Modeling and Estimation of Dynamic Tire Properties

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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 tekniska högskola

Linköpings universitet Linköpings universitet

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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: Thomas Schön

isy, Linköpigs universitet

Anders Stenman

NIRA Dynamics AB

Examinator: Fredrik Gustafsson

isy, Linköpigs universitet

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Avdelning, Institution Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet S-581 83 Linköping, Sweden Datum Date 2006-02-26 Språk Language ¤ Svenska/Swedish ¤ Engelska/English ¤ £ Rapporttyp Report category ¤ Licentiatavhandling ¤ Examensarbete ¤ C-uppsats ¤ D-uppsats ¤ Övrig rapport ¤ £

URL för elektronisk version http://www.control.isy.liu.se http://www.ep.liu.se/ ISBN — ISRN LITH-ISY-EX--06/3800--SE Serietitel och serienummer Title of series, numbering ISSN_—

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 esti-mating 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

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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 esti-mating 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.

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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.

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Notation

Symbols

FxXX longitudinal tire force from wheel XX FyXX lateral tire force from wheel XX

FN normal force

FD air resistance force

Ff frictional force

r vehicle yaw rate (turning rate around COG)

r0 wheel radius

vx longitudinal velocity at COG

vy lateral velocity at COG

ax longitudinal acceleration at COG

ay lateral acceleration at COG

δ steering angle of front wheels

δ slip offset

Jz vehicle yaw moment of inertia around COG

Jw wheel moment of inertia

a distance from front axle to COG

b distance from rear axle to COG

hf front axle length

hr rear axle length

µx normalized traction force

µy normalized lateral force

µmax coefficient of friction

µxmax longitudinal coefficient of friction µymax lateral coefficient of friction

µC Coulomb coefficient of friction

µV coefficient of viscous friction

µS coefficient of stiction

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x

r0 wheel radius

Cx longitudinal tire stiffness

Cy lateral tire stiffness

CxN normalized longitudinal tire stiffness

CyN normalized lateral tire stiffness

CF front lateral tire stiffness

CR rear lateral tire stiffness

CD drag coefficient

CAIR total air resistance coefficient

Cδ ratio between steering wheel angle and steering angle

s longitudinal wheel slip

α slip angle

ρair air density

ω 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 Society of Automotive Engineers

GPS Global Positioning System

ABS Anti-lock Braking System

ESP Electronic Stability System

RFI Road Friction Indicator

TPI Tire Pressure Indicator

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Introduction

1.1 Background

Friction is the basic principle upon which all wheel driven vehicles rely. The max-imum 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, mo-torcycle 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 esti-mating 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: www.niradynamics.se.

1.3 Objectives

The objectives of this thesis are:

• To implement and evaluate new methods of estimating the longitudinal tire-road stiffness.

• To develop, implement and evaluate methods for estimating the lateral tire-road stiffness.

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2 Introduction

• To develop, implement and evaluate a method for using estimated longitu-dinal 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 µmax. The effects which surface

conditions have on slip-slope are discussed in Chapter 7. In Chapter 8 methods of estimating µmax directly, without going via tire-road stiffness are described and

evaluated. Chapter 9 discusses what limits there are to effect based methods of estimating µmax. Finally Chapter 10 sums up the results of the thesis.

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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. Model-based 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 differ-ential/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:

˙x(t) = f (x(t), u(t)), (2.1a)

y(t) = g(x(t), u(t)), (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]), (2.2a)

y[t] = g(x[t], u[t]). (2.2b)

Systems which can be described by a linear system of differential/difference equa-tions are of special importance in system theory and control theory as there exists

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4 Model Based Signal Processing

many methods for analyzing such systems. A linear continuous-time system can be written as:

˙x(t) = Ax(t) + Bu(t), (2.3a)

y(t) = Cx(t) + Du(t). (2.3b)

A linear discrete-time system can be written as:

x[t + 1] = Ax[t] + Bu[t] (2.4a)

y[t] = Cx[t] + Du[t] (2.4b)

If the system functions/matrices do not vary with time the system is called time-invariant.

2.1.1 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

˙x(t) = Ax(t) + Bu(t) (2.5a)

y(t) = Cx(t) + Du(t) (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 x[kT + T ] = F x[kT ] + Gu[kT ], (2.6a) y[kT ] = Hx[kT ] + Ju[kT ], (2.6b) where F = eAT_{, G =} T Z 0 eAt_{Bdt, H = C, J = D.} _(2.6c)

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2.2 Observers and Observability 5

2.1.2 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 (x0, u0) 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 (x0, u0) and disregard higher-order terms. This

gives for a time-discrete system on form (2.6).

x[t + 1] ≈ f (x0, u0) + A(x[t] − x0) + B(u[t] − u0), where A = df (x, u) dx ¯ ¯ ¯ ¯ x=x0,u=u0 , B = df (x, u) du ¯ ¯ ¯ ¯ x=x0,u=u0

With the approximation x0= f (x0, u0) the variable substitution ˜x[t] = x[t] − x0,

˜

u[t] = u[t] − u0 gets us back to the linear case;

˜

x[t + 1] = A˜x[t] + B ˜u[t]. (2.7)

2.2 Observers and Observability

2.2.1 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:

˙x(t) = Ax(t) + Bu(t), (2.8a)

y(t) = Cx(t) + Du(t) + v(t). (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) − C ˆx(t) − Du(t) which gives us the observer:

˙ˆx(t) = Aˆx(t) + Bu(t) + K(y(t) − C ˆx(t) − Du(t)), (2.9)

where K is the observer gain matrix. Let us study the estimation error: ˜

x(t) = x(t) − ˆx(t) (2.10)

From (2.8), (2.9) and (2.10) we now get

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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.

2.2.2 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∗ _{6= 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 beobservable 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 matrixO.

O(A, C) =      C CA .. . CAn−1      (2.12)

This means that the system is observable if and only if O has full rank.

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 esti-mation error? The filter which solves this problem is called the Kalman filter [11].

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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] = Atx[t] + Bv,tu[t] + Bu,tv[t] (2.13a)

y[t] = Ctx[t] + e[t] (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, (2.14a)

Cov{v[t]} = Qt, (2.14b)

Cov{e[t]} = Rt. (2.14c)

(2.14d) Let the initial state x[0] of the system have the following properties:

E{x[0]} = x0, (2.15a)

Cov{x[0]} = Π. (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:

ˆ

x[t + 1|t] = Atx[t + 1|t] + Bˆ u,tu[t], (2.16a)

It can also be proved that if the process and the measurement noise are Gaus-sian 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

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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].

2.4.1 Discretized Linearization

Assume that we have a non-linear time-continuous system on the form

˙x(t) = f (x(t), u(t)) (2.17a)

y(t) = g(x(t), u(t)) (2.17b)

Taylor expansion of (2.17a) around the point (ˆx, u0) give when disregarding

higher-order terms:

˙x(t) = f (ˆx, u0) + fx(ˆx, u0)(x − ˆx) + fu(ˆx, u0)(u − u0), (2.18)

where fx and fu denote the derivative of f(x, u) with respect to x and u,

respec-tively. Rearrangement of (2.18) give us the linear system:

˙x(t) = fx(ˆx, u0)x + u0, (2.19)

where

u0= f (ˆx, u0) + fu(ˆx, u0)(u − u0) − fx(ˆx, u0)ˆx (2.20)

We now discretize the system as explained in Section 2.1.1. This yields: ˆ x(t + T ) = efx(ˆx,u0)T_{x(t) +} T Z 0 efx(ˆx,u0)τ_{dτ u}0 (2.21)

With the approximation x(t) ≈ ˆx(t) we get the linear discrete-time system: ˆ x(t + T ) ≈ efx(ˆx,u0)T_ˆ_{x(t) +} T Z 0 efx(ˆx,u0)τ_{dτ u}0 (2.22)

Although the extended Kalman filter is widely used it is hard to analyze its per-formance analytically, it is by no means optimal as is the case with the standard Kalman filter.

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2.5 Adaptive Filtering 9

2.5 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.

2.5.1 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] = φT_{[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], (2.24a)

y[t] = φT[t]θ[t] + e[t], (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.

2.6.1 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:

gt= gt−1+ st− ν, (2.25a)

gt= 0, if gt< 0, (2.25b)

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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

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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 µmax which is the highest normalized friction force which is possible for a

specific surface-surface combination. The coefficient of friction µmax is a

unit-less 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.

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12 Friction and Tire-Road Friction Estimation Methods

F

N

F

f

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 phenom-ena. In this section some of these models are presented and described.

3.2.1 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 µC.

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 µV. Viscous friction is illustrated in Figure 3.2(b).

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3.2 Friction Models 13

F

v

(a) Coulomb friction.

F v (b) Viscous friction. F v (c) Stiction F v

(d) Combined model illustrating the Stribeck effect.

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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 µCand

the coefficient of stiction µS, where µS > µC.

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 de-scribed by the characteristic Stribeck velocity vstwhich denotes the sliding

veloc-ity 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).

3.2.2 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

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

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3.3 Cause Based Tire-Road Friction Estimation Methods 15

not model static friction or the Stribeck effect. It is formulated as: dF dx = σ0 µ 1 − F µmaxFN sgn(vr) ¶α , (3.1)

where x is displacement, σ0 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 = σ0z,

where z is a measure of deflection in the surface (3.1) can be written as: dF dz = vr− σ0 z µmaxFN vr (3.2a) F = σ0z (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 µmax 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 µmax

the most are road parameters and most work on cause based tire-road friction estimation methods have been done on estimating these.

3.3.1 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 an-gular 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 µmax. 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 classifi-cation of the road surface.

3.3.2 Lubricant Based methods

Lubricant based methods try to identify lubricants e.g., water, snow etc., present on the road. From this conclusions regarding µmax can be drawn. In [6] optical

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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.

3.4.1 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 − µxcurve at an

arbitrary point, see Section 4.3 for a description of the s − µxcurve. This value is

then said to be correlated with µmax.

3.4.2 Acoustic Based Methods

Acoustic based methods use acoustic sensors to collect tire sound data. By ana-lyzing the sound made by the tire when rolling over the road surface conclusions can be drawn regarding µmax. Acoustic based methods are investigated in for

example [6, 22].

3.4.3 Slip Based Methods

Slip based methods of estimating µmax use tire models which model the relation

between tire slip and tire forces. By calculating longitudinal and lateral slip and corresponding tire forces µmaxcan 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 µmaxby calculating the slope of the s − µxcurve 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 − µx 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 µmax.

3.4.4 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

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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 con-tact 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 µmaxamong other things, see [21] for a theoretical

explanation. Tire-tread deformation based approaches for tire-road friction esti-mation uses sensors embedded in the tire to measure tire-tread deforesti-mation. This information can then be used to estimate µmax. 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 µmax 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.

3.4.5 Hard Braking

The classical way of measuring tire-road friction is to brake hard and to calculate µmax 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 µmax.

3.4.6 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 = s0 where s0 = argmaxsµx(s). If the normal force is known the normalized

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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( ˙vx− vyr) = (FxF R+ FxF L) cos(δ) − (FyF R+ FyF L) sin(δ) + FxRR+ FxRL− FD, (4.1) and in the y direction:

m( ˙vy+ vxr) = (FxF R+ FxF L) sin(δ) + (FyF R+ FyF L) cos(δ) + FyRR+ FyRL. (4.2) τ = J ˙r gives:

Iz˙r = a ((FxF R+ FxF L) sin(δ) + (FyF R+ FyF L) cos(δ)) − b ((FxRR+ FxRL) + +hf

2 ((FxF L− FxF R) cos(δ) + (FyF L− FyF R) sin(δ)) . (4.3) Here vxand vy 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

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20 Vehicle and Tire Dynamics

h

_F

h

_R

δ

F

xFL

F

yFL

F

xFR

F

yFR

F

xRL

F

yRL

F

xRR

F

yRR

v

y

v

x

r

a

b

F

D

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4.2 Wheel Modeling 21

Fx/yXX are tire forces and δ is the front wheel steering angle. Air resistance can be modeled as FD= 1₂CDρairAvx2= CAIRv2x[2]. With a slight rearrangement of

the equations we get the system in state-space form: ˙vx= vyr+

1

m((FxF R+FxF L) cos(δ)−(FyF R+FyF L) sin(δ)+FxRR+FxRL)−CAIRv

2 x (4.4a) ˙vy= −vxr + 1 m(FxF R+ FxF L) sin(δ) + (FyF R+ FyF L) cos(δ) + FyRR+ FyRL (4.4b) ˙r = 1 Iz

(a((FxF R+ FxF L) sin(δ) + (FyF R+ FyF L) cos(δ)) − b((FxRR+ FxRL)+ +hf

2 ((FxF L− FxF R) cos(δ) + (FyF L− FyF 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 Jw

(τ − r0Ff). (4.5)

Here, Jw is the moment of inertia of the wheel, τ is torque applied to the wheel,

r0is wheel radius, ω is the wheel’s angular velocity, FN is the normal force acting

upon the wheel and Ff the longitudinal frictional force of the wheel.

v r0 ω τ FN F

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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.

4.3.1 Longitudinal Tire Modeling

In this section the important variables for longitudinal tire modeling are defined.

Normalized Traction Force

The normalized traction force µx for a tire is defined as

µx=

Fx

FN

, (4.6)

where Fx 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 µx ≤ µxmax where µxmax 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 =ωr0− vx vx

. (4.7)

Here ω is the angular velocity of the wheel, r0 is wheel radius and v is wheel

longitudinal velocity.

4.3.2 Lateral Tire Modeling

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4.3 Tire Modeling 23

Road Tire

Roll direction

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: µy = Fy

FN

, (4.8)

where Fy is lateral force and FN is the normal force acting upon the wheel. The

normalized lateral force always complies to the relation µy ≤ µymax, where µymax 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 longi-tudinal 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 an-gle 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 = arctanvy vx

, (4.9)

where vx is the longitudinal velocity and vy is the lateral velocity of the wheel.

An illustration of the slip angle can be seen in Figure 4.4.

4.3.3 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.

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α

v

y

v

x

v

Figure 4.4. Definition of the slip angle.

4.3.4 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 tire-road 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.

4.3.5 Longitudinal Tire Models

From Section 3.2.2 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

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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

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µx µ_x max = 1.0 µ_x max = 0.5 µ_x max = 0.1

Figure 4.5. The magic formula longitudinal tire model for different values of µmax.

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 µx-s

function into two parts we get the model µx=

(

CxNs, s ≤ µ_Cxmax_xN

µxmax, otherwise

, (4.11)

where Cx is referred to as the longitudinal tire stiffness. This name is somewhat

misguiding as Cx 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

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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

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µx µ_x max = 1.0 µ_x max = 0.5 µ_x max = 0.1

Figure 4.6. Piecewise linear longitudinal tire model for different values of µmax.

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 = 2cpa2s −4 3 (2cpa2s)2 µFN + 8 27 (2cpa2s)3 (µFN)2 , (4.12)

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4.3 Tire Modeling 27 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µx

Magic Formula Model Piecewise Linear Model

Figure 4.7.Comparison between the piecewise linear tire model and the magic formula tire model

where cp is a stiffness parameter, a is the contact patch length and µ is the

coeffi-cient 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 µmax

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 3.2.2. This yields the following tire model

dz dt = vr− σ0 z µmaxFN |vr|, (4.13a) F = σ0z, (4.13b)

where vr= ωr0− v and σ0 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.

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28 Vehicle and Tire Dynamics 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µ µ_max=1.0 µmax=0.5 µ_max=0.2

Figure 4.8. s-µx curves for the brush model.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µ µ_max=1.0 µ_max=0.5 µ_max=0.2

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4.3.6 Lateral Tire Models

Lateral tire models are to great extent analogous to longitudinal tire models al-though the parameters can differ. The displacement of the contact patch in the lateral direction is proportional to the slip angle. This makes it natural to develop models of the longitudinal tire force as functions of tire slip angle. Here we present and describe the lateral versions of the two models described in Section 4.3.1.

The Magic Formula Model

The magic formula model is the most commonly used empirical tire model in the literature also in its lateral version. Plots of the normalized lateral force versus slip angle generated with the magic formula tire model for different coefficients of friction are illustrated in Figure 4.10. Notice that the difference in slope at α = 0 is exaggerated in this plot.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 α µy µ_y max = 1.0 µ_y max = 0.5 µ_y max = 0.1

Figure 4.10. The magic formula lateral tire model for different values of µmax.

Piecewise Linear Tire Model

The lateral piecewise linear tire model is analogous to the longitudinal case. By dividing the µx-α function into three parts we get the model:

µy = ( CyNα, −µ_Cymax_yN ≤ α ≤ µ_Cymax_yN −µymax, α ≤ − µ_ymax CyN µymax, otherwise (4.14)

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Plots of the normalized lateral force versus slip angle generated with the piecewise linear tire model for different coefficients of friction can be seen in Figure 4.11. Notice that the difference in slope at α = 0 is exaggerated in this plot.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 α µy µ_y max = 1.0 µ_y max = 0.5 µ_y max = 0.1

Figure 4.11. Piecewise linear lateral tire model for different values of µmax.

4.3.7 Combined Tire Model

The longitudinal and lateral tire forces are not independent of each other. The resulting force can for obvious reasons not be larger than the normal load times the coefficient of friction. As µymax and µxmax generally differ due to different tire characteristics in the lateral and the longitudinal directions we get:

µ µx µxmax ¶2 + µ µy µymax ¶2 ≤ 1 (4.15)

The possible area of the tire force resultant is called the friction circle [16], despite the fact that it generally is described by an ellipse. The friction circle is illustrated in Figure 4.12.

4.4 Sensors and Measurements

In a modern car there are a number of sensors available. The sensors used in the methods presented and evaluated in this thesis are:

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4.4 Sensors and Measurements 31

µxmax µymax

Figure 4.12. Illustration of the friction circle.

• Wheel angular velocity sensors • Lateral accelerometer

• Yaw rate gyro

• Steering wheel angle sensor

These are standard sensors which are present in modern day cars equipped with ABS and ESP systems. A description of how these sensors work can be found in for example [8]. In this section we will discuss how the available sensors can be used in conjunction with the previously derived vehicle model.

4.4.1 Wheel Angular Velocity Sensors

The wheel angular velocity sensors can be used together with the vehicle model in two ways: directly as measurements of ωXX or to calculate an estimate of vx. In

this thesis an estimate of vxis calculated according to:

ˆ vx=

r0(ωRL+ ωRR)

2 . (4.16)

4.4.2 Lateral Accelerometer

The lateral accelerometer gives measurements of ˙vy+ vr. This yields the

measure-ment equation:

y = 1

m(FxF R + FxF L) sin(δ) + (FyF R+ FyF L) cos(δ) + FyRR+ FyRL (4.17) If a tire model is used to model the tire forces as functions of slip and slip angle this yields a measurement equation which can be used with the state space vehicle model (4.4).

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4.4.3 Yaw Rate Gyro

The yaw rate gyro yields measurements of r which can be used directly together with the vehicle model (4.4).

4.4.4 Steering Wheel Angle Sensor

The vehicle model (4.4) uses the steering angle δ as input signal. As standard cars are not equipped with a sensor to measure the steering angle the steering angle has to be estimated from measurements of the steering wheel angle β. In this thesis the following simple static model of the relation between β and δ is used:

δ = Cδβ, (4.18)

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Chapter 5

Stiffness Estimation

The combined stiffness of the tire road contact patch can be used to estimate the maximum tire-road friction. In this thesis we divide the methods of estimating the combined stiffness into two groups:

• Stiffness estimation using regression models.

• Stiffness estimation using parameter identification in state space models. This grouping is made due to the fact that when estimating lateral stiffness using regression models the estimation has to be preformed in two steps, whereas when using parameter identification methods in state space models the estimation is performed in one step. This will be further explained in this chapter. In this chapter the two approaches will be explained in detail. In order to maximize the use of the available tire-force excitation, see Section 9.1, estimation of both longitudinal and lateral stiffness is considered.

5.1 Tire Stiffness or Slip-slope?

As explained in Chapter 3, the way in which frictional forces between two surfaces arise can be divided into two phases. In the pre-sliding phase the magnitude of the frictional force is governed by the relative displacement and the stiffness of the two surfaces and in the sliding phase the frictional force is dependent on the sliding velocity and the coefficient of friction µmax. As explained in Section 4.3,

the frictional phenomena of pre-sliding displacement and sliding give rise to the slip and the slip-angle of a tire. If slip is plotted against normalized traction force for a tire this yields a curve which can be divided into two parts. As can be seen in Figure 5.1 the slip curve is approximately linear for s < 0.015. For s > 0.015 the slip curve starts to bend, it reaches the peak friction value µmax at s = 0.030 and

then stabilizes at a static friction value. Note that these values may vary with the tire-road combination. The slope of the slip curve in the approximately linear low slip region of the curve is commonly called tire stiffness. This name is somewhat misguiding as the slope of the slip curve also depends on surface properties [17].

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34 Stiffness Estimation 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.2 0.4 0.6 0.8 1 s µx

Figure 5.1. Typical s-µxcurve.

In [10] it is proposed that the slope should simply be called slip-slope instead. In this thesis the term slip-slope is used when discussing the slope of the s-µxor α-µy

without interpreting the slope as a measure of stiffness. When interpreted as a stiffness measure, slip-slope is referred to as combined stiffness or simply stiffness. Tire stiffness is used when talking about the stiffness of the tire alone.

5.2 Stiffness Estimation using Regression Models

The longitudinal stiffness Cxand the lateral stiffness Cy can be seen as the slopes

of the linear parts of the s-µx and the α-µy curves respectively, see Figure 5.2.

When using regression models for stiffness estimation we calculate these slopes from measured data in order to calculate the stiffness parameters.

5.2.1 Estimation of Longitudinal Stiffness

In [10] a method for estimating the longitudinal tire stiffness from measurements of wheel velocities is proposed. This method is used in this thesis and is outlined here.

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5.2 Stiffness Estimation using Regression Models 35 0 0.01 0.02 0.03 0.04 0 0.2 0.4 0.6 0.8 1 1.2 s µx −0.2 −0.1 0 0.1 0.2 −1 −0.5 0 0.5 1 α µy C_x∆ s ∆ s ∆α C_y∆α

Figure 5.2. Combined stiffness as the slope of a slip curve.

Calculation of Slip

The definition of slip is stated in Section 4.3.1. As we have no reference speed available we calculate a reference speed by using the wheel angular velocity of the non-driven rear wheels:

vxF X = wRXr0 (5.1)

Where wXX is the angular velocity of wheel XX and r0 is the nominal wheel

ra-dius. The slip can now be calculated according to the definition in Section 4.3.1.

Calculation of Normalized Traction Force

The definition of normalized traction force is stated in Section 4.3.1. The traction force can be calculated by using measurements of injection time and engine speed using a tabulated model for a specific car and engine [10], but is considered to be known in this thesis. The normal force distributes itself over the four wheels depending on the forces acting on the vehicle [16], but is in this thesis approximated with the static normal force distribution:

FNXX = mg

4 (5.2)

Algorithm for Estimation of Longitudinal Stiffness

The linear part of the slip curve is approximated with a line with slope k and offset δ, see (5.3). The offset is necessary to compensate for the difference in wheel radius between the front wheel for which the slip is calculated and the rear wheel which is used to calculate the longitudinal velocity.

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36 Stiffness Estimation

It is advantageous, see [10], to rewrite (5.3) as s = µ1

k+ δ. (5.4)

We introduce measurement noise and rewrite the model on linear regression form: s[t] =¡µ[t] 1¢ µ 1/k δ ¶ + e[t] (5.5)

We can now apply the Kalman filter to adaptively estimate the parameters 1/k and δ for each driven wheel according to the method described in Section 2.5.1. To be able to get both good tracking performance and high noise suppression we apply a CUSUM detector according to Section 2.6.1. The distance measure used in the CUSUM test is the estimation error s − ˆs. Upon alarm the Kalman filter design variable Q is multiplied with a pre-defined factor to increase amplification for one sample. This method yields good results as the tire-road friction coefficient often is piecewise close to constant with sudden jumps.

Results

Figure 5.3 shows a typical plot of the estimated longitudinal stiffness when driving from asphalt to gravel and back on asphalt again using the method outlined in Section 5.2.1. As can be seen the decrease in slip-slope when going from asphalt to gravel at time t = 170 is quickly detected. As can be seen in Figure 5.3 there is an initial transient before reaching a stable estimate of the slip-slope. This is due to the initial state of the Kalman filter not being equal to the actual state of the system which in this case is the linear tire model used. A more thorough discussion of the results of this method can be found in [10].

5.2.2 Estimation of Lateral Stiffness

Estimation of lateral stiffness is to a great extent analogous with estimation of longitudinal stiffness.

Normalized Lateral Force

The normalized lateral force µyXX for wheel XX is defined as µyXX =

FyXX NXX.

(5.6) Here Fy is the lateral force and N is the normal force acting upon the wheel. As

we have measurements of the lateral acceleration we can calculate the total lateral force acting on the vehicle as:

Fy = may (5.7)

We assume that the side force distributes over the four wheels in the same way as the normal forces do, see Section 5.2.1 for a description of normal force distribution.

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5.2 Stiffness Estimation using Regression Models 37 0 100 200 300 400 500 600 700 800 900 1000 40 50 60 70 80 90 100 110 120 t k

Figure 5.3. Estimated longitudinal stiffness.

This yields: µyXX = FyFNXX_N FNXX = Fy FN (5.8) Slip Angle

The definition of slip angle for a wheel is stated in Section 4.3.2. From the geometry in Figure 4.1 we get: αF X = − arctan(vy+ ar vx ) + δ, (5.9a) αRX= − arctan(vy− br vx ). (5.9b) Lateral Velocity

We see in (5.9) that knowledge of the lateral velocity of the center of gravity of the vehicle is necessary in order to calculate the slip angles. As standard vehicles are not equipped with sensors which can measure the lateral velocity we are left with two options:

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38 Stiffness Estimation

1. Assume that the lateral velocity is so small that it can be set to zero without losing too much accuracy.

2. Construct an observer which uses available sensors to estimate the lateral velocity.

Both these possibilities have been evaluated in this thesis. There are many ways to construct an observer to estimate the lateral velocity. In this thesis we evaluate three different methods of estimating the lateral velocity which all are based upon the vehicle model derived in Chapter 4.

Method 1: Integration of Measurement Signals We have the vehicle model

presented in Chapter 4. We can estimate the longitudinal velocity as vx≈

r0(ωRL+ ωRR)

2 . (5.10)

If we reduce the vehicle model by replacing the longitudinal velocity vx with the

estimate calculated according to (5.10), the yaw rate r with its measured counter-part and lateral tire forces with measured lateral acceleration ay we obtain:

˙vy= −vxr + ay (5.11)

We can now approximate the derivative with an Euler difference approximation. This yields:

vy[t + 1] ≈ vy[t] + T (−vx[t]r[t] + ay[t]). (5.12)

An estimate of the lateral velocity can now be calculated using (5.12).

Method 2: Kinematic Model Observer This method relies on a

simplifica-tion of the vehicle model presented in Chapter 4 and measurements of yaw rate, lateral acceleration and longitudinal acceleration. From Section 4.1 we have our standard vehicle body model (4.4). We replace the state r with the measured yaw rate r. Then F = ma is used to replace lateral tire forces with measured lateral acceleration. The longitudinal acceleration is approximated as:

ax[t] = vx[t] − vx[t − 1]

T (5.13)

We then use F = ma to replace longitudinal tire forces with the approximated longitudinal acceleration. We also neglect the effects of drag. This yields the model: ˙x = µ ˙vx ˙vy ¶ = µ 0 r −r 0 ¶ | {z } At µ vx vy ¶ + µ ax ay ¶ (5.14a) We use measurements of the longitudinal velocity vx to construct an observer.

This gives us the measurement equation. y =¡1 0¢

| {z }

C

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5.2 Stiffness Estimation using Regression Models 39

We study the observability matrix: det µµ C CA ¶¶ = det µµ 1 0 0 r ¶¶ = r (5.15)

We see that the model is observable only when r 6= 0. In [7] this model is used with an observer designed with pole-placement such that the amplification is zero when r = 0 and then increases as |r| increases. Here we discretize the model analytically and apply a Kalman filter to the discretized model. We have:

eAt= L−1{(sI − A)−1} = L−1 (µ s −r r s ¶−1) = = L−1 ½ ₁ s2_{+ r}2 µ s r −r s ¶¾ = µ cos(rt) sin(rt) − sin(rt) cos(rt) ¶

Using the method described in Section 2.1.1 we get: F = eAT = µ cos(rT ) sin(rT ) − sin(rT ) cos(rT ) ¶ (5.16a) G = T Z 0 eAτ_{dτ = A}−1_(eAT _{− I)B =} 1 r µ sin(rT ) 1 − cos(rT ) cos(rT ) − 1 sin(rT ) ¶ (5.16b) H = C =¡1 0¢ (5.16c)

Method 3: Linear Tire Model Kalman Filter Again we start with the

vehicle model derived in Section 4.1. We group the wheels on the front and rear axle together, this yields what is called a bicycle model. We then model the lateral tire forces with a linear tire model in accordance with Section 4.3.2. This yields:

FF = 2CFαF = 2CF(− arctan(vy+ ar

vx

) + δ) ≈ −2CFvy+ ar

vx

+ δ, (5.17a)

for the front axle and

FR= 2CRαR= 2CR(− arctan( vy− br vx )) ≈ −2CR vy− br vx , (5.17b)

for the rear axle. If δ is sufficiently small we can make the approximation sin(δ) ≈ 0, cos(δ) ≈ 1. If we now eliminate the state vxby instead using vxcalculated from

measurements according to (5.10) as a model parameter we get the model: ˙x = µ ˙vy ˙r ¶ = Ã −_m1(2CF+2CR vx ) −vx+ 1 m( −2aCF+2bCR vx ) 1 Jz( −2aCF+2bCR vx ) 1 Jz( −2a2 CF−2b 2 CR vx ) ! x + µ2CF m 2aCF m ¶ δ (5.18a) y = r = µ 0 1 −_m1(2CF+2CR vx ) 1 m( −2aCF+2bCR vx ) ¶ x (5.18b)

This is a common model in vehicle dynamics used in, for example, [24, 3]. We discretize the model at each time instant according to Section 2.1.1 using Matlab. We can now apply the Kalman filter on this model to estimate the lateral velocity.

Modeling and Estimation of Dynamic Tire Properties

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Modeling and Estimation of Dynamic Tire

Properties

Modeling and Estimation of Dynamic Tire

Properties

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Acknowledgements

Notation

Symbols

Acronyms

Contents

Chapter 1

Introduction

1.1

Background

1.2

Company

1.3

Objectives

1.4

Limitations

1.5

Thesis Outline

Chapter 2

Model Based Signal

Processing

2.1

State-Space Models

2.1.1

Sampling of Continuous-Time Systems

2.1.2

Linearization

2.2

Observers and Observability

2.2.1

Observers

2.2.2

Observability

2.3

Kalman Filter

2.4

Extended Kalman Filter

2.4.1

Discretized Linearization

2.5

Adaptive Filtering

2.5.1

Kalman Filter for Adaptive Filtering

2.6

Change Detection

2.6.1

CUSUM Test

2.7

Grey Box Modeling and System Identification

Chapter 3

Friction and Tire-Road

Friction Estimation Methods

3.1

What is Friction?

F

F

F

3.2

Friction Models

3.2.1

Static Friction Models

3.2.2

Dynamic Friction Models

F

F

3.3

Cause Based Tire-Road Friction Estimation

Methods

3.3.1