Institutionen för systemteknik

Transcription

1 Institutionen för systemteknik Department of Electrical Engineering Examensarbete Modeling and Estimation of Dynamic Tire Properties Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Erik Narby LITH-ISY-EX--06/3800--SE Linköping 2006 Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden Linköpings tekniska högskola Linköpings universitet Linköping

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3 Modeling and Estimation of Dynamic Tire Properties Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Erik Narby LITH-ISY-EX--06/3800--SE Handledare: Examinator: Thomas Schön isy, Linköpigs universitet Anders Stenman NIRA Dynamics AB Fredrik Gustafsson isy, Linköpigs universitet Linköping, 26 February, 2006

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5 Avdelning, Institution Division, Department Division of Automatic Control Department of Electrical Engineering Linköpings universitet S Linköping, Sweden Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LITH-ISY-EX--06/3800--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Modellering och skattning av dynamiska däcksegenskaper Modeling and Estimation of Dynamic Tire Properties Författare Author Erik Narby Sammanfattning Abstract Information about dynamic tire properties has always been important for drivers of wheel driven vehicles. With the increasing amount of systems in modern vehicles designed to measure and control the behavior of the vehicle information regarding dynamic tire properties has grown even more important. In this thesis a number of methods for modeling and estimating dynamic tire properties have been implemented and evaluated. The more general issue of estimating model parameters in linear and non-linear vehicle models is also addressed. We conclude that the slope of the tire slip curve seems to dependent on the stiffness of the road surface and introduce the term combined stiffness. We also show that it is possible to estimate both longitudinal and lateral combined stiffness using only standard vehicle sensors. Nyckelord Keywords sensor fusion, system identification, tire stiffness, slip, vehicle dynamics

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7 Abstract Information about dynamic tire properties has always been important for drivers of wheel driven vehicles. With the increasing amount of systems in modern vehicles designed to measure and control the behavior of the vehicle information regarding dynamic tire properties has grown even more important. In this thesis a number of methods for modeling and estimating dynamic tire properties have been implemented and evaluated. The more general issue of estimating model parameters in linear and non-linear vehicle models is also addressed. We conclude that the slope of the tire slip curve seems to dependent on the stiffness of the road surface and introduce the term combined stiffness. We also show that it is possible to estimate both longitudinal and lateral combined stiffness using only standard vehicle sensors. v

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9 Acknowledgements I would like to thank my supervisor at NIRA Dynamics Anders Stenman for his help and advice especially during the initial phase of the work with the thesis. I would also like to thank my supervisor at Linköping University Thomas Schön for his support during the project and for taking the time to proofread the report. Thanks also to Urban Forssell CEO of NIRA Dynamics and to my examiner Fredrik Gustafsson for their input to and their profound interest in this thesis. Special thanks to Peter Lindskog at NIRA Dynamics for his help with the non-linear system identification parts of the thesis. Finally I would like to thank everyone else at NIRA Dynamics for showing interest in my work and for all fruitful discussions. vii

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11 Notation Symbols longitudinal tire force from wheel XX lateral tire force from wheel XX normal force air resistance force frictional force r vehicle yaw rate (turning rate around COG) wheel radius longitudinal velocity at COG lateral velocity at COG longitudinal acceleration at COG lateral acceleration at COG δ steering angle of front wheels δ slip offset J z vehicle yaw moment of inertia around COG J w wheel moment of inertia a distance from front axle to COG b distance from rear axle to COG h f front axle length h r rear axle length µ x normalized traction force µ y normalized lateral force µ max coefficient of friction µ xmax longitudinal coefficient of friction µ ymax lateral coefficient of friction µ C Coulomb coefficient of friction µ V coefficient of viscous friction µ S coefficient of stiction F xxx F yxx F N F D F f r 0 v x v y a x a y ix

12 x r 0 C x C y C xn C yn C F C R C D C AIR C δ s α ρ air ω wheel radius longitudinal tire stiffness lateral tire stiffness normalized longitudinal tire stiffness normalized lateral tire stiffness front lateral tire stiffness rear lateral tire stiffness drag coefficient total air resistance coefficient ratio between steering wheel angle and steering angle longitudinal wheel slip slip angle air density wheel angular velocity For all symbols the index XX denotes a wheel index and can be: FR Front Right FL Front Left RR Rear Right RL Rear Left Acronyms SAE GPS ABS ESP RFI TPI COG Society of Automotive Engineers Global Positioning System Anti-lock Braking System Electronic Stability System Road Friction Indicator Tire Pressure Indicator Center Of Gravity

13 Contents 1 Introduction Background Company Objectives Limitations Thesis Outline Model Based Signal Processing State-Space Models Sampling of Continuous-Time Systems Linearization Observers and Observability Observers Observability Kalman Filter Extended Kalman Filter Discretized Linearization Adaptive Filtering Kalman Filter for Adaptive Filtering Change Detection CUSUM Test Grey Box Modeling and System Identification Friction and Tire-Road Friction Estimation Methods What is Friction? Friction Models Static Friction Models Dynamic Friction Models Cause Based Tire-Road Friction Estimation Methods Roughness Based Methods Lubricant Based methods Effect Based Tire-Road Friction Estimation Methods Vibration Based Methods Acoustic Based Methods xi

14 xii Contents Slip Based Methods Tire-Tread Deformation Sensors Hard Braking Extra Wheel Vehicle and Tire Dynamics Vehicle Body Model Wheel Modeling Tire Modeling Longitudinal Tire Modeling Lateral Tire Modeling Empirical Tire Models Analytical Tire Models Longitudinal Tire Models Lateral Tire Models Combined Tire Model Sensors and Measurements Wheel Angular Velocity Sensors Lateral Accelerometer Yaw Rate Gyro Steering Wheel Angle Sensor Stiffness Estimation Tire Stiffness or Slip-slope? Stiffness Estimation using Regression Models Estimation of Longitudinal Stiffness Estimation of Lateral Stiffness Stiffness Estimation using Parameter Identification in State-Space Models Linear Tire Model Extended Kalman Filter Non-Linear Tire Model Off-Line System Identification Mapping Stiffness to Friction Calculating µ max from Braking Data Mapping Functions and Curve Fitting Linear Function Structure Quadratic Function Structure Results Surface Effects on Slip-Slope The Secant Effect Sliding in the Pre-Sliding Displacement Phase Tire Stiffness and Surface Stiffness

15 8 Direct Estimation of µ max Direct Friction Estimation using the Brush Tire Model Results Wheel Model with Dahl Tire Model Results Limiting Factors in Effect-Based Tire Road Friction Estimation Excitation Noise Conclusions Estimating Stiffness Surface Effects on Slip Using Estimated Stiffness to Estimate µ max Direct Estimation of µ max Bibliography 73 A Vehicle Simulation Model 75 B Test Cases 76 B.1 Sim B.2 Sim B.3 Sim

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17 Chapter 1 Introduction 1.1 Background Friction is the basic principle upon which all wheel driven vehicles rely. The maximum tire-road friction has heavy impact upon how a vehicle behaves in different situations. This is something which everyone who has ever driven a car, motorcycle or bicycle on a slippery surface has experienced. Knowledge about the maximum tire-road friction has been important for the driver for as long as wheel driven vehicles have existed. With the increasing amount of systems in modern vehicles designed to measure and control the behavior of the vehicle, information regarding the maximum tire-road friction becomes even more important. As this information has grown more and more important efforts to find a method of estimating the available friction have increased accordingly [1]. Many methods have been proposed, but so far no final solution to the problem has been given. 1.2 Company This master s thesis has been performed at NIRA Dynamics AB. NIRA Dynamics AB is a company active in the area of safety enhancing software for the automotive industry. The company currently has 17 employees and has offices in Linköping and Gothenburg. More information about NIRA Dynamics AB can be found at the company s web site: Objectives The objectives of this thesis are: To implement and evaluate new methods of estimating the longitudinal tireroad stiffness. To develop, implement and evaluate methods for estimating the lateral tireroad stiffness. 1

18 2 Introduction To develop, implement and evaluate a method for using estimated longitudinal and lateral tire stiffnesses to estimate the tire-road friction coefficient. 1.4 Limitations To limit the scope of this thesis some limitations are made. The vehicle is assumed to front wheel driven. Only methods using standard vehicle sensors i.e., ABS, Anti-lock Braking System, and ESP, Electronic Stability Program, sensors, are considered. 1.5 Thesis Outline Chapter 2 contains an introduction to model based signal processing and describes the theory used in this thesis. The concept of friction and methods of estimating tire-road friction are discussed in Chapter 3. Chapter 4 contains a description of vehicle and tire dynamics and modeling. Chapter 5 describes and evaluates methods of estimating tire-road stiffness and Chapter 6 explores whether it is possible to map values of stiffness to values of µ max. The effects which surface conditions have on slip-slope are discussed in Chapter 7. In Chapter 8 methods of estimating µ max directly, without going via tire-road stiffness are described and evaluated. Chapter 9 discusses what limits there are to effect based methods of estimating µ max. Finally Chapter 10 sums up the results of the thesis.

19 Chapter 2 Model Based Signal Processing In model based signal processing a model of the system which generated a signal is used. This makes it possible to analyze the signal and the system which generated the signal with methods developed in system and control theory. The method of using measurements from multiple sensors to get measurements with higher precision or estimates of non-measurable quantities is called senor fusion. Modelbased signal processing and the Kalman filter in particular are the foundations of sensor fusion. 2.1 State-Space Models Many systems can mathematically be expressed as a system of first-order differential/difference equations. A system which is described by a system of first-order differential/difference equations is said to be on state-space form. Knowing the state of a system makes it possible to calculate all future output signals if all future input signals are known. A general continuous-time system in state-space form can be written as: ẋ(t) = f(x(t),u(t)), y(t) = g(x(t),u(t)), (2.1a) (2.1b) where x is the state vector, u is the input signal vector and y is the output signal vector. A general discrete-time system in state-space form is often written as: x[t + 1] = f(x[t],u[t]), y[t] = g(x[t],u[t]). (2.2a) (2.2b) Systems which can be described by a linear system of differential/difference equations are of special importance in system theory and control theory as there exists 3

20 4 Model Based Signal Processing many methods for analyzing such systems. A linear continuous-time system can be written as: ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t). (2.3a) (2.3b) A linear discrete-time system can be written as: x[t + 1] = Ax[t] + Bu[t] y[t] = Cx[t] + Du[t] (2.4a) (2.4b) If the system functions/matrices do not vary with time the system is called timeinvariant Sampling of Continuous-Time Systems The modeling of a physical system in many cases lead to a continuous-time model i.e., a system of differential equations. When implementing a controller or an observer based on a system model in a computer it is necessary to have the system model in discrete-time form. Therefore it is often necessary to approximate the continuous-time system model with a discrete-time system model. This process is called discretization or sampling of a system. For a linear system this can be done analytically using he following theorem: Theorem 2.1 (Discretization of linear systems) If the system ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (2.5a) (2.5b) is controlled with an input signal which is piecewise constant during the sample interval T, then the relation between the input signal, the system state and the value of the output signal in the sample moments is given by the discrete-time system where x[kt + T] = Fx[kT] + Gu[kT], y[kt] = Hx[kT] + Ju[kT], T F = e AT,G = e At Bdt,H = C,J = D. 0 (2.6a) (2.6b) (2.6c) See for example [9] for a proof of the theorem.

21 2.2 Observers and Observability Linearization Many methods in system and control theory are based upon the theory of linear differential/difference equations. In many cases when we have a non-linear system it useful to approximate the system with a linear system around a certain point in (x 0,u 0 ) in state-input space. We can then analyze/control the system close to this point using linear methods. The standard way of doing this is to do a Taylor expansion of the system around (x 0,u 0 ) and disregard higher-order terms. This gives for a time-discrete system on form (2.6). where x[t + 1] f(x 0,u 0 ) + A(x[t] x 0 ) + B(u[t] u 0 ), A = df(x,u) dx, x=x0,u=u 0 B = df(x,u) du x=x0,u=u 0 With the approximation x 0 = f(x 0,u 0 ) the variable substitution x[t] = x[t] x 0, ũ[t] = u[t] u 0 gets us back to the linear case; x[t + 1] = A x[t] + Bũ[t]. (2.7) 2.2 Observers and Observability Observers Often we are in the situation that we have a system which we want to control or analyze without having the possibility to measure all states. In this case we may want to estimate the states which we do not have the possibility to measure. The most common way of doing this is to use an observer. We start with a system in state-space form with measurement noise: ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) + v(t). (2.8a) (2.8b) Here v(t) is measurement noise. We start with a simulation of the system with the known input signals: ˆx(t) = Aˆx(t) + Bu(t). We now feed back the output signal error y(t) Cˆx(t) Du(t) which gives us the observer: ˆx(t) = Aˆx(t) + Bu(t) + K(y(t) Cˆx(t) Du(t)), (2.9) where K is the observer gain matrix. Let us study the estimation error: From (2.8), (2.9) and (2.10) we now get x(t) = x(t) ˆx(t) (2.10) x(t) = (A KC) x(t) + Kv(t) (2.11)

22 6 Model Based Signal Processing We see now that as long the poles of the observer eig(a KC) lie in the stability region and the system is observable, the estimation error will decay towards zero. Furthermore it can be seen that the choice of K is a trade-off between having the estimation error decaying quickly to zero and sensitivity to measurement noise Observability In order to be able to estimate a state from measured signals a change in the state must be visible in the measured signals. A state which has this property is said to be observable. Definition 2.1 (Observability) The state x 0 is said to be non-observable if, when u(t) = 0, t 0 and x(0) = x the output signal y(t) = 0, t 0. The system is said to be observable if it has no non-observable states. Observability of Linear Systems The observability of a linear system can be analyzed using the following theorem. Theorem 2.2 The space of non-observable states of a linear system is the null space to the observability matrix O. O(A,C) = C CA. CA n 1 This means that the system is observable if and only if O has full rank. (2.12) Observability of Non-Linear Systems Observability is a much more complicated matter for non-linear systems than for linear. There exists theory which extends the concept of linear observability to non-linear systems, see for example [9]. This theory is however hard to apply for practical systems which are subject to noise. In real world applications the observability of non-linear systems is often decided by comparing state estimates from non-linear observers with measured or simulated data. Good tracking ability of the observer indicates that the states which are estimated are observable at least in the tested signal region whereas bad tracking ability indicates the opposite. However no definite conclusions can be drawn from such a test. 2.3 Kalman Filter How should the observer amplification K be chosen in order to minimize the estimation error? The filter which solves this problem is called the Kalman filter [11].

23 2.4 Extended Kalman Filter 7 Suppose that we have a linear discrete time, time variable system in state space form with state and measurement noise: x[t + 1] = A t x[t] + B v,t u[t] + B u,t v[t] y[t] = C t x[t] + e[t] (2.13a) (2.13b) Assume that the process noise v and the measurement noise e are non-correlated, white noise random processes with the following properties: E{v[t]} = E{v[t]} = 0, Cov{v[t]} = Q t, Cov{e[t]} = R t. (2.14a) (2.14b) (2.14c) (2.14d) Let the initial state x[0] of the system have the following properties: E{x[0]} = x 0, Cov{x[0]} = Π. (2.15a) (2.15b) Then the optimal linear observer in the least-squares sense is called the Kalman filter, after its founder, and is given by the equations: ˆx[t + 1 t] = A tˆx[t + 1 t] + B u,t u[t], ˆx[t t] = ˆx[t t 1] + K t (y[t] C tˆx[t t 1]), K t = P t t 1 Ct T (C t P t t 1 Ct T + R t ) 1, P t+1 t = A t P t t A T t + Q t, P t t = P t t 1 P t t 1 Ct T (C t P t t 1 Ct T + R t ) 1 C t P t t 1, ˆx[0 0] = x 0, P t t = Π. (2.16a) (2.16b) (2.16c) (2.16d) (2.16e) (2.16f) (2.16g) It can also be proved that if the process and the measurement noise are Gaussian random processes the Kalman filter is not only the optimal linear filter but also optimal when non-linear filters are taken into account. The Kalman filter also solves the problem of choosing the observer amplification for a time-varying system. In most cases Q and R are not known exactly but are used as tuning parameters. A large Q/R ratio puts high emphasis on tracking ability but less on noise suppression and vice versa. 2.4 Extended Kalman Filter In many situations we have a non-linear model of a system which we want to analyze. If we do not have the possibility to measure all states it is, as in the linear case, desirable to be able to estimate the non-measured states using an observer. The extended Kalman filter is a method for constructing observers

24 8 Model Based Signal Processing for non-linear systems. The idea is simply to linearize the system around the previously estimated state and then apply the Kalman filter in each step. If the system is given in continuous-time we have two possibilities: 1. To first linearize the model around the previously estimated state and then discretize the linear model, so called discretized linearization. This is the method used in this thesis. 2. To first discretize the model and then linearize the discrete model around the previously estimated state, so called linearizied discretization. This method is not used in this thesis. For a thorough explanation of this method, see [11] Discretized Linearization Assume that we have a non-linear time-continuous system on the form ẋ(t) = f(x(t),u(t)) y(t) = g(x(t),u(t)) (2.17a) (2.17b) Taylor expansion of (2.17a) around the point (ˆx,u 0 ) give when disregarding higherorder terms: ẋ(t) = f(ˆx,u 0 ) + f x (ˆx,u 0 )(x ˆx) + f u (ˆx,u 0 )(u u 0 ), (2.18) where f x and f u denote the derivative of f(x,u) with respect to x and u, respectively. Rearrangement of (2.18) give us the linear system: ẋ(t) = f x (ˆx,u 0 )x + u, (2.19) where u = f(ˆx,u 0 ) + f u (ˆx,u 0 )(u u 0 ) f x (ˆx,u 0 )ˆx (2.20) We now discretize the system as explained in Section This yields: T ˆx(t + T) = e fx(ˆx,u0)t x(t) + 0 e fx(ˆx,u0)τ dτu (2.21) With the approximation x(t) ˆx(t) we get the linear discrete-time system: T ˆx(t + T) e fx(ˆx,u0)t ˆx(t) + 0 e fx(ˆx,u0)τ dτu (2.22) Although the extended Kalman filter is widely used it is hard to analyze its performance analytically, it is by no means optimal as is the case with the standard Kalman filter.

25 2.5 Adaptive Filtering Adaptive Filtering In model-based signal processing we are dependent on a good model of the system from which our signals originate. In many cases it is not possible to model the system once and for all and then use that model as the system might be changing with time. When this is the case we need to adapt the model to the changes in the system. Using adaptive models for filtering is called adaptive filtering Kalman Filter for Adaptive Filtering Suppose that we have a system which can be written as a linear regression with time-variable coefficients: y[t] = φ T [t]θ[t] + e[t], (2.23) where y[t] is the observed signal, φ[t] is the regression vector, θ[t] are the system parameters and e[t] is measurement noise. If the coefficients are modeled as a random walk we can get a state-space description of the system with the process coefficients as states: θ[t + 1] = θ[t + 1] + w[t], y[t] = φ T [t]θ[t] + e[t], (2.24a) (2.24b) where w[t] is process noise. The Kalman filter can now be applied in order to estimate the parameters. 2.6 Change Detection It is often of great importance to be able to detect a change in the characteristics of a signal. When a change is detected in a signal necessary actions can be performed by the system. These actions can be to notify a user or another system or to change the amplification of a filter CUSUM Test One of the most commonly used methods for detecting a change in a signal or system is the CUSUM test [11]. From the signal which is to be examined a distance measure s is calculated. The calculation of the distance measure depends on the kind of change which is to be detected, e.g., a change in mean value or a change in variance. The distance measure is then averaged in a certain manner, the averaged distance measure is called g. The averaged distance measure is then compared with a predefined threshold h. If g exceeds the threshold the CUSUM test indicates that a change has been detected. The CUSUM test is defined as: g t = g t 1 + s t ν, g t = 0, if g t < 0, g t = 0 and alarm if g t > h > 0. (2.25a) (2.25b) (2.25c)

26 10 Model Based Signal Processing 2.7 Grey Box Modeling and System Identification Physical system modeling often leads to a model where one or more parameter values are unknown i.e. a grey box model. In these cases it is necessary to estimate the unknown parameter values from measured data. The parameter values can be estimated by minimizing an error function with respect to the parameters. For linear and some non-linear systems this can be done analytically [13]. For general non-linear systems this can be done with numerical algorithms. In this thesis a version of the Gauss-Newton algorithm described in [13] is used. To calculate the necessary derivatives a standard central difference is used. To solve the system differential equations for necessary values used in the difference approximation Simulink (ODE45) is used.

27 Chapter 3 Friction and Tire-Road Friction Estimation Methods This chapter starts by giving an introduction to friction, then the modeling of friction is discussed. The last part of the chapter discusses different methods of estimating and measuring the maximum tire-road friction. The methods of estimating and measuring the maximum tire-road friction can be divided into two main groups: effect based methods and cause based methods [17]. 3.1 What is Friction? Friction is the resistive force which occurs when two surfaces which travel along each other are pressed together. The frictional force is always exerted in a direction which opposes movement. The frictional force is dependent on the microscopic properties of the two surfaces at the area of contact, see Figure 3.1. Knowledge of the frictional force is essential within many engineering disciplines. Friction is caused by a wide range of physical phenomena including plastic and elastic deformation, fluid mechanics and wave mechanics [18]. This makes the modeling of frictional forces a daunting task. Often we talk about the coefficient of friction or µ max which is the highest normalized friction force which is possible for a specific surface-surface combination. The coefficient of friction µ max is a unitless quantity which summarizes the microscopic properties of the two surfaces in the area of contact. It is important to note that the coefficient of friction is a simplification of the real world and that the value of the coefficient of friction only can be found empirically. A common misconception is to talk about the coefficient of friction of a surface. There is no such thing. The coefficient of friction is defined only between two surfaces. 11

28 12 Friction and Tire-Road Friction Estimation Methods F f F F N Figure 3.1. The frictional force is dependent upon the microscopic properties of the contact area and the force pressing the surfaces together. 3.2 Friction Models Many physical phenomena are involved in how frictional forces arise. There are many different models of friction which incorporate one or many of these phenomena. In this section some of these models are presented and described Static Friction Models Static friction models are used to model the frictional force as a function of the force pressing the two surfaces together, usually referred to as the normal force, and sliding velocity. Coulomb Friction The simplest and the most commonly used friction model is Coulomb friction or kinetic friction [18]. The frictional force is said to be dependent only on the direction of the sliding velocity. Coulomb friction is illustrated in Figure 3.2(a). Coulomb friction is specified by the Coulomb coefficient of friction µ C. Viscous Friction Viscous Friction is the frictional force originating from the viscosity of lubricants in the contact area. This force is modeled to be proportional to the sliding velocity with proportionality constant µ V. Viscous friction is illustrated in Figure 3.2(b).

29 3.2 Friction Models 13 F F v v (a) Coulomb friction. (b) Viscous friction. F F v v (c) Stiction (d) Combined model illustrating the Stribeck effect. Figure 3.2. Static friction models.

30 14 Friction and Tire-Road Friction Estimation Methods Stiction Stiction or static friction states that the force needed to initiate slide often is higher than the frictional force once sliding is taking place. Stiction is illustrated in Figure 3.2(c). Stiction is specified by the Coulomb coefficient of friction µ C and the coefficient of stiction µ S, where µ S > µ C. The Stribeck Effect The Stribeck effect causes the frictional force to decrease continuously from static to kinetic friction as the sliding velocity is increases. The Stribeck effect is described by the characteristic Stribeck velocity v st which denotes the sliding velocity where 37% of the static friction is active. Coulomb friction, viscous friction, stiction and the Stribeck effect combined yield a static friction model which can be seen in Figure 3.2(d) Dynamic Friction Models Static friction models imply that there is no displacement of the contact area until sliding occurs. This is in reality not the case. The contact area shows spring like behavior until sliding starts taking place at what is called break away [18]. This phenomenon is called pre-sliding displacement and is illustrated in Figure 3.3. Dynamic friction models are used to model pre-sliding displacement and other F F Figure 3.3. The contact area shows spring-like behavior until break-away. dynamic effects of friction, while also taking one or more static friction phenomena into account. Common dynamic friction models are the Dahl model which models pre-sliding displacement and Coloumb friction and the LuGre model which extends the Dahl model by also modeling stiction, viscous friction and the Stribeck effect. See for example [18] for a description of these models. The Dahl Model The Dahl model was introduced by P.R. Dahl in 1976 and is based upon the stress-strain curve in classical solid mechanics [5]. It was developed to simulate control systems with friction and is a generalization of Coloumb friction. It does

31 3.3 Cause Based Tire-Road Friction Estimation Methods 15 not model static friction or the Stribeck effect. It is formulated as: ( ) α df dx = σ F 0 1 sgn(v r ), (3.1) µ max F N where x is displacement, σ 0 is material stiffness in the contact patch and α is a design parameter. With α = 1, which is the value normally used [5], and F = σ 0 z, where z is a measure of deflection in the surface (3.1) can be written as: df dz = v z r σ 0 v r (3.2a) µ max F N F = σ 0 z (3.2b) For a more thorough explanation of friction and friction models see for example [18, 5]. 3.3 Cause Based Tire-Road Friction Estimation Methods Cause based methods try to measure and identify parameters which have impact on the maximum tire-road friction. From this information conclusions regarding maximum tire-road friction can be drawn. The parameters influencing µ max can be divided into three groups: vehicle parameters e.g., speed and wheel load, tire parameters e.g., tire material and tread depth and road parameters e.g., road type, presence of lubricants and temperature. The parameters influencing µ max the most are road parameters and most work on cause based tire-road friction estimation methods have been done on estimating these Roughness Based Methods Roughness based methods measure the roughness of the road. From the measured roughness it can be possible to draw conclusions regarding the available tire-road friction. In [10] a roughness measure calculated from measurements of wheel angular velocities is used to complement an effect based method by allowing better classification of road surfaces. In [6] an optical surface roughness sensor is used in combination with a wetness sensor to produce estimates of µ max. The general conclusion which can be drawn from these tests is that a road surface roughness measure can be used to complement other methods and allows for better classification of the road surface Lubricant Based methods Lubricant based methods try to identify lubricants e.g., water, snow etc., present on the road. From this conclusions regarding µ max can be drawn. In [6] optical wetness and roughness sensors are combined to produce an estimate of µ max.

32 16 Friction and Tire-Road Friction Estimation Methods 3.4 Effect Based Tire-Road Friction Estimation Methods Effect based methods try to measure the effects of friction on measured signals. The methods and sensors used to do this vary. There are four main groups of effect based methods Vibration Based Methods The idea of vibration based methods is that a change in tire-road friction causes a change in the frequency characteristics of measured wheel speed signals. In [23] a vibrational model of a tire is used to estimate the slope of the s µ x curve at an arbitrary point, see Section 4.3 for a description of the s µ x curve. This value is then said to be correlated with µ max Acoustic Based Methods Acoustic based methods use acoustic sensors to collect tire sound data. By analyzing the sound made by the tire when rolling over the road surface conclusions can be drawn regarding µ max. Acoustic based methods are investigated in for example [6, 22] Slip Based Methods Slip based methods of estimating µ max use tire models which model the relation between tire slip and tire forces. By calculating longitudinal and lateral slip and corresponding tire forces µ max can be estimated. Different methods of curve fitting or parameter identification schemes are used to map the measured/estimated data to the available tire-road friction. This area can be divided into two subgroups: longitudinal and lateral methods. The longitudinal methods map longitudinal slip when applying positive, when driving, or negative, when braking, torque to the wheels. Lateral methods try to estimate the side slip angle during turning. There are an abundance of articles within this area. In [10] a method of detecting changes in µ max by calculating the slope of the s µ x curve when applying driving torque is presented. The method is complemented by a road surface roughness measure to improve classification of the road surface. In [17] a method of calculating the slip of the s µ x curve when braking is presented. In [17] and [21] methods of adapting non-linear tire models to measured slip data are presented. In [19] a non-linear vehicle observer coupled with a Bayesian hypothesis selector is used to estimate µ max Tire-Tread Deformation Sensors When a tire is subject to driving or braking torque the tire-tread in the contact patch between tire and road starts to deflect slightly. As the torque increases

33 3.4 Effect Based Tire-Road Friction Estimation Methods 17 parts of the contact patch starts to slide. This sliding occurs in parts of the contact patch even with a low friction demand and long before the entire tire starts to slide. Where in the contact patch sliding occurs and the amount of sliding taking place is determined by µ max among other things, see [21] for a theoretical explanation. Tire-tread deformation based approaches for tire-road friction estimation uses sensors embedded in the tire to measure tire-tread deformation. This information can then be used to estimate µ max. A method based on tire-tread deformation sensors is described in [17]. In the recent APOLLO project in which tire sensors are developed it is concluded that even using in-tire sensors it is hard to estimate µ max in a reliable way [1]. Furthermore the need of sensors in the tires and self-powered wireless data links between the tire sensors and the vehicle makes this approach costly and complicated Hard Braking The classical way of measuring tire-road friction is to brake hard and to calculate µ max from the average deceleration. This approach is described in more detail in Section 6.1. This is obviously not a very convenient way of measuring tire-road friction and it is only a viable solution for making reference measurements of µ max Extra Wheel Systems using an extra wheel have also been designed, see for example [12]. On this extra wheel the applied braking or driving torque can be regulated so that s = s 0 where s 0 = argmax s µ x (s). If the normal force is known the normalized traction force µ x and hence µ max can be calculated.

34 18 Friction and Tire-Road Friction Estimation Methods

35 Chapter 4 Vehicle and Tire Dynamics In order to to simulate a vehicle and also to be able to design observers to estimate non-measured quantities it is necessary to develop a vehicle model. The vehicle model developed consists of three subsystems: vehicle body model wheel model tire model In this thesis only front wheel driven cars are considered in the modeling. 4.1 Vehicle Body Model In this section a vehicle body model will be derived from standard rigid body mechanics. We use a planar model of a four wheel two axis vehicle illustrated in Figure 4.1. We neglect roll and pitch motion. We also neglect lateral air resistance. F = ma gives in the x direction: m( v x v y r) = (F xf R + F xf L )cos(δ) (F yf R + F yf L )sin(δ) + F xrr + F xrl F D, (4.1) and in the y direction: m( v y + v x r) = (F xf R + F xf L )sin(δ) + (F yf R + F yf L )cos(δ) + F yrr + F yrl. (4.2) τ = Jṙ gives: I z ṙ = a((f xf R + F xf L )sin(δ) + (F yf R + F yf L )cos(δ)) b((f xrr + F xrl ) + + h f 2 ((F x F L F xf R )cos(δ) + (F yf L F yf R )sin(δ)). (4.3) Here v x and v y are the vehicle s longitudinal and lateral velocities respectively at the vehicle s Center Of Gravity (COG). r is the vehicle s yaw rate around COG. 19

36 20 Vehicle and Tire Dynamics h F δ Fx FL Fx FR F D Fy FL Fy FR a v x r v y b Fx RL Fx RR Fy RL Fy RR h R Figure 4.1. Planar four wheel vehicle model.

37 4.2 Wheel Modeling 21 F x/yxx are tire forces and δ is the front wheel steering angle. Air resistance can be modeled as F D = 1 2 C Dρ air Av 2 x = C AIR v 2 x [2]. With a slight rearrangement of the equations we get the system in state-space form: v x = v y r+ 1 m ((F x F R +F xf L )cos(δ) (F yf R +F yf L )sin(δ)+f xrr +F xrl ) C AIR v 2 x (4.4a) v y = v x r+ 1 m (F x F R +F xf L )sin(δ)+(f yf R +F yf L )cos(δ)+f yrr +F yrl (4.4b) ṙ = 1 I z (a((f xf R + F xf L )sin(δ) + (F yf R + F yf L )cos(δ)) b((f xrr + F xrl )+ + h f 2 ((F x F L F xf R )cos(δ) + (F yf L F yf R )sin(δ))) (4.4c) 4.2 Wheel Modeling We want to model the rotational dynamics of a wheel when torque and frictional force is applied to the wheel, see Figure 4.2. J ω = τ yields: ω = 1 J w (τ r 0 F f ). (4.5) Here, J w is the moment of inertia of the wheel, τ is torque applied to the wheel, r 0 is wheel radius, ω is the wheel s angular velocity, F N is the normal force acting upon the wheel and F f the longitudinal frictional force of the wheel. ω τ r 0 v F F N Figure 4.2. Longitudinal wheel model.

38 22 Vehicle and Tire Dynamics 4.3 Tire Modeling We want to model the forces exerted by a tire when torque, steering angle and normal forces are applied to the wheel on which the tire is mounted. We divide the modeling of the tire forces into two parts: longitudinal and lateral forces. There are many similarities between the modeling of the two forces as they both are frictional forces. They differ in variables and parameters by which they are governed. Common for all models for tire-road interaction is that they use dynamic friction models, see Section 3, as tire road interaction exhibit significant pre-sliding displacement due to the relatively low stiffness of rubber Longitudinal Tire Modeling In this section the important variables for longitudinal tire modeling are defined. Normalized Traction Force The normalized traction force µ x for a tire is defined as µ x = F x F N, (4.6) where F x is the traction force and N is the normal force acting upon the wheel. The normalized traction force is also called utilized friction and always obeys the relation µ x µ xmax where µ xmax is the coefficient of friction in the longitudinal direction. Slip When driving torque is applied to a wheel this results in a force being applied to the contact area between tire and road surface. This force is opposed by a frictional force which causes the wheel and hence the vehicle to accelerate. The pre-sliding displacement phenomenon, see Section 3.2.2, causes a deformation and hence a displacement of the tire-road contact patch, see Figure 4.3. As the tire rotates each new part of the tire tread which enters the contact patch is slightly displaced. This causes the tire to rotate slightly faster than what the longitudinal velocity of the wheel hub would indicate. The relative difference in a wheel s angular velocity times it s radius compared to it s longitudinal velocity is called slip. The slip s is according to Society of Automotive Engineers (SAE) defined as: s = ωr 0 v x v x. (4.7) Here ω is the angular velocity of the wheel, r 0 is wheel radius and v is wheel longitudinal velocity Lateral Tire Modeling In this section important variables in lateral tire modeling are defined.

39 4.3 Tire Modeling 23 Roll direction Tire Road Figure 4.3. Tire deformation leads to slip. Normalized Lateral Force The normalized lateral force of a tire is analogously to (4.6) defined as: µ y = F y F N, (4.8) where F y is lateral force and F N is the normal force acting upon the wheel. The normalized lateral force always complies to the relation µ y µ ymax, where µ ymax is the coefficient of friction in the lateral direction. Slip Angle During cornering lateral force is applied to the tires. This lateral force is opposed by a frictional force in the tire-road contact patch. Analogously with the longitudinal case the pre-sliding displacement phenomenon, see Section 3.2.2, causes a displacement of the tire-road contact patch in the lateral direction. As the tire rotates each new part of the tire tread which enters the contact patch is slightly displaced. This causes the tire to be displaced in the lateral direction. The faster the longitudinal velocity of the tire, the higher the lateral displacement. The angle between the heading direction of the tire and the velocity vector of the tire is called the slip angle α and is defined as: α = arctan v y v x, (4.9) where v x is the longitudinal velocity and v y is the lateral velocity of the wheel. An illustration of the slip angle can be seen in Figure Empirical Tire Models Empirical tire models try to adjust functions to measured data without giving a physical explanation to the origin of the tire forces. Examples of empirical tire models are the magic formula tire model and the piecewise linear tire model both which will be presented in this chapter.

40 24 Vehicle and Tire Dynamics v x v α v y Figure 4.4. Definition of the slip angle Analytical Tire Models In contrast to empirical tire models analytical tire models model tire forces from a physical point of view using knowledge about how frictional forces arise in the tireroad contact patch. These models are not always able to explain all phenomena which can be encountered in reality but in return their parameters have physical explanations. Common analytical tire models models in the literature are the brush model [21], and tire models based on the Dahl model and the LuGre friction models [18]. The two latter models use general dynamic friction models to model tire-road friction Longitudinal Tire Models From Section we know that the frictional force can be described as a function of the displacement of the contact patch. The displacement of the contact patch in the longitudinal direction is proportional to the slip. This makes it natural to develop models of the longitudinal tire force as functions of the tire slip. Here we will present and describe two longitudinal tire models: the magic formula model and a simpler piecewise linear tire model. As no slip data from braking is used in this thesis only positive values of longitudinal tire slip are considered in this section. The Magic Formula Model The magic formula model was introduced by Pacejka et al. in [4]. It is defined as F = D sin(c arctan(bλ E(Bλ arctan(bλ)))), (4.10) where B, C, D and E are model parameters. Plots of the normalized traction force versus slip generated with the magic formula tire model for different coefficients

41 4.3 Tire Modeling 25 of friction can be seen in Figure 4.5. Notice that the difference in initial slope between the curves is exaggerated in this plot. There is no physical explanation of µ xmax = µ xmax = 0.5 µ xmax = µ x s Figure 4.5. The magic formula longitudinal tire model for different values of µ max. the parameters of the magic formula tire model. The parameters must be identified from measurement data for a specific tire-surface combination. The magic formula tire model has shown to be able to adapt to measured data very accurately [5]. However, there are drawbacks with the magic formula model. The structure of the model makes it unsuitable for on-line parameter estimation as it is hard to evaluate the gradient with respect to the parameters. Piecewise Linear Tire Model Tire forces are by nature highly non-linear. In many cases reasonable performance can be obtained with a model which is piecewise linear. By dividing the µ x -s function into two parts we get the model { C xn s, s µxmax µ x = C xn µ xmax, otherwise, (4.11) where C x is referred to as the longitudinal tire stiffness. This name is somewhat misguiding as C x is not solely dependent on the tire but also on the road surface, this will be further explained in Chapter 7. Plots of the normalized traction force versus slip generated with the piecewise linear tire model for different coefficients

42 26 Vehicle and Tire Dynamics of friction can be seen in Figure 4.6. Notice that the difference in initial slope is exaggerated in this plot. This model can be made to resemble the magic formula µ xmax = µ xmax = 0.5 µ xmax = µ x s Figure 4.6. Piecewise linear longitudinal tire model for different values of µ max. model quite well for many tire-surface combinations. As can be seen in Figure 4.7 the model yields very similar results to the magic formula model, especially in the low slip region of the slip-normalized traction force curve where the magic formula is very close to being linear. The Brush Tire Model The Brush Tire Model assumes that slip is caused by deformation of the tire tread. The tire tread is said to consist of small brush elements attached to the tire carcass. A brush element behaves like a linear spring which creates frictional force as it deforms up to a friction dependent break-away force where the brush element starts to slide. This divides the tire-road contact patch into two sections: one sliding and one non-sliding. When parts of the contact area starts to slide the slip curve deviates from the tangent to the curve in the origin. This explains the shape of the slip-curve theoretically. The brush model depends on the vertical pressure distribution in the contact patch, the rubber stiffness and the tire-road friction coefficient. In [21] the following version of the brush model is derived F = 2c p a 2 s 4 (2c p a 2 s) (2c p a 2 s) 3 3 µf N 27 (µf N ) 2, (4.12)

43 4.3 Tire Modeling 27 1 Magic Formula Model Piecewise Linear Model 0.8 µ x s Figure 4.7. Comparison between the piecewise linear tire model and the magic formula tire model where c p is a stiffness parameter, a is the contact patch length and µ is the coefficient of friction. In Figure 4.8 three curves of the normalized traction force plotted against wheel slip generated with the brush tire model for different values of µ max is shown. This simple form of the brush model lacks modeling of static friction and the Stribeck effect. However its relative simplicity compared to the magic formula model and the physical interpretation of the model parameters makes the model attractive within some applications such as on-line tire-road friction estimation. The Dahl Tire Model The Dahl Tire model is based entirely on the Dahl friction model presented in Section This yields the following tire model dz dt = v z r σ 0 v r, µ max F N (4.13a) F = σ 0 z, (4.13b) where v r = ωr 0 v and σ 0 is a tire-road stiffness parameter. Plots of the steady state normalized traction force against tire slip are given in Figure 4.9. The main advantage of the Dahl tire model is that it is stated as a linear dynamic model. This makes it suitable for control and simulation applications since there are many established methods for controlling and simulating linear dynamic systems.