Tire and force distribution modeling and validation for wheel loader applications
John Spencer and Bernhard Wullt
Mechanical Engineering, master's level 2017
Luleå University of Technology Department of Engineering Sciences and Mathematics
Acknowledgements
We would like to thank our supervisors, Andre Fernandez and David Berggren, at Volvo CE as well as Jan-Olov Aidanp¨a¨aat LTU. Your help and support during the thesis work has been of high importance for our work. We would also like to thank Lennart Skogh and his test department, who helped us out a lot with the measurements and the testing. Last but not least, we would like to give a big thank you to Kausihan Selvam and Auayporn Elfving at Volvo CE for giving so freely of their time. You have been a great help to us! Thanks!
Abstract
This thesis describes the development of a machine force distribution estimator and the calibration of tire models for wheel loader applications.
Forces generated in the contact patch between the tires and the ground are crucial for under- standing and controlling machine dynamics. When it’s not possible to directly measure these contact patch forces they are estimated from other sensor data. Validated models of the contact patch forces are also used in machine dynamics simulations and are very relevant to model based development. Vehicle dynamics is of crucial importance to the automotive industry. In contrast the modeling of these forces has not been very important to the construction equipment industry and as such wheel loaders haven’t been studied as much as conventional cars.
In order to model the forces in the contact patch, suitable tire models have been studied and calibrated. The tire models have been calibrated by using two different sources, field data and test rig data. Two steady state tire models were chosen for the field data. These were the brush model and the Magic Formula. The resulting fit from the data was not ideal, but this was due to that the given data were of low accuracy. The Magic Formula was used for the test rig data, which gave a good overall fit. The results from the test rig measurements were then used in a transient model, the single contact point model, and the MF-Tyre software. The models were implemented in Simulink and were validated against experimental data. They showed good correspondence, but deviated for some levels of slip.
Another important aspect of the wheel loader is the force distribution over the entire machine. Two estimators have been developed, one to estimate the vertical forces on each tire, the normal force estimator and one to calculate the turning behavior due to different force outputs on the tires, the turning torque estimator. The normal force on each tire is information that is important for the tire model, but it can also be used to estimate when the wheel loader risk tipping on its side. The turning torque estimation is useful for control systems to optimize the driving behavior of the machine.
Compared against measured data from an actual wheel loader the normal force estimator showed a high accuracy in estimating the individual wheel vertical forces. The turning torque estimator could estimate the behavior of the torque but had problems when estimating the magnitude. Table of Content
1 Introduction 1 1.1 Thesis work ...... 1 1.1.1 Scope ...... 2 1.2 Literature review ...... 3
2 Theory 4 2.1 Tire model: Tire fundamentals ...... 4 2.1.1 Introduction to tire quantaties ...... 4 2.1.2 Tire forces and moments ...... 5 2.2 Tire model: Tire models ...... 7 2.2.1 The Magic Formula ...... 7 2.2.2 MFSWIFT ...... 9 2.2.3 The brush model ...... 11 2.2.4 Single contact point model ...... 13 2.3 Tire model: Algorithms for parameter estimation ...... 13 2.3.1 Differential evolution ...... 14 2.4 Force distribution: Normal force ...... 14 2.4.1 Defining the basis for the normal force distribution model ...... 14 2.4.2 Finding the center of gravity ...... 15 2.4.3 Shifts in center of gravity ...... 16 2.4.4 Normal force on each tire ...... 20 2.4.5 Roll-over ...... 25 2.4.6 Pile Entry ...... 25 2.5 Force distribution: Acceleration from wheel sensors ...... 26 2.6 Force distribution: Turning torque ...... 27
3 Method 30 3.1 Tire model: Test vehicle ...... 30 3.2 Field measurments ...... 30 3.2.1 Tire used for measurements ...... 30 3.2.2 Measurement systems ...... 30 3.2.3 Experiments ...... 32 3.2.4 Complementary field test data ...... 32 3.3 Tire model: Test rig measurements ...... 33 3.3.1 Tire used for measurements ...... 33 3.3.2 Main setup ...... 33 3.3.3 Pure longitudinal force characteristics ...... 35 3.3.4 Relaxation length ...... 36 3.3.5 Validation ...... 36 3.4 Tire model: Optimization routine for parameter identification of tire models . . . 36 3.5 Tire model: Parameter and model identification of field measurements ...... 36 3.5.1 Reference tire model ...... 37 3.5.2 Magic Formula parameters ...... 37 3.5.3 Brush model ...... 38 3.6 Tire model: Data processing of test rig measurements ...... 38 3.6.1 Processing of force characteristics measurements ...... 38 3.6.2 Processing of relaxation length measurements ...... 38 3.7 Tire model: Parameter and model identification of test rig measurements . . . . . 39 3.8 Tire model: Model validation of test rig measurements ...... 39
4 Results 41 4.1 Tire model: Processed field measurements ...... 41 4.2 Tire model: Parameter and model identification from field measurements . . . . . 41 4.2.1 Magic Formula parameters ...... 41 4.2.2 Brush model ...... 44 4.2.3 Comparison of tire models ...... 46 4.3 Tire model: Parameter and model identification from test rig measurements . . . 47 4.3.1 Magic Formula parameters for nominal load and inflation pressure . . . . . 47 4.3.2 Magic Formula parameters for variation of load and nominal inflation pres- sure ...... 48 4.3.3 Comparison of resulting fitted curves for variation of loads ...... 50 4.3.4 Magic Formula parameters for nominal load and variation of inflation pres- sure ...... 51 4.3.5 Comparison of resulting fitted curves for variation of inflation pressures . . 52 4.3.6 Relaxation length ...... 53 4.4 Tire model: Model validation of models from test rig measurements ...... 53 4.5 Force distribution: Normal force estimation results ...... 54 4.5.1 Normal force estimator vs Simulation ...... 54 4.5.2 Transient torque case ...... 54 4.5.3 Transient steering angle case ...... 55 4.5.4 Using the old measurement data data ...... 56 4.5.5 Using newer complimentary data ...... 58 4.5.6 Roll over ...... 60 4.5.7 Pile entry ...... 61 4.6 Force distribution: Turning torque results ...... 62
5 Discussion 64 5.1 Tire model: Field measurements ...... 64 5.2 Tire model: Test rig measurements ...... 64 5.3 Force distribution: Normal force ...... 65 5.3.1 Compared against simulation ...... 65 5.3.2 Compared against the old tests ...... 65 5.3.3 Compared against the new tests ...... 65 5.3.4 Roll over ...... 66 5.3.5 Pile entry ...... 66 5.4 Force distribution: Turning torque ...... 66
6 Conclusions 67 6.1 Tire model: Field measurements ...... 67 6.2 Tire model: Test rig measurements ...... 67 6.3 Force distribution: Normal force estimation ...... 68 6.4 Force distribution: Turning torque ...... 68
7 Future work 69 7.1 Tire model ...... 69 7.2 Force distribution estimator ...... 69
References 72
Appendices 73
A Tire model: Data processing for field measurements 73
B Tire model: Processed field measurement data 79
C Tire model: Structural properties from field measurements 83
D Tire model: Parameters and bounds for field measurements 84
E Tire model: Parameters and bounds for test rig measurements 85
F Kalman filter 86 Variable list
Name Symbol Unit Half contact patch length ac m Longitudinal distance between wheel and COG a m Lateral distance between wheel and COG b m Stiffness factor B - 2 Longitudinal stiffness cpx N/m Shape factor C - Longitudinal slip stiffness CF κ N Lateral slip stiffness CF α N Normalized change in load dfz - Normalized change in inflation pressure dpi - Peak factor D N Slope factor E - Unit vector e - Longitudinal Force Fx N Steady state force Fxss N Lateral Force Fy N Vertical Fore Fz N Nominal load Fzo N a Normal force from asphalt data Fz N g Normal force from gravel data Fz N Gravity constant g m/s2 Weighting function G - Population for differential evolution Gi - Cost function J - Stiffness Kxκ N Moment around the longitudinal axle Mx Nm Moment around the lateral axle My Nm Moment around the vertical axle Mz Nm Mass m kg Curbed mass, L-60 mc kg Bucket mass, L-60 mb kg Gross mass, L-60 mg kg Added mass, L-60 ml kg Mass of tire mt kg Revolutions per second of cardan shaft ncs rps Inflation pressure of back tire pb bar Pressure in piston pc bar Inflation pressure of front tire pf bar Inflation pressure tire pi bar Nominal inflation pressure pio bar Longitudinal parameters for shape factor PCx - Longitudinal parameters for peak factor PDx - Longitudinal parameters for slope factor PEx - Longitudinal parameters for stiffness factor PKx - Longitudinal parameters for inflation pressure PP x - Longitudinal parameters for horizontal offset PHx - Longitudinal parameters for vertical offset PV x - Loaded radius r m Unloaded radius rf m Rotation matrix R - Horizontal factor SH - Vertical factor SV - Slip point S - Longitudinal deflection u m Velocity v m/s Speed of wheel V m/s m/s Slip speed Vs m/s Linear speed of rolling Vr m/s Max speed for L-60 Vmax m/s Velocity cylinder Vcyl km/h Velocity vector V m/s Tire contact patch width w m Wheel position wp m Longitudinal position x m Longitudinal position of center of mass Xcog m Lateral position y m Lateral position of center of mass Ycog m Measured value in cost function Ymeasured N Model output in cost function Ymodel N Vertical position z m Vertical position of center of mass Zcog m
Side-slip angle α rad Camber angle of tire γ rad Steering angle δ deg Longitudinal composite function θx - Pitch θ deg Longitudinal slip κ - Location of peak longitudinal force κm - Transient longitudinal slip κ0 - Friction coefficient, asphalt µa - Friction coefficient, gravel µg - Friction coefficient, concrete µc - Peak friction µp - Friction variable in Magic Formula µx - Friction coefficient µ - Radial deflection of the tire ρ m Longitudinal theoretical slip σx - Longitudinal theoretical slip when sliding occurs σslx - Relaxation length σ m Roll ψ deg Rotational speed tire Ω rad/s 1 Introduction
Wheel loaders are used in diverse terrain types, they can be used on everything from concrete floors to muddy off-road work sites or deep inside mines running over gravel. All this while performing a range of operations from transporting huge stone blocks to stacking logs of timber. At present time Volvo Construction Equipment (VCE) has developed a functional prototype LX1, illustrated in Figure 1, which uses electric motors with individual wheel control. Conventional wheel loaders are typically slow in their dynamics due to that the drive line usually has high inertia. With individual control of each wheel the overall performance can be increased since the electric systems are de- coupled i.e. no connection between each motor. Thus each wheel can be controlled independently to achieve the maximum possible tractive force while not being affected by the ground conditions at the other wheels.
A problem for vehicles in general is unnecessary wheel spin which causes the tires to wear out faster and generates less tractive force. The tires of the wheel loader are a very expensive part, and is one of the parts that are associated with a high running cost. Thus it’s a cost that needs to be kept at a minimum. A way to control this phenomenon, is to use traction control, which has the aim to deliver the highest possible tractive force that the current condition allows. A necessary part to develop such a control algorithm is to have a simulation model that reflects the wheel loader accurately. Such a model can also be used to simulate the causes of undesirable and catastrophic effects, such as uncontrolled sliding or roll over.
A important aspect in traction control is the forces generated between the machine and the ground. These forces can be measured directly but the available sensors for this purpose are usually not robust enough for long time use and are also too expensive. Another way to get this information, is by estimating the forces from other sensor data by using mathematical models, which is what has been done in this thesis.
The following thesis report is a combined report of two parallel works, with the goal to cali- brate a tire model and a force distribution estimator. An import part in product development is by using what’s termed as Model Based Development (MBD). This is a working process to develop functions and products through simulation, and thereby decreasing the total developing time and cost by removing the need for physical tests and prototypes. This of course puts a high demand on the accuracy of the simulation model.
Figure 1: Illustration of the LX1 prototype [1].
1.1 Thesis work In 2015, measurements were performed where the dynamics of the wheel loader was measured in different situations by using a motion capturing system [2] together with force and moment transducers mounted on the wheels. The measurements were conducted with the primary goal to calibrate a tire model. The conducted measurement were raw field measurements and no pre- vious work at present time was known to have been performed in a similar way. Thus the used approached was an unconventional and unestablished way to perform the calibration, since its usually performed in special test rigs. However these test rigs and methods are mainly established for on-road vehicles and not for heavy off-road vehicles. Due to practical reasons, it was decided to conduct the tests as pure field tests. These measurements were used as a starting point, but further tests were conducted during the thesis work.
1 The normal load (vertical force) on the tires are an important input to the tire-model as it gives information regarding the available tractive forces. As the current machines are not equipped with load sensors the distribution must be estimated. This estimation must be able to handle shifts in the weight due to various dynamic phenomena. A reliable estimator of the normal load can also be used as an input for the tire model to give higher precision estimation of the forces acting on the surface. Since the wheel loader can articulate around its mid point, the normal car models used in most research papers can not be used. The center of gravity will significantly shift during turning as roughly half the machine will no longer align with the other half. Dynamic effects during a turning phase will create unbalanced weight shifts. To find the behavior of these shifts is the basis for this part of the thesis. To validate any developed model, data from wheel transducers in a com- bination of on-board systems can be used to show the real world distribution during operations. The distribution of the the normal load on the tires will also be used to determine the likelihood of the machine tipping over. Using the normal force distribution model as a base it is also possible to calculate the force trying to turn the machine due to the variation in force on the individual tires.
At the start of the thesis work, Volvo had a simulation model of a complete wheel loader. This model was used during the thesis work as a reference tool.
1.1.1 Scope Tire model Following list gives an overview of the different stages and goals for the tire modeling part of the thesis.
• Perform a literature review of tire kinematics and dynamics. Study tire models that has the potential to be used. • Process the raw data from the old and newly performed experiments. • From the processed data, perform a model and parameter identification, and develop a tran- sient and steady state tire model.
• Implement the tire models in the pre-existing simulation environment (Simulink). Perform simulations and validate the tire models. • Implement the tire model in the real time system (ECU) of the wheel loader and evaluate from test runs.
The preconditions of the tire model are as follows. • The model most be sufficiently fast in order to be used in the real time system. • The tire model must be allowed to be incorporated in the pre-existing simulation environment.
• The tire model is suppose to take slip and normal force as input. The output should be the tire forces and moments.
Force distribution estimator The work process of the force distribution estimator is: • Literature study of weight distribution and turning moment prediction for articulated vehi- cles.
• Develop a model for finding the center of gravity for any steering angle. • Include the dynamic shift of the gravity due to road topology and vehicle acceleration, using available sensors. • Evaluate any special drive cases.
• Utilize the developed model to find the turning-moment from all wheels combined • Validate the model against simulated and measured data • Transfer the model to Simulink and run it on the plant model.
2 • Run the model on the on-board ECU. • Set up plan for roll over prevention and feed-forward prediction using the steering wheel turning velocity. Following preconditions on the force distribution estimator are:
• The normal force estimator can not use the longitudinal or lateral forces as inputs as it will be used to calculate these. • Can not use any sensors except those for: steering angle, acceleration of vehicle, rotational wheel speed, motor torque, change in the machine articulation angle, angle of the lift arm and lift arm hydraulic cylinder pressure caused by the load
• Most be small enough in storage space to be able to be run on the wheel loaders on board computer.
1.2 Literature review A extensive literature review was performed at the start of the thesis. In the following sections, a small but a important selection will be discussed. However, the remainder of the sources are referenced within the relevant sections within the thesis.
Tire model The main source that was used for studying tire dynamics and models was [3]. The book gives detailed information regarding tire dynamics and different tire models. The author of the book, Hans Pacejka, is well known within the area of tire dynamics and is one the developer of the Magic Formula, which is a well known and established tire model. Due to its popularity and its abilities to describe the tire characteristics, the Magic Formula was chosen in the beginning of the thesis as the primary tire model. The book further contains useful empirical equations for the structural properties of the tire. Other books have also been used to expand the knowledge base in tire dynamics [4, 5, 6].
A big part of the thesis work has been to identify the parameters of the tire models from mea- surement data. A part of the literature review therefore dealt with finding suitable least square methods. A good article that deals with this is [7], where various algorithms were used to estimate the Magic Formula parameters. The algorithms were applied to different characteristics of the tire and the data that was used was also from different sources. The article gave a good overview and introduction to different algorithms that are suitable for the task. It also highlights some difficulties in the estimation process that were useful for the thesis work.
Force Distribution One of the most interesting articles when performing the literature review was [8]. The authors discuss that there has been very little research in this field regarding vertical forces and roll- over. To remedy this the authors develop a model using lagrangian mechanics defined in a global coordinate system. The result is a mathematical model suitable for simulating many different drive cases. Unfortunately the proposed model requires information that will not be available for the machines that are of interest in this thesis. The general idea of how to model a wheel loader and its behavior was however used. Most of the defining equations that are used to build the model comes from [4], where the author, Reza Jazar, gives easy to understand explanations of vehicle dynamics and their effects. While the book focus on the automotive applications, the books way of building a model for how the weight of a vehicle shifts during acceleration, braking and changes in road inclination how been incorporated in the developed wheel loader model.
3 2 Theory
This section presents the results from the literature review conducted at the beginning of the thesis work.
2.1 Tire model: Tire fundamentals The tire is a important component for machine dynamics. Section 2.1.1 introduces and defines some basic quantities and tire concepts. Next in section 2.1.2 the forces that act on the tire are described.
2.1.1 Introduction to tire quantaties A tire traveling with velocity V and an angular speed Ω is presented in its general configuration in Figure 2.
Figure 2: Illustration of the tire coordinate system [4].
In [4] the tire is defined to be travelling on a flat plane termed as the ground plane. Next, the tire plane is introduced which is defined as the mid-plane corresponding to the tire [4]. A coordinate system is next defined with the origin placed at the center point of the contact patch. This co- ordinate system represents the tires local coordinate system. The x-axis is defined to be aligned with the line of intersection between the tire plane and the ground plane. The z-axis points in the direction along the normal to the ground plane. Finally, the y-axis points in the direction such that a right handed coordinate system is given. The rotation about the x, y and z-axis is termed as the roll, pitch and yaw-angle.
The following forces and moments are applied to the wheel,
• Fx, longitudinal force.
• Fy, lateral force (cornering force).
• Fz, vertical force (load).
• Mx, roll moment (overturning moment).
• My, pitch moment.
• Mz, yaw moment (aligning moment). In [4], two different angles are presented which are the side slip and camber angle represented by α and γ respectively. The side slip angle is the angle the velocity vector, V makes with respect to the x-axis. The camber angle, γ, corresponds to the inclination of the wheel and defined as the angle the tire plane makes with the z-axis.
The loaded radius of the wheel is in [3] represented by the variable r and its unloaded radius by rf . With these radii the radial deflection, ρ, is defined [3] as
ρ = rf − r (1)
4 Another important radius is defined in [3], which is the effective rolling radius of the tire. It’s an important radius, and according to [3], defined as the radius which couples the longitudinal speed, Vx, and the angular speed, Ω, when the wheel is rolling freely, which means that no torque is being applied to it. From [3] it’s stated as follows V r = x (2) e Ω and the radius is under free rolling conditions bounded to be between the unloaded and loaded radius of the tire [6].
r < re < rf . (3)
For the free rolling case, the radius defines the instantaneous center of rotation of the wheel and defines an slip point S which is distanced re radially from the center of the wheel [3]. This is illustrated in Figure 3.
Figure 3: Presents the slip point S and the effective rolling radius re [3].
Longitudinal and lateral slip When a torque is applied to the wheel, the instantaneous center of rotation moves away from being located at the slip point. Relative motion is therefore given between the slip point and the wheel. A longitudinal slip speed is given which in [3] is calculated as
Vsx = Vx − Ωre (4) and longitudinal slip arises, which is a quantity defined by [3] as
V (V − r Ω) κ = − sx = − x e . (5) Vx Vx
When the wheel is driving forward, the sign of κ is positive, and positive longitudinal force, Fx, arises. If however the wheel is braked, the sign of κ turns negative and a negative longitudinal force is given. Since the previous definition of slip has the longitudinal velocity in the denominator, this gives rise to singularities when the wheel is standing still, Vx = 0, and the wheel spins, Ω 6= 0. An alternative definition of slip [5] can instead be applied for the case of a wheel driving forward, which is as follows Ωr − V κ = − e x . (6) Ωre If the side slip angle is non-zero, lateral slip or side slip is given which becomes, with the used coordinate system and the definition given in [3], as follows V tan(α) = y . (7) Vx
2.1.2 Tire forces and moments
The horizontal forces, Fx and Fy, and the aligning moment, Mz, are in steady state motion func- tions of the previously introduced slip quantities. Thus slip needs to be present in order for forces
5 and moments to exist. The wheel load and camber angle has also an influence of the forces and moments.
A turning wheel, under side slip forces causes the treads belonging to the tire to deflect both longitudinally and laterally [4]. This causes an lateral tangential stress distribution and the resul- tant lateral force gets displaced a distance behind the center line [4]. This distance is termed as the pneumatic trail, which gives rise to the aligning moment, Mz. The aligning moment tends to turn the tire towards the direction of the tires velocity vector, V, thus it aligns the wheel with the velocity vector hence its name.
Figure 4 presents typical force curves for pure longitudinal and lateral slip, as well as combined slip. Pure slip occurs when only one slip quantity is present.
Figure 4: Graphs showing typical force characteristics for pure and combined slip [3].
As can be observed in Figure 4, the relationship between the slip and forces is of non-linear nature. If the slip values are low, which is the case for normal driving conditions [9], a linear relationship between the slip values and forces are valid.
The constant of proportionality for the longitudinal force, Fx, and longitudinal slip, κ, is termed as the longitudinal slip stiffness, CF κ. Similarly the constant of proportionality for the lateral force is termed as the lateral slip stiffness or the cornering stiffness, CF α [3]. Thus for low slip values, following relationships are valid
Fx = CF κκ, (8)
Fy = CF αα. (9)
As further can be observed from the figure, when the situation is such that combined slip is occurring, the forces reduce in magnitude. This behavior is due to that the total horizontal force, F , cannot exceed the maximum friction force, µFz, which is a function of normal force and current surface friction [3]. By the assumption that the tire has isotropic adhesion properties, the maximum allowable horizontal force is given from [10] as follows
2 2 2 Fx + Fy = (µpmg) . (10) where µp is the peak coefficient of friction, m the mass and g the gravitational constant. Thus the horizontal force is constrained to be within what’s termed as a friction circle. However this is a rough simplification, and in the reality the tire has anisotropic adhesion properties. Tires which posses the non-linear nature presented in the figure, may be described by a set of empirical formulas called the Magic Formula which are addressed in more detail in section 2.2.1.
6 2.2 Tire model: Tire models As was discussed in previous section, the force characteristics of the tire are non-linear. In the most general case, the tire can be modeled as a non-linear system which is dependent on multiple inputs such as its slip quantities, normal load, camber angle etc. The outputs of this general tire model are the tire forces and moments. There exist many different models for tires which couples the previous mentioned quantities. One of the most established tire model is the Magic Formula, which is a semi-empirical tire model and is described in section 2.2.1.
Even though the Magic Formula serves as a satisfactory tire model, it has its limitations to quasi steady state conditions, [11]. Thus when the use case is of transient nature and conditions where the surface is non-flat with short wavelength, the Magic Formula is not applicable. An improved model to the Magic Formula is the MFSWIFT model, which handles transients and is further discussed in section 2.2.2. The brush model is a theoretically derived tire model, and relatively simple in its formulation. The model is described in section 2.2.3. Section 2.2.4 introduces a simple transient tire model, which uses the Magic Formula to calculate the forces.
2.2.1 The Magic Formula As discussed in previous section, the slip and force relationship for a tire is highly non-linear. A widely used and well established tire model is the Magic Formula (MF). The MF has been published in many different versions, however the MF used for this thesis work and described in the following section is given from [3], which introduces the general appearance of the MF for the case of pure slip as
y =D sin(C arctan(Bx − E(Bx − arctan(Bx))) (11)
Y (X) =y(x) + Sv (12)
x =X + SH (13)
In the above formula, Y represents the output variable which could either be Fx, Fy or Mz. As input the formula takes the corresponding steady state slip quantitiy, which either is κ or α. The coefficients in the formula represents a certain characteristic in the curve and some of these coefficients are presented in Figure 5. The coefficients are dependent on normal load, Fz, and inflation pressure, pi. The non-dimensional parameters, dfz and dpi, are introduced in [3], and are as follows
Fz − Fzo dfz = , (14) Fz pi − pio dpi = . (15) pi
The parameters Fzo and pio are the nominal load and the nominal inflation pressure.
7 Figure 5: The figure illustrates some of the implications that some of the coefficients in the MF has [5].
To explain the relationship between Fz, pi, and the MF coefficients, the longitudinal version of the MF given by [3] will be presented.
The product BCD corresponds to the slope of the linear part of the function. In [3] it’s given a dependency on both load and inflation pressure, and expressed as follows 2 2 Kxκ = Fz(PKx1 + PKx2dfz ) exp(PKx3dfz)(1 + ppx1dpi + ppx1dpi ). (16) The parameter C in Eq (11), controls the range of the sine function which has the implication of controlling the shape of the curve. It’s therefore named as the shape factor, and it can analytically be calculated [3] as 2 y C = 1 ± (1 − arcin( a )). (17) π D
In the above equation, ya is the force which the curve approaches asymptotically as the slip grows. The coefficient D is the peak value, and controls the peak tractive/braking force of the curve. The peak force is limited by the friction coefficient, µx, and dependent on the normal load. In [3], this is expressed as
Dx = µxFz. (18) where the variable µx further is given a load and inflation pressure dependency as follows 2 µx = (PDx1 + PDx2dfz)(1 + PP x3dpi + PP x4dpi ). (19) The factor E is termed as the slope factor, which can be calculated by using the information of the location of the peak force, κm. In [3], a analytical expression for the slope factor is given as π Bxκm − tan( ) E = 2Cx , if C > 1 (20) Bκm − arctan(Bκm) and its dependency of normal force is in [3] expressed as follows 2 Ex = (PEx1 + PEx2dfz + PEx3dfz )(1 − PEx4sign(κx)). (21)
The coefficient Bx is the only free coefficient to determine the initial slope, thus its termed as the stiffness factor and calculated as
Bx = Kxκ/(CxDx). (22) For some cases, the curve passes through the origin, but this is not always the case since effects such as conicity and ply-steer, which are effects connected to non-symmetry in the tire construction, offset the curve [3]. The rolling-friction also has this effect. To allow the formula to describe this behavior the coefficients SHx and SV x are introduced, which offset the curve horizontally and vertically. They are in [3] expressed as
SHx = (PHx1 + PHx2dfz), (23)
SV x = Fz(PV x1 + PV x2dfz). (24)
8 Combined slip For the case of combined slip (α 6= 0 and κ 6= 0), a weighting factor is introduced [3]. It’s multiplied with the pure slip force to give the effect of a combined slip condition. In [3], following weighting function is presented
G = D cos(C arctan(Bx)). (25)
The coefficients in the function have similar interpretations as the previously introduced. In [3] the coefficients are presented as follows. D is the peak value, C determines the height and B the sharpness of the function. In the ideal case, the behavior of the function is such that the function equals unity in the case of pure slip which gradually decreases as the situation turns into combined slip [3]. The function therefore takes the form of a hill. Figure 6 and 7 illustrates the weighting function and the effects it has on the lateral and longitudinal force when combined slip is present.
Figure 6: Graphs the weighting function (left graph) and the influence is has on the lateral force (right graph) [3].
Figure 7: Graphs the weighting function (left graph) and the influence is has on the longitudinal force (right graph) [3].
2.2.2 MFSWIFT The MF is accurate and robust, but is limited to describe the forces under quasi steady state conditions. To describe the dynamics in more detail, the MFSWIFT model could instead be used, which is a fusion of the MF together with the SWIFT model. A conceptual overview of the model is presented in Figure 8.
9 Figure 8: Illustrates the concept of the MFSWIFT model [12].
The model consists of four important elements [13], which are the MF (described in previous section), a rigid ring to approximate the dynamics of the tire belt, the contact patch slip model, and finally a enveloping model to handle obstacles and arbitrary 3 D roads.
Rigid ring dynamics To properly describe the dynamics of the wheel, a rigid ring is included into the model. The purpose of this is to include the inertia of belt as well and the tire sidewalls [12]. By including the inertia of the belt, the wheel is split into two parts, rim and a tire belt, which are interconnected with each other by a set of dampers and springs in all directions. The belt is further connected to the contact patch by a residual spring. By dividing the wheel into two parts, the model becomes fairly accurate in the frequency range (60-100Hz [3]), due to that the first mode shapes are rigid vibration modes, which makes the rigid ring approximation valid.
The contact patch slip model A certain distance is needed for the tire to be travelled before the steady state forces are reached. The distance needed to be travelled is termed as the relaxation length and can be interpreted as a spatial time constant. From experiments [13] it has been observed that the relaxation length is dependent on the load, magnitude of slip and inflation pressure [14]. An example of this dependency on the side slip is shown in Figure 9, where a step response in side slip is introduced. It’s seen that with increasing side slip the tire responds quicker, thus the relaxation length decreases with increasing side slip.
Figure 9: Illustrates how the magnitude of side slip affects the relaxation length [13].
To describe these tire transients, a contact-patch model which accounts for the transient behavior is used. The MFSWIFT uses the brush model [15] together with transient slip quantities. The
10 analytical model that describes the non steady state response to slip variations is approximated by a set of first-order differential equations, where the transient slip values are given as outputs from these sets of equations. To calculate the transient slip forces, the transient slips are used as inputs in the MF. The forces generated are then applied to the contact patch which further propagates to the wheel axle through the set of dampers and springs, which connects the different parts [13].
Enveloping model The previous discussion has been limited to a flat and smooth plane road or a road where the wavelength of the vertical road height is sufficiently large. In these situations the tire is excited by axle motion, braking or steering. However the tire can also be excited via the road, examples are road unevenness with short wavelengths (two to three time the contact length [15]) or when traversing obstacles. This gives rise to non-linear behavior of the horizontal and vertical force as well as changes of the effective rolling radius re [13].
To incorporate this excitiation effect into the model, the actual road is filtered by using a set of elliptical cams which are discretized [12] and used to sense the modeled road, this is illustrated in Figure 10.
Figure 10: Presents the concept of elliptical cams [14] .
2.2.3 The brush model The brush model is a physical tire model, which approximates the tire as a rigid ring where brushes extend radially outward from its circumference [6]. The bristles are given a linear stiffness and are supposed to model the elasticity of the carcass, belt and treads [3]. A schematic layout of the brush model is presented in Figure 11.
Figure 11: Illustration of the brush model [6].
If the tire rolls freely, without any presence of slip, the bristles remain vertical when entering the contact region. No horizontal deflection is given, why no forces are introduced. If however slip is present, the brushes deform horizontally in the contact area which generates the tire forces and moments [3].
In the contact area, two regions are defined, which are the adhesive and sliding region [3]. Within the adhesive region the bristles follow the direction dictated by the velocity vector, which by def- inition is the side slip angle α. For the case of pure lateral slip, the tip of the bristle enters the
11 contact region with zero lateral deflection which then grows linearly as the tip progresses through the contact area. The maximum possible deflection is however limited by the present friction co- efficient, the vertical force distribution and the stiffness of the element [3]. A transition from the adhesive region to the sliding region occurs when the limit is reached, and the deflection instead follows a pattern dictated by the previous mentioned quantities. This is illustrated in Figure 12 for the case of a parabolic pressure distribution. If the situation is such that pure longitudinal slip occurs, the bristles move rearwards in the case of braking and forward in the case of driving [6].
Figure 12: The left illustration shows how the bristles deflects as side slip increases. The plot to the right plots the corresponding lateral forces [3].
Pure longitudinal slip The brush model in a state of braking and under pure longitudinal slip is illustrated in Figure 13.
Figure 13: The brush model for the case of pure longitudinal slip [3].
In the model, the base points of the bristles move rearwards with the velocity Vr termed as the linear speed of rolling, which is as follows
Vr = Ωre. (26)
The elements tip points adheres to the ground which results in a longitudinal deflection if slip is present. In [3] this is expressed as
Vsx u = −(ac − x) . (27) Vx − Vsx
In [3], the theoretical longitudinal slip, σx, is introduced and defined as
Vsx κ σx = − = . (28) Vr 1 + κ By the introduction of the theoretical slip, the longitudinal deflection may instead be expressed in terms of this quantity, which by [3] becomes
u = (ac − x)σx. (29)
12 A composite function θx is in [3] introduced
2 2 cpxac θx = (30) 3 µFz and total sliding occurs at 1 σslx = ± . (31) θx If a parabolic force distribution of the normal load is assumed in the contact area, then following expression for the longitudinal force is given [3] 1 2 3µFzθxσx(1 − |θxσx| + (θxσx) ) , if |σx| ≤ σslx Fx = 3 (32) µFzsgn(κ) , if |σx| ≥ σslx.
2.2.4 Single contact point model The single contact point model (SCP) [3] is a transient tire model, which works together in com- bination with the steady state MF to obtain the tire forces. It’s limited to low levels of slip [11]. The model is illustrated in Figure 14.
Figure 14: Illustration of the single contact point model [3].
From the description given in [3], two different points are used, the slip point S which is attached to the rim and the contact point S0. These points move with different velocities, and the difference between them give a longitudinal deflection, u, and lateral deflection, v. The differential equation for the longitudinal case reads according to [3] as du 1 + |V |u = −|V |κ = −V (33) dt σ x x sx where σ is the longitudinal relaxation length. Next, the transient slip quantity is calculated [3] as u κ0 = . (34) σ This transient slip quantity is then used as input in the MF in order to get the forces acting on the tire
0 Fx = MF (κ ,Fz) (35) where MF is the Magic Formula for the pure longitudinal slip case.
2.3 Tire model: Algorithms for parameter estimation The MF is a widely used and established tire model, but it possesses a certain level of complexity due to its non-linear relationship and vast number of parameters that it contains. These param- eters are usually determined by least square fitting techniques [7]. The techniques uses what is termed as a cost function, which defines how good of a fit some parameters give to observed data. The problem then becomes to find the parameters which gives the best fit.
In [7] various algorithms were applied to estimate the parameters from the MF. The used al- gorithms, included both gradient based and derivative-free methods as well as unconstrained and constrained methods. The algorithms were applied to lateral force, longitudinal force and self- aligning torque data, all under pure slip conditions. The data that were used was also from
13 different sources. The three methods that gave best fits were the trust region reflective, differential evolution and bounded cuckoo search. A important conclusion from [7] was that even though an algorithm converges to a certain set of parameters, the calculated parameters can differ between different algorithms. A recommendation from [7] was to compare the results of a number of algo- rithms before choosing the parameters. If the estimated parameters are all the same, then clearly this is a good indication.
The differential evolution algorithm showed good results in [7], why the algorithm was deemed as being suitable for the thesis work. The following section gives a brief description of how the algorithm works.
2.3.1 Differential evolution Differential evolution is based on evolutionary optimisation techniques [7], which means that it mimics the way evolution works. The algorithm works by defining a initial population, Gi. Each inhabitant is a vector and its elements contains the parameters that are suppose to be estimated. These are in the beginning chosen randomly [16]. A new generation is created by a process termed as mutation [16]. Practically this is performed by combining randomly chosen inhabitants in the current generation, to create a new mutated vector. As a next step, the mutated vectors parameters are mixed with a chosen inhabitant in the current generation, Gi, to give diversity in the solutions [16]. This process is termed as crossover. By performing these steps, a new generation is given, Gi+1, which contains new potentially better parameters. The process of deciding if a a inhabitant in the new generation, Gi+1, should replace a corresponding inhabitant in the current generation, Gi, is termed as the selection process [16]. The previously mentioned steps are then iterated until a solution is found.
2.4 Force distribution: Normal force The theory regarding normal force distribution and estimation is quite extensive for the automo- tive industry. There are some papers regarding articulated machines [17, 18] but when looking for articles on modelling articulated machines most articules focuses on trucks. The articulation system for wheel loaders differ quite a bit from trucks as the wheel loaders articulation joint is unable to turn by itself as it is held stiff and controlled by two hydraulic cylinders. However when including tractors and general articulated machines as a base for modelling wheel loaders there are some articlesthat are useful [19, 20, 8, 21]. These paper however do not describe the load distribution in a way that is useful for this thesis. The limit on any model created is to only use available sensors which can measure: steering angle, acceleration of the machine, rotational wheel speed, motor torque, change in the machine articulation angle, angle of the lift arm and lift arm hydraulic cylinder pressure caused by the load. Anything else is considered unavailable and any model requiring more inputs than these, unusable. As such it was determined to build a model from the ground up, using simple physics in order to have a better understanding of the model and be able to improve it as time went on. When deciding how to model the system, inspiration was collected from several sources, including models for cars [22, 8, 20]. Due to the available sensor a local coordinate system was chosen, this means the machine only know its position with regards to its own origin point placed on the machine, but has no idea about its position relative the global world. General vehicle dynamics effects were also investigated [23, 3, 24] when building this model.
With the introduction of powerful electric motors the wheel loaders are able to perform in ways not previously possible. This creates situations that were previously safe which can now, with for example faster acceleration, lead to roll over. For inexperienced drivers it becomes extra important to prevent these situations from arrising. When it comes to roll-over prediction it has been pro- posed that looking at the difference between left and right side wheel load can be used to estimate when roll over is likely to happen [8, 18] so this will be the approach for this thesis as well.
2.4.1 Defining the basis for the normal force distribution model How the center of gravity of a machine affects the normal force on the tires is the basis for the model to be used. When calculating the distribution, the origin is placed at the articulation joint. The x-direction is defined positive in the front-facing direction of the machine, the longitudinal direction. The y-direction is positive to the right of the machine, in the lateral direction and the z-direction is positive upwards, in the vertical direction. Figure 15 shows a simple illustration of a wheel loader from above with the coordinate system set at the articulation point.
14 Figure 15: Wheel loader as seen from above facing ”up-ward”.
2.4.2 Finding the center of gravity The wheel loader is able to turn at the middle due to its articulation joint. In order to estimate the normal forces on each tire it is important that the static center of gravity is accurately calculated. The static center of gravity refers to when the machine is in a steady state and accounts only for the articulation angle δ, the lift arm angle φ and the bucket load. The machine is divided into separate parts: The rear, the front, the lift arm and bucket. The lift arm and bucket will collectively be named simply as bucket from here on out. The center of gravity is calculated independently in the x, y and z directions by taking how far away the rear, front and bucket center of gravity is from the articulation point and weighting the values with the parts different masses. Figure 16 shows the distance in the x-direction for a machine that is turning.
Figure 16: Distance between masses and origin in the x-direction.
The machine center of gravity is calculated in the following way
1 Xcog = mfrontxfront cos(δ) + mbucketLoadxbucketLoad cos(δ) + mbackxback . (36) mmachine
Xcog is the distance in the x-direction from the articulation point to the center of gravity. m represents mass and x the longitudinal distance for the given part. The weight of the bucket with
15 arm-linkage and the payload has been combined, while accounting for the fact that the change in arm position will happen some distance from the origin. This then becomes
1 xbucketLoad = (mbucket(xfront + (xbucket − xfront) cos(φ) mbucketLoad (37) +mload(xfront + (xload − xfront) cos(φ)). mbucketLoad is the combined weight of the bucket, arm-linkage and load
mbucketLoad = mbucket + mload (38) and mmachine is the total mass and is the sum of the individual masses
mmachine = mback + mfront + mbucketLoad. (39) In the y-direction it becomes 1 Ycog = (mfrontxfront sin(δ) + mbucketLoadxbucketLoad sin(δ) + mbackxback) . (40) mmachine Since the contact is along the ground the vertical distance will be of importance for the moment equilibrium equations and is 1 Zcog = (mfrontzfront + mbucketLoadzbucketLoad + mbackzback) (41) mmachine and
zbucketLoad = mbucket(zbucket + (xbucket − xfront) sin(φ)+ 1 (42) mload(zload + (xbucket − xfront) sin(φ) . mbucketLoad The distribution of the load on the wheels is assumed to be inversely proportional to the distance from the center of gravity. The first step is then to find the position of the wheels , wp, with regard to the origin. These are calculated as vectors for the x- and y-coordinates to make the equations more clean. The wheel loader is symmetric for the wheels along the x-axis when straight, so the following equations hold true: xfrontAxle cos(δ) − yfrontAxle sin(δ) wpfront left = , (43) −yfrontAxle cos(δ) − xfrontAxle sin(δ) xfrontAxle cos(δ) + yfrontAxle sin(δ) wpfront right = , (44) yfrontAxle cos(δ) − xfrontAxle sin(δ) xrearAxle wprear left = , (45) −yrearAxle and xrearAxle wprear right = (46) yrearAxle where the longitudinal distances are between the articulation point and the middle of the axles and the lateral distances are to the end of the axle, at a point in the middle of the tire.
2.4.3 Shifts in center of gravity To account for shift in the center of gravity due to change in speed of the machine, road inclination or rocking of the machine, equations of equlibrium will be used. As the normal force estimator will be an input to the tire model it can not use the longitudinal or lateral forces, since those are calculated in the tire model. Instead the acceleration will be used since that can be measured or calculated from avaliable sensors. For the change in angle with regards to inclination of the road or rocking of the machine, this data is avalaiable from gyroscopes installed on the machine. To calculate the shifts in center of gravity, the theory used for cars will be used as the base and then later adopted for the center of gravity model. All equlibrium equations are derived from [4].
16 Steady-state longitudinal case A stationary machine is a simple system but will be used to define the system used for weight distribution calculations. The system is shown in Figure 17.
Figure 17: Stationary machine with the front to the right.
The longitudinal distance between wheel position and center of gravity is represented by a and index f refers to the front of the machine while b refers to the back of the machine. The forces on the machine when stationary can be described by two equilibrium equations
X Fz = 0, X (47) My = 0.
In this case it is assumed that the weight is roughly in the middle of the left and right side. The sum of the normal forces will equal the weight of the machine X Fz = 2Fzf + 2Fzb − mg = 0. (48)
The moment around the y-axis comes from the vertical forces acting on the tires X My = 2Fzf af − 2Fzbab = 0. (49) Rewriting the equation to find the relation between the vertical forces yields
af Fzb = Fzf . (50) ab Substituting into Eq (48) gives
af 2Fzf + 2Fzf − mg = 0. (51) ab
Simplifying and solving for Fzf produces
1 ab Fzf = mg . (52) 2 af + ab
The force on the back tire is then calculated from Eq (50).
Accelerating in a straight line The next step is to look at the how the vertical forces shifts between the tires when the machine is accelerating. The forces are shown in Figure 18.
17 Figure 18: Accelerating machine with the front to the right.
This time the equilibrium equations become
X Fx = mv,˙ X Fz = 0, (53) X My = 0.
The longitudinal forces will be equal to the acceleration times the mass of the machine X Fx = 2Fxf + 2Fxr = mv˙ (54) where v is the velocity of the machine andv ˙ is its time derivative. The sum of the normal forces will still be the total weight of the machine X Fz = 2Fzf + 2Fzb − mg = 0. (55)
This time the moment around the y-axis is caused both by the vertical forces at the tires, but also by the longitudinal forces X My = −2Fzf af + 2Fzbab − (2Fxf + 2Fxr)Zcog = 0. (56)
Substituting Eq (54) into Eq (56) gives X My = 2Fzf af − 2Fzbab + mvZ˙ cog = 0. (57)
Solving for Fzb
mvZ˙ cog + 2Fzf af Fzb = (58) 2ab and substituting into Eq (55) gives the solution for Fzf
mvZ˙ cog + 2Fzf af 2Fzf + 2 − mg = 0, 2ab (59) 1 ab Zcog Fzf = mg − mv˙ . 2 af + ab af + ab
The back tire force is calculated from Eq (58).
18 Stationary machine on inclined road When looking at the case of a machine on an inclined road, the stationary case is shown in Figure 19.
Figure 19: Inclined machine with the front to the right.
The angle θ shown in the figure is the pitch angle. The equilibrium looks similar to the accelerating case, except the sum of the longitudinal forces are zero X Fx = 0, X Fz = 0, (60) X My = 0.
The longitudinal forces must resist the part of the weight which is trying to move the machine backwards X Fx = 2Fxf + 2Fxr − mg sin(θ) = 0. (61) The vertical forces only takes a portion of the total weight, the part that is pushing against the ground X Fz = 2Fzf + 2Fzb − mg cos(θ) = 0. (62) The moment around the y-axis is by itself independent of the angle of the incline. Any variance in the moment comes from the fact that the effecting forces change with the angle X My = 2Fzf af − 2Fz2ab + (2Fxf + 2Fxr)Zcog = 0. (63)
Inserting Eq (61) into (63) and simplifying in the same way as before gives
1 af Zcog Fzf = mg cos(θ) − sin(θ) , 2 af + ab af + ab (64) 1 ab Zcog Fzb = mg cos(θ) + sin(θ) . 2 af + ab af + ab
Stationary machine on banked road A shift of the weight in the lateral direction can happen for example when the machine is on a banking road as shown in Figure 20.
19 Figure 20: Machine on a banking road, seen from behind.
Index r refers to the right side and l to the left while ψ is known as the roll angle and b is the lateral distance between wheel and center of gravity. For the equilibrium equations the moment will be calculated around the y-axis instead of the x-axis
X Fx = 0, X Fz = 0, (65) X My = 0.
The lateral forces counteract the lateral portion of the weight which can be seen in X Fy = 2Fyr + 2Fyl − mgsin(ψ) = 0. (66)
The vertical forces correspond the the part of the weight that is directed into the inclined ground as shown in X Fz = 2Fzr + 2Fzl − mgcos(ψ) = 0. (67) The moment around the y-axis is positive in the clock-wise direction as follows X Mx = 2Fzlbl − 2Fzrbr − (2Fyr + 2Fyl)Zcog = 0. (68)
The forces on each tire is calculated in the same way as prevision examples resulting in
1 b 1 Z F = mg l cos(ψ) − mg cog sin(β) zr 2 b + b 2 b + b r l r l (69) 1 br 1 Zcog Fzl = mg cos(ψ) + mg sin(ψ). 2 br + bl 2 br + bl
2.4.4 Normal force on each tire Adapting the equation above for an articulated wheel loader is possible when all distances between the wheels and the center of gravity are determined. To reduce the size of the equations and make them easier to follow each of the four tires will be given a separate numerical index and the mass will be designated mt. The numeration convention is as follow
1. Front left tire. 2. Front right tire.
3. Back left tire. 4. Back right tire.
20 Thich is illustrated in Figure 21
Figure 21: Labelling of wheels on a wheel loader.
To find the forces on each tire, five equilibrium equations can be used to evaluate the effect off the normal forces
X Fz = 0, X Fx = mtv˙x, X Fy = mtv˙y, (70) X My = 0, X Mx = 0.
But before those equations are defined it is necessary to look at the rear axle. As there is no damping in a wheel loader the rear axle can rotate independently of the rear body, which is where the cabin and driver are located. This changes the way the load is distributed. Figure 22 shows a detailed view of a wheel loaders rear axle.
Figure 22: Rear axle that can rotate independent of the body [25].
The mounting points are attached to the wheel loader body and join together to create a single contact point for the rear. This means that in order to get a accurate machine model, the wheel loader has to be modeled as a tricycle. This set up is shown in Figure 23.
21 Figure 23: Tricycle configuration of load distribution.
Using the tricycle model where mt is the mass of that portion of the machine, the sum of the vertical forces becomes X Fz = Fz1 + Fz2 + FzR − mtg cos(ψ) cos(θ) = 0 (71) where index R refers to the forces on the rear contact point. The longitudinal forces are X Fx = Fx1 + Fx2 + FxR − mtg sin(θ) = mtv˙x (72) and the lateral forces are X Fy = Fy1 + Fy2 + FyR − mtg sin(ψ) = mtv˙y. (73) Before calculating the moment around the x- and y-axis it is important to look at where the rear forces act. The tire forces act on the ground at the contact points between the wheels and ground, but the rear forces acts where the support attaches to the rear body, as can be seen in Figure 22. The configuration instead becomes the one seen in Figure 24.
Figure 24: Contact points for the various forces for the tricycle model.
With this in mind the moment around the y-axle is X My = Fz1a1 + Fz2a2 − FzRaR + Zcog(Fx1 + Fx2) + (Zcog − ZR)FxR = 0 (74) which can be rewritten using Eq (72) as
22 X My = Fz1a1 + Fz2a2 − FzRaR + Zcog(mtv˙x + mtg sin(θ) − FxR) + (Zcog − ZR)FxR = (75) Fz1a1 + Fz2a2 − FzRaR + Zcogmt(v ˙x + g sin(θ)) − ZRFxR = 0 The moment around the x-axis is only dependent on the front wheels as the forces from the back wheels go trough the rear axle attachment point which lies along the x-axis X Mx = Fz1b1 − Fz2b2 − Zcog(Fy1 + Fy2) − (Zcog − ZR)FyR = 0 (76) which, using Eq (73) can be rewritten as X Mx = Fz1b1 − Fz2b2 − Zcogmt(v ˙y + g sin(ψ)) + ZRFyR = 0. (77)
Because the forces FxR and FyR are unknown they are approximated by assuming that all tires have an equal longitudinal force acting on them as well as an equal lateral force. This would make FxR and FyR take half the force from the weight and acceleration acting on the machine
1 FxR = mt(v ˙x + g sin(θ)), 2 (78) 1 F = m (v ˙ + g sin(ψ)). yR 2 t y This is a big simplification that needs to be done in order for the model to work. The alternative is to artificially set the contact point for the rear on the ground, in line with the front wheels contact point. This will enable all vertical and longitudinal forces to be substituted for the acceleration. The problem with this is that for the rear axle, the forces will then only account for the acceleration acting on the rear axle mass which is only a fraction of the total mass. The second approach will simplify the final equations quite a bit, but the more complex setup was chosen as it is still a closer representation of the real world behavior.
Substituting Eq (78) into Eq (75) and (77) yields
X 1 My = Fz1a1 + Fz2a2 − FzRaR + Zcogmt(v ˙x + g sin(θ)) − ZRmt(v ˙x + g sin(θ)) = 2 (79) 1 F a + F a − F a + (Z − Z )m (v ˙ + g sin(θ)) = 0, z1 1 z2 2 zR R cog 2 R t x and
X 1 Mx = Fz1b1 − Fz2b2 − Zcogmt(v ˙y + g sin(ψ)) + ZR mt(v ˙y + g sin(ψ)) = 2 (80) 1 F b − F b − (Z − Z )m (v ˙ + g sin(ψ)) = 0. z1 1 z2 2 cog 2 R t y The rear tires have their own set of equations with regards to the rear vertical forces and the weight of the real axle ma. The normal force on the rear axle will be called Fza and all other forces acting on the axle will have a similar index
X Fya = mav˙y, X Fza = 0, (81) X Mxa = 0.
The load distribution is a function of both the forces on the rear axle’s support and the weight of the real axle itself X Fza = Fz3 + Fz4 − FzR − mag cos(ψ) = 0. (82) As the rear tires are aligned along the y-axis, the moment around this axis is of no interest. The lateral forces are dependent on the weight of the rear axle X Fya = Fy3 + Fy4 − FyR − mag sin(ψ) = mav˙y. (83)
23 The moment around the x-axis is X Mxa = Fz3b3 + Fz4b4 − Za(Fy3 + Fy4) − (ZR − Za)FyR = 0 (84) where Za is the distance between the ground and the center of gravity for the rear axle. b3 and b4 are the lateral distances to the center of gravity of the rear axle for each wheel. Substituting Eq (83) and (78) into (84) gives
X 1 M = F b + F b − Z m (g sin(ψ) +v ˙ ) − Z m (v ˙ + g sin(ψ)) = 0. (85) xa z3 3 z4 4 a a y 2 R t y To be able to solve any equilibrium equations the distances between each wheel and the center of gravity needs to be calculated:
a1 = |Xcog − wp1(1)| ,
a1 = |Xcog − wp2(1)| , (86)
aR = |Xcog − wp3(1)| . The rear point is aligned with the rear wheels and because the rear will always be stationary according the to the local coordinate system the distance in the longitudinal direction will always be the same for the right and left side and as such it does not matter if wp3(1) or wp4(1) is used when calculating the distance for the rear. For the distances that are of interest, the lateral distances are:
b1 = |Ycog − wp1(2)| ,
b2 = |Ycog − wp2(2)| , (87) b3 = |Ycoga − wp3(2)| ,
b4 = |Ycoga − wp4(2)| . As has been mentioned a is the distance along the x-axis and b is along the y-axis. This is illustrated in Figure 25 to further clarify how they are calculated.
Figure 25: Wheel position relative the center of gravity.
To find the equations for Fz1, Fz2, FzR, Fz3 and Fz4 MATLABs solve() function is used, which produces the following equations
Fz1 = mt 2Zcoga2v˙y + 2ZcogaRv˙y − ZRa2v˙y − ZRaRv˙y − 2Zcogv˙xb2 + ZRv˙xb2−
2Zcogb2gsin(ψ) + ZRb2gsin(ψ) + 2Zcoga2gsin(θ) + 2ZcogaRgsin(θ) − ZRa2gsin(θ)− (88) ZRaRgsin(θ) + 2aRb2gcos(ψ)cos(θ) /(2(a1b2 + a2b1 + aRb1 + aRb2)),
Fz2 = mt ZRa1v˙y − 2ZcogaRv˙y − 2Zcoga1v˙y + ZRaRv˙y − 2Zcogv˙xb1 + ZRv˙xb1−
2Zcogb1gsin(ψ) + ZRb1gsin(ψ) − 2Zcoga1gsin(θ) − 2ZcogaRgsin(θ) + ZRa1gsin(θ)+ (89) ZRaRgsin(θ) + 2aRb1gcos(ψ)cos(θ) /(2(a1b2 + a2b1 + aRb1 + aRb2)),
24 FzR = mt 2Zcoga1v˙y − 2Zcoga2v˙y − ZRa1v˙y + ZRa2v˙y + 2Zcogv˙xb1 + 2Zcogv˙xb2−
ZRv˙xb1 − ZRv˙xb2 + 2Zcogb1gsin(ψ) + 2Zcogb2gsin(ψ) − ZRb1gsin(ψ) − ZRb2gsin(ψ)+ (90) 2Zcoga1gsin(θ) − 2Zcoga2gsin(θ) − ZRa1gsin(θ) + ZRa2gsin(θ) + 2a1b2gcos(ψ)cos(θ)+ 2a2b1gcos(ψ)cos(θ) /(2(a1b2 + a2b1 + aRb1 + aRb2)),
Fz3 = 2FzRb4 + mtZRv˙y + 2Zav˙yma + mtZRgsin(θ)+ (91) 2b4gmacos(θ) + 2Zagmasin(θ) /(2(b3 + b4)) and
Fz4 = − mtZRv˙y − 2FzRb3 + 2Zav˙yma + mtZRgsin(θ)− (92) 2b3gmacos(θ) + 2Zagmasin(θ) /(2(b3 + b4)).
2.4.5 Roll-over If the machine is, for example, accelerating to rapidly when turning or turning sharply when driving along a step banking surface, the machine risks tipping over on its side, this is known as roll-over. A first step in actively preventing roll-over is to observe the vertical force on each wheel and either warn the driver or activate limitations on the machine behavior when the vertical force on any of the tires approaches zero. For roll-over this means that the shift of the weight from the left to the right side or the reverse is the cause. By looking at the moment around the x-axis in Eq (80) the main contributor of the change in the distribution of load is the roll-angle and the lateral acceleration. If sensors can accurately measure those values, it should then be possible to measure when the weight approaches zero. But, as it would update in real-time a counter measure would need to start before the estimator shows the roll-over point, since then it would already be to late.
If the roll-over could instead be predicted it would be possible to much more safely operate the machine in difficult environments. Also, in the cases where a sudden movement causes the roll-over the risk of the estimator registering a roll-over too late is also diminished. Due to the 3-point load distribution configuration of today’s machines it should be sufficient to look at the front wheels as they should be the ones that lifts first, with the rest of the machine tipping after.
A simplified model for looking at this could be to take Eq (88) and (89) that shows how the tires vertical forces are calculating and simplify it to only shown the effects of the lateral acceler- ation and roll-angle. These equations have the potential to give a good way to predict how the likely a roll-over case is to happen with regards to a change in roll-angle or lateral acceleration. But the best way would still be to use the full equations if possible.
2.4.6 Pile Entry A special case for the load distribution is the load entry case. When the bucket is filled the center of gravity will shift towards the front and more closely follow the movement of the front body. However while the wheel loader bucket is entering a pile of material to fill the bucket, another kind of load distribution happens. As the machine forces the bucket into the pile it will move material in-front of the bucket while simultaneously pushing down on the material under it. Eventually the bucket can no longer push the material away from its path and the pile will act as a new contact point of the wheel loader and as such take some of load, mainly from the front. In this case the load is originally distributed between the four wheels but when pushing on the pile it will instead be distribution over 5 points. Because the hydraulics of the machine are so strong the machine might actually lift the front or rear wheels slightly from the ground.
This can be modeled by having the force on the bucket as negative. In the wheel loader the lifting arm has a hydraulic pressure sensor, it should therefore be possible to screen for a decrease in the pressure value as an indication that the machine is being lifted when pushing into by the pile.
25 2.5 Force distribution: Acceleration from wheel sensors The acceleration needs to be measured in real-time in order to be useful for the normal force estimator. If there is no sensor on the machine that can reliably record the acceleration, it must be estimated. On the wheel loaders there are currently vehicle velocity estimators that utilizes the cardan shaft speed and the current gear ratio and gives a reasonably reliable reading. The velocity can be derived in real time making it an excellent source for the acceleration. The vehicle velocity is the same for the front and rear axle as they are connected with a stiff axle. This poses a problem when the machine is turning, in which direction is the velocity vector pointing? Figure 26 shows such a situation.
Figure 26: Velocity vector for front and rear wheels while turning.
As can be seen above, the velocity vector of the front and rear tires points in different direction. To find the machine direction the direction at the articulation point needs to be found. To help with this the trigonometric relation of the machine while turning is explored. The wheel loaders articulated turning behavior enables two lines following each axle to intersect as seen in Figure 27 where the machine turning radius is illustrated.
Figure 27: Turning radius of vehicle.
26 As can be seen in Figure 26 the velocity vectors are tangent to circular path which makes them normal their respective axle intersection lines. This is only strictly true in the static case where there is no slip, but since that information is unavailable it is assumed that this holds true at all times. The angle of the complete machines velocity vector will be the angle between the tangent of the circular path at the origin and the x-axle. This angle will be the same as between one of the axles and the turning radius R. Because the articulation is in the middle of the machine the two triangles created from the axle intersection lines and the turning radius are symmetrical. Utilizing that the articulation angle is known and that the front and rear bodies are orthogonal to their respective intersecting axle lines, some trigonometrical relations can be used as shown in Figure 28.
Figure 28: Trigonometrical relations of a turning vehicle.
The blue lines are only a visual help. Since the upper body and triangle 1 shares a side and are both right triangles, the top corner of triangle one must have the same angle, δ. Because the upper corner of the big triangle is 90◦ the upper corner of triangle 2 must be 90 − δ and as triangle two is also a right triangle the left corner has the angle δ. This means that the velocity vector has an angle of δ/2 with regards to the x-axis. This means that the longitudinal and lateral acceleration are
δ v˙ =v ˙ cos , x 2 δ v˙ =v ˙ sin . y 2
2.6 Force distribution: Turning torque For control purposes it is important to estimate the turning moment, which is the moment along the z-axis that tries to turn the machine. This can be useful in cases where one tire start slipping and the other tires increase their force outputs to compensate. If all tires do not increase the same amount there will be a turning moment that tries to shift where the machine is going, this can be compensated for if the effect of the turning moment from each tire is known. To do this the machine is modeled as a stiff body as shown in Figure 29.
27 Figure 29: Vehicle seen from above.
As the front tire forces are not aligned with the coordinate system they have to be converted using vector calculations as illustrated in Figure 30
Figure 30: Forces acting on the front left tire split into vector components.
The front tire forces then becomes, in the local coordinate system denoted with ◦
◦ Fx1 = Fx1sin(90 − δ) − Fy1sin(90 − δ), ◦ Fy1 = Fx1cos(90 − δ) + Fy1cos(90 − δ), ◦ (93) Fx2 = Fx2sin(90 − δ) − Fy2sin(90 − δ), ◦ Fy2 = Fx2cos(90 − δ) + Fy2cos(90 − δ). Using the center of gravity model from ??, the turning moment then simply becomes
X ◦ ◦ ◦ ◦ ◦ ◦ Mz = −Fx1b1 + Fx2b2 − Fx3b3 + Fx4b4 − Fy1a1 − Fy2a2 + Fy3a3 + Fy4a4 (94) ◦ ◦ where b3 and b3 are the lateral distances to the main body center of gravity as that is where the forces are acting and they are calculated as following
◦ b3 = |Ycog − wp3(2)| , ◦ (95) b4 = |Ycog − wp4(2)| . Since the rear axle mass is so much smaller than the main body, it will be disregarded here and unlike for the vertical forces the fact that the rear axle can move up and down independent of
28 the rear body will not affect the turning moment. Neither will the angle at which the machine is turning since the maximum turning angle is less than 90 degrees, making the forces always trying to turn the machine in the same direction regardless. With theses points in mind, the above equations should suffice to calculate the turning moment of the machine.
29 3 Method 3.1 Tire model: Test vehicle For all the measurements that were conducted, a wheel loader of type L-60 was used which is shown in Figure 31. Data specification regarding the wheel loader is presented in Table 2.
Table 2: Presents numerical values for the L-60. Parameter Description Value
mc curbed weight 11842 kg mb bucket load 3204 kg mg gross weight 15046 kg Vmax max speed 35 km/h
Figure 31: Volvos L-60.
3.2 Field measurments To capture the slip and force relationship, a series of field measurements were conducted by Volvo CE in 2015. The conducted measurements were not performed as a part of the thesis work, instead the raw data were given for evaluation and processing. This process is described in this section. The data processing is presented in Appendix A.
3.2.1 Tire used for measurements The tire that was used during the measurements was a Bridgestone tire, model VJT with tire dimensions 20.5R25. Table 3 summarizes data of relevance for the used tire.
Table 3: Presents parameters and their numerical values for the used tire. Parameter Description Value r Overall tire radius 0.740 m re Effective rolling radius 0.708 m w Overall tire width 0.530 m mt Mass of tire 426 kg pf Inflation pressure front tire 3.5 bar pb Inflation pressure back tire 2.5 bar
3.2.2 Measurement systems Two different measurement systems were used to sample data during the measurements. One system measured the vehicle externally, the Vicon system, while one system measured the vehicle internally, the Somat system.
Vicon system In order to get the kinematic description of the vehicle a motion sensing system called Vicon was used [2]. It consisted of a set of cameras which captured the motion of a point through a set of markers which were attached to the body of the wheel loader in a irregular pattern. For the
30 conducted measurements, four points on the right side of the vehicle were studied. One located on to the back wheel (bw), one at the back part of the chassis (ba), one at the front wheel (fw) and the last one at the front part of the chassis (fa). The Vicon system captured the information of all degrees of freedom for the points in a global coordinate system. Thus all the translational degrees of freedom Xvicon ,Yvicon and Zvicon with its corresponding rotational degrees of freedom Uvicon ,Vvicon and Wvicon were captured. During the measurement the cameras were located along a arc in order to measure the depth correctly. A conceptual overview of the setup during the measurements is presented in Figure 32. As seen in the figure, the kinematic information of the markers were only given within the scope of the Vicon system. An example of how the markers were placed in a irregular pattern is presented in Figure 33. For some measurements, the ground was measured to capture the vertical height of the markers in relation to the ground.
Figure 32: Shows the Vicon setup for one of measurements.
Somat system To measure the resulting forces and moments that were introduced when slip was present, trans- ducers with model name SWIFT 50 GLP S were mounted onto the wheels right front and rear hubs, bw and fw. The transducer contained a digital encoder, which measured the angular displacement of the wheel, vsomat, with 2048 points per revolution. As oppose to the Vicon system, the measured quantities were measured in the local coordinate system of the tire. Stationary vertical rods were placed outside of the wheels as a point of reference for the hub transducers coordinate systems. The transducer is shown in Figure 33 together with the Vicon markers for the wheel.
Figure 33: The image shows the SWIFT transducer together with highlighted Vicon markers mounted on the wheel.
The signal from the transducers were sampled by a data acquisition device called Somat. In addition to the signals from the hub transducers, real time data, such as vehicle speed and brake
31 pressure, were also sampled. All data sampled internally from the Somat system will from now on be termed as Somat data.
Overview A summary of all the most important measured variables are summarized in Table 4.
Table 4: Presents measured variables together with a brief description and what sampeling fre- quency was used. Variable Description Location Device Unit Frequency
Xvicon Global x position ba, bw, fa, fw Vicon m 200 Hz Yvicon Global y position ba, bw, fa, fw Vicon m 200 Hz Zvicon Global z position ba, bw, fa, fw Vicon m 200 Hz Uvicon Global roll angle ba, bw, fa, fw Vicon rad 200 Hz Vvicon Global pitch angle ba, bw, fa, fw Vicon rad 200 Hz Wvicon Global yaw angle ba, bw, fa, fw Vicon rad 200 Hz vsomat Local pitch angle fw, bw Somat rad 500 Hz Fx Local longitudinal force fw, bw Somat N 500 Hz Fy Local lateral force fw, bw Somat N 500 Hz Fz Local load fw, bw Somat N 500 Hz
3.2.3 Experiments The measurements that were conducted were standard field tests, where the wheel loader was measured in typical working situations. Measurements on the tires structural properties were also performed where the tires vertical stiffness and contact patch behavior were studied.
The general measurement procedure was performed by positioning the vehicle outside of the scope of the Vicon system. The vehicle then entered the scope, performed the action specified by the measurement protocol, and then left the scope. One such entering and leaving defines an event during the measurement and it’s the time when the Somat and Vicon system measure together. Generally one measurement consisted of several events. The reason for making a definition of an event was to extract the correct data sets from the Somat and the Vicon data, since they measured independently of each other.
The field tests were conducted on different surface types. These were concrete, asphalt and gravel. The friction coefficients that were assumed for these surfaces during the thesis work were given from [26] and are presented in Table 5.
Table 5: Presents the used friction coefficients given from [26]. Variable Description Numerical value
µa Asphalt 0.75 µg Gravel 0.6 µc Concrete 0.75
Since the friction coefficient for asphalt and concrete are the same in the table, they will for sim- plification be assumed to be of the same surface type.
Additional field tests were performed in the thesis work, which mimicked the older tests. This was done in order to expand the data set.
3.2.4 Complementary field test data In order to increase the amount of data available for model creation, newer, complementary data was provided by Volvo’s test engineers. Theses measurements utilized the SWIFT 50 GLP S system to record forces and moments, but did not use the Vicon system. However, in addition to force measurements the machine was equipped with a GPS as well as accelerometers and gyroscopes capable of measuring in all directions.
The GPS-signal was only sampled at 5 hertz, making the data unreliable for speed measurement,
32 but could be used to describe the driving behavior during a test. The accelerometers where at this time uncalibrated. Uncalibrated accelerometers have a tendency to drift in their measurements due to the effect of gravity, as such they should not be used to represent the acceleration of the machine. The gyroscopes were able measure the changes in angles of the machine, which meant that the measurements contained information about how the machine tilts between the right and left side as well as between the front and back while driving.
The measurements were all field test performed in similar fashion as the 2015 measurements. The data did not need to be processed as both the force and moment sensor and the gyroscopes logged their data in the storage unit, which mean they where already synchronized.
3.3 Tire model: Test rig measurements This section addresses the measurements that were performed during the thesis work in a estab- lished test rig, termed as a chassis dynamometer. The measurements had the aim to study the longitudinal stationary and transient behavior of the tire.
3.3.1 Tire used for measurements The tire that was used during the measurements was a Michelin tire, model XHA2 with tire dimensions 20.5R25. Table 6 summarizes data of relevance for the used tire.
Table 6: Presents parameters and their numerical values for the used tire. Parameter Description Value
rf Free tire radius 0.743 m w Overall tire width 0.528 m pf Inflation pressure front tire 3.5 bar pb Inflation pressure back tire 2.5 bar
3.3.2 Main setup The main setup consisted of a test vehicle (same as described in section 3.1) which was fixed in position. The front wheels were placed on a cylinder which acted as the ground. The velocity of the cylinder, Vcyl, was controllable. The rear part of the vehicle was lifted such that the rear wheels hanged freely. An overview of the installation is illustrated in Figure 34. The load on the front axle was adjustable by varying the pressure in a piston, pc, which was located underneath the vehicle and attached to the front axle. To fix the front axle in vertical position, a nominal pressure of pc=1 bar was applied. Furthermore, a hydraulic jack was placed underneath the center of the front axle and adjusted vertically such that it just barely touched the front axle. With this configuration, the front axle was modeled as being rigidly fixed.
33 Figure 34: The two upper images gives an overview of the test rig. The lower left image shows the piston that was used to vary the load. The lower right image shows how the front axle was locked vertically.
The throttle speed could be adjusted through a remotely controlled gas pedal. The installation allowed to send ramp signals, which were sent from a control central. For all acceleration mea- surements the vehicle was put in first gear. The rotation of the wheels could further be adjusted by applying the brakes. This was however conducted manually by a test driver. The rotation of the wheel was measured through the rotation of the cardan shaft, ncs. In the test rig, pre-existing transducers existed which measured the braking/tractive force, Fx, and the velocity of the cylinder, Vcyl. The setup is shown in Figure 35.
Figure 35: The left image shows the mounting that was used for the gas pedal. The right image gives a closer view on the cylinder and the tire.
All signals were sampled in a DAQ-box. A summary of the sampled signals are presented in Table 7.
Table 7: Presents measured variables together with a brief description and what sampeling fre- quency was used. Variable Description Unit Frequency
Vcyl Tangential speed of cylinder m/s 500 Hz Fx Tangential force applied to cylinder N 500 Hz ncs Revolution of cardan shaft rps 500 Hz
34 Estimation of load As was mentioned, the load on the wheels could be varied by changing the pressure in the piston. However, there was no way to directly measure the load that was applied to the wheels during the measurements, why a rough calibration had to be done in order to estimate what load was applied to the wheels for a certain pressure level in the piston. This was done by measuring the vertical distance of the tires center, r, as the load of the piston was varied from pc=0 bar to pc=4 bar in steps of 1 bar. The measurement of the vertical distance was conducted manually. Next, the structural data of the tires vertical stiffness from the measurement conducted in 2015 was used (see Appendix C). Even though it was a different tire, it was assumed that the structural properties were similar enough. With the structural properties, the deflected tire radius and the vertical load applied to the wheels could be coupled. The different pressure levels in the piston together with what load they corresponded to are presented in Table 8.
Table 8: Measured variables, pc, and calibrated load Fz. pc [bar] Fz [kN] 1 bar 34 2 bar 38 3 bar 44 4 bar 48
3.3.3 Pure longitudinal force characteristics To obtain the longitudinal tire characteristics, both acceleration and braking measurements were conducted to cover both the positive and negative slip domain. The acceleration measurements were conducted by fixing the cylinder velocity to Vcyl=1 km/h. The vehicle was then put in first gear and ramped from zero to full throttle speed in a time interval of 10 s. After full throttle speed had been established, the vehicle was de-accelerated from full to zero throttle speed. The ramp signals were sent from the control central as predefined signals to the installation for the gas pedal, this is illustrated in Figure 34. The acceleration measurements were suppose to capture the positive slip characteristics (κ > 0 and Fx > 0).
The braking measurements were performed by setting the cylinder velocity to Vcyl=5 km/h. The vehicle was put in idle mode. The brakes were then applied manually by a test driver, who ap- plied the brake pedal from zero to fully applied. The brake pedal was then let go. The braking measurements were conducted to capture the negative slip characteristics (κ < 0 and Fx < 0).
Both the accelerating and braking measurement were repeated a certain number of times to study the accuracy in the measurements. They were also performed for different loads and different levels of inflation pressures. The procedures for these measurements are discussed next.
The experiments which studied the load dependency of the tire were performed by varying the applied pressure in the piston underneath the vehicle (illustrated in Figure 35). The different pressure levels that were applied are presented in Table 9. For all these measurements, a nominal inflation pressure of pi=3.5 bar was used.
Table 9: Lists the different loads that were applied. Variable Value
pc1 1 bar pc2 2 bar pc3 3 bar pc4 4 bar
When the dependency of the tires inflation pressure was studied, a nominal load of pc=1 bar was applied. Two different inflation pressures apart from the nominal inflation pressure were studied, which are presented in Table 10.
35 Table 10: Lists the different inflation pressures that were used. Variable Value
pi1 2 bar pi2 5 bar
3.3.4 Relaxation length The engine of the vehicle was set to 1200 rpm and put in first gear. The velocity of the cylinder was initially set to Vcyl=5 km/h. The velocity of the cylinder was then stepped down in steps of 1 km/h, until it reached Vcyl=1 km/h from which it was stepped up to Vcyl=5 km/h again. The measurements were conducted with pressure levels of pc=1 bar (nominal condition) and pc=3 bar. The reason for this was to study any load dependency of the tires relaxation length.
3.3.5 Validation In order to validate the resulting tire models, a validation measurement was conducted. The cylinder attained a constant velocity of Vcyl=2.3 km/h and the velocity of the wheel was increased in six even steps, from zero to full throttle speed. The measurements were conducted under nominal conditions. The gas pedal was in these measurements applied manually by a test driver.
3.4 Tire model: Optimization routine for parameter identification of tire models For the parameter identification process of the measurement data from 2015, MATLABs optimiza- tion function fmincon(·) was used. The default settings were used for the function. The function constrains the search space, by setting a lower and upper bound on each parameter that is esti- mated.
For the parameter identification of the measurement data conducted in 2017, it was also decided to use the differential evolution (DE) algorithm in order to compare the different results and to rule out the possibility of obtaining a local minimum in the solution. A open source function, found in Mathworks file exchange library [27], was used for the optimization. The same constraints that were used for the fmincon(·) function was also used for the DE. The default settings had to be changed and these were • Number of population members: 100.
• Max iterations: 100. • Max time: 3600 s. Both optimization functions needed a cost function in order to perform the procedure. The sum square error (SSE) was used for this purpose and has following general appearance
n X i i 2 J(P1,P2, ..., Pm) = (Ymodel(P1,P2, ..., Pm, xi) − Ymeasured(xi)) . (96) i=1
i In the above equation, Ymodel, is the output from a general tire model for a certain general input i xi. Ymeasured(xi) is the corresponding measured output. The model consists of a certain number of parameters, P1,P2, ..., Pm, which are supposed to be estimated by the optimization routine. The objective therefore becomes to find the set of parameters which gives the smallest difference from the observed values.
3.5 Tire model: Parameter and model identification of field measure- ments This section presents the procedure that was performed to estimate the parameters in the MF and the brush model. The dimensions of the tires that were used during the calculations are presented in Table 3 and the nominal conditions are defined by Table 11. All parameters and their corresponding bounds that were used for the different tire models are presented in Appendix D.
36 Table 11: Presents the nominal parameters and their numerical values. Variable Description Numerical value
Fzo Nominal load 27 kN pio Nominal inflation pressure 3.5 bar
3.5.1 Reference tire model A set of MF-parameters from a FEM model were given. The resulting tire model will from now on be termed as the FEM model. The FEM model was modeled for a tire dimension of 26.5R25. Further assumptions and specifications that were given were as follows • A friction coefficient of 0.6 was used. • The camber angel was neglected.
• The used inflation pressure of the model was 4 bar. • The loads which the FEM model is valid within is from 30 kN to 200 kN. • The tire was mounted on a 25x22/3” rim.
• The MF is valid for longitudinal and lateral slip values between [–15%,+15%] As can be noted from the list of specifications, the FEM model deviates from the circumstances of the measurements. However, the FEM model will be used as a reference tire model. It’s therefore assumed that any difference between a FEM model of the tire used in the measurements and the given FEM model can be regarded as negligible.
The MF parameters from the FEM model will further be assumed to be valid for different friction coefficients and inflation pressures. It will further be assumed to be valid outside of its specified slip interval.
3.5.2 Magic Formula parameters This section presents the procedure that was conducted to identify and calibrate the MF parameters for pure longitudinal slip. The set of parameters that were estimated from the MF are described in section 2.2.1.
Parameter identification of gravel surface measurements The data for the gravel surface measurements only contained small slip values, why only the longitudinal stiffness of the MF was calibrated. The equation with its corresponding parameters given from [3], is presented without its pressure dependency
2 Kxκ = Fz(PKx1 + PKx2dfz ) exp(PKx3dfz). (97)
Since the longitudinal stiffness assumes small slip values, all data points inside of the interval [−5%, 5%] were deemed to be within the linear region. A least square fit of a linear curve passing through the origin was then conducted to the longitudinal forces and slip pairs for the different loads. The slope of the linear line gave the stiffness for that corresponding load and a relationship of the longitudinal slip stiffness, CF κ, as a function of load Fz was given. Due to the lack of non-linear information from the measurements, it was decided to use the non-linear information given by the FEM model. This was done by tuning the MF such that it by appearance resembled the FEM model as much as possible.
Parameter identification of asphalt surface measurements In contrast to the gravel measurements, the asphalt measurements contained non-linear information why more parameters in the MF could be identified. The procedure that was conducted was first to fit the nominal parameters in the MF for the nominal load case. When the nominal parameters had been identified, the parameters which gives MF its load dependency were identified. The parameters that were estimated and their corresponding bounds are presented in Appendix D.
37 3.5.3 Brush model This section presents the procedure that was conducted to identify the parameters for the brush model, Eq (32), from the measurement data. The section starts with presenting the procedure for the gravel measurements and then continuous with the asphalt data.
Gravel The data for the gravel measurements missed non-linear data, why a complete parameter identifi- cation wasn’t possible. This limited the optimization to only identify the longitudinal slip stiffness CF κ from the data. A friction coefficient of µg = 0.6 was assumed (see Table 5) and the calibrated empirical formula for the contact patch length, ac, was used, see Appendix C.
Asphalt Since the data for the asphalt case contained larger slip values, a parameter identification of both longitudinal slip stiffness, CF κ, and friction coefficient µa was performed. As for the gravel surface data, the calibrated empirical formula for the contact patch length, ac, was used, see Appendix C.
3.6 Tire model: Data processing of test rig measurements This section presents the data processing that was conducted for the test rig measurements, and discussed in section 3.3. Before any signals were processed a Butterworth low pass filter was ap- plied to all the signals with a cut off frequency of 5 Hz.
In order to calculate the slip of the tire, the measured cardan shaft signal needed to be scaled. This was performed by assuming that the slip was zero when the measurements started. The cardan shaft signal was then scaled with the corresponding scale factor, Kscale, such that the difference between the scaled cardan shaft and the measured velocity of the cylinder was zero in the beginning of the measurements. By scaling the cardan shaft signal with the corresponding scale factor, the linear speed of rolling was obtained Vr = ncs · Kscale. The longitudinal slip was then calculated by the definition in Eq (5), which was
−(V − V ) κ = cyl r . (98) Vcyl
3.6.1 Processing of force characteristics measurements From the force signals, the intervals where the ramp began and ended were manually identified and extracted from the data set. The data were sorted based on what load and inflation pressure was used.
3.6.2 Processing of relaxation length measurements The intervals were the step inputs occurred were identified and extracted from the data set. The scaled velocity of the wheel, Kscalencs, was next numerically integrated by using the MATLAB function cumtrapz(·), which gave the travelled distance of the wheel, x. Next the following model was fitted to the data
− x Fx = Fxss(1 − e σ ) (99) where σ is the relaxation length and Fxss is the steady state force. A least square fit of the measured step response in force to Eq (99) was then conducted by using the MATLAB function fminunc(·), where the relaxation length, σ was estimated. The steady state force, Fxss, was taken from the corresponding step response. The same cost function as described in section 3.4 was used. An example of how the step input in slip and step response in force could look like from the measurements is shown in Figure 36.
38 Figure 36: Gives an example of a resulting step input in slip from the measurements and a corre- sponding step response in force.
As seen in the figure, the resulting step input in slip is not ideal which of course affects the corresponding step response in force. The approximation of the step response as a first order system therefore becomes rather rough, but it was deemed to be good enough in order to get a rough estimate of the relaxation length. In order to couple the relaxation length, σ, with a corresponding slip, it was decided to take a mean value of the slip before and after the step input had settled to a steady state value.
3.7 Tire model: Parameter and model identification of test rig measure- ments This section presents how the parameter and model identification of the data collected from the measurements in 2017 was performed. Table 12 defines the nominal load and inflation pressure for the process.
Table 12: Lists the nominal load and inflation pressure. Variable Value
Fzo 34 kN pio 3.5 bar
When the parameter identification was performed, some of the data points from the measurements were excluded, which were data points that clearly showed a different behavior compared to the other data.
The parameters in the MF were estimated in different steps in order to break down the pro- cess. When the parameters had been estimated in one step, they were used in the subsequent step. The order in which they were estimated are presented in following list. The parameters that belongs to the different steps are presented in Appendix E with their corresponding bounds. • Nominal load and inflation pressure. • Varying load and nominal inflation pressure. • Nominal load and varying inflation pressure.
3.8 Tire model: Model validation of test rig measurements In order to validate the resulting tire model given from the test rig measurements, a simulation was performed in Simulink. A Simulink block (MF-Tyre [28]) that implements the MF was used as a tire model together with a implementation of the SCP-model, described in section 2.2.4. Both models used the same MF that resulted from the measurements. For comparison, the MF given by the FEM model was also implemented in the MF-Tyre block. Its friction parameter in the MF was adjusted such that it had the same friction parameter as that given from the measurements. The relaxation length used for the SCP-model, was taken from the measurements where the relax- ation length was studied. The relaxation length that occurred at the lowest level of slip was chosen.
The models were validated by using the validation measurement discussed in section 3.3. The models used the cylinder velocity, Vcyl, and the scaled cardan shaft signal, Vr, as inputs. The rota- tional speed of the tire was calculated by dividing the scaled cardan shaft signal with the effective
39 rolling radius Ω = Vr/re, where the calculated effective rolling radius from the field measurements were used (see Appendix A).
40 4 Results 4.1 Tire model: Processed field measurements The data obtained after processing the field data is located in Appendix B and the tires structural properties are presented in greater detail in Appendix C. The observations that were made will be summarized in this section.
The resulting data from the measurements gave only longitudinal slip, since the lateral and com- bined slip values had to be neglected. The main reason for neglecting these cases were due to that the rotational velocities from the different systems, deviated significantly from each other. This limited the model and parameter identification to only the longitudinal case.
The longitudinal slip measurements further failed to capture the whole force characteristics and the data was relatively small in size. Only small slip values within the linear region were given from the gravel surface measurements. The asphalt surface measurements resulted in small and large longitudinal slip values, but no data in between were given.
The given slip values were clustered within a reasonable domain and range, but were spread apart from each other within the clusters. Thus the accuracy in the data was low. It was also observed that the slip quantities that were formed from the different systems, Vicon and Somat, de- viated from each other. Thus no strong confirmation between the two different systems were given.
When the processed data was compared to the reference model, it showed a less stiffer behav- ior.
4.2 Tire model: Parameter and model identification from field measure- ments This section presents the results from the model and parameter identification procedure as de- scribed in section 3.5. The field measurement data is only used in this section. The first section presents the MF model given from the data. Section 4.2.2 presents the resulting brush model. The structural properties of the tire from the measurements are presented in Appendix C.
4.2.1 Magic Formula parameters This section presents the results of the parameter identification of the MF-parameters. The section starts with presenting the results from the data measured for a gravel surface. It then continuous with the data from the asphalt surface.
Gravel The resulting longitudinal stiffness as a function of load is presented in Figure 37 together with the resulting best fitted curve. If the data points are analyzed, a distinct outlier in the data can be identified at a load of Fz ≈ 50 kN. The resulting curve seems to give a fair description to data points.
Figure 37: Graph of the best fit to the longitudinal stiffness given from the measurement data. The surface type is gravel.
41 The resulting MF-curve with the stiffness parameters given from the optimization, and the resulting parameters given after manual tuning by using the FEM model, is presented for different loads in Figure 38. The FEM-model is also added for comparison. What’s observed is that, at nominal load, there is a large difference in stiffness. However with increasing load, the difference in stiffness seems to decrease. The stiffness of the calibrated curve increases faster compared to the FEM. A reason that a lower stiffness was given, could be due to that the gravel ground deforms.
Figure 38: A comparison between the longitudinal MF given by the FEM (solid line) and calibrated MF (dashed line) for different loads. Gravel surface.
In Figure 39, the resulting MF is presented together with the measurement data. By observing g g the results, it’s seen that the MF for Fz1 and Fz1 seems to fit the data fairly well. This is not the g g case for load Fz3, but the data points for that case are rather scattered. The MF for Fz4 is stiffer compared to the measurement data. Altogether, the quality of the fitted MF is rather low. This is due to that the measurement data are so scattered and that to little information about the whole slip range is given.
42 Figure 39: The resulting MF curves for all load cases together with the measurement points. The surface type is gravel.
Asphalt The MF with the resulting parameters from the optimization, is plotted together with the mea- surement data for all loads in Figure 40 for all the different load cases.
43 Figure 40: The resulting MF curves for all load cases together with the measurement points. The surface type is asphalt.
Even tough the data is rather spread out, the resulting MF seems to give a good general fit to the data points. However, it’s impossible to comment on the main shape of the MF since a lot of data in between the data clusters for small and large slip values are missing.
4.2.2 Brush model This section presents the result from the model identification of the brush model. The section starts with the results from the gravel data and then continuous with the results from the asphalt data.
Gravel The resulting brush model together with all the measurement data is presented in Figure 41.
44 Figure 41: Optimal brush model together with measurement data for different loads. The surface type is gravel.
As can be observed, the overall fitting is rather poor. A reason for this is that the quality of the data is low but also that the brush model is to simple.
Asphalt The brush model together with all the measurement data is presented in Figure 42.
45 Figure 42: Optimal brush model together with measurement data for the different loads. The surface type is asphalt.
By observing the results presented in the plots, it’s seen that the non-linear behavior seems to be handled in a good way even though a large spread in the data points are given. Concerning the a a fitting for small slip values (κ < 0.1) the quality is mixed. For lower loads, Fz1 and Fz2, the brush model seems to give a stiffer behavior compared to the measurement data. In contrast to this, the a behavior seems to be reversed for the highest load, Fz5. The reason for this behavior could be due to that the optimization was conducted over all loads which results in a trade off in the resulting stiffness.
4.2.3 Comparison of tire models The MF and the brush model for the gravel and asphalt surfaces are presented in Figure 43.
46 Figure 43: The resulting MF and brush model from the parameter identification. Left plot presents the curves from the gravel surface data and the right plot presents the curves from the asphalt surface data.
If the curves for the gravel surface are observed, it’s seen that the brush model gives a stiffer behavior compared to the MF. Despite this, the differences between the curves are rather small. When the slip grows, the brush model saturates and settles at a constant value. The MF attains the same peak force but instead of settling at this value it diminishes slowly in magnitude with increasing slip. The MF and the brush model attains their peak force at approximately the same slip value (κ ≈ 0.3). By the observations made from the previous results, the MF seems to give a better fitting compared the brush model.
In contrast to the curves for the gravel surface, the fitted curves for the asphalt data shows a large difference in behavior. The brush model is still stiffer compared to the MF, but the difference is much more distinct. It further attains its peak force at a much lower slip value (κ ≈ 0.07) than the MF (κ ≈ 0.4). The MF attains a higher peak force, but decays with increasing slip to the same longitudinal force as the brush model. This behavior is not possible to verify from the measurement data. By observing the results from the curve fitting in the previous section, the MF seems to give a much better fit compared to the brush model.
4.3 Tire model: Parameter and model identification from test rig mea- surements 4.3.1 Magic Formula parameters for nominal load and inflation pressure The resulting MF-curves from the different algorithms are presented in Figure 44. The measure- ment data show some minor spread, but the accuracy is overall deemed to be good. The resulting curves from the different algorithms match each other, which could point to that a global minimum is given in the estimation. The overall fit to the data is good. However, the curves seem to be off for the larger negative slip. This could be due to that the braking measurements gave a lower measured force compared to the acceleration cases, which the MF is not able to describe. Another reason could be that the data is biased to the acceleration data points (κ > 0), since it contains more data.
47 Figure 44: The resulting MF given by the measurement data for the nominal load.
4.3.2 Magic Formula parameters for variation of load and nominal inflation pressure
The MF-curves given from the measurement data when the load was varied, are presented in Figure 45-47 and will be commented one by one. Starting with Figure 45, it’s seen that resulting measurement data shows good repeatability compared to the nominal load measurements (Figure 44). There is no apparent difference between the MF-curves given from the different algorithms. The resulting MF-curves seems to give a good fit to the data points, but seems to be diverging slightly from the measurement points for larger slips, κ > 2.5. They further seem to fail to resolve the peaky part from the braking measurements.
Figure 45: The resulting MF curves from the measurements when a load of Fz=38 kN was applied.
Next the results in Figure 46 are analyzed. As can be seen in the figure, the accuracy in the measurements is somewhat reduced for both the braking (κ < 0) and accelerating (κ < 0) cases. No difference between the MF-curves can be seen. The resulting fitted curves seems to handle
48 the positive slip values in an intuitive way, by passing through the mean value of the data points. For larger slip values (κ > 2), the fitted curves and the data points deviate from each other. Concerning the negative slip values, the MF-curves seems to favor the measurements points that attain a larger braking force.
Figure 46: The resulting MF curves from the measurements when a load of Fz=44 kN was applied.
Figure 47 presents the results from when the highest load was applied to the tires. No difference in behavior is seen between the MF-curves. The deviating behavior for larger positive slips that has been observed for all measurements is also present for this case. The MF-curves seem to fail to fit the data in the curved region (−0.5 < κ < −0.2), but converge to the data afterwards.
Figure 47: The resulting MF curves from measurements when a load of Fz=48 kN was applied.
49 4.3.3 Comparison of resulting fitted curves for variation of loads An overview of all the resulting fitted curves with the parameters given from the fmincon(·) function are presented in Figure 48. As can be seen from the figure, with increasing load the tractive/braking force increases which is an expected behavior.
Figure 48: The MF-curves given by the parameters from fmincon(·) when the load is varied.
50 4.3.4 Magic Formula parameters for nominal load and variation of inflation pressure
The resulting MF-curves are plotted in Figure 49 together with the measurement data from when an inflation pressure of pi=2 bar was used. By observing the figure, it’s seen that there is a minor issue in accuracy for positive slip values, while the negative slip values show good repeatability. No differences between the MF curves are seen. They further seem to give a good fit to the data.
Figure 49: The resulting MF curves given by the measurement data when a inflation pressure of pi=2 bar was used.
In contrast to the previous results, the measurements where an inflation pressure of pi=5 bar was used (Figure 50) doesn’t show the same quality in terms of accuracy and curve fitting. If the braking measurements are addressed first, it’s seen that they give a large variation in what braking force was measured. A reason why this result is given could be that the brake pressure had to be applied manually and could therefore not be controlled in the same manner as the acceleration case. However, this large variation in brake force hasn’t been observed in any other measurement, why the result is unexpected.
The resulting fits from the MF-curves are not ideal. The curves for the accelerating cases fit the upper bound of the measurement points. They further doesn’t seem to resolve the peaky region in the non-linear region (0.1 < κ < 0.4) and they also deviate from the measurement data for larger slips, κ > 2.7. The same failure in resolving the peaky region for the braking cases is also seen.
51 Figure 50: The resulting MF curves given by the measurement data when a inflation pressure of pi=5 bar was used.
4.3.5 Comparison of resulting fitted curves for variation of inflation pressures To give a feeling of what behavior has been captured from the measurements, the MF-curves given from the fmincon(·) parameters with the same inflation pressures used in the measurements, are presented in Figure 51. Despite the poor quality of the pi=5 bar measurements, a intuitive behavior is given, where the highest tractive/braking force is given with lowest inflation pressure.
Figure 51: The MF-curves given by fmincon(·) for different inflation pressures.
52 4.3.6 Relaxation length The resulting relaxation lengths that were calculated for the two different loads are presented in Figure 52. As can be seen from the plot, the relaxation length decreases with increasing slip, which is an expected outcome by [29]. There seems to be a slight dependency of load in the relaxation length. The relaxation length seems to be higher for a larger load when the slip is small, but as the slip grows the relaxation length decreases more rapidly for the higher load compared to the lower load.
Figure 52: The measured relaxation lengths for different loads.
4.4 Tire model: Model validation of models from test rig measurements
The results when the MF-Tyre model and the single contact point model (SCP) were simulated with the validation measurement as input, are presented in Figure 53. The FEM model is also added for reference. The set of parameters for the MF-curve was taken from the results of the fmincon(·).
Figure 53: The slip input is shown in the left plot. The resulting responses in force from the different models are shown in the right plot.
The left plot of the figure shows the slip input to the models, and the right plot shows the resulting force responses to the slip input. By observing the responses from the models, it’s seen that the calibrated models (MF-Tyre and SCP model) differ little from each other. This is reasonable, since they both use the same MF-curve. However, there is a distinct difference in behavior. The MF- Tyre gives a more unsteady and oscillating behavior compared to the SCP-model, which attains
53 a more filtered and smoothed behavior. The reason for this difference is believed to be due the relaxation length. The SCP-model uses the relaxation length given by the measurements. The relaxation length of the MF-Tyre model can’t be defined as an input in the software, instead the software defines the relaxation length to be ten times smaller than what was measured, which is a rather large difference. The FEM model shows a big difference in magnitude compared to the calibrated models and to the experimental data.
If now instead the simulated results from the calibrated models, are compared to the experi- mental data, it’s seen that they correspond well with each other for low slip values (κ < 0.04). When the slip increases, a distinct difference in steady state forces are given, which points to that the calibrated MF-curve doesn’t accurately describe the steady state response of the tire. This difference increases with increasing slip, but when a slip of κ ≈ 0.25 is reached, this difference de- creases. Thus the accuracy of the MF-curve is better for larger slip values, κ > 0.25. When a slip is stepped, and a transient response is given, it’s seen from the results that both the SCP-model and the MF-Tyre model overshoot which is not observed at all in the experimental result. Fur- thermore, the MF-Tyre model is seen to overshoot much more compared to the SCP-model, which makes the SCP-model correspond better to the experimental observed value. This points to that the calculated relaxation length could be more reasonable despite it being larger in magnitude.
4.5 Force distribution: Normal force estimation results Here the results from using the normal force model to estimate the load distribution on the wheels will be presented using both simulated and measured values to see if it can reliably predict the machine behavior. The estimated values comes from Eq (88) - (92) in section 2.4.4. The parameters used to calculate the center of gravity model can be found in appendix ??.
4.5.1 Normal force estimator vs Simulation In an effort to check the validity of the model a drive case was simulated in Volvos existing simulation environment. Two test cases were simulated. One where the output torque of the wheels varies in the shape of a sinusoidal signal while the steering angle is ramped and then held mostly constant and one case where the torque is slowly increased while the machine articulates from side to side.
4.5.2 Transient torque case Figure 54 and 55 shows the steering angle and velocity for the drive case with the transient torque.
Figure 54: Steering angle vs time. Figure 55: Velocity vs time.
The result from both the simulation and the estimator are shown in Figure 56.
54 Figure 56: Vertical forces on the wheels in the transient torque case.
There is a clear offset for the front wheels. The offset seems to come when the machine articulates rapidly causing the estimator to give a faulty base value for the normal load. It does however follow the shape of the remainder of the curve. The rear tires are closer in amplitude to the simulated values but seem to have a slight phase difference.
4.5.3 Transient steering angle case Figures 57 and 58 show the steering angle and velocity for the drive case with the transient steering angle.
Figure 57: Steering angle vs time. Figure 58: Velocity vs time.
The result from both the simulation and the estimator in this case are shown in Figure 59.
55 Figure 59: Vertical forces on the wheels in the transient articulation angle case.
In this case there seems to be a somewhat steady offset of the amplitude on the estimator. When looking close the offset seems to be the same for all speeds indicating that it is a base offset, that shifts depending on steering direction. For the rear tire the estimator is much closer in amplitude to the simulated.
4.5.4 Using the old measurement data data Using the data logged by the Somat-system during the 2015 tests, the model was compared against a real life field test. There is no logged information regarding the pitch and roll angles and the acceleration is derived from the wheel position sensor and as such are not completely accurate. In these measurements only the front tires are driving as the rear cardan shaft had been disconnected.
Three different cases where compared. • A unloaded case shown in Figure 60. • A loaded case shown in Figure 63.
• A unloaded case with hard braking as shown in Figure 64. Each case is done at three different speeds and the forces are measured on the front right and rear right tires.
56 Figure 60: Vertical forces in the unloaded case.
The calculated values seem to have a somewhat constant offset from the signal while following the trend of the machine. The offset might be explained by the fact that the position of the lifting arm and bucket is unknown, which will affect the position of the center of gravity. The high frequency wave-form might be the driving-mode frequency or some other phenomena. When these calculations were performed there was no information regarding the changes in pitch or roll angle. Later models of the machine are assumed to have angle sensor and it is possible that they will be enough to cover the behavior displayed. To get a feeling for the the drive case the steering angle and velocity of the vehicle for the bottom graph are shown in Figure 61 and 62.
Figure 61: Steering angle vs time. Figure 62: Velocity vs time.
The steering angle plot shows that there are no hard turns and the velocity plot shows that the vehicle is moving both forwards and backwards. Next up is the loaded drive case shown in Figure 63.
57 Figure 63: Vertical forces in the loaded case.
In the loaded case the offset seem to have been reduced which could indicate the the set-up of the center of gravity is more correct this time. As in the unloaded case the rear tire corresponds better with the measurement, this could be because the rear tire had no torque applied to it so there are less dynamic effects acting on it. The final case is the brake case shown in Figure 64.
Figure 64: Vertical forces in the unloaded case with braking.
On the braking case the model does a poor job of catching the high spikes. Most likely this is because the acceleration is calculated from the wheel position. If the wheel locks them self, the model will interpret it as if the machine has stopped decelerating although that might not be the case. The spikes also move rapidly up and down, this could be because the machine starts rocking back and forth, if that is the case it could possible be covered by an angle sensor as well.
4.5.5 Using newer complimentary data Even tough no new field tests could be performed during this thesis there was some newer mea- surements had been done since 2015 where roll and pitch angles could be measured from installed gyroscopes. There was still however no way to accurately measure the acceleration. For a breaking and acceleration case Figure 65 and 66 shows the velocity from the wheel sensor and the articulation angle.
58 Figure 65: Steering angle vs time. Figure 66: Velocity vs time.
As can be seen there are hard acceleration and braking forces being applied to the machine. This can be seen in Figure 67.
Figure 67: Vertical forces on the tires during hard acceleration and braking.
The areas that the estimator are unable to match the measured values against, are most likely the result of the inaccurate measurement of the acceleration as they coincide with either abrupt acceleration or braking. Another interesting case is when there is a high articulation angle. The articulation angle and velocity for just such a case can be seen in Figure 68 and 69.
Figure 68: Articulation angle vs time. Figure 69: Velocity vs time.
For roughly half the measured time the machine will turn harshly and the effects of this can be seen in Figure 70.
59 Figure 70: Vertical forces on the tires during hard turning.
The estimator somewhat matches the behavior of the load transfer when turning but the accuracy drop off for such an extreme case. How important this miss-match is depends on how likely this event is to happen, but it does show that the model have cases where it works less well.
4.5.6 Roll over There are no measurements for the behavior of the machine for drive cases leading to roll-over. So instead the simulation environment was used. The machine was made to roll-over by forcing a very abrupt articulation angle change. The forces as functions of time are shown in Figure 71.
Figure 71: Forces on the tires during the time leading up to roll-over.
As can be seen it is the front left tire that lift from the ground first. The simulated value level out when it reaches zero. The estimator seems to roll-over after the actual event occurs, but not by much. How likely these drive cases are in the real world is unknown as there are no test cases. But with the different result shown, for the safety aspect of it, this should be studied more but the estimator shows promises for being useful to warn about impending roll-over.
60 4.5.7 Pile entry The model for the estimator does not take into account the change of the load distribution while a pile entry case is performed. Due to time constraint this was never adequately studied. However Figure 72 shows the both the forces and pressure in the lifting arm during pile entry.
Figure 72: Forces and pressures during pile entry.
As can be seen from the two top graphs the estimator can not predict the load distribution. The other four plots are the different pressure signal data values that are collected from the wheel loader. When looking at the pressure values logged in the machine, there are several and exactly where each pressure is measured was never researched. However by looking at the shape of the graphs it should be possible to account for the pile entry load distribution fairly accurately. Another example is Figure 73.
Figure 73: Forces and pressures during pile entry.
Here the sensor P LP seems like a good match of the behavior but when looking at Figure 74
61 Figure 74: Forces and pressures during pile entry. sensor PTP is closer in shape. This shows that to account for the pressure sensor is not trivial, but should not be impossible either.
4.6 Force distribution: Turning torque results The turning torque is the torque around the body of the machine that is trying to force it to turn. This is due to differences in tire force outputs, creating an unbalance between the right and left side of the machine. As there are no sensor on the machine that can measure turning torque, this was simulated in the simulation environment. The torque was measured on the articulation point while the calculated lateral and longitudinal tire forces where given as inputs to the turning torque estimation model. The model calculates the torque at the center of gravity but the articulation point is the closest point where a sensor can be placed. For all calculations Eq (94) in section 2.6 is used. When trying to estimate the turning torque, the behavior of the simulation case shown in Figure 54 and 55 were used. The results are illustrated in Figure 75. For the other simulated case with the zigzag motion the result is shown in Figure 76.
Figure 75: The case with the transient articu- Figure 76: The case with the transient drive lation angle. torque.
Clearly the estimator over-estimates the turning moment around the vertical axis. To be able to see the behavior of the simulations, the first simulation case is given a gain of 12 which can be seen in Figure 77. The second case is given a gain of 15, the result of which is shown in Figure 78.
62 Figure 77: The case with the transient articu- Figure 78: The case with the transient drive lation angle and a gain of 12 for the simulation. torque and a gain of 15 for the simulation.
A offset can be seen here while there also seems to be a phase shift in the beginning of the drive cases.
63 5 Discussion 5.1 Tire model: Field measurements The field measurements were first believed to be a valid way to capture the force characteristics of the tire. The used approach had a lot of advantages and were able to capture the dynamics of the wheel loader on different surface types. It could measure a lot of different driving cases and typi- cal working situations for the wheel loader which are valuable for other purposes than tire modeling.
Despite the advantages, the way the measurements were conducted had clear drawbacks. The two different systems needed to be synced, which was observed to not be successful in all mea- surements. The Vicon scope was also rather small, why the measurements had to be conducted as events. These events needed later to be extracted from the raw data, which was a rather com- plicated and time consuming procedure. In a controlled environment the Vicon system has high accuracy, but during the measurements it was observed that the outdoor environment introduced considerable noise and corrupted some data points.
The nature of the measurements also made the whole post processing difficult. The normal load varied by a considerable amount, which made it difficult and rather gave rather rough results when the slip and force pairs were associated to a certain normal load. This also resulted in that the data were spread out on many different loads, why a few number of data points were given per load. Another problem from the measurement was that it failed to cover the whole force characteristics. The resulting data was mostly contained within the linear and beginning of the semi-linear region, with little non-linear information for the asphalt case and no non-linear information from the gravel case. Thus, a lot of information of the characteristics were missed from the measurements.
When the raw data was inspected, it was observed that the Vicon system had sampled at an uneven sampling rate and produced duplicates in times stamps. The team who performed these measurements were consulted regarding these issues and they were surprised by the results. The Vicon data were also seen to be delayed in time relative to the Somat data. The previously men- tioned issues are believed to be the cause for that the post processed data showed low quality in terms of accuracy.
5.2 Tire model: Test rig measurements At first, the SWIFT transducers were supposed to be used during the measurements in the test rig. These were however short-circuited at the beginning of the measurements and could therefore not be used. This had the implication that the vertical force had to be estimated, and that the angular speed of the tire had to be measured through the cardan shaft. This was not the ideal situation, but it was deemed to be the best solution at that time. The estimation of the vertical force introduced a rather large uncertainty into the results and the measurements should ideally be repeated with the SWIFT transducers. The variation of the vertical force during the braking measurement could then be further investigated to see if it varies.
The acceleration measurements in general showed good accuracy. This is believed to be due to it being conducted in a controlled fashion. The braking measurements didn’t show the same quality in accuracy. A reason for this could be that the brake pedal was applied manually and was therefore not conducted in the same controlled and predefined manner as the accelerating cases. Another reason could be that the load applied on the tire was somehow changed during the tests, which seems to be a bit unreasonable since the tire was mounted such that it should be completely locked vertically. Despite this, it would be good to rule this out. Furthermore, if the measurements will be repeated in the future, it’s advised to run the measurements with the same velocities for both braking and accelerating measurements, to rule out any velocity dependent effects.
The results from the relaxation behavior are somewhat questionable, since the resulting step re- sponse in force didn’t attain the characteristics of a first order system. It instead seemed to possess the characteristics of a second order system. The reason for this is believed to be due to that the step input in slip was not ideal. The given numerical values were also questionable, since it was almost ten times larger than what was computed by the MF-Tyre software. However, when it was compared in the model validation, it showed a much better behavior compared to the relaxation behavior given by the MF-Tyre software.
64 Overall, the test rig showed big potentials. However in its current form it has some clear drawbacks and limitations which will be discussed next. The used setup limited the measurement to only capture the pure longitudinal force characteristics of the tire. The acceleration could be controlled in a good way through installation of the gas pedal. However this was not possible for the braking cases, were the brake pedal had to be applied manually. An improvement to the test rig would therefore be to find a way to control the brake pressure in a more controlled way. Another limi- tation with the test rig was the surface, which doesn’t reflect the typical working condition of a wheel loader.
5.3 Force distribution: Normal force The various results will be discussed separately with regard to the normal force estimation model.
5.3.1 Compared against simulation The simulation environment was set up for a different kind of wheel loader than the one used with the measurement devices. The normal force model used the same defining parameter for positions and weight as the simulation model so the comparison would be valid between the two. However, the simulation model was never evaluated during the thesis work against the measured data. When evaluating Figure 56, the case with transient speed, there was a clear offset that come from when the machine abruptly turned. That the model was unable to handle this abrupt change was understandable but it changed the base value of the normal load to another level than the simulation, most likely due to the new center of gravity. For the rear wheels both tires seemed to have a slight phase difference, but matched the amplitude of the simulation better. If the simula- tions were assumed to be accurate it would show that the estimator still needed some work, but by catching the overall behavior it might be enough to tweak some parameters.
When looking at Figure 59, the case with transient articulation angle a similar offset on the front wheels was seen, one that seemed to change its sign depending on whether the machine was turning right or left. But the overall shape was definitely there. For the rear tire it matched to a high degree the simulated values.
5.3.2 Compared against the old tests The old cases lacked data regarding the acceleration, pitch and roll but still managed to capture the behavior of the machine, altough with some offset. The biggest discrepancy was in the first drive case where there was a peak at around 80 seconds, this peak corresponded to a sharp turn of the machine but seemed to not affect the load distribution of the actual machine. In the acceleration and brake case the estimator completely missed the peaks. But as they are most likely caused by braking or acceleration, this was not surprising. Overall, these comparisons showed that the model was quite capable of estimating the general load on each tire.
5.3.3 Compared against the new tests For the new test data, both pitch and roll was available from sensors and in the first case the model seemed to show a close estimation of the real values but with a constant offset. This offset is either from a misrepresentation of the weight of different parts of the machine as inputs to the model or from the fact that the position of the bucket was unknown. Unfortunately there was no information regarding the rear tires to check whether the model held up for the rear as well in a real life application.
For the second case there was again some offset, but the wave-pattern seems to be mirrored between the right and left when comparing the estimator with the measured values. Even tough the offset was larger here, it does a decent job of estimating the load even in the more extreme case.
Overall the result for the model are viewed as good and it should be fit to run on a wheel loader. The model consist of a simple foundation that should also make it robust. However, when devel- oping the model, simpler version with less emphasis on the distribution in the rear gave similar results which meant that the model could likely be simplified further if it is needed to run on the on-board computer.
65 5.3.4 Roll over The roll over case was simulated and has not been verified against real world data. If the simulation is assumed to be accurate, the estimator showed a good estimation for detecting roll over. The event happened fairly quickly, in only about 10 seconds. The model correctly estimated which tire would lose all load and lift. After that the model quickly deviates, but at this point the machine will likely have rolled over so the data points to the right of where it crosses zero are not as important. The estimator estimates the roll over event after it has happened, this indicates that it needs to be further tested and worked on. But the difference in time between when the estimator and simulation estimates roll over is less than one second. Only this roll-over case was tested so it is unknown how well it would behave in other situations or how long the time difference would be on a roll-over case that happened over a longer time. But with the information available, the potential for using the model for roll-over protection purposes seems promising.
5.3.5 Pile entry Normal force on the tires during pile entry is important for a wheel loader as the risk for slip is high. The model has been built with transportation and movement in mind but when comparing the load distribution with the pressure sensor data clear similarities could be seen. However which of these sensors best correspond with the measured values is harder to say. Most likely it is a combination of several sensors that should be used. Due to time constraints this was not further looked into but should be one of the first things to investigate if any further work is performed on the normal force estimator model. If the load pressure sensor measurements are correctly added to the model, it should also be able to account for any load in the bucket. This would make the model able to handle most situations wheel loaders drive in. This would then be a robust model that gives the individual load on each tire that can be used in conjunction with the rest of the wheel loaders system.
5.4 Force distribution: Turning torque The estimator over-estimated the torque acting on the machine. This is clearly something that needs to be looked over. Unfortunately this was never compared to a real life test for validation. The offset that was seen in both figures could be because the measurement point for the simulation is the articulation point. The estimator calculates the turning torque on center of gravity which moves as the machine is driving. To better determine the validity of the model it would either need to be changed to calculate the torque at the articulation point or the torque around the center of gravity would need to be measured. The shape of the figures seemed correct which mean the model could still be valid but needs a lot of tweaking.
66 6 Conclusions 6.1 Tire model: Field measurements The processed data from the field measurements gave two different data sets based on what surface the measurements had been conducted on. These were gravel and asphalt surface. Only pure lon- gitudinal slip values resulted from the field measurements. Both the data sets showed low quality, in the sense that the data were spread out and the assumed pattern from the data didn’t show a distinct force characteristics. The reason for this is believed to be that some error had been made in a previous step when the raw data was gathered. Another issue with the data was that only small regions of the longitudinal force characteristics were covered and the amount of data points were low. The gravel surface data only contained small longitudinal slip values. The asphalt surface data contained small and large longitudinal slip values, but nothing in between.
Both the brush model and the MF didn’t give a ideal fit to either the gravel or asphalt sur- face data. This was due to the low quality in the data. When the models given from the gravel surface data were compared between each other, it was observed that the MF gave the best fit even though they didn’t differ much from each other.
The MF fitted by the asphalt surface data, gave a better fit to the data compared to the brush model. When the models were compared, they differed much from each other within the linear region, where the brush model showed a stiffer behavior and attained its peak force at a lower slip. For larger slip values they corresponded well with each other.
6.2 Tire model: Test rig measurements The measurements that were conducted in the test rig captured the whole force characteristics for the case of pure longitudinal slip. From the measurements when the tires load dependency was studied, the results showed an expected behavior, where the longitudinal force increased with increasing load. Overall, the measurements showed good accuracy. The MF-curves given by MAT- LABs fmincon() and the differential evolution algorithm [27] gave the same curves for all load cases, which could point towards that a global minimum in the solution was given. Furthermore, the resulting MF-curves gave a good fit to the data points but failed to fit some details in the measurements. These were the peaky regions from the braking measurements and the regions where large positive slip values were given from the acceleration measurements.
The measurements when the inflation pressure was varied showed mixed results in accuracy. No difference in the MF-curves between the algorithms were given. The measurements when a inflation pressure of pi=2 bar was used, showed good quality in terms of accuracy between the measure- ments and the fit of the MF-curves. When a inflation pressure of pi=5 bar was used, the resulting accuracy in the braking measurements were of low quality. The resulting fit of the MF-curves to the measurement data were also not ideal, since it fitted the upper bound of the measurement points for the positive slip values.
The results from the measurements when the relaxation length was studied, was that the re- laxation length decreased with increasing magnitude of slip. Furthermore, the relaxation length showed a small load dependency. The general trend that was observed was that for low slip values the relaxation length was smaller for the lower load. When the slip increased, the relaxation length for the lower load decayed more rapidly compared to the higher load.
From the validation simulations, it was observed that the MF-Tyre and the SCP-model overall corresponded well with each other. The difference that was observed was that the MF-Tyre model showed a unsteady behavior, while the SCP-model attained a smoother behavior which corre- sponded better to the experimental data. The difference in behavior is believed to be due to the difference in relaxation length, where the relaxation length of the MF-Tyre was roughly ten times smaller than the relaxation length used for the SCP-model. Since the behavior of the SCP-model corresponded better with the experimental data this could point towards that the measured value is more reasonable. When the models were compared to the experimental data, it was seen that for low and high slip values, the models matched well with the experimental data. But for slip values in between, the models showed a distinct difference from the experimental data. The reason for this could be due to that the measurements might not have fully captured the tires steady state behavior. Furthermore, the given reference model showed a clear difference between the calibrated
67 models and the experimental data.
Overall, the measurements conducted in the test rig proved to be a valid way in order to ob- tain the force characteristics of the tire.
6.3 Force distribution: Normal force estimation A mathematical model, to calculate the vertical force (normal load) on each of a wheel loaders tires, was set up based on the machine center of gravity position. This model was then expanded to include shifts in the load distribution of the machine as a function of changes in speed and angle of the machine relative to the ground. The developed model was built with the focus of being run on-board the machine which limited the amount of data that was available to the model.
When testing the model in a pre-existing simulation environment it was able to decently esti- mate the normal load on each tire, with some offset on the front tires.
When comparing the estimator to the older measured data it managed to estimate the general load on the tires. As data regarding acceleration and angle with respect to the ground was un- available it could not estimate the smaller more frequent shifts in load.
With a gyroscope available for the complimentary field test data the estimator could now bet- ter account for shift in the load on the wheels. With this data the model can accurately estimate the load on each tire as can be seen in the figure below taken from, which shows the vertical force as a function of time.
In simulations, the model could also estimate the behavior of the machine in a drive case leading to the machine tipping over to its side, a so called roll over case. It showed the roll over point, the point when one of the tires have zero load and starts to lift from the ground, later than what the simulations showed. The difference in time was however small and thus the model can be useful for estimating roll over.
From a review of available sensors on the machines hydraulic lift arm cylinder, it can be seen that it is possible to use this information to account for the shift in load on each tire on a wheel loader when it performs a pile entry event.
6.4 Force distribution: Turning torque The turning torque model was developed using information that is available from the tire models, longitudinal and lateral forces. It uses the center of gravity model, to estimate the turning torque around center of gravity. Without any sensors capable of measuring turning torque the model was tested in the simulation environment. It estimated an amplitude several times higher than the simulated value but did capture the behavior of turning torque. It is therefore not ready to be used as is, but has good potential to be developed further to create an accurate model.
68 7 Future work 7.1 Tire model • One of the goals was to compile the model into the ECU. Time didn’t allow for this, but it’s something that should be done. • A recommendation to Volvo would be to build a dedicated test rig for tire modeling. This is believed to be the most effective way in order to get the most reliable tire model. The test rig could be implemented in the current chassis dynamometer, likely with electric motors directly connected to the tire or just a brake system. Such a test rig would most probably be quite easy to design and should allow the measurement of lateral and combined slip. Another idea would be to construct a mobile test rig, like a trailer, which then could be used for testing the tires characteristics in an typical working environment for the wheel loaders. • The first approach to conduct measurements on the tire characteristics in the thesis work, was by performing field tests with a high precision GPS and the hub transducer. This approach was inspired by the works conducted in [9] and [30], but had to be disregarded due to software issues with the GPS. It’s however a approach that’s believed to have potential in capturing the tire characteristics and should be studied more. • Model the tire in a advanced simulation software. • Perform a frequency response analysis of the tire, by for instance use small brake pressures as disturbances or different brake pressures with different frequencies. This is conducted in [29]. • In the thesis work, only one tire dimension was studied. Different tire dimensions available at VCE could be investigated to see how this changes the behavior of the MF. Another aspect that should be studied is the velocity dependence of the force characteristics of the tire. This aspect has been disregarded in the thesis. In [31] this aspect is analyzed and discussed. • The results from when the relaxation length were measured are some what questionable. It’s therefore advised to conduct more transient measurement. • The thesis work was limited to longitudinal tire dynamics. A natural continuation would be to study lateral dynamics and combined slip cases.
• Look into more complex tire models, such as the MFSWIFT model.
7.2 Force distribution estimator Normal force • Include the pressure sensor to account for pile entry and load in the bucket. This will greatly increase the versatility of the model. • Compare the estimator and the simulator against real world data and use that comparison to find where the estimator model is lacking. If the estimator can be calibrated against the simulation it would then be possible to look at predicting distribution behaviors. • Update the estimation model to be able to predict future shift in the vertical forces of the tires. This can be done by mapping the effects of rate of change in the gas pedal, brake pedal or angle of the steering wheel.
• Look into different roll-over cases in order to validate the models roll over protection capa- bilities.
Turning torque • Find the cause of amplitude offset. This is crucial in making the turning torque estimator useful. • Determine at what point on the machine the turning torque should be calculated. When this information for control purposes, where does it make the most since to measure the turning torque?
69 • When the measuring point on the machine is found, tune it against the simulation model. This tuned version can then be used for testing of different control system for control the turning torque.
70 References
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[19] T. Nayl, G. Nikolakopoulos, and T. Gustafsson, “Path following for an articulated vehicle based on switching model predictive control under varying speeds and slip angles,” in Emerging Technologies & Factory Automation (ETFA), 2012 IEEE 17th Conference on, pp. 1–7, IEEE, 2012. [20] A. Oida, “Turning behaviour of articulated frame steering tractor,” tech. rep., DTIC Docu- ment, 1984. [21] P. M. Leucht, “The directional dynamics of the commercial tractor-semitrailer vehicle during braking,” tech. rep., SAE Technical Paper, 1970. [22] J. G´omezFern´andez,“A vehicle dynamics model for driving simulators,” 2012.
71 [23] A. Albinsson, F. Bruzelius, M. Jonasson, and B. Jacobson, “Tire force estimation utilizing wheel torque measurements and validation in simulations and experiments,” in 12th Inter- national Symposium on Advanced Vehicle Control (AVEC’14), Tokyo Japan, pp. 294–299, 2014. [24] K. Bayar, J. Wang, and G. Rizzoni, “Development of a vehicle stability control strategy for a hybrid electric vehicle equipped with axle motors,” Proceedings of the Institution of Mechanical Engineers, Part D: Journal of automobile engineering, vol. 226, no. 6, pp. 795–814, 2012.
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[27] “Differential evolution algorithm.” https://se.mathworks.com/matlabcentral/ fileexchange/18593-differential-evolution. [28] “Mf-tyre/mf-swift, software.” https://www.tassinternational.com/ delft-tyre-mf-tyremf-swift. [29] P. W. A. Zegelaar, “The dynamic response of tyres to brake torque variations and road unevennesses,” 1998. [30] A. Jonson and E. Olsson, “A methodology for identification of magic formula tire model parameters from in-vehicle measurements,” Master’s thesis, 2016. [31] R. Uil, “Tyre models for steady-state vehicle handling analysis,” Eindhoven Univerity of Technology, Eindhoven, 2007. [32] A. F. Andreev, V. I. Kabanau, and V. V. Vantsevich, Driveline Systems of Ground Vehicles. Crc Press, 2010.
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72 Appendices
A Tire model: Data processing for field measurements
Algorithm to split measurements into events The Somat and Vicon systems sampled independently from each other and were not synced. There- fore processing was needed to sync and split the data into events were the systems sampled simul- taneously. A MATLAB script was given that performed this and its actions are presented below. However the script was incomplete and contained lots of bugs why it needed further improvement and development.
The given MATLAB scripts synced the two different data streams, by using the information given from a sync vector, illustrated in Figure 79. The Vicon system started to sample before the So- mat system. When the measurements started, a rectangular wave was generated from the Somat system which then was transmitted to the Vicon system. The first pulse that was detected in the Vicon sync vector was therefore the time when the measurements were supposed to start in sync. Thus the index of when the first pulse was detected, was located and all the information up until that index were suppressed in the Vicon data. This resulted in that the two different data sets were synchronized.
Figure 79: Left image shows the unsynced signals. The right image shows the synced signals.
Furthermore, the Vicon data had to be split into its events, meaning the time where the vehicle was within the Vicon scope. The developed MATLAB script used the information from the measured Yvicon of the Vicon system. When the vehicle markers were outside of the scope, the measured value was zero. However when the vehicle entered, the measured value was non-zero, why a beginning of an event could be identified. To identify when the vehicle left the scope, the last non-zero value in the event was located. Thus the script found the indices were the first non-zero value and last non-zero value were registered and saved these indices, which is illustrated in Figure 80.
Figure 80: Figure shows how the raw data was splitted into its corresponding events.
Trough the starting and ending indices of the events, the corresponding times could be found by using the given time vector from the Vicon data. The times when the event began and ended were
73 then matched with the Somat time vector, since these were synced after the previously mentioned procedure. This information made it possible to collect the correct Somat and Vicon data set.
Processing of ground maps The ground of the Vicon scope was measured in order to calculate the height of the Vicon markers ∗ in relation to the ground, Zvicon. The vertical positions of the ground points at the same horizontal positions given by the Vicon data, Zground, were obtained by interpolation. MATLABs function griddata(·) was used for this. Some of the beginning and ending points were not within the grid spanned by the measured ground points, why they were extrapolated by fitting a polynomial of degree 2 to the interpolated values. MATLABs function polyfit(·) was implemented for this. An example of the inter- and extrapolation is presented in Figure 81. The vertical height in relation ∗ to the ground, Zvicon, was then calculated as
∗ Zvicon = Zvicon − Zground. (100)
Figure 81: The figure shows how the grounds vertical positions were calculated through the mea- sured horizontal positions of the markers.
Signal processing of Vicon data During the data processing of the Vicon data it was observed that the given time vectors from the raw data showed signs of have being corrupted in a previous processing step, since the time vectors were sampled with uneven time intervals and contained duplicates in time stamps. This caused the signal to has been sampling at a lower frequency than the given frequency of 200 Hz.
To overcome this problem, all the signals were reconstructed by collecting all the unique time stamps and their corresponding elements. The unique set of data points were then interpolated into a 200 Hz signal. To calculate the velocity components, a numerical differentiation was applied by using the central difference method together with an Euler forward on the first sample and a Euler backward on the last sample point. This is as follows
x −x i+1 i , i = 1 , dy h ≈ xi+1−xi−1 , i ∈ [2, n − 1] , (101) dt 2h xi−xi−1 h , i = n . In the above equation, n, is the total number of samples and h is the sample period. An example of the used method on a interpolated signal is presented in Figure 82.
74 Figure 82: An example of a differentiated signal.
An obvious observation from the figure is that the differentiation results in a signal with high noise and large spikes, even tough the undifferentiated signal was rather smooth. This was the general tendency for the given data. A Kalman filter was used to filter the data. This was the recommended filtering technique by the team who performed the Vicon measurements which provided a script for a Kalman filter. The given Kalman filter applied to the signal is presented in Figure 83.
Figure 83: The figure shows the effect of filtering the differantiated signal by a Kalman filter
As seen in the very beginning of the plot, the Kalman filter needs a certain amount of samples before it assumes reasonable velocities. This behavior was the general trend for the measurements. Since these velocities were obviously wrong and easily spotted, they were simply suppressed in the post data processing.
The given Kalman filter contained a tuning parameter named sigma vel which had to be tuned for every signal. The parameter tuned how small accelerations the model were supposed to assume. An initial value of 10−2 was used, which then was tuned by manually inspecting the signal. For cases when the velocity of the vehicle was constant, the parameter was tuned with a fairly low tolerance (10−5-10−3). However, if acceleration or deceleration occurred, the tuning parameter had to be tuned in a somewhat higher interval,( 10−2-100). Since the Kalman filter was the rec- ommended approach from the team, it was used in the majority of cases. However, for some cases the Kalman filter was not able to give reliable results, why instead a manual filtering with a low pass filter was performed. A description of the Kalman filter is given in Appendix F.
Signal processing of Somat data An example of the given signal from the angular transducer is presented in the left graph of Figure 84. As seen the signal involves discontinues when the signal jumps from π to −π. To map this signal into a continuous signal, the MATLAB function unwrap(·) was used. The result of the applied signal is presented in the right graph of Figure 84.
Figure 84: The unprocessed signal to the left and the processed signal to the right.
75 Next, to calculate the angular speed the transformed signal was differentiated through the numer- ical differentiation given by Eq (101). The result is presented in the left plot in Figure 85.
Figure 85: An example of the differentiated signal to the left. The corresponding frequency spectrum to the right.
As seen from the figure, the differentiation introduces large spikes which occur when the signal makes a jump between π and −π. To cut off these spikes, MATLABs median filter medfilt1(·) was used. The interval used varied from case to case, but was for most cases within 7-23 elements. Another issue that can be noted is the oscillating behavior that the differentiated signal possesses. The frequency spectrum of the signal is presented in the plot to the right. A distinct peak at 15 Hz is seen, which was observed for all cases. The reason for this is thought to be due some structural excitation from the tire or the vehicle. Since the oscillations were unwanted they we’re filtered away by using a 9th order low pass Butterworth filter, with a cutoff frequency of 5 Hz. As a final processing step the signal was smoothed to reduce any remaining oscillating behavior. The medfilt1(·) function applied to the signal in Figure 85 is presented in the above plot in Figure 86. Next below it, the signal passed through the low-pass filter is presented which presents the end result of the filtering process.
Figure 86: An example of how the median filter reduces the spikes in the above plot. The bottom plot shows the result of the lowpass filter.
Calculations of slip from Somat and Vicon data The translational and rotational velocities that were calculated from the Vicon data were expressed in terms of a global coordinate system. To calculate the slip quantities of the wheel, the spatial information of the markers needed to be transformed to the local coordinate system of the tire, which is illustrated in Figure 87.
76 Figure 87: The image illustrates the change of coordinates between the global coordinate system (X and Y) and the local coordinate system (x and y). The change of coordinate is performed through the global yaw angle W.
The transformation was carried out through a rotation of coordinates by the angle W , which is the angle the wheel makes with respect to the global coordinate system. Thus it’s the global yaw angle given by the Vicon system, Wvicon. The transformation is first performed by constructing the transformation matrix as follows cos(W ) −sin(W ) R(W ) = . (102) sin(W ) cos(W )
The unit vectors, eX and eY , for the global coordinate system were then transformed to the corresponding unit vectors in the local coordinate system of the wheel, ex and ey, through the rotation matrix. Mathematically this was performed as follows
ex = R(W )eX , (103)
ey = R(W )eY . (104)
To describe the velocity in terms of the local coordinate system of the wheel the velocity vector, V, was projected onto the unit vectors, ex and ey, by a scalar projection to yield the longitudinal and lateral velocities, Vx and Vy. This was performed as
V · ex Vx = , (105) |ex| V · ey Vy = . (106) |ey| The rotation vector in the horizontal plane W, was transformed in the same way. This is as follows
W · ex Wx = (107) |ex| (108)
W · ey Wy = (109) |ey| which gives the roll and pitch velocities, Wx and Wy, expressed in the coordinate system of the wheel.
Since information of the pitch angle of the wheel was both given from the Somat and the Vi- con data, two different slip variables were defined. The first slip variable which was formed was
(Vx − reWy) κ1 = − (110) Vx where Vx is the longitudinal velocity and Wy is the pitch angular velocity of the wheels marker after transformation, given from the Vicon data. The next slip variable that was defined used the information of the angular velocity from the Somat data. It was defined as
(Vx − reWys) κ2 = − (111) Vx where Wys is the angular pitch velocity calculated by using the Somat data.
77 Unreasonable slip values For some measurements unphysical slips resulted, in the sense that if the studied measurement was known to be isolated positive longitudinal slip, negative slip could be calculated which is unphysical due to the prior information. This lead to that the effective rolling radius had to be recalculated. From one of the measurements, an approximately free rolling case (meaning that the applied torque to the wheel approximately was zero) could be identified.
From the processed ground maps (section A), the vertical height of the wheel marker in rela- tion to the ground could be calculated. This resulted in that the loaded radius was obtained, r. Next the following empirical formula given in [32] was used to calculate the effective rolling radius in the free mode 3r re = . (112) e 1 + 2r rf The deflection of the wheel varied during the measurement, why a mean value of the calculated effective rolling radius was performed. This recalculation of the rolling radius instead gave re = 0.72 m, which is approximately 1 % larger than the given value. The calculated rolling radius was instead used for all the measurements since it gave better results. However, negative slip could still result. To overcome this issue, the following implementation was performed. If the studied measurement was known to be of positive nature, the negative slips were simply ignored and vice versa if the studied case was of negative slip nature. A mean value of the slip values and the corresponding forces given by the Somat data was then performed. This coupling formed a slip and force pair which then was used for the parameter identification.
Sorting of data points Before any parameter identification could be conducted, the data needed to be sorted, which will be described briefly. Every measurement point that had been processed was associated with a friction coefficient, µ, longitudinal slip, κ, longitudinal stiffness, Fx, and load, Fz. Thus the following data set was defined and created for every measurement set di = µi, κi,Fxi,Fzi , i ∈ 1, 2, ...., n (113) where n is the total amount of data points. All data sets, di, were then sorted based on their friction coefficient, µ, which then gave two different sets, one for asphalt and one for gravel.
Next the data within the previous sets were divided into different groups based on their load. The following intervals for the different load groups were created for the asphalt case
a Fz1 = 19, 29 kN, a Fz2 = 29, 39 kN, a Fz3 = 39, 49 kN, (114) a Fz4 = 59, 69 kN, a Fz5 = 79, 89 kN.
For the gravel case, the load groups were as follows
g Fz1 = 19, 29 kN, g Fz2 = 29, 39 kN, g Fz3 = 39, 49 kN, (115) g Fz4 = 49, 59 kN, g Fz5 = 59, 69 kN. The intervals were chosen to be rather large but this was necessary in order to include enough data points for the parameter and model identification process. Next, for every group, a mean value of the loads within the group was conducted to quantify the load. Furthermore the load of the wheel was added Fzwheel=4.19 kN. All data sets within the corresponding group were then associated with that load.
78 B Tire model: Processed field measurement data
This section presents the results from the measurement performed in 2015 and the additional field test that was performed. The sections presents the calculated slips and forces plotted together with the given FEM MF curve (presented in section 3.5.1). The results will be commented and compared against the results given from the FEM model. For cases when small slips are present, a linear line will be fitted to the measurement points in order to approximate the stiffness given by the measurements.
Measurement 1 Figure 88 presents the slips and forces which resulted after processing. Its seen that all resulting longitudinal slip values are within the linear region, which is as expected. If the resulting line is compared to the FEM curve, the result is that a lower stiffness is given. Concerning the two different slip values, κ1 and κ2, they seem to be in the same range and in agreement to each other.
Figure 88: Slip and force values that were given from the measurements conducted on asphalt.
Moving on to a loaded case, which is presented in Figure 89. As seen the given slip values show a much less stiffer response compared to the FEM model. For some slip values, the two different slip quantities, κ1 and κ2, deviate quite much. All slip values are further within the linear region.
Figure 89: The figure presents the resulting slip and force values that were given from a loaded case. The measurements were performed on asphalt.
The result from the gravel measurements are presented in Figure 90. As seen the trend that the measurement points give a much less stiffer response continuous. The slip quantities deviate from each other.
79 Figure 90: Slip and force values for measurements conducted on gravel.
Measurement 2 The results is shown in Figure 91. The measurements were conducted on asphalt. The given slip values are at the border of the linear region (κ < −0.05), with one slip value at approximately -0.09 which deviates much from the other slip values. The outcome of the measurements are somewhat unexpected, since the expected results were to measure slip values within the non-linear region. If the resulting stiffness given by the straight line from the curve fitting is compared with the FEM model, it’s observed that measurements show a much less stiffer response, which has been the general trend so far.
Figure 91: Slip and force values from measurements conducted on asphalt.
The next result to be presented, is shown in Figure 92, which presents the slip values when a higher load is applied. The slip values are still within the linear region, and as the previous case this is somewhat unexpected. The slip values seem further to deviate from each other. A lower stiffness is still maintained.
80 Figure 92: Slip and force values that resulted when a higher load was applied. The measurements were conducted on asphalt.
The section is ended with the measurements conducted on gravel, presented in Figure 93. The resulting slip values are seen to be within the non-linear region, which is as expected. The slip value pairs, κ1 and κ2, seem further to deviate between each other for some cases. The measured braking force in the non linear region is further observed to be lower than the braking force given by the FEM model.
Figure 93: Slip and force values that were given on a gravel surface.
Measurement 3 The slip values from the measurements are presented in Figure 94. It was observed from the measurements that the Vicon and Somat data were unsynced for those measurements, why the second slip quantity κ2 was not used. If the slip values from the Vicon system are observed, κ1, it’s seen that they are both within the linear region and seem further to continue the trend that the given stiffness is less then the FEM model.
Figure 94: Force and slip values from the measurements were the vehicle drove upon a inclined hill.
81 Measurements 4 The slip values given after processing are presented in Figure 95 when the ground was made up of gravel. The given slip values are fairly large, which is an expected outcome, but the tractive forces seem to be much lower than expected.
Figure 95: Force and slip values for the pile entry measurements, gravel surface.
Figure 95 shows the results when the ground was made up of asphalt. As can be seen the in the figure, the resulting slips are rather high. Most of the data points give a higher force compared to the FEM, which suggests that the assumed friction coefficient should be higher.
Figure 96: Force and slip values for the pile entry measurements, asphalt surface.
Measurments 5 The results from the front wheel spin cases are presented in Figure 97. The measurement cap- tures the non-linear part of the curve and seems to be in agreement with the FEM. This gives a confirmation that the non-linear part is reasonable.
82 Figure 97: Force and slip values for the front wheel full spin measurements.
C Tire model: Structural properties from field measure- ments
The structural properties given from the field measurements are presented in this section.
Contact patch dimensions The measurements of the contact patch gave the total length and width of the contact patch, A and B, as a function of normal load, Fz. In [3] following sets of equations are given as constitutive relationship between normal load and contact patch dimensions. r Fz Fz ac = (qa1 + qa2 )r (116) Fzo Fzo r 3 Fz Fz bc = (qb1 + qb2 )r (117) Fzo Fzo where ac and bc are half the total contact patch length and width, A and B. Fzo is the nominal load. To calculate the parameters, a least square fit of the equations in Eq (116) and (117) were performed by the normal equation. The resulting fit of the empirical equations are presented in Figure 98.
Figure 98: Plot to the left presents the contact patch length, ac, as a function of normal load, Fz. The plot to the right presents the contact patch width, bc, as function of normal load, Fz.
Vertical stiffness The radial deflection of the tire was measured with the Vicon system as the load on the tire was varied. In [3], a empirical equation for the normal force of the tire, Fz, as a function of the tires radial deflection, ρ, is suggested. This equation is as follows ρ ρ F = F (q + q ( )2) (118) z zo F z1 r F z2 r
83 where Fzo is the nominal load. The parameters, qF z1 and qF z2, were then least square fitted to the measurement data by using the normal equation. The resulting fit of the empirical equation is presented in Figure 99.
Figure 99: The measurement data together with resulting fitted curve.
D Tire model: Parameters and bounds for field measure- ments
The following sections presents the parameters and their corresponding bounds that were used for the parameter and model identification process for the field measurement data.
Magic Formula parameters Parameters for gravel surface measurements The parameters that were estimated are listed below with their corresponding bound that were used in the parameter identification process PKx1 ∈ 1, 20 , PKx2 ∈ 1, 20 , (119) PKx3 ∈ −0.1, 0 .
Parameters for asphalt surface measurements The following parameters were estimated for the nominal load case PCx1 ∈ 1, 2 , PDx1 ∈ 0, 2 , PKx1 ∈ 0, 20 , (120) PEx1 ∈ −4, 1 .
The parameters below were estimated with the corresponding bounds for the data where a non- nominal load was applied PDx2 ∈ −2, 2 , PKx2 ∈ 0, 20 , PKx3 ∈ −1, 1 , (121) PEx2 ∈ −4, 4 , PEx3 ∈ −4, 4 .
84 Brush model parameters Parameters for gravel surface measurements The following bound was used for the longitudinal slip stiffness when it was estimated from the gravel data
7 CF κ ∈ 1, 10 . (122)
Parameters for asphalt surface measurements The following bounds were used for the longitudinal slip stiffness and the friction coefficient when it was estimated from the asphalt data
µ ∈ 0, 1 , (123) 7 CF κ ∈ 1, 10 . (124)
E Tire model: Parameters and bounds for test rig measure- ments
The following sections presents the different parameters that were estimated in the different steps in section 3.7.
Magic Formula parameters for nominal load and nominal inflation pressure The following parameters and bounds were used for the curve fitting for the nominal loads PCx1 ∈ 1, 3 , PDx1 ∈ 0, 2 , PKx1 ∈ 1, 20 , PEx1 ∈ −20, 1 , (125) PHx1 ∈ −1, 1 , 3 3 PV x1 ∈ −10 , 10 .
Magic Formula parameters for varying load and nominal inflation pressure The parameters below were estimated from the measurement data where the load was varied. They are presented below with their corresponding bounds. PEx2 ∈ −10, 2 , 2 PEx3 ∈ −10 , 2 , PEx4 ∈ −10, 2 , PDx2 ∈ −10, 2 , PKx2 ∈ 0, 50 , (126) PKx3 ∈ −10, 10 , PHx2 ∈ −1, 1 , PV x2 ∈ −10, 10 .
Magic Formula parameters for nominal load and varying inflation pressure The parameters and their corresponding bounds, that were estimated when the inflation pressure was varied are as follows
2 Ppx1 ∈ −10, 10 , 2 Ppx2 ∈ −10, 10 , (127) 2 Ppx3 ∈ −10, 10 , 2 Ppx4 ∈ −10, 10 .
85 F Kalman filter
Kalman Filter The Kalman filter is actually an estimator that, in its simplest form looks at both its model for prediction and the value of its connected sensors. It can be used to filter and estimate values from sensor with a lot of noise, or it can be used as a sensor fusor. The basic formula is [33, 34]
xk = axk−1 + buk + wk, (128) zk = cxk + vk.
The filter works in discrete time and k is the current time step. xk is the guess of the current state of the system, a is the model of change of the system. u is the user input to the system, for example the change in angle of a gas pedal and b is the model for the effect it has on the system. wk is the process noise and account for the forces not part of the model. z is the observation of the sys- tem, c is the model of the effects the changes have on the system, v is the noise in the measurement.
Ideally the current state of the system xk could be determined from the observer z, but due to noise the measurements might not be reliable enough. To help make a better estimate both the current and the previous estimates are used with the help of a gain factor g.
xˆk =x ˆk−1 + gk(zk − xˆk−1). (129) The gain term can have a value between 0 and 1. If it is zero the observer will have no effect on the estimation. If it is 1 the current state will be the same as the observed value.
This gain factor is calculate for each time step
pk−1 gk = (130) pk−1 + r where r is the pre-measured base noise value of the particular sensor and p is the prediction error that is calculated as: pk = (1 − gk)pk−1. (131) If the prediction error is 0, the current gain will be 0, meaning that the state estimator is accurate and no adjustment is needed. If the prediction error is 1 the gain will become 1/(1 + r), if r is small this will give a big weight to the observer, but if r is large the observer will affect the system very little. For the prediction error, if the gain is 0 there will be no update to the prediction error. If the gain is 1, the prediction error is set to zero and the system behaves just as the observer shows.
The completed kalman filter will run on a two-step sequence, first a prediction and then an update.
xˆk = axˆk−1, (132) pk = apk−1a,
pk gk = , pk + r (133) xˆk =x ˆk + gk(zk − xˆk),
pk = (1 − gk)pk. As it continuous to run the estimator will zero in on the correct values and eliminate noise.
Predicting other values than those measured. Another application of the kalman filter is to estimate values that are not directly measured. An example of this could be a way to measure velocity from distance. The model will then predict the next position using the current guess of the velocity and compare that with the position given by the sensor. Like the normal filter function, for every cycle the function tries to eliminate the error in its estimate until it gives an accurate value for something that is not actually measured. Using the stated example the current state of the system, xk, will contain two variables
position x = (134) k velocity.
86 When predicting the next step the the model for change, a becomes
1 δt a = (135) 0 1 where δt is the timestep between predictions. This matrix will update the position via the velocity but keep the value of velocity the same as the sensor will contain no information with which to update the value. However, when the state of the system is recalculated with the gain, the velocity will automatically be updated to better reflect the actual change in position. Because the calculation are now done with matrices Eq (132) and (132) becomes
xk = axk−1 T (136) pk = apk−1a and
gk = pkinv(pk + r)
xk = xk + gk(zk − xk) (137)
pk = (I − gk)pk. where I is the identity matrix.
87