ABS for FSAE-Thesis

7/21/2019 ABS for FSAE-Thesis

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

João Pedro Carrapiço Ferro

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisor: Prof. Paulo Jorge Coelho Ramalho Oliveira

Co-supervisor: Prof. João Miguel da Costa Sousa

Examination Committee

Chairperson: Prof. João Rogério Caldas Pinto

Supervisor: Prof. Paulo Jorge Coelho Ramalho Oliveira

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

May 2014



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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 I’ve 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 theinestimable 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 estabilidadesuficientes 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 veıculo, sustentado nos

princıpios da dinamica de veıculos. 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 tıpico, para os casos com e sem ABS.

Palavras-chave: sistema de travagem ABS, Formula Student , controlador fuzzy , controlador

PID, estimacao do escorregamento da roda, modelo de veıculo, 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 µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.2.2 Varying µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 MATLAB’s lsqcurvefit parameters [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 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 eS X and ∆eS X . . . . . 59

6.3 Takagi-Sugeno-Kang (TSK) consequent function values C k. . . . . . . . . . . . . . . . . . 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 µ. . . . . . . . 70

7.4 Braking distance and time results with and without ABS, with varying road µ. . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Planar tire forces on vehicle’s 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 T b(t) for τ = 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 Pacejka’s 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 F ZT = {50, 150, 250, 350} lbs, α = γ =

0◦ and p = 96.5 kPa (14 psi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 Plot of S X vs. F XT for four sweeps of different F ZT , with α = γ = 0◦ and p = 96.5 kPa

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

3.7 Pacejka Model vs. TTC Data for F ZT = {150, 250, 350} lbs. . . . . . . . . . . . . . . . . . 33

3.8 Burkhardt Model vs TTC Data for different velocities. . . . . . . . . . . . . . . . . . . . . . 33

3.9 Neural Network Model vs. TTC Data for F ZT = {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 f c = 50 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4 Open-loop schematic for pressure control. . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.5 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. 56

6.8 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 µ, with (solid) and without (dashed) ABS. . . 70

7.2 Comparative simulation results for varying µ, 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. . . . . . . . . . . . . . . . . . . . 747.6 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.

µ coefficient of friction; membership function parent.

Greekα tire slip angle .

β vehicle sideslip/attitude angle .

δ SW steering-wheel angle .

δ steer angle (road wheel steer angle).

γ inclination angle .

ωW wheel-spin velocity .

ωW 0 reference wheel-spin velocity .

φ roll angle .

ψ yaw/heading angle .

θ pitch angle .

∆spring spring deflection/compression.

κARB ARB torsional spring rate .

τ brakeline lag; time integration variable.

Latin

S X 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 .

f c modulator signal frequency.

f function.

h vehicle CG height above ground.

I xx vehicle roll moment of inertia .

I yy vehicle pitch moment of inertia .

I zz 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.

T s 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 Z V .

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, Dunlop’s 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 pilot’s 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 T ecnico , 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 the

ABS 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

vehicle’s 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 vehicle

model. 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 data

outsourced from a renowned tire test facility. Three different fit models were used and compared,

including the well-known and widely used Pacejka’s 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 PD

controller 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 to

the 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 vehicle’s 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 car’s 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



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 S X (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 S X 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 parameters’s 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

(X E ,Y E )1 and its heading/yaw angle ψ2. Computation comprises three steps:

1. Resolution of individual tire longitudinal and lateral forces into a single force F = (F X , F Y ) and

moment M Z

acting on the vehicle’s 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 vehicle’s corner i [2].

F X =

i

F X,i =

i

F XT,i cos δ i − F Y T,i sin δ i (2.1a)

F Y =

i

F Y,i =

i

F Y T,i sin δ i + F Y T,i cos δ i (2.1b)

M Z =

i

M ZT,i −

i

F X,i.yi +

i

F Y,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 F L F R RL RR

xi a a −b −b

yiT F 2

−T F 2

T R2

−T R2

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

2. Calculation of resulting forces F X and F Y and moment M Z into linear and angular acceleration

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

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

F X = m

aX − ψ vY

(2.2a)

F Y = m

aY + ψ vX

(2.2b)

M Z = I zz ψ (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 ψ − sin ψ 0

sin ψ cos ψ 0

0 0 1

vX

vY

0

(2.3)

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

xE (t) = x0

E +

t0

xE (τ ) dτ (2.4a)

Y E (t) = y0

E +

t0

yE (τ ) dτ (2.4b)

ψ(t) = ψ0 +

t0

ψ(τ ) dτ (2.4c)

where x0E , y0

E and ψ0 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 car’s suspension and tires, as

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

• Roll angle6 of the sprung mass θ;

• 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 + xiθ + yiφ (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

∂ q r−

∂ K

∂q r+

∂D

∂ q r+

∂V

∂q r= f r 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

2 ms zs2

+ 1

2 I xx φ2

+ 1

2 I yy θ2

+ 1

2

mu,F

2 z2

u,FL + 1

2

mu,F

2 z2

u,FR + 1

2

mu,R

2 z2

u,RL + 1

2

mu,R

2 z2

u,RR (2.7)

V = 1

2kS,F

zu,FL − zs + aθ +

T F

2 φ

2

+ 1

2kS,F

zu,FR − zs + aθ −

T F

2 φ

2

+

1

2kS,R

zu,RL − zs − bθ +

T R2

φ

2

+ 1

2kS,R

zu,RR − zs − bθ −

T R2

φ

2

+

1

2kT (zr,FL − zu,FL)

2+

1

2kT (zr,FR − zu,FR)

2+

1

2 kT (zr,RL − zu,RL)

2

+

1

2 kT (zr,RR − zu,RR)

2

+1

2κARB ,F

φ −

zu,FL − zu,FR

T F

2

+ 1

2κARB ,R

φ −

zu,RL − zu,RR

T R

2

(2.8)

D = 1

2cS,F

zu,FL − zs + aθ +

T F

2φ

2

+ 1

2cS,F

zu,FR − zs + aθ −

T F

2φ

2

+

1

2cS,R

zu,RL − zs − bθ +

T R2

φ

2

+ 1

2cS,R

zu,RR − zs − bθ −

T R2

φ

2

+

1

2 cT (zr,FL − zu,FL)

2

+

1

2 cT (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 = kspring MR2

(2.10)

where MR stands for motion ratio , defined as

MR = zu − zs

∆spring

(2.11)

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

cS = cdamper

MR2 (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. The

vectors q , q and q represent the acceleration, velocity and position for each one of the 7 DOF’s.

[M] =

ms 0 0 0 0 0 0

0 I xx 0 0 0 0 0

0 0 I yy 0 0 0 0

0 0 0 mu,F

2 0 0 0

0 0 0 0 mu,F

2 0 0

0 0 0 0 0 mu,R

2 0

0 0 0 0 0 0 mu,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 T F 2

cS,F T F 2

−cS,RT R2

cS,RT R2

#1 0 #3 a cS,F a cS,F −b cS,R −b cS,R

−cS,F −cS,F T F 2

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

−cS,F cS,F T F 2

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

−cS,R −cS,RT R2

−b cS,R 0 0 cS,R + cT 0

−cS,R cS,RT R2

−b cS,R 0 0 0 cS,R + cT

(2.15)

#1 = 2 b cS,R − 2 a cS,F

#2 = cS,F

T F 2

2 + cS,R

T R2

2

#3 = 2 cS,F a2 + 2 cS,R b2

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 −κARB ,F

T F 2 0 0

−kS,F #4F a kS,F −κARB ,F

T F 2 #5F 0 0

−kS,R −#4R −b kS,R 0 0 #5R −κARB ,R

T R2

−kS,R #4R −b kS,R 0 0 −κARB ,R

T R2 #5R

(2.16)

#1 = 2 b kS,R − 2 a kS,F

#2 = kS,F

T F 2

2 + kS,R

T R2

2 + κARB ,F + κARB ,R

#3 = 2 kS,F a2 + 2 kS,R b2

#4i = kS,i

T i2

+ κARB ,i

T i

#5i = kS,i + kT + κARB ,i

T i2

The matrix of F is a 7-by-1 column matrix with the total forces on each DOF as a function of the

external forces F s,i and F u,i.

{F} =

−F s,FL − F s,FR − F s,RL − F s,RR

−F s,FLT F 2

+ F s,FRT F 2

− F s,RLT R2

+ F s,RRT R2

F s,FL a + F s,FR a − F s,RL b − F s,RR b − M θ

cT zr,FL − F u,FL + kT zr,FL

cT zr,FR − F u,FR + kT zr,FR

cT zr,RL − F u,RL + kT zr,RL

cT zr,RR − F u,RR + kT zr,RR

(2.17)

14



2.3.2 Vehicle Loads

At any point in time, the vehicle’s 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 = F sts,FR = ms g

2

b

L (2.18a)

F sts,RL = F sts,RR = ms g

2

a

L (2.18b)

F stu,FL = F stu,FR = mu,F g

2 (2.18c)

F stu,RL = F stu,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:

∆F wtxs = ms aX

hs

L (2.19a)

∆F xwtu = (mu,F + mu,R) aX

hu

L (2.19b)

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

∆F wtys,F = ms aY

b

L

hs

T F

(2.20a)

∆F wtys,R = ms aY

a

L

hs

T R(2.20b)

15



Figure 2.5: Longitudinal load transfer.

∆F wtyu,F = mu,F aY

hu

T F

(2.20c)

∆F wtyu,R = mu,R aY

hu

T R(2.20d)

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

masses:

∆F wts,FL = −

∆F wtxs

2 − ∆F wty

s,F (2.21a)

∆F wts,FR = −

∆F wtxs

2 + ∆F wty

s,F (2.21b)

∆F wts,RL =

∆F wtxs

2 − ∆F wty

s,R (2.21c)

∆F wts,RR =

∆F wtxs

2 + ∆F wty

s,R (2.21d)

∆F wtu,FL = −

∆F wtxu

2 − ∆F wty

u,F (2.21e)

∆F wtu,FR = −

∆F wtxu

2 + ∆F wty

u,F (2.21f)

∆F wtu,RL =

∆F wtxu

2 − ∆F wty

u,R (2.21g)

∆F wtu,RR =

∆F wtxu

2 + ∆F wty

u,R (2.21h)

16



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

vehicle’s CG:

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

F D = 1

2ρ A C D v2 = cD v2 (2.22)

where ρ is the air density, A is a reference frontal area and C L 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 .

F L = 1

2ρ A C L v2 = 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 vehicle’s CG.

M θ = 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:

Figure 2.7: Aerodynamic load.

∆F aeros,FL = ∆F aero

s,FR = −F D hs + F L b − M θ

2L (2.25a)

∆F aeros,RL = ∆F aero

s,RR = F D hs + F L a + M θ

2L (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:

F s,i = ∆F wts,i + ∆F aero

s,i (2.26a)

F u,i = ∆F wtu,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

q i((k + 1)T s) = [M]−1

[F (kT s)] − [C ] q i(kT s) − [K] q i(kT s)

(2.28)

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

q (t) =

q (τ )dτ (2.29a)

q (t) =

q (τ )dτ (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:

F ZT,i = F sts,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.

F ZT,i = f (F s,i, F u,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 ωW and velocity ωW . With the knowledge of the vehicle velocity,

Wheel Dynamics ultimately calculates and outputs the actual longitudinal slip ratio S X , 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

4 vX = F XT (2.32)

I W ωW = T W − RLF XT (2.33)

19



Figure 2.8: Freebody diagram of the wheel.

where the wheel torque T W can be positive or negative, either this comes from the motor or the brakes.

T W = T d, if T W > 0 (Driving Torque)

T W = T b, if T W < 0 (Braking Torque)

(2.34)

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

S X . 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 F bp into a brake torque T b 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 F bp 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).

F mc = F bp ∗ PR (2.35a)

F mc,F = F mc ∗BB

F (2.35b)

F mc,R = F mc ∗ BB R (2.35c)

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

kept constant:

pmc = pcal = p = F mc

Amc

= F cal

Acal

(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:

F cal = F mc

Acal

Amc

= F mc

dcal

dmc

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, µ pads, brake torque is finally calculated:

F pads = 2 F cal (2.38a)

T b = F b Rb = µ pads F pads Rb (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:

T b = f (F bp) = 2 µ pads P Rdcal

dmc2

· F bp · Rb

= K · F bp = 1.02 · F bp

(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 τ :

G = 1

τ 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 T b(t) for τ = 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

= pmc− p pmc

· Rinc −→ Pressure increase

u = (1, 0) −→ dp

dt

= 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 they’re 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).

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.

[F XT , F Y T , M XT , M Y T , M ZT ] = f (F ZT , S X ,α,γ, 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 structuresthat 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 − arctan Bx)}]

Y (X ) = y(x) + S V

x = X + S H

(3.2)

where Y is the output variable F XT , F Y T or M ZT and x is the main x-axis input variable, namely tan α or

S X . 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

S H horizontal shift

S V 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, q i, ri or si) whose inputs are the supplementary input variables F ZT , γ 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, F XT = f (S X , α = 0◦)

• Lateral Force for pure side slip, F Y T = f (S X = 0 , α)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: Pacejka’s Magic Formula curve and geometric parameters.

• Aligning Torque for pure side slip, M ZT = f (S X = 0 , α).

• Longitudinal Force for combined slip, F XT = f (S X , α)

• Lateral Force for combined slip, F Y T = f (S X , α)

• Aligning Torque for combined slip, M ZT = f (S X , α)

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

constant pressure p and inclination angle γ , the simplified equations in the Pacejka’s 2002 version aredisplayed next (scale factors λi are ignored).

F XT = Dx sin (C x arctan (Bxkx − E x (Bxkx − arctan (Bxkx)))) + S V x (3.3a)

kx = S X + S Hx ; (3.3b)

df z = F ZT − F 0ZT

F 0ZT

(3.3c)

C x = pCx1 (> 0) (3.3d)

Dx = µxF ZT (> 0) (3.3e)

µx = ( pDx1 + pDx2df z)µ (3.3f)

E x = ( pEx1 + pEx2df z + pEx3df z2)(1 − pEx4 sgn(kx)) (≤ 1) (3.3g)

K x = F ZT ( pKx1 + pKx2df z)exp( pKx3df z) (3.3h)

Bx = K xk

C xDx

(3.3i)

S Hx = pHx1 + pHx2df z (3.3j)

S V x = F ZT ( pV x1 + pV x2df z)µx (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 − e−Bx

− Cx

e−Dxv (3.4)

where (F, x) is any of the following combinations: (F XT , S X), (F Y T , α) or (M ZT , α).

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 F ZT or inclination angle γ . 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 node’s parameters. A learning rule specifies how these parameters should be updated to

minimize a prescribed error measure between the network’s 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., F XT = f (S X , F ZT ), or {F XT , F Y T } = f (S X , α, F ZT )).

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 thiswork 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., F ZT =

150 lbs, γ = 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 n − 1 variables so as to have a new combination (e.g., F ZT : 150 →

250 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 S X sweep for each one of the possible combinations of the other 4 variables, α, F ZT , γ and p, as

a series of for -cycles. In red is highlighted a sample of data for F ZT = {50, 150, 250, 350} lbs, α = γ = 0◦

and p = 96.5 kPa (14 psi). The output F XT of these data is plotted over S X in Fig. 3.6.

Figure 3.5: Excerpt data from a complete run. In red, data for F ZT = {50, 150, 250, 350} lbs, α = γ = 0◦

and p = 96.5 kPa (14 psi).

30



−0.3 −0.25 −0.2 −0.15 −0.1 −5 · 10−2 0 5 · 10

−2 0.1 0.15 0.2 0.25 0.3−4,000

−3,000

−2,000

−1,000

0

1,000

2,000

3,000

4,000

S X

F X T

( N )

F ZT = 350 lbs

F ZT = 150 lbs

F ZT = 250 lbs

F ZT = 50 lbs

Figure 3.6: Plot of S X vs. F XT for four sweeps of different F ZT , with α = γ = 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 MATLAB’s 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 × 10−6

TolX 1 × 10−6

MaxIter 1 × 10−6

Others default

Table 3.2: MATLAB’s 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 F ZT . Moreover, the structure of Eqs. 3.3 ensure that model extrapolation

beyond |S X | > 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 S X . For F ZT = 350 lbs, α = 0◦, γ = 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 for

reproducing 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 0.5−4,000

−3,000

−2,000

−1,000

0

1,000

2,000

3,000

4,000

S X

F X T

( N )

F ZT = 350 lbs

F ZT = 150 lbs

F ZT = 250 lbs

F ZT = 50 lbs

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

peak value of F XT decreases for higher velocities.

−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1−5 · 10−2 0 5 · 10−2 0.1 0.15 0.2 0.25 0.3−4,000

−3,000

−2,000

−1,000

0

1,000

2,000

3,000

4,000

S X

F X T

( N )

TTC Data

v = 40.2 km/h (25 mph)

v = 60 km/h

v = 80 km/h

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 S X values outside the experimental range.

−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1−5 · 10

−2

0 5 · 10

−2

0.1 0.15 0.2 0.25 0.3

−4,000

−3,000

−2,000

−1,000

0

1,000

2,000

3,000

4,000

S X

F X T

( N )

FZ = 350

FZ = 150

FZ = 250

FZ = 50

Figure 3.9: Neural Network Model vs. TTC Data for F ZT = {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 S X values, with specialimportance for the wheel-lock case S X = −1 (see Section 4.3.1). Mean squared error (MSE) results for

the 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 S X (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 = F X(µ)

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 µ. 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, F XT and

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

on the value of the wheel slip S X .

{F XT , F Y T } = f (S X , . . .) (4.2)

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

slip ratio S X 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:

ωW 0 = vX

Re

(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.

dRe

dt = 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 ωW is different from the reference wheel-spin velocity, i.e., the wheel and the vehicle

have different longitudinal velocities.

ωW = ωW 0 ⇔ ωW .RL = vX (4.5)

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

vs = ωW − ωW 0 = ωW − vX

RL

(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 S X :

S X = ωW − ωW 0

ωW 0

= ωW Re − vX

vX

(4.7)

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

ωW ωW 0 ⇔ S X → ∞ −→ Wheel is spinning

ωW > ωW 0 ⇔ S X > 0 −→ Tire driving force (F XT > 0)

ωW = ωW 0 ⇔ S X = 0 −→ Free-rolling tire (F XT = 0)

ωW < ωW 0 ⇔ S X < 0 −→ Tire braking force (F XT < 0)

ωW = 0 ⇔ S X = −1 −→ Wheel lock-up

(4.8)

When the car stops ωW = vX = 0, which leads to the mathematical indetermination 0

0. To avoid the

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

ment, where ≡ eps = 2−52 (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 S X 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 S X do not coincide for both F XT and F Y 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 ωW 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 µ), sensor noise w and others.

38



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.

39



Object variable S X Longitudinal slip ratio

Controlled variable ωW Wheel speed

Reference variable S ref X Reference longitudinal slip ratio

Manipulated variable pb Braking pressure

Disturbances zr, µ, 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 understood

area 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 are

likely 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 µ are hard

and 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 control

commands: 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.

40



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 ofABSs. 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 (λ1) 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 frequently combines and extends theories and methods from

areas such as control, computer science, and operations research. It includes areas of knowledge likeartificial neural networks (ANNs), fuzzy logic, machine learning, evolutionary computation, or genetic

algorithms (GAs). Its resourcefulness allied to the continuous advances in computing technology allows

it to deal with increasingly broad and complex control problems.

On what concerns the ABS application, intelligent control has been an emerging and effective solu-

tion to deal with the nonlinear, uncertain and varying dynamics of braking process. An optimized fuzzy

controller is proposed by Mirzaei et al [3] to maintain the wheel slip on a desired level. Using genetic

algorithms and an error-based optimization approach for all parameters of a Takagi–Sugeno–Kang FIS,

very good performance and smoothness of the controller is achieved for a variety of road conditions. An

adaptive PID-type fuzzy control algorithm is designed in [6] for the ABS control. Experimental results

evidence that the performance of the ABS controller is increased by adding slip ratio estimation, optimal

slip ratio search and road condition identification mechanisms to the control scheme.

The integration of ANN on fuzzy controllers allows the inclusion of important adaptive and/or self-

tuning algorithms that may significantly increase ABS performance and robustness. A neuro-fuzzy

adaptive control approach for nonlinear dynamical systems is proposed by Topalov et al [29] and used

to design a wheel slip regulating controller. Raesian et al proposes in [30] an adaptive neuro-fuzzy

self-tuning PID control scheme to overcome the nonlinear dynamics. A hierarchical Takagi–Sugeno

fuzzy-neural model is used in [31] to develop a method for identification and robust adaptive control of

wheel slip integrated with an active suspension system.

Ultimately, GA provide a powerful optimization tool for the design of ABS controllers. As an exam-

ple, fuzzy and GAs are used in [32] to accomplish self-tuning on a PID control scheme and overcome

parameter variations and uncertainties and achieve the performance consistency.

43



44



Chapter 5

Proposed Approach

This chapter starts by outlining the available data and the design requirements (Section 5.1). A control

scheme is then proposed for the ABS (Section 5.2), including the selected control methods (Sections

5.2.1 and 5.2.2) based on the review included at the end of the previous chapter. Finally, the design

methodology for the following chapter is described (Section 5.3).

5.1 Problem data and requirements

Prior to any design decision, information regarding the prototype and the competition characteristics

should be taken into account and may affect the choice of the ABS control scheme. Moreover, there are

certain requirements with which the controller must comply, namely the competition rules.

5.1.1 FS Rules

Every FS competition must obey a set of rules defined by SAE International1. Beyond the main rules

whose purpose is to guarantee the safety of every member involved, including the pilot, team members,

organisational staff and public, the 2014 version of FSAE Rules 2 establishes a set of safety and technical

requirements that specifically concerns the brake system:

• The car must be equipped with a braking system that acts on all four wheels and is operated by a

single control.

• It must have two independent hydraulic circuits such that in the case of a leak or failure at any point

in the system, effective braking power is maintained on at least two wheels.

1Formerly the Society of Automotive Engineers , SAE International is a globally active professional association and standardsorganization for engineering professionals in various industries, namely automotive, aerospace and commercial-vehicle.

2Availabe at fsaeonline.com.

45



• The brake system must be capable of locking all four wheels during the Brake Test, i.e., one must

be able to switch off the ABS.

• Brake-by-wire systems are prohibited.

5.1.2 Competition characteristics

Since most of the dynamic events are regulated and the layout of circuits are, at least partially, known

from previous competitions, it is possible to accurately reproduce the conditions of the competition,

including the load cases the car will be subjected to and the corresponding dynamic behaviour. The

following information about the competition circuit is also known a priori :

• Pavement type may only vary between cement, normal asphalt, and circuit asphalt.

• Pavement condition may be dry, damp, or wet, but will generally be uniform in the entire layout and

remain constant throughout the event.

• The vertical profile of the circuit is roughly plane.

Figure 5.1: Formula Student Germany 2013 endurance event.

5.1.3 FST 05e

In addition to the known parameters of the target vehicle, listed in Table 7.1, specifications on the brake

system architecture of the FST 05e, available hardware and sensors onboard are to be considered.

46



Brake system architecture

The prototype has two independent hydraulic brake circuits for the front and rear wheels, respectively.

The brake pedal acts on the master cylinders, one for each circuit, and the hydraulic pressure is trans-

ferred through rigid brake lines to a floating caliper on each wheel. The calipers act on self-made brake

steel brake discs. Pressure distribution for the front and rear circuits (brake bias) is adjustable.

Available hardware and sensors

The following hardware and sensors are installed on the FST 05e prototype:

• Encoders for angular speed measurement in all 4 wheels;

• Three-dimensional accelerometer near the CG of the car;

• Pressure sensors for brakeline pressure, mounted on the wheel calipers;

• LVDT3 for linear travel measurement of the brake pedal.

5.2 Proposed Control Structure

For the ABS problem, a cascade control structure with two feedback loops is proposed and displayed

in Fig. 5.2. The main ABS controller, the wheel slip controller situated on the outer loop, compares the

actual wheel slip S X with a given reference slip S ref X and outputs a reference pressure pref b accordingly.

This reference pressure is then sent to a brake pressure controller, in the inner loop, which in turn sends

a command signal u to the hydraulic modulator (actuator) so the reference pressure is matched.

Additionally to the controllers, a wheel slip estimator is included in the outer loop. Having in mind the

physical implementation of the proposed structure, the estimator ensures an accurate estimation of the

wheel slip based on noisy measurements from the vehicle accelerometer and wheel speed encoders.

Figure 5.2: Proposed control structure.

3Linear Variable Differential Transformer

47



Three types of blocks are depicted:

Controller (red) Two controllers will be designed to track reference signals of brake pressure and wheel

slip. Further details on the chosen control methods are addressed in sections 6.1 and 6.2, respec-

tively.

Model (black) Contain the theoretical and experimental models built upon on the dynamic equations

of Chapter 2. Brake system and wheel model are representations of the brakeline (Section 2.5)

and wheel dynamics (Section 2.4), respectively. The vehicle and tire model includes the remaining

horizontal, vertical dynamics, as well as the tire model described in Chapter 3.

Estimator (blue) Includes the aforementioned wheel slip estimator, described and designed in Sec-

tion 6.3.

This cascade arrangement means that the pressure controller deals only with the fast brakeline dy-namics, allowing the wheel slip controller to exclusively deal with the slower remaining vehicle dynamics.

The combined effectiveness of the controllers is, thus, enhanced. Moreover, with this type of structure

the global ABS problem may be split into several well-defined sub-problems, one for each loop, which

can be analysed and synthesized separately and systematically, as seen in Section 5.3.

5.2.1 Inner Loop - Brake Pressure Controller

The controller present in the inner loop should regulate the command signal u sent to the hydraulic

modulator (Section 2.5.3) so that pressure on the caliper pcal matches the reference pressure pref b

determined by the slip controller in the outer control loop (Fig. 5.3).

Figure 5.3: Schematic of brakeline dynamics and pressure controller.

The pressure controller has the following requirements:

• Fast response, near the physical capacity of the system actuator, i.e. the hydraulic modulator.

• No overshoot, which could lead to overbraking and consequent location of the wheel on the unsta-

ble region of the tire friction curve (Section 4.3.3).

48



Object variable pcal Pressure on wheel caliper

Controlled variable pcal Pressure on wheel caliper

Reference variable pref Reference pressure

Manipulated variable p Brake line pressure

Disturbances w p, . . . Pressure sensor noise, etc.

Table 5.1: Closed-loop variables for pressure control.

• Must be stable.

Since no particularly demanding specifications are required, the design simplicity and well-known

principles of a PID-type controller makes this the preferred choice. Through the use of MATLAB’s PID

Tuner tool for automatic tuning, the design process is even simpler and intuitive. Additionally, ready-to-

run hardware implementations of a PID controller are accessible and straightforward to work with.

5.2.2 Outter Loop - Wheel Slip Controller

The wheel slip controller, in the current cascade structure, has the function of calculating a reference

pressure based on the wheel slip error eS X to a given reference S ref X (Fig. 5.2). It can be considered the

main controller in the complete ABS scheme.

A PID-type fuzzy controller with a Takagi–Sugeno–Kang (TSK) fuzzy system is proposed here to

maintain the wheel slip to a desired value (Fig. 5.4). By using this type of hybrid control, the intuitive

design and computational efficiency of a FIS is joined with the well-known control PID theory. Moreover,

fuzzy algorithms are nowadays simple to implemented in hardware via digital control.

Figure 5.4: PID-type fuzzy controller scheme [6].

Using the TSK method for the FIS, instead of the more common Mamdani’s FIS for example, offers

the following advantages [1]:

• It is computationally efficient, since the algorithm is simpler and the output does not need defuzzi-

fication.

• It works well when combined with linear techniques like the PID control proposed here.

• Its parameters are suitable for optimization and adaptive techniques.

49



• It guarantees smoothness of the output signal.

As depicted in Fig. 5.4, the input variables of the fuzzy controller (also the linguistic variables) are the

wheel slip error eS X = S ref X − S X and its variation ∆eS X within a time step T s. The output variable is the

reference pressure variation ∆ pref

b

. Choosing pressure variation instead of directly inferring reference

pressure is preferred, as its optimal value varies with vehicle speed. Moreover, the designed controller

will be more robust against parameter uncertainty and disturbances that, otherwise, would require an

adaptive controller.

The PID-type classification is related to the fact that, before entering the FIS, both input variables are

multiplied by the gains K e and K ∆e, which equivalent to the proportional and derivative gains of a PID

controller. On the other side, the output of the FIS is multiplied by the gain K ∆ p before integration.

5.3 Design Approach

Due to the nature of the proposed control scheme, the ABS design process may is streamlined into the

following steps [33]:

1. Brake system and tire/vehicle dynamics modelling and validation.

2. Brake pressure controller design, assuming an adequate reference pressure pref b signal. In be-

tween, a conversion mechanism is also needed to convert the analog command signal u to the

open-close valve commands, as described later on Section 6.1.2. The controller design comprises

the PID tuning solely.

3. Wheel slip controller design, assuming that the actual wheel slip S X is directly known from the

vehicle model. The controller design comprises the FIS derivation and the tuning of the outside

gains.

4. Development of an algorithm for the wheel slip estimation.

5. Final validation and fine-tuning of model and controllers via simulation on the Simulink environ-

ment.

While the dynamic models, brake pressure controller and wheel velocity controller must be designed

in this order, vehicle speed estimator could be carried out separately or in parallel. Iterations of the

controller design, the simulation evaluation, the testing and the controller fine-tuning may be necessary.

50



Chapter 6

ABS Design

The current chapter describes the design process of the brake pressure PID controller (Section 6.1),

including the pulse-width modulation (PWM) mechanism (Section 6.1.2); the wheel slip PID-type fuzzy

controller (Section 6.2); and the wheel slip estimator (Section 6.3). Brief theoretical introductions are

given in each case. At the end of the chapter, some ABS trigger algorithms are outlined (Section 6.4).

6.1 Brake Pressure Controller

As described in Chapter 5, a PID controller is proposed for the control of the brake pressure pcal, given

a reference pressure pref b . The design process includes the modelling and validation of a PWM on-off

conversion mechanism. Afterwards, open and closed step responses of the inner loop are simulated for

the design of the PID gains.

6.1.1 PID Overview

A proportional-integral-derivative (PID) controller is a control loop feedback mechanism that outputs a

control signal u(t) as a function of a given error e = yref (t) − y(t), its time derivative e and integral ∫ e

(Fig. 6.1). In the ideal form :

u(t) = K P .e(t) + K I

t0

e(τ ) dτ + K D e(t) (6.1)

or, after Laplace transform, written as a transfer function:

GPID (s) = U (s)

E (s) = K P +

K I s

+ K D.s (6.2)

where K P , K I and K D designate the tuning parameters of proportional, integral and derivative gains,

respectively.

51



Figure 6.1: PID controller scheme.

The most common form seen in the industry (including MATLAB) and the one most relevant to tuning

algorithms is, however, the so-called standard form :

u(t) = K P

e p(t) +

1

T I

t0

e p(τ ) dτ + T D e p(t)

(6.3)

with proportional gain K P applied also to the integral and derivative terms, and the parameters K I and

K D being replaced by the more meaningful time constants T I = K P K I

and T D = K DK P

.

6.1.2 PWM Conversion

Since the hydraulic modulator (actuator) only receives on-off signals for the inlet and outlet valves (Sec-

tion 2.5.3), the analog control signal u outputted by the pressure controller must be converted into two

binary signals u01 = [uin uout]. A technique known as pulse-width modulation (PWM) will be used for

the effect.

PWM is a method that converts an analog signal into a pulse signal with variable duty ratio, which

is the duration ratio of on-signal to off-signal . PWM scheme determines that an on-off switch occurs for

every intersection between the original analog signal and a high frequency periodic signal (modulator

signal), as depicted in Fig. 6.2.

A sawtooth wave with frequency f c is used here as modulator signal. Since there must be two on-

off signals retrieved from the PWM conversion, two modulator signals are used: a sawtooth ranging

between 0 and 1 for the inlet valve and a sawtooth ranging between -1 and 0 for the outlet valve. This

way, the inlet valve is energized (closed) according to the first modulator signal or whenever the analog

signal is negative, whilst the outlet valve is energized (open) according to the second modulator signal.

Figure 6.3 shows the Simulink implementation of a PWM conversion, using an example analog control

signal and a 50 Hz modulator signal.

52



Figure 6.2: PWM conversion scheme [6].

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.5

0

0.5

1

1.5

Time [s]

uin

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

u

Sawtooth [01]

Sawtooth [ − 10]

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−0.5

0

0.5

1

1.5

Time [s]

uout

i i l i i

Figure 6.3: PWM implementation with f c = 50 Hz.

6.1.3 Open/closed loop step response

Open loop

Prior to any design, it’s important to understand the dynamic behaviour of the brake system in an open-

loop architecture (Fig. 6.4).

Step response shown in Fig. 6.5 allows a few nonlinearities can be noticed. For a positive/negative

command signal u, outlet pressure p will increase/decrease according to Eq. 2.41. This also means that

53



Figure 6.4: Open-loop schematic for pressure control.

pressure saturates on pmc or 0. Due to the PWM conversion (Section 6.1.2), the actual increase/decrease

rate will be as follow:

• u > 1 — hydraulic modulator will always be on increase mode and pressure increase rate is

pmc− p pmc

· Rinc, regardless of the magnitude of the command signal.

• 0 < u < 1 — hydraulic modulator alternates between increase and hold mode. In practice, this will

be equivalent to a smaller increase rate R∗

inc = u · Rinc.

• −1 < u < 0 — as above, hydraulic modulator alternates between decrease and hold , resulting in

a smaller decrease rate R∗

dec = |u| · Rdec

• u < −1 — hydraulic modulator will always be on decrease mode and pressure decrease rate is

Rdec, regardless of the magnitude of the command signal.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

0

1

2

Time (s)

u i n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−5

0

5

10

Time (s)

P r e s s u r e ( M P a )

pref

pcal( pmc = 5)

pcal( pmc = 10)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

0

1

2

Time (s)

u o u t

Figure 6.5: Open-loop variable-step response.

54



Closed loop

When the feedback loop is closed (Fig. 5.3), the system behaves as having a pole in the origin (integra-

tor) and the reference pressure is followed with zero steady-state error, as long as pref ≤ pmc. Figure

6.6 also shows that the response is more aggressive (fast response and oscillatory behaviour) for higher

values of pmc, as result of Eq. 2.41. Nevertheless, the response time varies between 0.2 and 0.4 s,

which can be improved for faster dynamics of the inner loop control. The overshoot verified may be also

undesired and lead the slip to the unstable region of the tire F (µ) curve, with consequent wheel lock-up.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

−1

0

1

2

Time (s)

u i n

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

0

5

10

Time (s)

P r e s s u r e

( M P a )

pref

pcal( pmc = 5)

pcal( pmc = 10)

u ( pmc = 5)

u ( pmc = 10)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

0

1

2

Time (s)

u o u t

Figure 6.6: Closed-loop variable-step response.

6.1.4 PD Design

Linearization

The nonlinear behaviour seen before means that Ziegler-Nichols methods for both open and closed loop

PID tuning cannot be applied:

• Reaction Curve (Open loop) — Steady-state value of pcal is pmc, instead of being proportional to

u. Therefore, the system cannot be approximated by a first-order system.

55



• Critical Gain (Closed loop) — The integral action cannot be dissociated from the system, thus the

system never becomes unstable.

An alternative method is to approximate both PWM conversion and hydraulic modulator models by a

linearisable function that receives the command signal u and outputs the pressure p. Assuming pmc =

∞:

d pdt

= Rinc for u ≥ 1

d pdt

= u · Rinc for 0 ≤ u < 1

d pdt

= |u| · Rdec for −1 < u < 0

d pdt

= Rdec for u ≤ −1

d p

dt = a · tanh(b · u + h) + v (6.4)

The resulting linearisable function d pdt

= f (u) is plotted in Fig. 6.7. The step response of the approx-

imated system (the transfer function on Eq. 2.40 is still used to model transient dynamics between p

and pcal) is depicted in Fig. 6.8. Although the approximation quality decays over time, it is sufficiently

accurate within the desired response time.

−4 −2 0 2 4−400

−200

0

200

400

600

u

d p d t

Original ( pmc = ∞)

Approximation

Figure 6.7: Linearisable approximation of brakeline system (a = 400, b = 1.28, h = −0.22 and v = 100.

PD Tuning

Since the approximated system can now be linearised, Matlab’s PID Tuning tool can be used to automat-

ically calculate proportional and derivative gains K P and K D, as well as the filter coefficient N , according

to the desired fastness and robustness of the step response (Eq. 6.5). Recalling the closed-loop step

response characteristics on Section 6.1.3 (no steady-state error), the integral gain is not included. After

calculating the values in Table 6.1 using the approximated system, the designed PD is used to control

the original system. Figure 6.9 displays the controlled step response of the original system and shows

56



Figure 6.8: Step response of original and approximated system.

that it was possible to eliminate the overshoot whilst maintaining a very fast response.

GPID = K P + K D · N

1 + N 1s

(6.5)

K P 225

K D 1.46

N 1000

Table 6.1: Tuned PD gains for pressure controller.

0 5 · 10−2 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (s)

P r e s s u r e

( M P a

)

pref

pcal(Original)

pcal(PID)

i i ll

Figure 6.9: Step response with PD controller for pcal.

57



6.2 Wheel Slip Controller

Following the proposed approach in Chapter 5, the wheel slip controller is built upon a PID-type fuzzy

control structure, which combines the already introduced PID control principles with the fuzzy inferencesystem (FIS) theory.

For the wheel slip controller design, one will for now assume that the actual wheel slip S X is directly

and accurately known from the vehicle dynamic model.

6.2.1 FIS Overview

Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy

logic. The basic structure of a fuzzy inference system consists of three conceptual components: a

rule base, which contains a selection of fuzzy if-then rules; a database, which defines the membership

functions used in the fuzzy rules; and a reasoning mechanism, which performs the inference procedure

upon the rules [15].

Figure 6.10: Fuzzy inference system diagram.

Takagi-Sugeno-Kang Fuzzy Inference System (FIS)

The proposed controller uses a Takagi-Sugeno-Kang (TSK) FIS is selected. The main difference be-

tween the more general Mamdani and TSK fuzzy models lie on the nature of the consequent, which are

respectively a fuzzy set or a crisp value given by a constant or linear function of the inputs [34].

TSK fuzzy system rules Rr have the general form

Rr: If xi is Aij then yr = f k(xi) = C 0,k + C 1,kx1 + . . . + C n,kxn

58



where xi is the input i = 1, . . . , m, Aij is the fuzzy set j = 1, . . . , s for the input i, and yr is the output of

the rule r, given by one of the k functions f k(xi), which are linear combinations of the inputs.

The procedures involved in the computation of the overall output signal y is also distinct between

Mamdani and TSK fuzzy systems. In the former case, it is computed as a weighted average1 of all rule

outputs yr:

y =

r yr.wr

r wr

(6.6)

The weights wr stand for the firing strength of each rule, which result from the application of the min

operator to the membership function values µAij, as the AND (T-norm) operator:

wr = mini

{µAij(xi)} (6.7)

6.2.2 FIS Design

Fuzzy sets

Has described in Section 5.2.2, the controller has two input variables eS X and ∆eS X . Three fuzzy sets

are defined for each input variable, according to Table 6.2. Similarly, five zero-order functions for the

output ∆ pref b are defined, as listed in Table 6.3.

Fuzzy Set Classical Set c σ Description

A1 Negative eS X < 0 -1 0.425 S X is below reference slip, within thestable region of the tire friction curve(brake pressure too low)

A2 Zero eS X = 0 0 0.425 S X is equal to the desired referenceand F XT is maximum

A3 Positive eS X > 0 1 0.425 S X beyond reference slip, within theunstable region of the tire friction curve(brake pressure is too high)

B1 Decreasing ∆eS X < 0 -1 0.425 S X is increasing, thus evolving towards

or away from the reference value,whether S X is within the unstable orstable region, respectively

B2 Steady ∆eS X = 0 0 0.425 S X is not decreasing nor increasing

B3 Increasing ∆eS X > 0 1 0.425 S X is decreasing, thus evolving to-wards or away from the referencevalue, whether S X is within the stableor unstable region, respectively

Table 6.2: Fuzzy sets and MFs parameters for inputs eS X and ∆eS X .

1The weighted sum(numerator of weighted average) is also an alternative method for output calculation with even more reducedcomputational effort

59



k Function Name Ck Description

1 Decrease-fast -1 Decrease reference pressure rapidly

2 Decrease-slow -0.5 Decrease reference pressure slowly

3 Hold 0 Hold reference pressure

4 Increase-slow 0.5 Increase reference pressure slowly

5 Increase-fast 1 Increase reference pressure rapidly

Table 6.3: TSK consequent function values C k.

Membership Functions

Membership functions (MFs) for the input variables are similar and defined by Gaussian curves (Eq. 6.8)

whose parameters are the variance σ and center of gravity c, also listed in Table 6.2.

µAi = µBj

= Gaussian(x; c, σ) = e−(x−c)2

2σ2 (6.8)

The choice of Gaussian curves over, e.g. triangular or trapezoidal MFs, ensures a smoother transition

between fuzzy sets, which translates into a continuous output signal. MFs for each fuzzy set, as depicted

by Fig. 6.11, are evenly distributed within the normalized interval [−1 1] (universe of discourse). Doing

so allows the controller dynamics to be quantitatively adjusted solely by tuning the PID gains outside the

FIS, whereas the qualitatively aspect is ensured inside the FIS.

Figure 6.11: Membership functions of FIS.

60



Rule Set

For this specific problem, a set of rules is established as follow:

Rij : If eS X is Ai and ∆eS X is Bj then yij = f k = C k

where the correspondence between the rule output yij and the function f k is given by Table 6.4. The

respective constant value C k is as listed previously in Table 6.3.

Ai

Bj Decreasing Steady Increasing

Negative Increase-fast Increase-slow Hold

Zero Increase-slow Hold Decrease-slow

Positive Hold Decrease-slow Decrease-fast

Table 6.4: Fuzzy rules for wheel slip controller.

Output surface

The overall output of the given FIS is, therefore, calculated as follow:

∆ pref b =

ij yij .wij

ij

wij

(6.9)

where wij = min{µAi, µBj

} for each rule.

The non-linear output surface of the designed FIS displayed in Fig. 6.12. An almost linear relation

between inputs and output is achieved when the wheel slip error and its variation are near zero, i.e.,

the system is within the desired region depicted in Fig. 4.3. Outside this region the controller experi-

ences saturation phenomena, as the reference pressure variation ∆ pref b is limited to prevent excessive

overshoot when the system is evolving far from the reference slip or within/towards the stable/unstable

region of the friction curve.

6.2.3 PID Gains Optimization

The input and output gains K e, K ∆e where firstly tuned so the signal values entering the FIS would

range within the normalized interval [−1 1]. As for the output gain K ∆ p, the value is primarily chosen

according to the actuator limits of pressure increase/decrease rate (Table 7.2). The three parameters

where then optimized, as to minimize the MSE to the reference S ref X signal. Optimization is performed

using the Simulink Design Optimization tool, with the default method Gradient descent and algorithm

Sequential Quadratic Programming . The final values are listed in table Table 6.5

61



Figure 6.12: Output surfaces of FIS models for wheel slip controller.

K e 1.92

K ∆e 0.08

K ∆ p 1159

Table 6.5: Tuned PID gains for fuzzy wheel slip controller.

6.2.4 Reference Wheel Slip

The reference wheel slip that is fed to the newly designed controller is the last decision variable. From

the tire friction model curves attained in Chapter 3, tire braking force magnitude (F XT < 0) is maximized

near S X = −0.2. After some trial-and-error iterations during the simulation phase, the reference wheel

slip S ref X is fixed in

S ref X = −0.16 (6.10)

62



6.3 Wheel Slip Estimator

Recalling Eq. 4.7 (rewritten below), S X calculation depends on both wheel velocity ωW and longitudinal

vehicle velocity vX :

S X = ωW Re − vX

vX

= vS

vX

(6.11)

Wheel velocity is available through direct, yet noisy measurements from the equipped wheel en-

coders (Section 5.1.3). On the other hand, vehicle inertial velocity can only be accurately measured

by means of special instruments based on optical principle, Doppler or GPS, which are expensive and

therefore not practical for must applications [35]. Hence, the problem of estimating the longitudinal wheel

slip S X is crucial for an effective design of ABS, especially in the present case where the control problem

is formulated as the regulation of the wheel slip itself.

A systematic, intuitive and effective estimation method based solely on wheel speed measurements

using an adaptive nonlinear filter was proposed by [36]. However, this type of estimation tends to be

inaccurate as the wheel speed differs from the actual vehicle speed ( S X = 0). A coarse estimation

coupled with a slip controller can even lead to closed-loop instability [ 37]. An alternative method is

to use the direct integration of the longitudinal acceleration measurement provided by the mounted

accelerometer as displayed in Fig. 6.13. The initial velocity value is given by the wheel velocity just

before the driver starts braking, i.e., while the wheel is still in free-rolling condition (S X = 0).

Figure 6.13: Block diagram of vS estimation.

Kalman filter algorithms have been widely used in vehicle navigation and control and proved to be and

accurate estimation method for vehicle velocity [38,39]. Kalman filtering can also be used to estimate

measurement offset caused by temperature drift, inclination of the road surface and vehicle pitch motion

[35]. This method reveals to effectively reduce estimation errors accumulated by the integration process.

Another estimation method using complementary filters is described in [40] and applied on a similar

problem where yaw angle ψ of a vehicle is estimated based on measurements ψm and rm = ψ. This

method will be addressed in the following section.

63



6.3.1 Complementary Filter

A complementary filter, i.e., a Wiener filter equivalent to the stationary Kalman filter solution, arise nat-

urally in the context of signal estimation based on measurements provided by sensors over distinct, yet

complementary regions of frequency. For the specific problem of slip velocity estimation, complemen-

tary filter combines low frequency information from accelerometer with high frequency measurements

from wheel speed encoders. The relative amount of information from each sensor is adjusted through

the Kalman gain K .

ˆvS = K (ωmRe − (vS + vm)) (6.12)

where

ωW,m ≡ ωm = ωW + wω (6.13)

vX,m ≡ vm = ∫ (aX + wa) (6.14)

Figure 6.14: Block diagram of complementary filter.

The estimate longitudinal slip ratio is then calculated by dividing the newly estimated slip velocity vS

by the

S X = vS

∫ am

(6.15)

As depicted in Fig. 6.15, a good estimation is achieved despite the noisy measured data. More-

over, different choices of gain K translate in a compromise between fast convergence and steady-state

smoothness. Higher values favor high frequency measurements from wheel encoders and thus a noisier

response is attained.

64



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Times (s)

v S

( m / s )

Real (vS = −5 m/s)Measured

Estimated (K = 100)

Estimated (K = 10)

Figure 6.15: Slip velocity vS estimation results with complementary filter.

6.4 ABS Triggers

To ensure the correct operation of the ABS, the following on and off thresholds are defined.

6.4.1 Minimum brake pressure

A minimum brake pressure is necessary to guarantee that the ABS system is active when, and only

when the brake pedal is stepped. A threshold of 0.1 MPa for the caliper pressure pcal is chosen.

6.4.2 Wheel slip and wheel slip rate trigger

The combined information of wheel slip and wheel slip rate provide an indication on whether or not the

wheel may lock. If wheel slip is close or beyond reference slip (unstable region), or decreasing at a high

rate (towards lock-up), ABS should start its action. Recalling Table 6.4, these conditions correspond

also to the rules where the pressure should be decreased, initiating the control cycle. Trigger values are

defined as S X = −0.1 and ∆S X = −2.

65



6.4.3 Low velocity trigger

Rewriting Eq. 4.7 in the form

S X = ωW .RL − vX

vX

⇔ ωW = vX(1 + S X)RL

(6.16)

and assuming a constant wheel slip ( dS Xdt

= 0), as the vehicle tends to complete stoppage, both vehicle

velocity and wheel velocity tend, naturally, to zero.

limvX→0

ωW = 0 (6.17)

In practice, this means that the S X calculation/estimation will tend to a zero-by-zero indetermination and

may lead to unstable computation.

limvX→0

S X = 0 − 0

0 (6.18)

Even though FS prototypes rarely run slow enough so this problem may occur, a vehicle velocity thresh-

old of 20 km/h (5.6 m/s) is enforced, below which the ABS controller is turned off. Increasingly bigger

oscillations around the reference wheel slip are thus avoided, which exceedingly degraded ABS effi-

ciency and could even drive the system towards instability.

66



Chapter 7

Simulation Results and Analysis

This chapter contains the final simulation results. Firstly, simulation parameters are listed (Section 7.1).

Then, a set of braking simulations are performed and compared between a vehicle with and without the

designed controller, for a road with constant µ (Section 7.2.1) and real-time varying µ (Section 7.2.2).

The role of the inner brake pressure controller on the ABS performance is also tested (Section 7.2.3).

Finally, a complete lap around a typical Autocross circuit provides the information on the real advantage

of a ABS-equipped FS prototype (Section 7.3).

7.1 Simulation Parameters

7.1.1 FST 05e Parameters

The most relevant FST 05e data needed for modelling and simulation purposes are available via CAD

tools or physical measurements. The complete set of parameters’ values is listed in Table 7.1. Other

model and simulation parameters are listed in Table 7.2.

7.2 Straight Line Hard Braking

The first set of simulations to be performed are simple hard braking manoeuvres: initially the vehicle

travels at constant velocity vX = v0; at t = 0.5 s, the driver steps hard on the pedal brake (so that

T b F XT .RL) and keeps it until the vehicle stops. The term acceleration refers to the magnitude of the

acceleration henceforth.

67



Variable Symbol Value Unit

Mass and Inertia

Vehicle total mass m 251.3 kg

Total sprung mass ms 214.63 kg

Front sprung mass ms,F 95.52 kg

Rear sprung mass ms,R 119.11 kg

Total unsprung mass ms 36.67 kg

Front unsprung mass ms,F 17.82 kg

Rear unsprung mass ms,R 18.85 kg

Vehicle roll moment of inertia I xx 13.68 kg.m2

Vehicle pitch moment of inertia I yy 74.62 kg.m2

Vehicle yaw moment of inertia I zz 76.00 kg.m2

Wheel rolling moment of inertia I W 0.29 kg.m2

Distance

Wheelbase L 1.59 m

CG to front axle distance a 0.82 m

CG to rear axle distance b 0.77 m

Front track T F 1.20 m

Rear track T R 1.17 m

Car CG height h 0.29 m

Sprung mass CG height hs 0.30 m

Unsprung mass CG height hu 0.26 m

Tire loaded radius RL 0.26 m

Suspension and Chassis

Front wheel rate kS,F 18580 N/m

Rear wheel rate kS,R 23900 N/m

Tire spring rate kT 168000 N/m

Front ARB torsional spring rate κARB ,F 306 N.m/ ◦

Rear ARB torsional spring rate κARB ,R 242 N.m/ ◦

Front suspension damping coefficient cS,F 3200 N.s/m

Rear suspension damping coefficient cS,R 3200 N.s/m

Tire damping coefficient cT 250 N.s/m

Table 7.1: FST 05e parameters.

68



Variable Symbol Value Unit

Initial vehicle velocity v0 100 km/h

Pedal force F bp 600 N

PWM modulator signal frequency f c 100 Hz

Pressure increase rate Rinc 800 MPa/s

Pressure decrease rate Rdec 500 MPa/s

Hydraulic lag τ 0.02 s

Sample time T s 0.002 s

Table 7.2: Other model and simulation parameters.

7.2.1 Constant µ

A constant value for the road coefficient of friction is firstly tested, with µ = 0.80 (dry road). Figure 7.1

plots over time the comparative simulation results with (solid line) and without (dashed line) the designed

ABS scheme. Plots depict, from top to bottom, vehicle position (xE ), vehicle and wheel velocities (vX

and ωW ), followed by the vehicle longitudinal acceleration (aX), the wheel slip (S X) and its reference

value, and the pressures on the master cylinder ( pmc), wheel caliper ( pcal) and reference pressure pref b .

The simulation case without ABS serves both as validation for the vehicle and tire dynamic models and

a term of comparison for further simulations with the designed ABS.

Up to t = 0.5 s, wheel tangential velocity ω.RL equals the vehicle velocity for both cases, i.e., wheel is

in free-rolling condition (S X = 0). When the driver starts braking without ABS, wheel rapidly decelerates,

until it locks-up. This is also shown by the value of S X heading towards −1. If the vehicle is equipped

with the ABS, wheel velocity decreases as fast, until the reference slip S ref X = −0.16 is matched and

efficiently tracked.

Regarding the aX plot, it is noticeable that, for both cases, a maximum deceleration of |aX | 30 m/s2

is soon reached. As expected, this occurs when S X = S ref X , i.e., when S X crosses the optimal value for

which peak longitudinal force is attained, as shown in Fig. 4.3 and described in Section 4.3.2. However,

the ABS is able to keep maximum acceleration thenceforth, thus resulting in a steeper velocity plot, and

a smaller braking time and distance. Final numeric results are listed on Table 7.3.

Observing the pressure plot for a braking without ABS, pressure on the caliper quickly matches the

pressure on the master cylinder (a certain dynamic can be noticed). With ABS, wheel slip and wheel

slip rate triggers (Section 6.4) allow the initial pressure increase to be near the maximum rate allowed

by the actuator, i.e., similar to the case without ABS. At t 0.6 s, wheel slip approaches the reference

value and the ABS is turned on, noticeable by the sudden pref b discontinuity and subsequent pressure

decrease.

69



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

V e l o c i t y ( m / s

)

ω.RLvX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

x E

( m / s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−40

−20

0

20

a X

( m

2

/ s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

−0.8

−0.6

−0.4

−0.20

S X

S refX

S X

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

5

10

15

20

Time (s)

p c a l ( M P a )

pref

pmc pcal

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

Without ABS With ABS ∆ %∆

Braking Distance (m) 21.87 15.21 -6.66 -30.5 %

Braking Time (s) 1.63 1.17 -0.46 -28.2 %

Table 7.3: Braking distance and time results with and without ABS, with constant road µ.

70



7.2.2 Varying µ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

V e l o c i t y ( m / s

)

ω.RLvX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

10

20

30

40

x E

( m / s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−40

−20

0

20

a X

( m

2 / s )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4−1

−0.8

−0.6

−0.4

−0.20

S X

S refX

S X

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

5

1015

20

Time (s)

p c a l ( M P a )

p

ref

pmc pcal

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

To test the adaptivity of the proposed ABS controller to real-time variations of the surface type, the

following values for the road coefficient of friction µ have been introduced as input parameter on the

Pacejka tire model (Section 3.1.1):

µ = 0.8 for xE < 22 m

µ = 0.5 for 22 ≤ xE < 28 m

µ = 0.8 for xE ≥ 28 m

(7.1)

The resulting plots on Fig. 7.2 evidence a fast adaptation of the controller to sudden changes on

the road µ. At xE = 22 m, µ suddenly decreases, which also decreases the available tire force that is

reacting the brake torque. Consequently, wheel slip enters the unstable region of tire friction curve and

decays abruptly towards lock-up. Since pressure must be released to counter-act this effect, the slip

71



controller immediately diminishes the reference pressure. After a brief undershot, reference pressure

stabilizes near 5 MPa in about 0.1 s and wheel slip is restored to the reference slip value. At xE = 28 m,

the opposite phenomenon occurs and brake pressure can again be increased to maximize tire force.

Since wheel slip is now within the stable region, the reference pressure is increased slower and tracked

without overshoot, with wheel slip settling in 0.2 s.

When compared with the simulation without ABS, namely the velocities and acceleration plots, the

effectiveness of the ABS on reducing braking distance is once again demonstrated. Even for varying

road conditions, the adaptivity of the proposed controller allows vehicle deceleration to be kept at a

maximum, regardless of the mu value. Final results for this simulation are listed in Table 7.4.


Braking Distance (m) 24.61 18.42 -6.19 -25.2 %

Braking Time (s) 1.77 1.37 -0.40 -22.6 %

Table 7.4: Braking distance and time results with and without ABS, with varying road µ.

7.2.3 Without pressure controller

To simulate the ABS system without the inner loop, the reference pressure variance ∆ pref b is directly feed

into the hydraulic modulator as the command signal u, as depicted in Fig. 7.3. Figure 7.4 evidences that

the pressure PID controller plays a major role on ABS performance. Even though wheel slip controller

gains are not optimized, since it cannot cope with the hydraulic lag dynamics, simulation results show

continuous oscillation of the wheel slip around the reference value.

Figure 7.3: Control scheme without pressure inner loop.

With PD Without PD ∆ %∆

Braking Distance (m) 15.21 16.14 +0.93 +6.11 %

Braking Time (s) 1.17 1.28 +0.11 +9.40 %

Table 7.5: Braking distance and time without PD pressure controller.

72



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−40

−20

0

20

a X

( m

2

/ s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

10

20

30

V e l o c

i t y ( m / s

) ω.RLvX

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−1

−0.5

0

0.5

S X

( −

)

S refX

S X

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0

20

40

60

Time (s)

p c a l ( M P a )

pref

pmc pcal

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

7.3 Complete Lap

The absolute advantage of the ABS on a FS prototype is more evident within a complete Autocross1 lap,

simulated with modified GG diagrams for the cases with and without ABS (Fig. 7.5). The GG diagram is

a plot used to describe the performance of a car, since it contains the envelope of combined longitudinal

(driving and braking) and lateral accelerations attainable by the tires, in units of g = 9.8 m/s2. As

described in chapters 3 and 4, the absence of an ABS may lead to wheel lock, i.e. S X = −1, for which

no lateral force is developed by the tires and longitudinal force is not maximum. On the other hand, with

ABS not only the driver benefits from maximum braking force on every corner, even if he steps too hard

on the brake pedal, but steerability is also guaranteed. This features translate in the GG diagrams as

a narrower and more cross-shaped envelope without ABS, whilst with ABS it is wider and more round,

due to combined longitudinal and lateral acceleration that is thereby achievable.1The Autocross Event consists of a timed lap around a tight course to evaluate the car’s maneuverability and handling qualities,

combining the performance features of acceleration, braking, and cornering.

73



0 2 4 6 8 10 12 14 16 18 20−25

−20

−15

−10

−5

0

5

10

15

20

ay (m/s2)

a x

( m / s 2 )

Without ABS

With ABS

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

Figures 7.6 compare the actuation of the brake pedal (in red) within the perimeter of a typical Au-

tocross circuit, when this is made with and without ABS. In the former case, one can noticed that the

driver is able to brake later, noticeable by shorter trajectory sections in red at the end of straights, and

while cornering. The resulting velocity plot over the lap is shown in Fig. 7.7, on which the driver reaches

higher velocities with the ABS turned on. This is also shown in Fig. 7.8 by longer parts of the trajectory

in yellow/red, namely inside corners.

Lap time results for each simulation are listed in Table 7.6, with a significant difference of nearly 4 s

per lap. After a complete Endurance Event2, this would represent a major improvement in the final time.

Moreover, tire wear would be also reduced and more even along tire surface.

Lap Time (s)

Without ABS 52.17

With ABS 48.31

∆ 3.86

%∆ 7.4%

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

2The Endurance Event is the main FS event, designed to evaluate the overall performance of the car and to test the car’sdurability and reliability. It consists of a 22 km race around the same track as the Autocross Event.

74



−40 −20 0 20 40 60 80 100 120 140 160

−120

−100

−80

−60

−40

−20

0

20

xE (m)

y E

( m )

(a) Without ABS.

−40 −20 0 20 40 60 80 100 120 140 160

−120

−100

−80

−60

−40

−20

0

20

xE (m)

y E

( m )

(b) With ABS.

Figure 7.6: Brake pedal actuation within lap perimeter.

75



0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

Lap percentage (%)

v

( k m / h )

Without ABS

With ABS

Figure 7.7: Velocity plot over an Autocross lap.

−40 −20 0 20 40 60 80 100 120 140 160

−120

−100

−80

−60

−40

−20

0

20

xE (m)

y E

( m )

0

10

20

30

40

50

60

70

80

90

100

v

( k m

/ h )

i .(a) Without ABS.

−40 −20 0 20 40 60 80 100 120 140 160

−120

−100

−80

−60

−40

−

20

0

20

xE (m

y E

( m )

0

10

20

30

40

50

60

70

80

90

100

v

( k m

/ h )

i .(b) With ABS.

Figure 7.8: Velocity distribution within lap perimeter.

76



Chapter 8

Summary and Conclusions

This thesis reported the design of an ABS application for a Formula Student (FS) prototype, with the

objectives of reducing braking distance while ensuring steering stability. For this purpose, a cascade

control architecture containing two feedback loops was proposed for enhanced performance. The outer

loop and main controller makes use of a PID-type fuzzy system for wheel slip regulation, allying the well-

known control PID theory with the intuitive design and computational efficiency of a FIS, whilst the inner

loop resorted to a conventional and easily implementable PD controller for brake pressure reference

tracking.

For the ABS design process and further simulation, a self-developed vehicle model with 14 DOFs

was built upon the principles of vehicle dynamics, mainly under braking (Chapter 2). The full modelwas divided into the following sub-models: horizontal model for the position ( xE and yE ) and orientation

(yaw angle ψ) of the vehicle’s CG in the inertial reference plane, as it is subjected to tire forces and

moments; a 7-DOFs vibrational model of the vehicles submitted to aerodynamic and load transfer loads,

which allowed the real-time computation of transient tire vertical loads, as well as the vertical displace-

ments of the sprung zs and unsprung masses zu,i, roll angle φ and pitch angle θ ; wheel dynamics, for

wheel kinematics (ωW and ωW ) as balance between an applied brake torque and the tire longitudinal

force; brakeline dynamics, namely the hydraulic lag phenomenon and a transient model of the hydraulic

modulator.

An accurate tire friction model is of absolute importance for ABS applications. With that in mind,

special attention has been given to this subject in Chapter 3. Taking advantage of reliable experimental

data from a renowned tire test facility, three types of tire models were designed and compared: the

well-known and widely used Pacejka’s Magic Formula (MF), the Burkhardt velocity-dependant model,

and an self-developed new approach using artificial neural network (ANN). Fitting results showed that

Pacejka was the most suitable tire model, providing accurate longitudinal tire force F XT mapping for

a wider range of possible S X , including the wheel lock-up case S X = −1. The accurateness of the

other two models was compromised for extrapolation beyond the limits of experimental data availability.

Later simulation results allowed the validation of the complete vehicle and tire model integration, namely

vehicle and wheel kinematics throughout braking.

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The ABS problem was then introduced from the control theory point of view: the object variable S X ,

mathematically defined as the longitudinal (wheel) slip ratio should be kept within a range for which

longitudinal and lateral tire forces are near its peak values (Chapter 4). Some of the the most common

and relevant solutions were then reviewed, namely threshold control, PID control, sliding-mode control

(SMC) and the emerging new approaches on intelligent control. Threshold control applies a basic, yet

widely used control algorithm, mainly in the early versions of the ABS. Since the control algorithm is

solely based on binary relations between input variables and threshold values, this method is nowadays

a limited solution and lacks some of the controllability and efficiency of other control methods. PID

control is a very versatile and well-known control method for numerous applications. Its hybridization

with other control techniques provided the necessary robustness, self-tuning or adaptability features

for ABS employment, which naturally enhanced the already satisfactory controller performance. SMC

is a more complex but particularly effective method for ABS control, given the robustness properties

and applicability to nonlinear plants. Published results on this subject exhibit excellent performancefor a wide variety of conditions and reinforced the suitableness of SMC for wheel slip manipulation.

Finally, a research has been put through on the later intelligent control solutions for the ABS problem.

Though diversity of resources and applications is, in fact, very broad and arbitrary complex, a few notable

publications were considered and revised, namely on the areas of fuzzy or neuro-fuzzy theory and

genetic algorithm (GA) optimization.

Considering the characteristics of the reviewed methods, as well as the available information on the

ABS target vehicle and the FS competition requirements, the aforementioned proposal was introduced

(Chapter 5). Then, the ABS design was streamlined into the following components: brake pressure

controller, wheel slip controller, and wheel slip estimator (Chapter 6).

Due to the dynamic characteristics of the inner loop plant, i.e., brakeline and hydraulic modulator

system, the regular Ziegler–Nichols methods for open and closed loop could not be applied. Therefore,

a linearisable approximate model of the plant was arranged, so that Matlab’s PID Tuning tool could be

used. PD gains were tuned in order to achieve the desired fast dynamics, near the actuator limit, without

overshoot, which could let the wheel slip to the unstable region of the tire friction curve. As for the

PID-type fuzzy controller, a Takagi-Sugeno-Kang (TSK) fuzzy inference system (FIS) was selected and

modelled with two inputs variables for the slip error eS X and its time-step variance ∆eS X . A rule set for

the output reference pressure variance ∆ pref b was selected based on conventional ABS increase-hold-

decrease cycles. The external PID gains were then iteratively tuned for optimal controller performance,

i.e., braking distance minimization with reduced tracking oscillations.

In what concerns wheel slip estimation, a solution using a complementary filter, i.e., a Wiener filter

equivalent to the stationary Kalman filter solution, was successfully implemented. Test results with noisy

measurements of the acceleration and wheel speed confirmed the efficiency of the filter on reducing

signal noise while ensuring good estimation accuracy. Moreover, this approach is simpler than the

stochastic setting proposed by Kalman, which requires a complete characterization of process and noise

observation.

A set of hard braking simulations followed the design phase. Results shown in Chapter 7 and sum-

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marized in Table 8.1 evidence a significant reduction of the braking time and distance. Furthermore,

the proposed controller exhibit a fast response and relatively smooth tracking of the reference wheel

slip. The adaptivity of the controller to sudden variations of the road coefficient of friction µ, both high-

to-low and low-to-high, has also been successfully demonstrated. Further simulations without the inner

pressure manipulation testify the relevance of the PID controller on achieving the aforementioned per-

formance, namely by independently dealing with the hydraulic dynamics. A final simulation within a

complete lap around an typical Autocross circuit conclusively confirmed the competitive advantage of an

ABS-equipped FS prototype, by deducting nearly 4 seconds to the final lap.


Braking Distance, constant µ (m) 21.87 15.21 -6.66 -30.5 %

Braking Time, constant µ (s) 1.63 1.17 -0.46 -28.2 %

Braking Distance, varying µ (m) 24.61 18.42 -6.19 -25.2 %

Braking Time, varying µ (s) 1.77 1.37 -0.40 -22.6 %

Lap Time (s) 52.17 48.31 3.86 7.4%

Table 8.1: Results summary with and without ABS.

8.1 Future work and research

As stated in the thesis’s objectives in Section 1.3, physical implementation of the proposed controller was

a concern. Having that in mind during the modelling and design phases, special attention has already

been given to some important practical issues: sampling time, PWM conversion and noise filtering were

all considered. For that reason, hardware-in-the-loop validations and hardware implementations on a

real FS prototype are the natural steps to follow.

Besides that, further research on the following topics is recommended:

• Although the braking distance reduction and steerability objectives were already acomplished,

the stability of the vehicle on a asymmetric µ surface is not ensured (Section 4.1). A balancingalgorithm should then be implemented so that even tire forces on the right and left side of the car

are always a priority.

• As an electric vehicle, FST 05e makes use of an electric motor to drive each of the rear wheels,

which may introduce important inertial effects during braking that were not studied. Moreover, an

electric motor has the ease and flexibility to implement any control algorithm that may work closely

or instead the proposed ABS controller [41] and increase braking performance and/or stability.

• Oscillations around the reference wheel slip, mainly during the transient period, correspond to

oscillations in the acceleration of the car’s CG, which in turn are transmitted to the suspension

due to weight transfer. A comparative study between the frequency of these oscillations and the

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natural frequency of the car’s suspension may be carried out using the vibrational model described

in Section 2.3. Results may be evaluated either in terms of the driver comfort or ABS enhancement

with an active suspension system, as proposed in [31].

80



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Appendix A

SAE Definitions

This chapter outlines the main definitions from standard SAE J670 (2008) [ 7] used throughout this work.

Definitions for terms in italic font can be found elsewhere either in this appendix or in the above docu-

ment.

A.1 Axis Systems

A.1.1 Earth-Fixed Axis System

This axis system is fixed in the inertial frame with zero linear and angular acceleration and zero angular

velocity.

Origin Fixed at arbitrary point on ground plane .

XE, YE Parallel to ground plane , arbitrary orientation.

ZE Up direction, aligned with gravitational vector.

A.1.2 Vehicle Axis System

This axis system is fixed in the reference frame of the vehicle sprung mass .

Origin At vehicle reference point .

XV Parallel to the vehicle plane of symmetry . Is substantially horizontal and points forward (with the

vehicle at rest).

YV Perpendicular to the vehicle plane of symmetry , pointing to the left.

ZV Parallel to the vehicle plane of symmetry , upward direction.

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Figure A.1: Vehicle Axis System [7].

A.1.3 Intermediate Axis System

The intermediate axis system is used to facilitate angular rotations using the vehicle Euler angles and

the definition of angular orientation terms, the components of force and moment vectors, and the com-

ponents of translational and angular motion vectors.

Origin At vehicle reference point .

X Parallel to the ground plane , aligned with the vertical projection of XV axis onto the ground plane .

Y Parallel to the ground plane , orthogonal do X.

Z Parallel to the ZE

, upward direction.

A.1.4 Tire Axis System

Origin At contact center .

XT Parallel to local road plane , defined by the intersection of the wheel plane and the road plane .

YT Parallel to local road plane , orthogonal do XT.

ZT Normal to local road plane , upward direction.

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Figure A.2: Tire and Wheel Axis System [7].

Figure A.3: Tire Force and Moment Nomenclature [7].

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A.2 Angles

The sign of angles resulting from angular rotations is determined in accordance with the right-hand rule.

A.2.1 Vehicle Orientation

Yaw Angle (Heading Angle), ψ The angle from the XE axis to the X axis, about the ZE axis.

Pitch Angle, θ The angle from the X axis to the XV axis, about the Y axis.

Note: The pitch angle is not measured relative to the road plane , thus a vehicle at rest on an

inclined planar road surface will have a non-zero pitch angle .

Roll Angle, φ The angle from the Y axis to the YV axis, about the XV axis.

Vehicle Sideslip Angle, β The angle from the X axis to the vertical projection of vehicle velocity onto

the ground plane , about the Z axis.

A.2.2 Tire Orientation

Wheel Plane Orientation The angular orientation of the wheel plane with respect to the road plane andthe tire trajectory velocity . It is expressed in terms of inclination angle and slip angle .

Slip Angle, α The angle from the XT axis to the normal projection of the tire trajectory velocity onto the

XT –YT plane.

Inclination Angle, γ (ε) The angle from the ZT axis to the ZW axis.

Camber Angle The angle between the ZV axis and the wheel plane , about the XV axis. It is considered

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