Model-Based Brake Control including Tyre Behaviour
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 4 december 2012 om 12.30 uur door
Edwin John Henry DE VRIES
werktuigkundig ingenieur geboren te Amsterdam.
Dit proefschrift is goedgekeurd door de promotoren: Prof. dr. ir. D.J. Rixen
Prof. dr. ir. M. Verhaegen
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. ir. D.J. Rixen, Technische Universiteit Delft, promotor Prof. dr. ir. M. Verhaegen, Technische Universiteit Delft, promotor Prof. dr. H. Nijmeijer, Technische Universiteit Eindhoven Prof. J. Deur, Ph.D. Cveučilište u Zagrebu, Kroatië
Prof. dr. ir. P. van der Jagt, Loughborough University of Technology, GB Em. Prof. dr. ir. H.B. Pacejka, Technische Universiteit Delft
Prof. dr. ir. E.G.M. Holweg, Technische Universiteit Delft
Copyright © 2012 by Edwin de Vries
–All rights reserved – No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, without the prior permission of the author.
Samenvatting
Model-gebaseerde Rem Regeling met inbegrip van Band Gedrag
Het doel van het onderzoek is een nieuwe methode te ontwikkelen voor het geregeld remmen van een voertuig. De rempedaalindrukking wordt als maatgevend geacht voor de gewenste remvertraging. De remregelaar beoogt niet cruise-control te vervangen, maar vervult wel een anti-blokkeer (ABS) functie. Het onderzoek heeft rechtuit remmen als basis,
maar laat zich door beheersing van de remkracht ook toepassen in bochten. De remregelaar wordt als continue regelaar ontworpen met een continu uitgangssignaal voor het remmoment, wat weer voor de aansturing van echte actuatoren gebruikt kan worden.
Om de regelaar te ontwikkelen wordt een model opgesteld van een kwart-voertuig met een
LuGre band model. Door slechts een kwart voertuig te beschouwen blijft het model compact
en kunnen 4 remregelaars onafhankelijk werken. Het nadeel van deze modelvorming is dat de andere 3 wielen geen invloed uitoefenen de vertraging van het kwart-voertuig dat beschouwd word.
Het verdeelde LuGre band-model is een variant van het borstel model dat de vervormingen
en de schuifkrachtverdeling over de contactlengte beschrijft. Voor het regelaarontwerp is een lumped-LuGre model gebruikt dat slechts een vervormingsgrootheid kent die
maatgevend is voor de remkracht. De relatie tussen het verdeelde en lumped-LuGre model
kan analytisch afgeleid worden. In dit proefschrift wordt een empirische parameterisering voor het LuGre model voorgesteld.
De kern van de regelstrategie is het succesvol inverteren van de niet-lineaire kwart-voertuig dynamica. Deze inverse wordt vervolgens gebruikt om een aansturing (feed-forward) te bepalen voor het remmoment. Hiervoor wordt de methode van ‘flatness’ gebruikt. Een systeem is ‘vlak’ (flat) als een uitgangsgrootheid gevonden wordt zodat alle systeemvariabelen (toestanden) en de ingangsgrootheid bepaald kunnen worden met niet-lineare functies van deze uitgangsgrootheid en een aantal van zijn tijdsafgeleiden. Als de vlakke uitgang bekend is, kan de bijbehorende ingang berekend worden, precies wat in de aansturing beoogd wordt. Voor het kwart voertuig is deze uitgang de voertuigsnelheid vx, De ‘vlakke’ toestanden zijn dan de snelheid versnelling en ruk. Als onafhankelijke input is nog een extra tijdsafgeleide nodig in de ‘vlakke’ toestandsruimte.
Voor het gebruik in de aansturing moeten deze hogere afgeleiden van het gewenste gedrag bekend en beschikbaar zijn. Hiertoe wordt bij machine aansturing vaak de baan van setpunt naar setpunt gepland. Voor het aansturen van de rem heeft het gebruik van vormende filters de voorkeur zodat een instantane reactie op een gewijzigde remwens gewaarborgd is. Het proefschrift laat zien dat de berekening van het remmoment op basis van de gewenste remvertraging goed werkt voor de nominale situatie; waarbij de kennis van de systeemparameters volledig en exact is. Om de robuustheid tegen parameteronzekerheid en externe verstoringen van de remregelaar te vergroten wordt ook twee typen terugkoppeling aan de regelaar toegevoegd. We onderscheiden een binnen en een buiten regellus.
Een probleem van de inverse functie op basis van flatness is dat het geldig domein begrensd is. Bij steeds hogere remkrachten zal de bandkracht verzadigen en uiteindelijk zijn wrijvingslimiet bereiken. Het kart-voertuigmodel heeft daardoor een eindig bereik van haalbare remvertagingen, dit weerspiegelt zich in een begrensd domein voor toelaatbare remvertraging in de voorwaartse aansturing. De ruwe remwens wordt gelimiteerd, maar doordat die discrete ingreep voor het vormend filter geplaatst is, is continuïteit van de hogere afgeleiden nog steeds gewaarborgd.
De beperkte bandkracht is de oorzaak van het verliezen van de open lus (lokale) stabiliteit, de waarneembaarheid, en de regelbaarheid van het systeem. Door het toevoegen van de binnenlus regeling die de wielrotatiesnelheid meet en vervolgens afregelt op de gewenste wielrotatiesnelheid ontstaat een gesloten lus die stabiel is. Deze binnenlus verzorgt zo een functie die vergelijkbaar is met ABS. De referentiewielsnelheid kan het best berekend worden gebaseerd op de werkelijke voertuigsnelheid, in plaats van de geplande snelheid van het kwart-voertuig. Deze extra wielsnelheidsmeting garandeert ook de waarneembaarheid van het systeem. De regelbaarheid blijft echter onder invloed van de fysica: het moet geaccepteerd worden dat bepaalde voertuigvertragingen niet behaald kunnen worden. Voor het bepalen van de terugkoppelversterking van de binnenlus is er een minimale waarde nodig om lokale stabiliteit in alle werkpunten te beiden. Vervolgens kan de versterking opgevoerd worden om een vlottere respons van de regeling na te streven. In een aparte sectie van het proefschrift wordt getoond hoe een bovengrens voor de terugkoppelversterking bepaald wordt als er een realistisch actuatormodel opgenomen wordt in de regelkring.
Het ontwerp van de buitenlus regeling profiteert van de toestandstransformatie op basis van flatness. Na de transformatie bestaat het systeem uit drie gestapelde integratoren zodat het lineair of zelfs triviaal genoemd kan worden. Door het kiezen van terugkoppelversterkingen voor alle vlakke toestanden worden de polen van het geregeld systeem eenduidig bepaald. Echter niet alle vlakke toestanden zijn meetbaar, en de wenssnelheid hoeft niet gevolgd te worden. Twee alternatieven om de buitenlus regelaar te ontwerpen op basis van gemeten remvertraging worden gepresenteerd.
De parametergevoeligheidsstudie laat zien dat het geregeld systeem vrij ongevoelig is voor 10% verkeerd bepaalde parameters. Echter wanneer er hoge remvertragingen gesimuleerd worden kunnen binnen en buitenlus contraproductief worden. Omdat geaccepteerd is dat de gewenste remvertraging mogelijk niet behaald wordt bij een gegeven wrijvingscoëfficiënt wordt in dat geval de buitenlus beëindigd. Het beheersen van de wielslip wordt als schakelcriterium gebruikt.
De parameters die in gebruik zijn in de ‘begrenzer’: wrijvingscoëfficiënt, massa, verticale bandkracht hebben een sterk effect op de remprestatie: te behoudend begrenzen van de ruwe vertragingswens resulteert in sub-optimale prestaties van het voorgestelde remregelsysteem.
Een van de tekortkomingen van de kwartvoertuiganalyse is dat de gewichtsoverdracht tijdens remmen genegeerd wordt. Tijdens het bepalen van de inverse voor de aansturing kan de verticale bandkracht als variabele meegenomen worden. Vervolgens kan een dynamisch dompmodel van het voertuig de wisselende verticale belasting beschrijven en op basis van de remactie voorspellen. Het is vervolgens van belang om de rembijdrage van het voorwiel en het achterwiel af te stemmen op deze veranderende kracht. Om deze verdeling te ontkoppelen van het inverse model zijn twee ‘vlakke’ uitgangsgrootheden nodig. Als ‘vlakke’ uitgangsgrootheden voor een half-voertuig model worden de stoot van de remkracht op het voorwiel en de stoot van de remkracht op het achterwiel gedefinieerd. Om de feed-forward aansturing in de praktijk te testen is een laboratorium test opstelling met een enkel geremd wiel op een stalen trommel gebruikt. Een vloeiende respons van de remkracht op een voorgevormd remmoment signaal is niet aangetoond met deze opstelling. De conventionele schijfrem, waarvan de remdruk met een servoklep geregeld kan worden heeft daarvoor onvoldoende bandbreedte in combinatie met de amplitudes die gebruikt
Summary
Model-Based Brake Control including Tyre Behaviour
The objective of the thesis is to develop a method for controlled braking of a vehicle. The brake pedal depression has been considered to be proportional to the intended deceleration. The brake controller is not aimed to replace a cruise control; it will have an anti-lock braking (ABS) function. The brake control is essentially designed to be a continuous control
law with a continuous brake torque output suitable for using it on real world actuators. To develop the continuous control and taking into account the friction and elastic properties of the tyre, a quarter-car model with LuGre tyre has been derived. Considering only a
quarter car keeps the model compact and by having 4 brake controllers each wheel can be controlled individually. The drawback is that the quarter car ignores a possible different contribution to the vehicle deceleration of the other 3 wheels.
The distributed LuGre tyre model can be considered as a brush model describing the
deformation and shear force of the tread elements in the tyre contact. For the control purpose in this thesis the lumped LuGre model has been employed. It considers one internal
tyre deformation to be representative for the tyre friction force. The relation between distributed and lumped tyre models can be derived analytically. In this thesis an empirical parameterisation of the lumped LuGre model has been proposed.
The core of the control philosophy is the successful inverse of the non-linear dynamics of the quarter car, and the use of this inverse as a feed-forward control for braking the quarter car. The concept of flatness has been employed to determine this inverse dynamic model. Flatness states that after having determined the flat output, the system states and the input are a function of the flat output and a number of its time derivatives. The flat output quarter-car is the velocity vx. Using this output and two higher time derivatives: acceleration and jerk the new state vector is constituted. The next higher derivative will take the role vx
of independent input in the flat state space.
For the use in feed-forward control the desired flat output and its derivatives should be known and available. In machine control a trajectory from one to the next set-point can be planned beforehand. For the control of a brake system shaping filters are preferred, so that instantaneous response to a change in brake wish is guaranteed
The thesis shows that feed-forward control using the flatness based inverse works flawlessly on the simulated nominal plant. To improve the robustness of the control law against parameter uncertainty and external disturbances two types of feedback branches have been added to the control scheme: an inner and an outer branch.
A problem of the flatness based inverse function is that its domain is restricted. For increasing levels of brake torque the brake force will saturate to its friction limit. The quarter car model has a limited attainable deceleration, which is reflected in a restricted domain for the inverse model in the feed-forward. A saturation function cuts the raw input to fit in the domain. Since this intervention takes place before the shaping filter, this discrete action still leads to continuous higher derivatives after filtering.
The saturating tyre force is the cause of loosing local open loop stability, loosing observability and loosing controllability. It was proven that demanding the state transformation to be a diffeomorphism is equivalent to requiring observability in its flat state. Observability on the full physical domain can be preserved by allowing the additional measurement of the wheel rotational velocity. With this measurement, a reference tracking feedback loop has been devised, where the reference can best be calculated with the actual vehicle speed. This inner loop is essential for preventing wheel lock (comparable to
conventional ABS) and for bringing the wheel speed back in domain where is also observable in the flat states.
Controllability has not been regained with the extra wheel-speed measurement. Given the physical system with limited friction force it must be accepted that some decelerations cannot be attained. For tuning the inner loop control gains a minimal value can be found to offer local stability for all operating conditions. This control gain can be increased to improve the response time of the controlled system. A separate section of the thesis shows that there also is an upper limit for the control gain when considering a more realistic actuator model in the control chain.
The design of an outer feedback loop benefits from the system in a set of linearised states. With the transformation the dynamic system can be linearised or even ‘trivialised’ to a series of three stacked integrators. By specifying the proportional gains for all flat states the poles of the system can be placed in the complex plane. In our problem not all flat states are considered measurable. Two alternative approaches have been presented to design the outer loop based a longitudinal deceleration measurement.
Although the parameter sensitivity study shows that the controlled system is rather insensitive to wrongly estimated parameters. However at large decelerations the inner and outer loop can be counter-effective. Since it is accepted that the desired deceleration can not always be reached for a given friction coefficient, the outer loop is terminated in when the loops start counter-acting. This termination is triggered when the slip in the simulated plant exceeds a threshold. Meanwhile wheel-lock needs to be prevented and the inner loop remains active.
The parameters used in the limiter: the friction coefficient, mass and normal load have a strong effect on the brake performance. Over-conservative limitation of the raw input based wrong parameters results in suboptimal performance of the quarter car braking.
One of the shortcomings of considering just a quarter-car is that the wheel load transfer during braking is ignored. Including the normal load as free variable allows anticipating on the load transfer that is induced by the braking. A dynamic pitch model of the vehicle is used to calculate these vertical load variations. It is important to adjust the brake force of the front and rear wheel to its varying vertical force. To avoid that this front rear distribution is merged with the flat inverse model, two flat outputs need to be defined. In the thesis the impulse of the front brake force and the impulse of the rear brake force have been proven to serve as flat outputs.
When implementing the feed-forward control strategy in a laboratory test rig with a single braked wheel and a conventional friction brake, we did not succeed in shaping the brake torque such that a smooth change in brake force is generated. The required bandwidth of the brake actuator cannot be achieved with a conventional hydraulic brake for the amplitude of brake torque variations that we calculated.
Acknowledgements
During the course of this PhD research project I have come to appreciate the interest, advice and support of many persons, some of whom I would like to acknowledge explicitly.
First of all I wish to express my gratitude to my promoter prof. Daniel Rixen, who has supported and encouraged me continuously and who has been inspiring me with many fruitful discussions throughout the years.
To promoter prof. Michel Verhaegen I also owe gratitude, for our cooperation, for the careful reviewing of the manuscript and for his attempts to bring rigor to the project.
I want to thank Erik Vermeer for pioneering in this research direction in his MSc. thesis and Achim Fehn for initial guidance in finding a fruitful approach to the technical problem. The material support from Robert Bosch GmbH and the cooperation during the initiation of the research is also highly appreciated.
I want to thank Yannick Postelmans, Marien van Ditten and Thijs van Keulen for their contribution to the research, and all other students that have contributed.
I want to thank William Passillas-Léphine for the cooperation we had on ABS braking
research during his visits to Delft, and for helpful discussions on my work. Also thanks to Mathieu Gerard, for the good times we shared in the lab and during conferences.
I am grateful for the opportunity that has been offered to me by TU Delft, specifically the department Precision and Microsystems Engineering (PME) of the Faculty of Mechanical, Maritime and Materials Engineering 3mE). I wish to thank all colleagues who supported the research project in the higher management. I also wish to thank my former colleague Ton van der Weiden for his inspiring companionship.
I thank my family for always having provided a warm and supportive home. My parents have always expressed their continuous faith in me. My daughters Melissa and Joëlle have been bringing much joy into my life.
And especially I wish to express my deepest gratitude to my wife Marjolein, who has encouraged and motivated me, which I have welcomed even more during the final stage of the project.
Contents
Samenvatting III Summary V Acknowledgements VII Chapter 1 Introduction 1 1.1 History of ABS 11.2 The research problem 2
1.3 Contribution to the state of art research 2
1.4 Structure of the thesis 3
Chapter 2 Models for tyre brake force, and a quarter vehicle model 5
2.1 Definition of slip variables 5
2.2 The brush model 8
2.2.1 Adhesion forces 9
2.2.2 The size of the adhesion region 10
2.2.3 The combined slip characteristic 13
2.3 LuGre friction model 14
2.3.1 Introducing the Dahl friction model for a single bristle 14
2.3.2 Introduction of the internal deformation state z 17
2.3.3 LuGre model for sliding contact 18
2.3.4 Using the true sliding speed in the Stribeck friction formula 21 2.4 Distributed tyre models based on LuGre friction 23
2.4.1 The distributed LuGre model 23
2.4.2 Analytical Steady-state characteristics with uniform normal load distribution 25 2.4.3 The deformation pattern of a distributed LuGre tyre model with constant shear
stiffness and parabolic normal load distribution 27
2.5 The Lumped LuGre model 29
2.5.1 The lumped state for a shear stiffness depending on normal load 29 2.5.2 A lumped model based on constant shear stiffness 31 2.5.3 The lumped LuGre tyre model in steady-state 32
2.5.4 Determining the convection term by equating the slip stiffness in steady state 34 2.5.5 Lumped LuGre parameters from empirical stationary characteristic 35
2.6 The quarter car model 38
2.6.1 Adding dynamic tyre modes by including the rigid ring model 40 Chapter 3 Outline of a control strategy for braking a (quarter) car 43
3.1 Tracking or stabilizing 44
3.2 Feed-forward and feedback 46
3.3.1 Lyapunov stability for nonlinear control design 48
3.3.2 Robust Control 50
3.3.3 Adaptive control 50
3.3.4 Feedback linearization 51
3.4 Control Approach 54
Chapter 4 Flatness-based feed-forward control 59
4.1 Differentially flat systems 59
4.1.1 Definition of flatness 60
4.1.2 Linearization 61
4.1.3 Controllability 62
4.1.4 Flatness and controllable realizations for LTI systems 62 4.2 Flatness of the quarter-car model with the LuGre tyre Model 64
4.2.1 Flatness of the LuGre model with constant 65
4.2.2 Flatness of the rigid wheel LuGre model 68
4.2.3 The domain of the flat state and its derivatives 73 4.2.4 Definition of the domain using the inverse function theorem 74 4.2.5 Application of the theorem to the quarter-car with LuGre tyre. 74
Chapter 5 Trajectory planning and constraint handling 79 5.1 Generation of the desired trajectory for the flat output and its derivatives 79
5.1.1 Polynomial planning 80
5.1.2 Shaping filters 86
5.2 Physical constraints on the input Mb 89
5.2.1 Response of the inverse system on a polynomial transition 90 5.2.2 Response of the inverse system to a filtered signal. 91 5.3 Physical constraint on planned deceleration 93
5.4 Braking to a full stop 98
5.5 Brake release 99
Chapter 6 Feedback 102
6.1 Outer loop feedback control 103
6.1.1 PID control by state feedback with feed-forward action 104
6.1.2 PD control using acceleration only 104
6.1.3 Assigning the jerk by feedback of the acceleration 108
6.2 Feedback in the inner loop 112
6.2.5 PI feedback 122 6.3 Combined operation of inner and outer loop feed-back 123
6.3.1 The event for switching off the outer loop 125
6.3.2 The method to switch off the outer loop 126
6.4 Inner loop feedback control including actuator dynamics 132 6.4.1 Classical feedback design applied to a reduced order system 132
6.4.2 Feedback design with actuator dynamics 135
Chapter 7 Parameter sensitivity 140
7.1 Performance metrics 140
7.2 Parameters of the baseline model and test cases 141 7.3 The parameter-sensitivity of feed-forward control 143
7.3.1 The effect on the control signal Mb 143
7.3.2 The effect on the open-loop FF controlled system performance 145 7.4 The parameter-sensitivity of FF & inner-loop feedback controlled system 148 7.5 The parameter-sensitivity with outer-loop feedback control 154 Chapter 8 Introduction of dynamic wheel-load variation 160
8.1 Simple pitch model 160
8.2 Application of dynamic wheel-load in quarter-car models 161
8.3 Flatness of a half car model 166
8.3.1 Proof of flatness of half-car for a simplified case 167
8.3.2 The fatness of a half car model 169
Chapter 9 Conclusion 172
9.1 Summary 172
9.2 Conclusions 173
9.3 Discussion 174
References 176
Appendix I Feedback linearization on a Quarter car model 181
I.1 Input output Feedback linearization 181
I.2 Mathematics 181
I.3 Input state feedback linearization 182
Appendix II Definition of the domain using the implicit function theorem 186
II.1. Theorem implicit function theorem 186
II.2. Theorem inverse function theorem 187
II.3. Observability by the flat output 187
Appendix III Reduced order local linear (Jacobian) model 192
Introduction
Chapter 1 Introduction
1.1 History of ABS
Most modern cars are equipped with electronic ABS. ABS can enhance the safety of a vehicle
in extreme situations, since it can optimize the use of the limited tyre road friction by trying to maximize the tyre longitudinal force whilst keeping sufficient lateral force to ensure stability and controllability of the handling of the car.
The design of automatic brake control systems is clearly highly dependent on the brake system characteristic and the actuator performance. As it is well known standard ABS
systems for wheeled vehicle equipped with traditional hydraulic actuators mainly are rule based control logics [19],[20]. Writing the discrete control interventions as a hybrid automaton puts this heuristic control in a formal and transparent framework [23],[24],[52],[57].
Recent technological advances in actuators have led to electro-hydraulic and electro mechanical brake actuation systems. These enable a continuous modulation of the brake torque, thereby allowing formulating active braking control as a classical regulating problem. In the field of automatic braking control a large number of methods and approaches have been proposed in the last decade ranging from classical regulation to sliding model, fuzzy neural, or hybrid architectures.
This chapter will provide and overview of ABS systems both in terms of historical
development and of the results in this field in the scientific literature. Further the main properties of the brake system will be described and the physical architecture in its components. Also the role of the brake actuation in a global chassis control (GCC)
architecture will be described.
The current ABS were conceived from system that were developed for trains in the early
1900’s. The first patents for ABS on cars date from 1932 ‘An improved safety device for
preventing the jamming of running wheels of automobiles when braking’ and a similar patent was filed in 1936 as a ‘Apparatus for preventing wheel sliding’
Anti-lock braking systems were first developed for aircraft use in 1929, by the French automobile and aircraft pioneer, Gabriel Voisin. An early system was Dunlop's Maxaret system, introduced in the 1950s and still in use on some aircraft models. These entirely mechanical control systems use a flywheel-operated pressure-limiting valve. In testing, a 30% improvement in braking performance was noted. An additional benefit was the elimination of burned or burst tires. A fully mechanical system saw limited automobile use in the 1960s in the Ferguson P99 racing car and the Jensen FF. The system was soon abandoned as it was expensive and unreliable.
Chrysler, together with the Bendix Corporation, introduced a true computerized three-channel, four sensor all-wheel antilock brake system called "Sure Brake" on the 1971 Imperial. In 1971 Nissan offered EAL (Electro Anti-lock System) as an option on the Nissan President. This became Japan's first electronic ABS (Anti-lock Braking System).
Teldix, a joint venture of Bendix and Telefunken had put Tekline on the market.
The electronics of these early systems was vulnerable to magnetic interference, large operating temperature variation, humidity and vibrations. Durability, reliability and most of all the associated legal concerns, has put the ABS development on hold in the US while
Europe took the lead: Teldix and Daimler-Benz have been co-developing anti-lock braking technology since the early 1970s. Soon after they asked Bosch to join the ABS project for
introduced the first completely electronic 4-wheel multi-channel abs system in the Mercedes-Benz S-Class in 1978.
By 1985 Mercedes, BMW and Audi had their Bosch ABS, and Ford introduced its Teves
system. Today 50 years after the first appearance have become standard and mandatory by law in Europe. Notwithstanding the historical development the research is far from complete [58]. Every step in brake actuation technology or availability of new sensors asks for a serious reconsideration of the control algorithms. In particular Force Sensing Bearings on the sensor developments side and Electro mechanical Friction brake, and in-wheel electromotor on the actuator side will trigger the next (r)evolution in active brake control. The modern ABS system applies individual brake pressure to all four wheels through a control system of hub mounted sensors and a dedicated micro-controller. ABS is offered, or comes standard, on most road vehicles produced today and is the foundation for ESC systems, which are also rapidly increasing in popularity due to the vast reduction in price of vehicle electronics over the years.
1.2 The research problem
After mass introduction of ABS the safety benefits of having Electronic Stability Program (ESP) also became apparent to the market and governments. Therefore ESP also became common and is now obligatory for all new cars to the EU market
ESP requires to brake individual wheels in order to adjust the yaw moment of the vehicle when a dangerous situation is detected. It therefore has authority over the brakes of individual wheels using the already present ABS Valves. The brake pressure that is needed for actuation is supplied by a pump in those instances that the driver is not braking. So ESP requires individual wheel brake control.
More recent developments include ‘City-safety’ systems that enable fully autonomous braking of the car from a moderate initial speed. These systems use optical laser-distance and radar sensors to recognize the other traffic and other obstacles. Hill-decent systems brake the individual wheels while driving slowly down a steep hill. Even though driving this steep hill with a fixed desired speed involves strong braking, the driver does not need to apply force on the brake pedal. These recent developments show that there is an increasing demand for brake control systems that are more versatile than just the ABS basic function of preventing wheel-lock.
The topic of the research presented is to develop a brake controller, or a brake control strategy, that can cooperate with human driver or other driver assistance system acting as supervising control. With a strong background in tyre modeling it was intended to include the knowledge of the tyre behaviour in the development of this controller. Including the dynamics of the tire contact in combination with the wheel and car body dynamics allows optimizing the braking control.
Furthermore the switching on and off of pressure and release valves in conventional ABS may be perceived as heuristic control. We try to investigate whether continuous control for braking is possible and therefore focused on continuous control development methods as a prerequisite.
The communication with higher hierarchy controllers should preferably use variables that represent physical quantities. And to be compatible with yaw control systems such as ESP, individual wheels should preferably be actuated independently.
Introduction quarter car (model) was devised. The inversion of nonlinear system dynamics based on a concept called ‘flatness’ has been introduced and applied for the purpose of feed-forward control of the brake torque. This idea was also patented by Bosch [29].
In this PhD thesis the use of flatness to linearise the quarter car system dynamics has first been employed to apply feedback on the linearised system. In Chapter 6 this leads to the development of an outer-loop feedback branch that in cooperation with the inner-loop feedback provides the required robustness. Furthermore the concept of shaping filters for the determination of smooth desired trajectories of the control inputs is a new contribution presented in Chapter 5 of this thesis. A thorough analysis of the allowed domain where the state transformations are applicable is a recent contribution to the research and is presented in Chapter 4, the method to enforce the input to be inside the domain is described in Chapter 5. A local linear stability analysis on a quarter-car model with LuGre tyre is
presented to asses the stability of the open and closed-loop system in Chapter 6 of this thesis.
Taking into account the full vehicle under the assumption of left right symmetric actuation of brakes, inspires our half car approach; In Chapter 8 the fore-aft wheel load transfer is first introduced in the equations and replaces the assumed constant normal. A framework in which the front and rear brake torques can be calculated and implemented as feed-forward control signals based on desired brake force output is developed.
1.4 Structure of the thesis
Chapter 2 of the thesis describes the derivation and selection of the suitable model to be used in the thesis research. The control approach will be outlined in Chapter 3, after a short introduction of control methodologies suitable for non linear systems. In Chapter 4 the Flatness based system inverse is presented. Also the domain where this inverse can be made is defined in this Chapter 4. In Chapter 5 some practical implementation tasks are described. These include making the control input smooth by trajectory planning or application of a shaping filter. Chapter 6 describes the feedback control that consists of an inner and outer control loop. A sensitivity analysis of the control method on the system parameters is presented in Chapter 7. In Chapter 8 the feed-forward control using a system inverse is developed for a half car leading to independent front and rear axle brake control. Finally we draw the conclusions in Chapter 9. At the end of Chapter 3 a more detailed reading guide is presented in Figure 3.8 based on the description of the control approach in that Chapter.
Tyre- and quarter car models
Chapter 2 Models for tyre brake force, and a quarter
vehicle model
This chapter introduces the model we have worked on to develop the control for braking a vehicle. Since the tyre plays an essential role in the force generation between the vehicle and the road, most of this chapter is devoted to tyre modelling. During the last decades an enormous amount of research has been invested in tyre modelling. In literature on tyre models for control purpose we see a strong preference for limited complexity.
This allows us to understand the physical phenomena through the describing equations. To have an analytical model allows the control law to be model-based. The possibility to run simulations with the tyre model in real-time is also a strongly preferred property for the applicability in control problems.
Limiting the number of parameters is efficient in establishing the control law; it also improves the physical insight. If the tyre road contact properties change, it is useful if the parameters in the tyre model are closely (preferably uniquely) related to these changes in tyre- or environmental properties. This allows for a predictable change in the control action. So we essentially expect the following properties from a tyre model to be used in the quarter car:
1. Understandable physics, parameters with a clear interpretation.
2. The typical tyre force characteristic that shows linear increasing force for small slip and a saturating tyre force due to limited tyre-road friction should be represented by the model.
3. Though the parameters in the tyre model originate from tyre tests, its physical structure should allow the tyre model to be used in operating conditions that were not identically measured. The model structure should facilitate inter- and some extrapolation.
4. The model should be able to describe the force response for moderate frequency inputs (<50 Hz), and model the transient tyre force response.
5. The tyre model should be compact, preferably with single point contact and should introduce a small number of additional tyre states.
In the subsequent section 2.1 we will introduce some definitions in tyre modelling. In section 2.2 the brush model will be introduced as the oldest and most elementary tyre model. Section 2.3 and 2.4 treat the LuGre model. It is first presented as a sliding friction model,
then by using that friction law in a brush model, the distributed LuGre tyre model is
established. The distributed model is condensed to a lumped LuGre model with one internal
deformation state in 2.5. The transition from distributed to lumped can be made analytically. Also we can find the equivalent lumped model (-parameters) in 2.5.4 by requesting equivalent axle forces. The lumped LuGre model has been applied as the road contact force
model in two quarter car plant models in 2.6; a simple 2 degree of freedom (DOF) model
with a rigid wheel. An elaborated quarter car with a so-called rigid-ring tyre model is introduced in 2.6.1
2.1 Definition of slip variables
This section describes the relevant tyre kinematics and introduces definitions that are used in the following paragraphs. Vectors have two components as the brush model will be developed in the x-y plane parallel to the road surface, and will be denoted by an overbar as in u The corresponding components are denoted by . u u Its modulus will be denoted by x, .y
the slip angle α. The wheel heading or local x-direction is defined to be perpendicular to the axle direction and road normal vector. It is typically pointing in the rolling direction of the wheel and taken positive for forward rolling. In the thesis the assumption has been made that all vehicles drive forward so vx is guaranteed. So by definition: 0
tan y x v v (2.1)Close analysis by Zegelaar [33] of a free rolling wheel has shown that a wheel with a deformable rubber tyre is not rolling on its deformed radius; neither is it rolling on its undeformed radius as if it were a rigid wheel. In fact the rolling radius is an empirical result from a dedicated experiment where a loaded wheel rolls freely, without braking.1
The effective rolling radius re is defined as:
0 b x e M v r (2.2)
where Ω denotes the wheel angular velocity.
c vx
v
x vr s r0 re vr W g rl r0 c sFigure 2.1 Kinematics of rolling of a deformable body (rubber tyre wheel)
1 Viscoelastic properties of the rubber in the contact area give rise to asymmetry in the normal pressure distribution, which leads to rolling resistance. For a non-driven (towed) wheel, this rolling resistance moment is counteracted by a couple supplied by a small friction force and a pulling force at the axle. This small friction has to be generated by a minimal amount of slip. In the thesis this determination of rolling radius is employed on a towed wheel; it thus defines the slip that generates the force that makes equilibrium with the rolling resistance to be zero slip. It is fundamentally better to measure on a driven tyre at constant speed: the drive torque provides the equilibrium with the rolling resistance torque, so no friction force is needed to provide a couple about the wheel axle.
Md Fx
Tyre- and quarter car models
Figure 2.1 shows a side view of a wheel during braking. The centre of the contact c is illustrated. The contact point c is on the intersection of the road normal vector through the wheel centre and road surface plane. The point s is introduced. It is an imaginary point attached to the wheel rim. The point s is the (virtual) centre of rotation of the wheel body in the global reference system. The location of the point s is so that the distance to the wheel centre equals the effective rolling radius re. The axle height or loaded radius is denoted rl. The inclination of the wheel plane with respect to the road normal is given by the camber angle .
By definition the circumferential velocity vc is equal to the product of angular wheel velocity Ω and the effective rolling radius:
c e
v (2.3) r
Using (2.2) it is easily shown that for free rolling vc=vx applies. The longitudinal slip velocity of the point s is defined as the difference between forward velocity of the wheel centre vx and the circumferential velocity of the wheelv . c
r x c x e
v (2.4) v v v r
During free rolling (no drive torque or braking), the longitudinal slip equals zero by definition (2.4). During braking the circumferential velocity is smaller than the axle velocity and thus there exists a positive relative velocity vr between the material point on the wheel body and the ground at s. The slip velocity or relative motion of the wheel body in the contact patch is thus: 1
,
s x c y
v v v v (2.5)
The tyre slip is defined by normalizing the slip velocity with a reference velocity. We distinguish three definitions that are commonly used; first the practical slip (- ratio) is obtained by dividing the slip velocity by the axle forward velocity.
s x v v
(2.6)
Secondly the theoretical slip: it is the slip velocity normalized by the rolling velocity v . c
This definition is favourable in theoretical analyses.
s c v v
(2.7)
Note that the slips are collinear with the slip velocityv . It is the custom to describe tyre s
forces as function of slip rather than slip velocity. This convention will prove its value in the development of the brush model.
There are several conventions on how to define the tyre slips, e.g. the ISO [46] and SAE standards [47] use -100κx [%] to represent the longitudinal slip, and α [deg.] for the lateral slip. In the paper of Gävert [34] the slips are defined such that the signs are consistent for
1there exists no circumferential speed in y direction when the wheel plane remains vertical or at a fixed ‘camber’ angle.
all slip definitions and such that the generated force has opposite sign to the slip. The slip ratio λ is also commonly used to denote longitudinal slip as
x
(2.8) It is straightforward to translate between the different slip representations:
, tan( ) / 1
1 x (2.9a)
, tan( )
1 x (2.9b)At low levels of slip the stationary force versus slip characteristics may be represented by a linear function.
x F
F C (2.10)
where Fx is the force in the x-direction on the wheel produced by the friction and where the slip stiffness CFλ denotes the local derivative of the tyre force characteristic:
0 x F F C (2.11)
For the sign convention chosen, the slip velocity vr during braking is positive, while the longitudinal force Fx is negative. For convenience the brake force Fb is introduced as a positive force in negative x-direction: Fb Fx
For increasing levels of slip the linearised relation (2.10) does no longer hold. The brake force is limited by the friction between tyre and road and thus saturates towards a maximum defined by friction and normal load. The subsequent section will introduce the brush model as a most simplified physical representation of the tyre road interaction.
2.2 The brush model
The brush model is a standard approach to model tyre forces developed first by Fromm [50], and widely used by others see e.g. [51],[30],[33],[34]. This section is kept brief and has the main purpose of introducing the notations and formulas needed in the succeeding sections. The brush model is a physics-based tyre model. These models describe the tyre kinematics of rolling, the contact force, and the compliance in each point of the contact area. Usually these types of models tend to a finite element approach and can only be solved with help of powerful computers. By taking for each model aspect the simplest approximation, the brush model has an analytical solution. The rubber volume between the tyre belt and road surface is partitioned into infinitesimal elements in the form of elastic bristles.
The tyre belt is assumed to deform in the contact area in a direction normal to the road, to allow contact with the road over a finite length; the contact width is disregarded completely. The vertical load is thus generated by distributed normal force qn acting over the contact length, defined as L. It is not needed that tread elements have thickness, but since the model is referred to as brush model, we imagine the bristles to have a finite size in the direction normal to the road.
Tyre- and quarter car models
bristles adhere to the road surface. Thus the deformation force is transmitted by static friction. In the sliding region the bristles move on the road surface under the presence of kinetic friction. Positions in the contact region are expressed in a local reference system [O,ξ,ψ] attached to the tyre belt, with its origin located in the centre of contact c. The contact length now spans the ξ interval [a, a], where a denotes half of the contact length L. The carcass and belt construction of the tyre are assumed to be infinitely stiff, so the effects of carcass deformations will be neglected here. To stay in the centre of contact, the origin of the local reference system has a global ‘propagation’ velocity that is equal to the longitudinal axle speed.1
a -a xs mc nq O Sliding Adhesion Road carcass rubber bristles vc vx x Fn
Figure 2.2 The Kinematic Brush Model
The brush model depicted in Figure 2.2 is an idealised representation of the tyre in the region of contact. The deformation of the tread elements is established by considering the velocity at both ends of the tread-element.
2.2.1 Adhesion forces
Regard a specific bristle in the adhesion region, which is attached to the tyre carcass at position ξ in the local coordinate system. The base point of a bristle is attached to the tyre belt and will move, relatively to the wheel axis, starting at the leading edge to the trailing edge with a velocity -v . The other end of the bristle, which is in adhesion with the road, c
has zero velocity in the global reference frame. Relatively to the wheel axis, the bristle tip moves rearward with the propagation speed of the contact centre, which itself is approximately equal to the forward speed of the axle vx. We introduce tc(ξ) the time elapsed since the bristle entered the contact region at the leading edge. For the development of the stationary brush characteristics the velocities vc, vx, and vy are assumed to be constant during the integration interval [0, tc], i.e. as a bristle travels through the adhesion region. Hence the
1 The propagation speed of the contact can be approximated by the longitudinal speed of the axle when the contact point is located precisely under the axle or the wheel is moving along a straight path. In other words, when the product of camber and turnslip is negligible [30]
bristle-base position is a v tc c( ) and tc
a
/ .vc Using this expression for tc(ξ), the deformation is the integral of velocity differences between bristle-base and -tip over the period tc(ξ). This directly leads to:
0tc
x c
x c x x c v v v v dt a a v
(2.12a)
0tc y
y y y c v v dt a a v
(2.12b)where slip definition (2.9a) is used in the last expression. This proportional dependence of deformation on theoretical slip supports its choice as tyre model input. With the assumption of linear elasticity the distributed shear force depending on deformation (2.12) becomes is:
ax px x dF c d (2.13a)
ay py y dF c d (2.13b)where cpx and cpy are the longitudinal and lateral bristle stiffness per unit length, and where the subscript a denotes the fact that we are dealing with the adhesion region. The strict assumption of having constant vc, vx, and vy over the interval [0, tc] could be relaxed to allow slow variations in ζx and ζy compared to the duration of the bristle travelling through the contact area tc. The total adhesion force is computed by integration of equation (2.13) over the contact region:
s s a a ax ax px x F dF c a d
(2.14a)
s s a a ay ay py y F dF c a d
(2.14b)The break-away point ξs is the point along the contact length that separates the regions of adhesion and sliding. To calculate the adhesion force it is necessary to know the location of ξs.
2.2.2 The size of the adhesion region
The size of the adhesion region is determined by the available static friction between the tyre and the road. Assume a parabolic pressure distribution
2 3 1 4 n n F q a a (2.15)
where Fn denotes the normal force acting from the road on the tyre to support the wheel weight and part of the weight of the vehicle. With using dFn
qn
d the integral of
n
q can be found equal to Fn. The deformation is limited by the largest force that can be transmitted by the friction. So the coordinate where sliding first occurs can be found s from demanding that the shear force generated by the deformations in the adhesion zone can actually be transferred to the road surface by the local friction force. This force is a
Tyre- and quarter car models
2 2 1 ay ax n sx n sy dF dF dF dF (2.16)Here the static friction s is assumed to be anisotropic, with the friction coefficients sx in x-direction and sy in y-direction. dFn(x) denotes the force acting on the infinitesimal bristle at position ξ. For isotropic friction sx = sy = s the constraint is often called friction circle, ‘circle of More’ or ‘Kammsher kreis’ and is characterised by
2
2
2 2ax ay n s
dF dF dF (2.17)
When dFa
exceeds the friction constraint, the bristle leaves the adhesion region andstarts to slide. While sliding, the bristle experiences the sliding- or ‘Coulomb’ friction c. This is illustrated in Figure 2.3 and will be further analysed in the sequel.
msx nq ( )x mcx nq ( )x zx pxc Fax( )zx Fsx( )|zx z =0y x zs( )|x z =0y -a 0 a x
Figure 2.3 Illustration of the partitioning of the contact area into regions of adhesion and sliding the case of pure longitudinal slip. The adhesion force of an element at is
cpxx (a - ) d. The sliding force for an element at is determined by the pressure
distribution as cx qn() d. The vertically and horizontally hatched areas represent the total
forces of adhesion and sliding.
Pure brake slip characteristic
The deformation pattern will first be determined for zero sideslip and stationary conditions. The pure brake slip characteristic can then be derived from the shear deformation over the contact length.
The coordinate ξs for the transition from adhesion to sliding can be found as the ultimate ξ for which inequality (2.17) holds after substitution of the adhesion force (2.13a):
2 2 2 3 1 1 4 n s x px F c a a a a
Besides the solution , the following non-trivial root of the above quadratic function a
gives ξs:
s x 2aa
where the tyre parameter is introduced as: 2 2 3 px s n c a F (2.18)
From this equation the slip at which the tyre is sliding over its complete contact length can be determined, by setting : s a 1 x
The brake force for pure longitudinal slip, (lateral slip equals zero) can be found from the following compound integral over the contact length, where Fax denotes the adhesion force and Fsx the sliding force contribution:
sgn 3 1 2 4 s s a n x ax sx px x c x a F F F F c a d d a a
With the simplifying assumption that static- and sliding friction are equal
s c
we solve the integral and find in analogy to [33] [30]:
2 3
3 3 sgn x n x x x x F F if x x
sgn x n x F F if x x (2.19)Figure 2.4 gives the pure slip characteristic of the brush model.
0 0.05 0.1 0.15 0.2 0.25 0 1000 2000 3000 4000 5000 ζx [ ] −F x [N]
Tyre- and quarter car models
2 2 2 2 a x px x px x px a F c d c a c a
(2.20)And thus the practical slip stiffness (2.11) follows from the theoretical slip stiffness 0 / x x x C F : 2 2 x px C c a (2.21)
For pure sideslip the derivation of the stationary force characteristic will follow the same procedure. In (2.19) and (2.20) the suffix x can be exchanged for y and we find for the cornering stiffness:
2 2
y py
C c a (2.22)
In the next section the interaction between longitudinal and lateral slip will be studied when they occur simultaneously.
2.2.3 The combined slip characteristic
In [34] an extensive discussion has been presented on anisotropic brush models under combined slip. A compact model can be derived under the simplifying assumptions that cp denotes the isotropic tread element stiffness and denotes the isotropic friction coefficient, see [44]. The tread element deflections are directed opposite to the slip speed vector V , s
also in the sliding region. Again using theoretical slip (2.7) and the boundary condition 0
at the leading edge, the deformation in the adhesion zone can be found:
a
(2.23)
The corresponding horizontal tread element shear force reads
c
q c (2.24)
In the zone of sliding the force vector is determined as:
s n s V q q V Analogous to the case of pure slip in the previous section the transition point can be determined:
2 1 ,
s a
(2.25)
The slip where total sliding starts:
1
2
3 3 n F F if n F F if (2.26) The vector force components in longitudinal and lateral direction becomeF F
(2.27)
Using these equations we can add combined slip to the brush model. Let use the brush model for a (virtual) experiment where the slip angle is fixed and the longitudinal slip ζx
is varied from 0 to 0.5 due to braking. Instead of fixing a value for theoretical slip ζy , we choose constant lateral slip , since that is more in line with a real test scenario. Figure 2.5 shows both lateral and longitudinal force for this experiment.
0 0.1 0.2 0.3 0.4 0.5 0 1000 2000 3000 4000 5000 ζx [ ] − F [N]
stationary characteristic brush model, ζ
y=(1+ζx)tan(α) F x,α=5 o F y,α=5 o Fx,α=10o Fy,α=10o Fx,α=15o F y,α=15 o
Figure 2.5 The steady state brush model characteristic. The graph shows longitudinal force in combined slip condition. The brake force –Fxversus theoretical slip is presented for
three increasing side slip angles of α= 5° 10° 15° respectively. The corresponding theoretical slip y can be calculated according to (2.09b) as y =(1+x)tanα.
The main purpose of presenting the model for combined slip is to show that the performance in lateral direction deteriorates under braking conditions and vice versa. In
Figure 2.5 we notice that when the brake force (solid line) grows due to longitudinal slip,
immediately the lateral force (dashed line) decreases. Limiting this decay of lateral force is the main reason for existence of ABS brake control systems.
2.3 LuGre friction model
The abrupt transition from adhesion to sliding and the treatment of two separated intervals in the contact is considered as a disadvantage to derive a compact dynamic tyre model from the brush model. The application of a model that describes a smooth transition from adhesion (or presliding) to sliding is described in this section.
Dahl was one of the first to describe gradual saturation of the friction force up to the maximum attainable value; this theory is presented in section 2.3.1. The introduction of an internal deformation state is subject of section 2.3.2. Then researchers from universities of Lund and Grenoble have added the velocity dependence of friction to the model. The so called LuGre model is described in section 2.3.3. Additional considerations on the sliding
speed are presented in section 2.3.4. The distributed LuGre model is subject of subsection
2.4, where the lumped LuGre model will be explained in the next section 2.5. 2.3.1 Introducing the Dahl friction model for a single bristle
Let us first consider a single bristle in the brush model. The property of a brush model to be spatially distributed is closely related to the discontinuous transition of a single bristle from adhesion to sliding. Over the contact length the friction constraint (2.16) needs to be monitored for each bristle.
Tyre- and quarter car models
section 2.4. By first assuming a tyre to be standing and non-rotating, the convective term is avoided in the sliding friction models that are analysed in the current and following section 2.3.3. We first concentrate on the transition between adhesion and sliding, by monitoring a single bristle with finite size and its deformation δx in a conventional brush model. When the bristle is in adhesion the relative motion xr between the bristle base and the road, directly prescribes the deformation δx. The shear force linear elastically depends on δx according to:
x px x
f c
where the product cpx denotes stiffness of a finite single bristle cx, based on the
distributed brush model stiffness cpx. Furthermore we introduce fb, the force at the bristle tip in negative x-direction:
b x
f and f cxcpx
In adhesion we then have the simple linear force relation:
fb
cx x
(2.28)An instantaneous transition into sliding occurs where the increasing deflection reaches the friction bound fc . cqn f fc xr v >r 0 v <r 0 vr<0 -fc O
Figure 2.6 The friction force response with alternating relative motion applied to a single bristle
In Figure 2.6 an alternating relative motion with arbitrary stroke is used to visualize the
force response of a single bristle. The linear force increase during adhesion starts from the origin. At fc the force saturates abruptly, and the force f stays at the friction limit. At reversal of the imposed motion, immediately the brush tip gets in adhesion again, and the present deformation will be linearly reduced to zero, and will be increased in opposite direction subsequently. This linear proportional behaviour is displayed with the dashed lines in Figure 2.6. The force response on this cyclic input motion shows a hysteresis loop.
Despite of this kinked force deflection curve, the brush model becomes a continuous model due to the presence of a great number of bristles that all have their discrete transition from adhesion into sliding at various instances. When the brush model is solved analytically under the assumption of a predefined input, the model eventually becomes continuous. This can only be achieved for a limited class of input signals, e.g. harmonic or stationary constant. For a general solution one would assign a state to each bristle deformation, which leads to a discretized contact model.
In the remainder of this section the slip model for an individual bristle will be reconsidered. When the force deflection curve smoothly saturates the creation of a compact lumped tyre model is facilitated. LuGre and Dahl friction models share this property of a smooth
saturation of the friction force.
The LuGre model was based on the friction model developed by Dahl [40] for simulating
control systems with friction. The Dahl model was developed to explain the hysteresis behaviour of precision ball bearings undergoing very small amplitude oscillations. The model is widely used for the simulation of motion control considering friction effects. The model adopts the stress-strain curve from solid mechanics and describes friction contact by the differential equation (2.29). The friction force on a single bristle will be denoted by f. The global tyre friction force can be found by a summation of many bristle forces. The force response is presented in Figure 2.7
1 bsgn( ) b r Dahl c r f df v f dx (2.29)
where fb is the friction force in negative x-direction, fc is the maximum (Coulomb) friction force and xr is the relative displacement, vr denotes the relative velocity between the parts. In Dahl’s model Dahl [Nm-1] denotes the pre-sliding stiffness. Suppose vr>0, then fb behaves as a first order system with space-constant fc/σDahl.
f fc xr v >r 0 v <r 0 vr<0 -fc
Figure 2.7 The typical stress strain curve of a solid in friction contact.
The stress-strain curve in Figure 2.7 represents the brush deformation during adhesion
phase and the gradual transition into the sliding phase, while respecting the friction force limit fc. To make the Dahl model equivalent to the shear stiffness of the brush model, we require the adhesion or ‘pre-sliding’ stiffness to be equal: Dahl cx
Since xr(t) is a function of time, the substitution of vr in (2.29) leads to an ordinary xr
differential equation:
r
1 b sgn( ) b b r r x c r dx t f df df v v c f dt dx t dt (2.30)The force function f monotonically approaches fc as long as the velocity vr is positive. When the direction of motion is reversed the force response f follows a mirrored trajectory
Tyre- and quarter car models
2.3.2 Introduction of the internal deformation state
z
For modelling the friction of a sliding bearing, Canudas de Witt et al. introduce in [35] a virtual deformation state z that replaces the force by using the relation:
0
b n
f (2.31) f z
where is the stiffness parameter, and f0 n is the normal load on the single bristle. When z is interpreted as the deformation of a bristle, a third notation for the pre-sliding stiffness is derived from (2.31):
0
x n
c (2.32) f
Note that in [35] Canudas de Witt et al. denote the presliding stiffness by 0 while in (2.32) we use cx. In [38] Canudas de Witt et al. normalize the presliding stiffness with the normal load and use 0 for this normalized stiffness. This notation is continued in [36],[37] and employed in this thesis.
Then equation (2.30) can be rewritten in the following form:
(2.33) (2.34) 0 0 0 0 0 1 n sgn( ) r n n r c n r r c r r c f z dz v f f v f dt f dz v z v dt f z v z v (2.35)
In order to illustrate (2.35) let us assume a time history for xr, xr vras desribed in the left
graph of Figure 2.8 and compute z from (2.35)
−0.1 −0.05 0 0.05 0.1 xr [m ] motion history −1 −0.5 0 0.5 1 t vr [m s − 1 ] −0.05 0 0.05 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 x r [m] z [m]
simulated Dahl friction
Figure 2.8 The left graphs show the displacement xr and velocity vr history used to
calculate the hysteresis loop in the right graph. The right graph shows the virtual deformation state z from (2.31) with σ0 = 50 m-1, c = 1. Note that initially the slope is 1:
during presliding (adhesion) all relative motion contributes to deformation.
In Figure 2.8 the deformation z is illustrated with an imposed motion with a 0.12 m stroke. The deformation in the presliding phase is equal to the imposed motion: for z0 we get from (2.35) that zvr xr, thus the slope equals +1 at the origin. z/ xr
In (2.32) the shear stiffness cx is proportional to normal load, a property that originates from the LuGre model and its application to sliding bearings. The bearing materials were
modelled with a brush model, where the bristles of unequal length represent the surface roughness of the material. In that brush-model increasing the normal load would lead to an increasing number of bristles that make contact and contribute to carrying the normal (and horizontal-) load. Under horizontal shear force the shear stiffness of all active bristles contribute to the overall shear (presliding) stiffness. The analysis of the sliding friction of this bearing material as a whole allows the hypothesis of a shear stiffness that is proportional to the normal load applied.
Remark
To model the contact of a bristle, it is our opinion that the model (2.31) is not appropriate. In the brush tyre model the bristles are imagined as ‘cylindrical’ rubber elements of equal dimensions. A friction model that considers only one bristle will not show the effect of the number of carrying bristles being proportional to the normal load. In fact it is expected that the single bristle has constant shear stiffness, denoted by cx0 and depending on the bristle dimensions and rubber properties. Starting from this hypothesis of constant shear stiffness and an arbitrary normal load distribution fn
, where the normal stress depends on the location of the bristle in the contact, it is impossible to define a constant σ0.
0 0 x n c f The authors of [37] and [42] require such constant σ0 to arrive at the common Dahl state equation (2.35), so in that case cx inevitably depends on the location ξ in the contact, and (2.32) becomes:
0
x n
c f
Let us however start again from (2.30) with the constant shear stiffness hypothesis, and derive equations similar to (2.33) - (2.35). By using cx0 to denote theconstant value of stiffness cx we get: (2.36) (2.37) 0 0 0 0 0 0 0 b b x r x r c x x x r x r c x r r c n df f c v c v dt f dc z c z c v c v dt f c z v z v f (2.38) In section 2.4.3 the preference to preserve a constant shear stiffness cx0 from a brush model perspective has been worked out. Meanwhile we comply with the definition of a constant σ0 for the upcoming sections to allow analytical treatment of equation (2.35).
2.3.3 LuGre model for sliding contact
Both Canudas de Witt et. al. [36][37][38] and Deur et. al. [42][43][71] have been developing the LuGre friction model, and made it applicable for car tyre friction. The LuGre
