEx2dfz + pEx3dfz2)(1 pEx4 sgn(kx)) ( 1) (3.3g)

Kx = FZT (pKx1 + pKx2dfz) exp(pKx3dfz) (3.3h)

Bx =KxkCxDx

(3.3i)

SHx = pHx1 + pHx2dfz (3.3j)

SV x = FZT (pV x1 + pV x2dfz)x (3.3k)

27

3.1.2 Burkhardt Model

Burkhardt tire model [14] is a static parametric model given by the following velocity-dependent para-

metric expression:

F (x) =[A(1 eBx) Cx] eDxv (3.4)

where (F, x) is any of the following combinations: (FXT , SX), (FY T , ) or (MZT , ).

Although this model takes into account the influence of the vehicle velocity v on the friction curve,

there is no dependency on the tire vertical load FZT or inclination angle . For that reason, a given set

of coefficients only applies for a given set of rolling conditions.

3.1.3 Neural Network Model

Inspired by biological nervous systems, artificial neural network (ANN) is a network structure consist-

ing of a number of parametric nodes connected through directional links defining a causal relationship

between them. The outputs of each node, and thus the output of the network, rely on the adaptable

values of the nodes parameters. A learning rule specifies how these parameters should be updated to

minimize a prescribed error measure between the networks actual output and a desired output [15].

Figure 3.3: Artificial neural network (ANN) scheme.

Trying to take advantage of the effectiveness of ANN for nonlinear curve fitting, a tire model has been

developed using this type of soft-computing algorithm. A model of this kind is not only relatively fast to

develop, it also allows the use of any mapping combination of variables as in the Pacejka tire model

(e.g., FXT = f(SX , FZT ), or {FXT , FY T } = f(SX , , FZT )).

28

3.2 Experimental Data

Having tire data is a major asset for any racing team: being able to analyse and use them to gain

advantage against your competitors is what every team wants to do. The data used throughout this

work is made available by the Tire Test Consortium (TTC). TTC is a volunteer-managed organization of

Formula SAE teams who pool their financial resources to obtain high quality tire force and moment data.

All tests are conducted at Calspan TIRF, a renowned tire testing facility in Buffalo, NY. As a member of

this consortium, Projecto FST has been using this data as a starting point for its prototypes design.

Figure 3.4: Tire test at TIRF [4].

Up to now, 5 rounds of tire tests have been delivered since 2005, each of them testing a different

set of FS tires from different brands, sizes and/or compounds. Each round is composed of several runs

regarding different combinations of tires, rims, operating conditions, test type and/or procedures. In turn,

each run test the effect of n variables in a series of sweeps of the slip ratio or slip angle (either if it is

a drive/brake or a cornering test) for a unique combination of a finite number of possible values for the

remaining n 1 variables. The example procedure for each run is presented next [4]:

1. Set run conditions (e.g., tire of brand A, size/compound B on a 7 rim width).

2. Select a combination of values for the n 1 variables except slip ratio or slip angle (e.g., FZT =150 lbs, = 0 deg and p = 12 psi).

3. Sweep a range of values for the slip ratio/slip angle and retrieve output data (tire forces, moments,

temperatures, etc.).

4. Change the value of one of the n1 variables so as to have a new combination (e.g., FZT : 150250 lbs).

5. Sweep a range of values for the slip ratio/slip angle and retrieve output data (tire forces, moments,

temperatures, etc.).

29

6. If all combinations have been tested, stop run. Else, return to step 4.

Figure 3.5 shows the value of 5 variables during a complete example drive/braking run. As described,

there is SX sweep for each one of the possible combinations of the other 4 variables, , FZT , and p, as

a series of for -cycles. In red is highlighted a sample of data for FZT = {50, 150, 250, 350} lbs, = = 0and p = 96.5 kPa (14 psi). The output FXT of these data is plotted over SX in Fig. 3.6.

Figure 3.5: Excerpt data from a complete run. In red, data for FZT = {50, 150, 250, 350} lbs, = = 0and p = 96.5 kPa (14 psi).

30

0.3 0.25 0.2 0.15 0.1 5 102 0 5 102 0.1 0.15 0.2 0.25 0.34,000

3,000

2,000

1,000

0

1,000

2,000

3,000

4,000

SX

FX

T(N

)

FZT = 350 lbsFZT = 150 lbsFZT = 250 lbsFZT = 50 lbs

Figure 1.6: Pacejka Model vs. TTC Data for FZT = {50, 150, 250, 350} lbs.

1.3 Data Fitting

The methodology used for fitting the experimental data depends on the model. Hence,

Pacejka and Burkhardt models make use an optimization technique that computes the model param-

eters that minimize the error to the real data.

Neural Network model fitting is achieved by network training using MATLABs Neural Network toolbox.

1.3.1 Methodology for Pacejka and Burckhardt models

The development of the empirical models described before is related with the estimation of the set of

coefficients pi for the Pacejka Model and the coefficients {A,B,C,D} for the Burkhardt model that best

7

Figure 3.6: Plot of SX vs. FXT for four sweeps of different FZT , with = = 0 and p = 96.5 kPa(14 psi).

3.3 Data Fitting

The methodology used for fitting the experimental data depends on the model. Hence,

Pacejka and Burkhardt models make use an optimization technique that computes the model param-

eters that minimize the error to the real data.

Neural Network model fitting is achieved by network training using MATLABs Neural Network toolbox.

3.3.1 Methodology for Pacejka and Burckhardt models

The development of the empirical models described before is related with the estimation of the set of

coefficients pi for the Pacejka Model and the coefficients {A,B,C,D} for the Burkhardt model that best

31

fit the TTC data shown in Section 3.2. It is therefore an optimization or nonlinear regression problem

that is adressed using the least squares method approach.

Least Squares Method

The best fit in the least squares sense is the column vector of parameters p = pi that minimizes the sum

of the squared residuals, i.e., the difference between an experimental value and the fitted value provided

by a model.

minp

i

r2i =i

(F (p, xi) yi)2 (3.5)

In MATLAB, this problem is solved using either the Curve Fitting Toolbox or the lsqcurvefit func-

tion with the parameters listed in Table 3.2. To avoid the algorithm to get stucked in local minima or

take too much iterations, a starting set of Pacejka and Burkhardt parameters is given by [13] and [16],

respectively.

Algorithm Levenberg-Marquardt

TolFun 1 106

TolX 1 106

MaxIter 1 106

Others default

Table 3.2: MATLABs lsqcurvefit parameters [1].

3.3.2 Results

Pacejka

Figure 3.7 plots the sample TTC data from Fig. 3.6 and the final optimized MF curve. As expected, a very

good fit is achieved for every FZT . Moreover, the structure of Eqs. 3.3 ensure that model extrapolation

beyond |SX | > 0.2, for which no experimental data is available, is relatively accurate, at least qualitatively[13].

Burkhardt

Since the Burkhardt model does not allow a change in the rolling conditions, Eq. 3.4 can only fit a single

TTC sweep of SX . For FZT = 350 lbs, = 0, = 0 and p = 14 psi, Fig. 3.8 shows the achieved fit.

As in the Pacejka model, the parametric expression defining the Burkhardt model (Eq. 3.4) is specific to

tire fitting, thus qualitatively correct. However, poor fitting on the limits of the experimental data suggest

that extrapolation is not advised in this case.

Nonetheless, using the same values for the coefficients {A,B,C,D}, Burkhardt model is useful forreproducing the influence of the velocity on the tire force. As expected from [17], Fig. 3.8 shows that the

32

0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.54,000

3,000

2,000

1,000

0

1,000

2,000

3,000

4,000

SX

FXT(N

)

FZT = 350 lbsFZT = 150 lbsFZT = 250 lbsFZT = 50 lbs

Figure 1.7: Pacejka Model vs. TTC Data for FZT = {50, 150, 250, 350} lbs.

9

Figure 3.7: Pacejka Model vs. TTC Data for FZT = {150, 250, 350} lbs.

peak value of FXT decreases for higher velocities.

0.4 0.35 0.3 0.25 0.2 0.15 0.15 102 0 5 102 0.1 0.15 0.2 0.25 0.34,000

3,000

2,000

1,000

0

1,000

2,000

3,000

4,000

SX

FX

T(N

)

TTC Datav = 40.2 km/h (25 mph)v = 60 km/hv = 80 km/h

Figure 1.8: Pacejka Model vs. TTC Data for FZT = {50, 150, 250, 350} lbs.

10

Figure 3.8: Burkhardt Model vs TTC Data for different velocities.

33

Neural Network

After several iterations of differently trained ANN, the network with the best fit was selected. Figure

3.9 shows the same experimental shown in Fig. 3.6 (also used for the Pacejka model fitting) and the

fitted Neural Network model. As in the Pacejka model, a very good fit is achieved for the experimental.

However, network outputs are not valid for SX values outside the experimental range.

0.4 0.35 0.3 0.25 0.2 0.15 0.15 102 0 5 102 0.1 0.15 0.2 0.25 0.34,000

3,000

2,000

1,000

0

1,000

2,000

3,000

4,000

SX

FX

T(N

)

FZ = 350FZ = 150FZ = 250FZ = 50

Figure 1.9: Pacejka Model vs. TTC Data for FZT = {50, 150, 250, 350} lbs.

11

Figure 3.9: Neural Network Model vs. TTC Data for FZT = {150, 250, 350} lbs.

Comparison

Considering the results achieved with the three tire models, a decision has been made on the Pacejka

model to be coupled with the full vehicle model for design and simulation of the ABS controller. In fact,

it is the only model that ensures accuracy within the complete range of possible SX values, with special

importance for the wheel-lock case SX = 1 (see Section 4.3.1). Mean squared error (MSE) results forthe three models are summarized in Table 3.3.

Model MSE

Pacejka 9594 N2

Burkhardt 26477 N2

NN (global) 8706 N2

Table 3.3: MSE fitting result for Pacejka, Burkhardt and Neural Network tire models.

34

Chapter 4

ABS Overview

This chapter gives a further overview of the ABS. It starts by outlining the ABS objectives (Section 4.1)

and describing its main components (Section 4.2). Afterwards, the problem of wheel slip control is

defined from the control theory point of view (Section 4.3), along with the definition of the longitudinal

slip ratio SX (Section 4.3.1). The chapter ends with a literature review on the more relevant control

methods (Section 4.4).

4.1 Objectives of ABS

Modern ABSs has three main goals that must be fulfilled, regardless of the application, system architec-

ture or chosen control algorithm:

Stopping Distance By maximizing the longitudinal frictional force under braking the stopping distance

will be minimized, for the same vehicle mass m and initial velocity v0.

dbraking = v20

2 aX where aX =FX()

m(4.1)

Steerability Available lateral friction force decays with wheel slip towards lock-up. To ensure steerability

during braking, the ABS should be able to maintain enough lateral friction force.

Stability Achieving the maximum friction level on all four wheels is not always desirable, specially for

sudden and uneven changes on the surface . Significant asymmetries between right and left side

braking forces result on a yaw moment and consequent vehicle instability, as depicted by Fig. 4.1.

Therefore, the ABS must keep balanced braking forces, even if the total force is not maximized.

35

Figure 4.1: Buildup of yaw moment induced by large differences in friction coefficients [5].

4.2 ABS Components

A typical ABS comprises a series of subsystems interconnected in a control loop (Fig. 4.2 and Fig. 4.4).

The main components of an ABS are described next:

Figure 4.2: ABS components diagram.

Electrical control unit (ECU) Usually microprocessor-based, it is the core component that receives,

filters and amplifies the signal from the sensors and performs the necessary calculations for ve-

hicle velocity and wheel slip estimation. The ECU then sends a signal to the hydraulic modulator

according to the implemented control algorithm.

36

Hydraulic Modulator An electro-hydraulic device that, depending on the control signal from the ECU,

is able to increase, hold or decrease the brakeline pressure downstream to the wheel calipers, by

energizing a couple of solenoid valves. The hydraulic modulator is described in more detail and

modelled in Section 2.5.3 as part of the Brakeline Dynamics.

Sensors These are connected with the ECU an provide feedback measurements of the rotational veloc-

ity of all wheels (normally an inductive or Hall-effect encoder mounted on the hub), brake pressure

and acceleration (accelerometer on the CG).

4.3 Wheel Slip Control

From Section 4.1, all ABS objectives rely on keeping tire longitudinal and lateral friction forces, FXT and

FY T , at desired levels. As described earlier in Chapter 3, these forces depend, among other variables,

on the value of the wheel slip SX .

{FXT , FY T } = f(SX , . . .) (4.2)

Before any formalization of the control problem, the mathematical concept of wheel slip or longitudinal

slip ratio SX is formally introduced in the next section.

4.3.1 Longitudinal Slip Ratio

A loaded tire is considered to be rolling freely if neither a driving nor a braking torque is applied (assuming

a linear path, zero slip angle and zero inclination angle). In this condition, the tire longitudinal velocity

and the vehicle longitudinal velocity are equal, and a reference wheel-spin velocity can thus be defined

as:

W0 =vXRe

(4.3)

where Re is the so-called effective rolling radius [13]. For simplification purposes, this will be considered

constant and equal to the tire loaded radius RL.

dRedt

= 0

Re(t) = RL

(4.4)

When a torque is applied about the wheel-spin axis, tire deformation is generated and causes a slip

to arise between the tire and the road [13]. This means that the actual wheel-spin velocity or wheel

37

angular velocity W is different from the reference wheel-spin velocity, i.e., the wheel and the vehicle

have different longitudinal velocities.

W 6= W0 W .RL 6= vX (4.5)

The difference between these velocities is called the longitudinal slip velocity vs:

vs = W W0 = W vXRL

(4.6)

Finally, the ratio of longitudinal slip velocity to the reference wheel-spin velocity is formally defined

in [7] as the longitudinal slip ratio, or simply wheel slip SX :

SX =W W0

W0=WRe vX

vX(4.7)

According to Eq. 4.7, to following situations may occur:

W W0 SX Wheel is spinning

W > W0 SX > 0 Tire driving force (FXT > 0)

W = W0 SX = 0 Free-rolling tire (FXT = 0)

W < W0 SX < 0 Tire braking force (FXT < 0)

W = 0 SX = 1 Wheel lock-up

(4.8)

When the car stops W = vX = 0, which leads to the mathematical indetermination 00 . To avoid the

consequent computacional error, the denominator is replaced by max(vX , ) in the Simulink environ-

ment, where eps = 252 (double-precision floating-point relative accuracy [1]).

4.3.2 Control Problem Definition

Recalling a typical tire friction curve from Chapter 3, fulfilling the ABS objectives means that the longitu-

dinal slip ratio SX should be kept within a range so that longitudinal and lateral friction are near its peak

values (Fig. 4.3). Hence, the ABS control problem is inherently related with the control of wheel slip. As

the optimum value of SX do not coincide for both FXT and FY T , a compromise is frequently necessary.

In practical terms, since vehicle speed vX cannot be directly controlled, wheel slip control is executed

by controlling wheel angular velocity W alone, which in turn is accomplished by manipulating the brake

pressure on the wheel caliper pcal. The complete set of closed-loop variables for wheel slip control is

listed in Table 4.1. Disturbances in the control loop may refer to, e.g. changes in the road conditions

(road profile zr, coefficient of friction ), sensor noise w and others.

38

0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 4,000

3,500

3,000

2,500

2,000

1,500

1,000

500

0

SX

F T(N

)

FXT( = 0.65, Pacejka)FXT( = 0.65, TTC)FXT( = 0.2, Pacejka)FYT( = 6o, TTC)

UnstableRegion

StableRegion

SXrefRegion

Figure 4.3: ABS action zones.

Figure 4.4: ABS control loop components [5]: 1-Hydraulic Modulator, 2-Master Cylinder, 3-Wheel caliper,4-ECU.

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Object variable SX Longitudinal slip ratio

Controlled variable W Wheel speed

Reference variable SrefX Reference longitudinal slip ratio

Manipulated variable pb Braking pressure

Disturbances zr, , w . . . Road conditions, sensor noise, etc.

Table 4.1: Closed-loop variables for wheel slip control.

4.3.3 Main difficulties

The main difficulties in the design of any ABS control arise the strong nonlinearity and uncertainty of the

system to be controlled, namely:

The interaction between the tire and the road surface is a very complex and hardly understoodarea of knowledge. Friction models like the ones described in Chapter 3 are experimental-based

approximations of highly nonlinear phenomena.

The dynamics of the whole vehicle are nonlinear and may even vary over time. Additionally, mod-els often have to deal with a certain amount of parametric or structural uncertainty (e.g. vehicle

parameters, differential exactitude, model simplifications, etc.).

ABS control requires significant insensitivity to the previous uncertainties and variations that arelikely to happen, e.g., changes of vehicle loading, unknown and changeable road conditions, com-

ponents wear, sensor noise.

Vehicle velocity, essential for wheel slip calculation, and road coefficient of friction are hardand expensive to measure directly. Reliable ABS performance depend on good estimates of this

variables.

ABS actuators are discrete and control precision must be achieved with only three types of controlcommands: build pressure, hold pressure or reduce pressure (see Section 4.2).

Another meaningful aspect is the fact that the system is unstable for wheel slip values beyond the

peak friction force seen in Fig. 4.3. Since longitudinal force diminishes from this point on, an increase in

wheel slip leads to a decrease in the reaction torque exerted by the road to counteract the brake torque,

thus continuously increasing wheel slip towards lock-up.

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4.4 Control Methods

From the detailed issues discussed above, wheel slip control is, indeed, a very complex and challenging

problem. Several approaches and constant improvements are still being released in newer versions of

ABSs. The predominant control methods existing in the literature can be categorized in four different

areas:

Threshold control

PID control

Sliding-mode control (SMC)

Intelligent control

The most relevant applications of each control method are reviewed in the next subsections.

4.4.1 Threshold Control

Threshold control is a simple control method that has been widely applied in the early ABS versions.

Generic threshold algorithm usually uses wheel acceleration and/or wheel slip as controlled variables

and defines threshold values, above/below which the pressure should be increased, held or decreased.

A typical control cycle earlier implemented by Bosch is described in [5] and uses three threshold values

for wheel acceleration (-a, +a and +A), as well as a slip-switching threshold (1) to manage pressure.

Control cycles with different threshold variables and criteria may also be used, with significant changes

on ABS performance, as tested and compared by Rangelov in [18].

Since the pressure is cyclically being increased, held and decreased based solely on binary states

of the input variables (Fig. 4.5), wheel speed oscillations over time are less controllable and ABS per-

formance is less effective than with other continuous control methods [19]. Nonetheless, as a simple

control algorithm, threshold control and its variants have been extensively applied in the literature over

the years. More recently, a four-wheel vehicle model with integrated vertical, horizontal, lateral and brak-

ing dynamics is used in [20] to evaluate the performance of a threshold ABS under various operating

conditions, with satisfactory results.

4.4.2 PID Control

Conventional PID is the most widely used control method in industry. Its simplicity makes it easy to

understand, design/tune and implement. It has been used for many years to successfully solve a diver-

sity of problems, from which ABS is no exception. Even for complex and changing surface types, good

results can be attained with conventional PID control algorithms [21].

41

Figure 4.5: ABS algorithm.

In the recent years, the well-known features of the conventional PID have been combined with the

robustness, self-tuning or adaptability to nonlinear models of other control methods, which highly en-

hanced ABS performance. A nonlinear PID control algorithm, similar to gain-scheduling algorithm, has

been applied in [22] to a class of truck ABS problems. Results show that ABS performance is increased

by adding robustness to the conventional PID. Lu et al [23] designed a fuzzy PID controller combined

with a simple automatic optimization algorithm for PID parameters. This improved PID showed better

control precision and robustness when compared with the conventional PID.

4.4.3 Sliding-mode Control

Sliding-mode control (SMC) is a form of robust control that provides an effective method to control

nonlinear plants. It consists of a sliding surface where a predefined function of error is zero, which

models the desired closed-loop performance; and a control law, such that the system state trajectories

are forced toward the sliding surface. Once the sliding surface is reached, the system state trajectories

should stay on it [24,25]. Due to the discontinuous switching of control force in the vicinity of the sliding

surface, SMC may, on the other hand, produce chattering effects which can cause system instability and

damage to both actuators and plants. In the development of SMC, the design of the sliding surface is of

significant importance, as it dictates the behaviour and performance of the overall control system [26].

A grey sliding-mode controller is designed by Kayacan et al [24] to maintain wheel slip at the desired

value. The proposed structure includes a grey predictor to anticipate the forthcoming velocity-dependent

values of wheel slip and the reference wheel slip. Simulation and experimental results that include sud-

den changes in road conditions indicate that the most attractive characteristic of a sliding-mode controller

is the robustness against the uncertainties in the system, such as noisy measurements or disturbances.

42

In [16], a sliding-mode controller is improved with integral switching surface to reduce chattering effects.

The robustness of the controller against nonlinearity and uncertainty is once again demonstrated by sim-

ulation results on a two-axle vehicle model. Patra et al proposes in [27] a recent sliding-mode controller

applied on a quarter car model with good slip tracking performance results. Yinggan Tang et al [28]

combine sliding mode controller with fractional order dynamics, in which fractional order proportional-

derivative (FOPD) sliding surface is adopted for faster track of the desired slip.

4.4.4 Intelligent Control

Intelligent control is an group of control techniques inspired on certain characteristics of intelligent bio-

logical systems. In fact, intelligent control frequen