IN DEGREE PROJECT VEHICLE ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Energy efficient corneringSimulation and verification

NICOLAS LUCO

KEREN ZHU

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Abstract

The purpose of this master thesis is to study the energy efficiency of a vehicle when it iscornering. To achieve this, a Simulink model was built from a simple basic bicycle modeland theoretically validated. This model was then analysed and successively improved byadding velocity and yaw moment control. A study of the vehicle model behaviour bychanging parameters such as cornering stiffness and centre of gravity position was thenconducted. The traction force needed for a constant radius was calculated and methodssuch as torque vectoring have been tested using the model to obtain the lowest tractionforce. The model was compared with different vehicle types and further validated bycomparing the simulation results with experimental data acquired from a field test. Therolling resistance and aerodynamic resistance were taken into account when the modelwas validated with the experimental data and the result suggest that by distributing therequired traction force (using torque vectoring between inner and outer driven wheels) theenergy efficiency could be improved by 10%. This report ends with recommendations forfuture work.Keywords: Bicycle model, Energy consumption, Vehicle cornering behaviour, Yaw momentcontrol

Acknowledgements

The thesis was carried out at the Division of Vehicle Dynamics, Department of Aero-nautical and Vehicle Engineering at KTH Royal Institute of Technology during 2015 andis the final part of the master’s program in Vehicle Engineering.

We would like to express our sincere gratitude to our supervisor and examiner of thethesis, Lars Drugge, Associate Professor at the Division of Vehicle Dynamics, Departmentof Aeronautical and Vehicle Engineering at KTH for his support, patience, and immenseknowledge. His guidance helped us through the research period and the writing of thethesis.

Our sincere thanks also go to Mats Jonasson, Associate Professor at KTH and Func-tional Developer of Vehicle Motion & Control at Volvo Cars, for his insightful help con-cerning vehicle steering and MATLAB/SIMULINK.

We would like to thank the Division of Vehicle Dynamics for its support and for pro-viding a friendly working environment.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Method 2

2.1 Bicycle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Equations of motion for the simple model . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Equations of motion with added torque and traction forces . . . . . . . . . . . . . . . 4

2.4 Steady state cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.5 Traction force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.6 Understeer gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.7 Cornering with traction forces and changing vx . . . . . . . . . . . . . . . . . . . . . . 8

2.8 Implementation of velocity and yaw rate controllers . . . . . . . . . . . . . . . . . . . . 8

2.9 Parameter strategy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Vehicle behaviour with basic bicycle model in SIMULINK 10

3.1 MATLAB/SIMULINK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Initial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Changing cornering stiffness analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Changing velocity and λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Vehicle behaviour with PID controller for the velocity 14

4.1 Simultaneous cornering stiffness change at front and rear . . . . . . . . . . . . . . . . . 15

4.2 Changing cornering stiffness non-simultaneously . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Vehicle behaviour simulation with λ as a factor of relevance . . . . . . . . . . . . . . . 19

5 Vehicle behaviour with PID controller for yaw moment 21

5.1 Traction force analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Results of traction force analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3 Parameter analysis with λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4 Parameter analysis with lateral acceleration, ay . . . . . . . . . . . . . . . . . . . . . . 27

5.5 Handling characteristics with constant velocity . . . . . . . . . . . . . . . . . . . . . . 28

5.6 Summary of baseline vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Comparison with other vehicles 30

6.1 Moment control of the bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2 Comparison of baseline vehicle with the bus for different velocities . . . . . . . . . . . 31

6.3 Comparison of baseline vehicle with the bus for different lateral accelerations . . . . . 32

7 Validation of the model with experimental data 33

7.1 Velocity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.2 Cornering with decreasing radius using cruise control . . . . . . . . . . . . . . . . . . . 34

7.3 Validation of the model using a slalom manoeuvre . . . . . . . . . . . . . . . . . . . . 35

7.4 Cruise controlled with constant radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.5 Resistance force assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.6 Slalom manoeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.7 Cruise control with different speeds and constant radius . . . . . . . . . . . . . . . . . 42

8 Simulation based on experimental results 44

9 Conclusion 45

9.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.2 Experimental data validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10 Recommendations for future work 46

11 References 46

12 Appendix 47

12.1 Radius, wheel angle and understeer gradient calculations for the simple model . . . . . 47

12.2 Model codes for vehicle dynamics part . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

12.3 Model codes for calculation of kus and the Yaw controller block . . . . . . . . . . . . . 48

12.4 Code to fit a smooth graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12.5 Filter code for experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1 Introduction

1.1 Background

Today there is one major topic in the world which is the question of how to preserve the world and howto protect it from further damage. This question is now more important than ever because society isbecoming more critical about environmental damage and production companies are also focused onkeeping the environment clean as well as sustainable. Vehicles are responsible for a great deal of theglobal warming and other problems caused by emissions. Vehicles produce exhaust gases, such as CO,HC, NOx, SOx and PM(Particle matter or SOOT) which besides affecting the environment are alsoa health problem, cause for example cancer or respiratory disorders. Knowing this problem relatedto vehicles, it is important to study and have selective options to lower the energy consumption ofa road vehicle which will also reduce the emissions. Even the smallest percentage can lead to a bigdifference for large numbers of vehicles driving worldwide.

The concept of yaw moment control was first presented by Mitsubishi Motors (Ikushima et al., 1995),and applied in the Mitsubishi Lancer Evolution IV [9]. Among different options to gain yaw moment,direct yaw moment control is mostly applied due to its effect in both the linear and the non linearregion of tyre characteristics. The theory of direct yaw moment control is to use the differencein driving or braking forces between left and right wheels to generate yaw moment. The controlmethod can be varied, including torque distribution control which is to distribute torque independentlybetween wheels, torque vectoring which is to transfer torque and braking forces control which appliesdifferent braking forces to left and right wheels (Takami et al., 2008). The yaw moment control allowsdistribution control of driving torques to obtain the desired vehicle performance. For instance, theundersteer characteristic can be modified by applying yaw moment control (Novellis, 2012).

The direct yaw moment control generated from braking, however, may slow the vehicle down, as wellas shorten tyre and brake life. The yaw moment therefore has to be limited to reduce undesirableeffects. A strategy has been proposed for minimum usage of external yaw moment control, includingyaw rate and side-slip angle control through LQ optimal control (Mizaei, 2009). In the thesis work,the yaw moment source is limited to tire force generation, braking distribution has not been discussed.In order to analyse vehicle dynamic behaviour, various models have been developed so far (Kim andRo, 2001). The bicycle model is a simple model that includes a rigid body and front/rear tyres. Themodel provides quick and fairly accurate analysis of vehicle motions, including yaw and lateral slip.In present thesis, the bicycle model is therefore chosen to be the main analysis model.

1.2 Objective

The objective of this master thesis is to study vehicle energy efficiency during cornering. The studyincludes simulation and calculation of the cornering resistance and analyse the effect of applying ayaw moment generated by the traction force on the cornering resistance. To validate the model,experimental data has to be analysed and compared with simulation results. The focus is on vehicledynamic performance with yaw moment control. A bicycle model with yaw moment applied at centreof gravity was therefore built and analysed in Matlab/Simulink. Applying such a system, howeverinduces costs in both the production and the economics of a vehicle. The increased energy efficiencyneeds to be compared with the system cost to determine the economic value.

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2 Method

To start the project, a simple model had to be made for this thesis. The model is built step bystep while being simulated to ensure that it behaves and works for each step. The simple model iscalled ”Bicycle model” and will be explained in section 2.1. The two-wheel model or one-track modelrepresents a four-wheel vehicle by ignoring the weight transfer and the difference in vehicle speedbetween lateral wheels. The bicycle model is accurate enough to represent the essential characteristics,such as yaw motion and lateral slip. This chapter will explain the theoretical part of the constructionof the model, which starts from a simple model to be expanded into a more complex model wheretorque vectoring is tested.

2.1 Bicycle model

A simplified vehicle model, also known as the bicycle model is presented in Figure 2.1. The yawmotion is indicated by yaw angle ψ. Other parameters concerned are steering angle δ, front and rearlateral tyre forces, F12/F34, and traction forces Ftf/Ftr.

Figure 2.1: Bicycle vehicle model with three degrees of freedom

The lateral tyre force is considered to have a linear relationship with slip angle α12 and α34 in this case.The tyre forces are calculated based on equations 2.1 and 2.2, where C12 and C34 are the corneringstiffness front and rear with corresponding slip angles, α12 and α34.

F12 = −C12α12 (2.1)

F34 = −C34α34 (2.2)

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Based on the geometry relation that can be seen in Figure 2.2, the slip angle can be calculated as inequations 2.3 and 2.4 where the slip angle is expressed with lateral velocity vy, vehicle velocity vx,yaw rate ψ, front (f) and rear (r) distance to centre of gravity position and wheel angle δ.

Figure 2.2: Slip angle and side force for a tyre [7]

α12 = arctanvy + ψ ∗ f

vx− δ (2.3)

α34 = arctanvy − ψ ∗ b

vx(2.4)

2.2 Equations of motion for the simple model

The model has three degrees of freedom, the longitudinal direction X, the lateral direction Y and yawwhich gives three equations of motion (see equations 2.5 to 2.7).

X direction:

m(vx − ψvy) = −F12sinδ (2.5)

Y direction:

m(vy + ψvx) = F34 + F12cosδ (2.6)

Yaw motion around CoG:

Jψ = fF12cosδ − F34b (2.7)

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Combining with the equations for the lateral forces (see equations 2.1 and 2.2), the equations of Yand yaw direction can be expressed in a matrix form as shown in equation 2.8 which makes it simplerto calculate the lateral velocity, vy, and the yaw rate, ψ, or implement it in Matlab code (see theappendix).

(mD + C12 cos δ+C34

vxvx + fC12 cos δ−bC34

vxfC12 cos δ−bC34

vxJzD + f2C12 cos δ+b2C34

vx

)(vyψ

)=(C12δ cos δfC12δ cos δ

)(2.8)

2.3 Equations of motion with added torque and traction forces

To be able to simulate the model when adding traction force and torque, the equation of motion hadto be derived by adding a traction force in front and rear, Ftf and Ftr, and a torque, M , in equations2.5 to 2.7. The upgraded equations can be seen below in equations 2.9 to 2.11 for each direction withtheir traction force and torque contribution.

X direction:

m(vx − ψvy) = −F12sinδ + Ftfcosδ + Ftr (2.9)

Y direction:

m(vy + ψvx) = F34 + F12cosδ + Ftfsinδ (2.10)

Yaw motion around CoG:

Jψ = fF12cosδ − F34b+ fFtfsinδ +M (2.11)

As before, the equation of motion is expressed in a matrix for simplicity of calculating the lateralvelocity, vy, and the yaw rate, ψ, when the equation of motion in each direction is combined withequations 2.1 and 2.2 (see equation 2.12 ).

(mD + C12 cos δ+C34

vxvx + fC12 cos δ−bC34

vxfC12 cos δ−bC34

vxJzD + f2C12 cos δ+b2C34

vx

)(vyψ

)=(

C12δ cos δ + FtfsinδfC12δ cos δ + fFtfsinδ +M

)(2.12)

2.4 Steady state cornering

For simplicity, several assumptions have been made to make the equations easier to calculate. Themodel is set to run in steady state cornering, meaning that it has to run at constant velocity, vx, andwith a constant radius, R, see the boundary conditions in Table 1.

Table 1: Boundary conditions for steady state cornering

vx (m/s2) vy (m/s2) ψ (rad/s2)0 0 0

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By considering a relatively low lateral velocity, vy, compared to the directional velocity, vx (vy << vx),the transient part of the equation is ignored, which gives the simplified equation of motion, equation2.13. The vehicle’s global coordinate system is shown in Figure 2.3.

(C12 cos δ+C34

vxvx + fC12 cos δ−bC34

vxfC12 cos δ−bC34

vx

f2C12 cos δ+b2C34vx

)(vyψ

)=(

C12δ cos δ + FtfsinδfC12δ cos δ + fFtfsinδ +M

)(2.13)

Figure 2.3: The bicycle model in a global coordinate system of X and Y

Knowing that the integration of the velocities in the X and Y direction gives the position of thevehicle in the global coordinate system (see equations 2.16 and 2.17). The velocities vX and vY arethen calculated as in equations 2.14 and 2.15.

vX = vxcosψ − vysinψ (2.14)vY = vxsinψ + vycosψ (2.15)

After obtaining the global coordinate system velocities vX and vY , the position is then calculated byintegrating the velocities within a time span from 0 seconds until it reaches the end time t, which canbe seen in equations 2.16 and 2.17.

X =∫ t

0vX (2.16)

Y =∫ t

0vY (2.17)

2.5 Traction force

To calculate the traction force needed to overcome the resistance force for the vehicle when drivingin a circular motion, simplifications of the equations had to be made. The new equation of motion isderived (see equations 2.18 to 2.20) with the assumption of having a small steering angle, sinδ = δ.

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Ftfcosδ + Ftr = F12δ −mψvy (2.18)

may = F34 + F12 (2.19)

M = −fF12 + F34b (2.20)

From the definition of the traction force Ftra in equation 2.21 and the body slip angle β, that isdefined in equation 2.22, also knowing that the yaw velocity ψ, can be expressed as equation 2.23, thetraction force is then able to be calculated.

Ftra = Ftfcosδ + Ftr (2.21)

β = vyvx

(2.22)

ψ = vx/R (2.23)

From equations 2.18 to 2.23, the traction force is then able to be expressed as in equation 2.24.

Ftra = F12δ −mayβ= F12(δ − β)− F34β

(2.24)

Based on geometry relations of tyre slip angles, α12 and α34, in equation 2.3 and 2.4 are expressed asbelow in equations 2.25 and 2.26.

α12 = β + f

R− δ (2.25)

α34 = β − b

R(2.26)

From this relations the body slip angle is able to be expressed as below (see equation 2.27 and 2.28)

δ − β = f

R− α12 (2.27)

β = b

R+ α34 (2.28)

With the body slip expressed in known variables, the traction force is then able to be calculated asfrom equation 2.29.

Ftra = F12( fR− α12)− F34( b

R+ α34)

= − 1R

(−F12f + F34b)− F12α12 − F34α34

= −MR− F12α12 − F34α34

(2.29)

As equation 2.29 indicates, there are two parts affecting the traction force, Ftra. The first has to dowith the added torque, M . The other contribution is due to the cornering forces, F12 and F34, andtheir respective slip angles, α12 and α34. Both parts vary depending on radius.

2.6 Understeer gradient

The understeer gradient can be seen as the slope of the curves given in Figure 2.4. It is a measurementof the steering needed when the vehicle has either an oversteer or an understeer characteristic. Theundersteer gradient is expressed in equation 2.30. The relationship between steering wheel angle andkus is shown in equation 2.31.As Figure 2.4 indicates, the line in the middle represents a neutralsteer vehicle with understeer gradientzero. One possible situation is the baseline vehicle while the centre of gravity is placed centrally (centre

6

of gravity coefficient λ is 0.5) and it has equal cornering stiffness. This means that the distance fromfront axle to centre of gravity is equal to the distance from rear axle to centre of gravity, and givesthat the understeer gradient becomes zero (as in equation 2.30).

Figure 2.4: Circular test for constant speed with a hypothetical, linear vehicle [7]

kus = m(bC34 − fC12)LC34C12

(2.30)

According to equation 2.31 and figure 2.5, when kus is positive, a larger steering wheel angle isneeded to follow a certain radius when the acceleration increases, which indicates that the vehicle isundersteering. An understeer vehicle steers less than the amount commanded by the driver, whilenegative kus makes the vehicle oversteer. Correspondingly, an oversteer vehicle steers more than theamount commanded by the driver.

δ = L

R+ kusay (2.31)

Figure 2.5: Circular test for constant radius with a hypothetical, linear vehicle

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2.7 Cornering with traction forces and changing vx

When the traction forces Ftf and Ftr are taken into consideration as well as the equation of motionin x direction (the vehicle path direction), the velocity, vx, will not stay the same. Calculation andsimulation of the velocity are therefore necessary. Matrix 2.13 concerning Y direction and yaw motionshas two variables: ψ and vy. The X direction motion equation 2.5 contains a two degree term ψvywhich is difficult to include in the matrix. In order to simplify the calculation, the equation in theX direction is expressed as in equation 2.32, while the changing values of vx can be applied to thematrix of Y and yaw direction equations of motion.

vx =C12 arctan( vy+ψf

vx)δ − C12δ sin(δ) + Ftf cos(δ) + Ftr

m+ ψvy (2.32)

2.8 Implementation of velocity and yaw rate controllers

Figure 2.6: PID controller system method example

To achieve a stable system, controllers have been implemented. In this project a PID controller wasimplemented. The PID controller corrects the error that is given by the difference between the ref-erence value and the actual value. The basics of the controller can be seen in Figure 2.7 but forsimplicity of being able to run the simulations faster and more smoothly without having to work toomuch when calibrating the controller, an internal PID controller was used here, shown in Figure 2.6.The basic maths behind this system are given in equation 2.33 where the equation is written in timedomain, t (s). Kp, Ki and Kd are the proportional, integral and derivative gain constants that tunethe controller by multiplying their values with the given error from the system.

u(t) = Kpe(t) +Ki

∫ t

0e(τ)dτ +Kd

de

dt(2.33)

8

Figure 2.7: PID example model

The PID controller is tuned using the Ziegler-Nichols method, starting with slowly changing the valuesof the proportional gain, Kp until the output of the control loop is stable with consistent oscillations,Ku(ultimate gain) with oscillating time Tu. With the help of the ultimate gain and the oscillationtime, the controller is tuned as shown in Table 2.

Table 2: Ziegler-Nichols method

Control type Kp Ki Kd

P 0.5Kp - -PI 0.45Kp Tu/1.2 -PD 0.8Kp - Tu/8

Classic PID 0.6Kp Tu/2 Tu/8With Overshoot 0.33Kp Tu/2 Tu/3

2.9 Parameter strategy analysis

To verify the model, a strategy for parameter study was applied. By changing the variables whilekeeping other variables constant, the behaviour can be plotted and analysed. The strategy for differentsettings is shown in Table 3.

Table 3: Parameter strategy analysis

Simulation ay V C12/C34 R λ1 changing constant constant changing constant2 changing changing constant constant constant3 constant changing constant changing constant4 constant constant changing constant changing

First, simulation was done by changing the lateral acceleration ay for different sets of radius R. Thesecond test had different lateral acceleration for changing velocities V. The third simulation kept thelateral acceleration constant and instead changed the velocity with different inputs of radius. Test 4was executed by trying different settings of the cornering stiffness, C12 and C34 on different positionsfor CoG, expressed by the factor λ.

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3 Vehicle behaviour with basic bicycle model in SIMULINK

The first step in constructing a model is to start from basics and move on to more complex systemsstep by step. The model is based on the equations from previous section and different simulationshave been performed for validation of the model such as in sections 3.3 to 3.5.

3.1 MATLAB/SIMULINK

There are different ways to obtain data on the behaviour of a vehicle while performing different kindsof excitations, i.e. experimental testing where sensors are inserted on the vehicle while driving ordifferent CAE programs like ADAMS/Car. This master thesis work was conducted by using thesimulation tool ′′SIMULINK ′′. The software is user-friendly, which benefits building a model andthe interface makes it easy to have an overview of the model. There are also advantages such asthe simplicity of changing the model and having a better graphical understanding when building themodel with predefined blocks that can be found in the tool library.

3.2 Initial parameters

The bicycle model runs in steady state and the initial parameters are set according to Table 4 forverification of the model’s behaviour.

Table 4: Parameters

Mass of the vehicle, m 1437 kgMoment of inertia, Jz 2380 kgm2

Wheelbase, L 2.55 mCentre of gravity coefficient, λ 0.5 (CoG in the middle)Front cornering stiffness, C12 90 kN/radRear cornering stiffness, C34 90 kN/radSteering gear ratio, Is 17Steering wheel angle, δ 0.08 radVelocity, V 20 m/s

3.3 Cornering

The first verification was made to be sure that the vehicle model built in ′′SIMULINK ′′(Figure 3.1)was able to handle a constant radius motion if all parameters were to be constant knowing that ifλ = 0.5 and C12 = C34, the radius will be independent of velocity. The steering wheel angle, δ, andthe understeer gradient, kus, were calculated in the grey subsystem that can be seen in the modelfrom equations 3.1-3.2.

δpre = L

R+Kusay (3.1)

With radius and velocity given, the wheel angle is calculated as equation 3.2.

δpre = L

R+ m(bC34 − fC12)v2

x

LC34C12R(3.2)

The vehicle dynamic equations (see Chapter, 2) have been applied in the yellow-coloured subsystemshown in Figure 3.1 .

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Figure 3.1: SIMULINK model for a simple vehicle model

The simplest form that was tested in this project gave the results of the vehicle path as expected (seeFigure 3.2). The path in the middle of the circles forms a constant radius for constant parameters,λ = 0.5 and the same cornering stiffness for front and rear. The simulation ran with a constantsteering angle, δ of 0.08 rad. It verifies that the model is functional under this circumstance.

3.4 Changing cornering stiffness analysis

One crucial parameter to be studied is the influence of the cornering stiffness on the behaviour of thevehicle. Simulations were made using the model with changing cornering stiffness at the front with thevalues shown in Table 5. The simulation was performed with a constant velocity of 20 m/s, λ = 0.5and δ = 0.08 rad which gives an radius of 31.875 m when both front and rear cornering stiffness is 90kN/rad. The steering angle was set to be constant to show how much the change in radius would befor a fixed steering angle.

Table 5: Tested values for front cornering stiffness

C12 kN/rad C34 kN/rad λ δ45 90 0.5 0.0890 90 0.5 0.08135 90 0.5 0.08

Figure 3.2 shows the position of the vehicle with a changing cornering stiffness in the front wheel. Itcan be seen that a vehicle with lower cornering stiffness has a larger radius compared to the vehicle withhigher cornering stiffness, meaning that the understeer gradient increases with lower front corneringstiffness.

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Figure 3.2: Vehicle path with different front cornering stiffness

3.5 Changing velocity and λ

Besides the cornering stiffness, two other parameters that impact the behaviour of the vehicle are tobe considered. These are velocity and centre of gravity position in the vehicle x-coordinate system,λ. Velocity was first analysed and the model was tuned to have constant parameters with a referencesteering wheel, δ, of 0.08 rad. Both cornering stiffnesses were set to be 90 kN/rad and λ = 0.4.

In the model a ramp was used as input on the velocity feed to analyse the change in radius from 15to 30 m/s.

Figure 3.3: Velocity ramp in SIMULINK

Figure 3.4 indicates that with increasing velocity the radius increases correspondingly, which can bestudied from equation 2.31, where the lateral acceleration is expressed as v2

x/R. This means thatradius will increase as shown in equation 3.3.

R = L

δ+ kus

v2x

δ(3.3)

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Figure 3.4: Radius change with increasing velocity

The model has been tested for different positions of the centre of gravity (see Figure 3.6). λ was setto five different values (see Table 6) to study how the vehicle would behave during constant steeringinput. The corresponding front and rear lengths are calculated as in Figure 3.5.

Table 6: Tested values for λ

λ 0.40 0.45 0.50 0.55 0.60

Figure 3.5: Simulink block design that calculates the front and rear length relative to the CoG

The distribution of vehicle weight, λ, has an important influence on vehicle cornering behaviour. Thecentre of gravity position λ influences the distribution of vehicle load on the front and rear wheels,affecting the cornering force. Figure 3.6 shows that the radius decreases as lambda increases, whichfits expectations.

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Figure 3.6: Vehicle path with lambda change

4 Vehicle behaviour with PID controller for the velocity

When the previous simulations had been verified, a PID controller was introduced to regulate thevelocity, as can be seen in Figure 4.1. Yaw rate is constant, making the radius constant as well. Thecontroller is built inside the blue-coloured subsystem where the vehicle mass, velocity reference, yawrate feedback and lateral velocity feedback are inserted.

Figure 4.1: Main view of the Simulink model with inserted control block

In Simulink, a PID controller is available in the library. This controller has the ability to easily changethe P, I and D-gain and also tune it in a productive way (see Figure 4.2). The feed that travels throughthe controller is the error between the reference velocity and the actual velocity, which gives out theforce in the x direction. From this, the actual velocity is calculated (equations 4.1-4.3) with the helpof the vehicle mass, the feedback from yaw rate, ψout, and the calculated lateral velocity feedback,Vy,out.

Vx,PID = Fxm

(4.1)

Vx,out = ψoutVy,out (4.2)

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Vx,out =∫

(Vx,PID + Vx,out)dt (4.3)

Figure 4.2: The velocity control block design with a PID-controller

The traction force distribution has been analysed for three conditions: front wheel drive (FWD),four wheel drive (4WD) and rear wheel drive (RWD). Both front and rear cornering stiffness are 90kN/rad and a reference velocity of 20 m/s and lateral acceleration, 4 m/s2, have been applied to thesimulations. As Table 7 indicates, the differences in cornering resistance results between drive modesare within 1%. The choice of drive mode in the model therefore has limited effect on the result. Forthe following analysis, four wheel drive has been chosen concerning force distribution. The front andrear traction forces are proportional to the lateral cornering force.

Table 7: Front/rear force distribution analysis

λ RWD (N) 4WD (N) FWD (N)0.4 191.1 190.7 190.60.45 185.6 185.4 185.20.5 183.6 183.4 183.4

The method to verify the model is to run simulations with different parameters and analyse if vehiclebehaves according to the theoretical expected behaviour. The chosen parameters that have beenstudied to analyse the vehicle’s behaviour are velocity, λ, and the cornering stiffness in both front andrear axle of the vehicle.

4.1 Simultaneous cornering stiffness change at front and rear

The model has been tested with equal front and rear cornering stiffness. The sets of values variedfrom 45 to 135 kN/rad with the values from Table 8. The velocity is set at a constant 20 m/s andthe centre of gravity coefficient, λ, is set to 0.5.

Table 8: Cornering stiffness values

C12=C34 (kN/rad) 45 60 75 90 105 120 135

Figure 4.3 shows the results for all the cornering stiffness values used in the model. The analysis showsthat the cornering resistance decreases as the radius increases. Since velocity remains constant duringsimulations, a higher radius means lower acceleration. Meanwhile, in steady state without momentcontrol, the traction force is proportional to the lateral forces according to previous discussion. Lowerlateral acceleration would result in lower traction force.

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Figure 4.3 also indicates that increasing the cornering stiffness decreases the necessary traction force.This trend fits the expectation that with higher cornering stiffness, the slip angle would be smaller toobtain the same lateral force. As a result, the traction force is reduced.

Furthermore, with the same cornering stiffness interval, the difference decreases as cornering stiffnessincreases. For instance, at a radius of 50 m, the increase between cornering stiffness 45 kN/rad and60 kN/rad is larger than between 120 kN/rad and 135 kN/rad. Apparently, at a certain radius, thelateral acceleration is constant because the velocity is stable (ay = v2

x/R). The slip angle is inverselyproportional to cornering stiffness, while the traction force is proportional to the square of the slipangle. The force change is thus smaller in a higher cornering stiffness range.

Figure 4.3: Force as function of radius with changing cornering stiffness

4.2 Changing cornering stiffness non-simultaneously

Another model test that has been studied is the behaviour of the vehicle when the cornering stiffnesschanges either front or rear but not at the same time. The wheels with constant cornering stiffnessare set at 90 kN/rad while both velocity and the centre of gravity are set to remain constant, V = 20m/s and λ = 0.5.

Because the model that has been developed in ′′SIMULINK ′′ only works for positive wheel angles,δ (rad), a prediction of the lowest possible cornering stiffness needed to be made. This was calculatedfrom equation 3.1. In order to have positive wheel angle, cornering stiffness was calculated fromthe equation when δpre > 0. This ultimately gives the equation for the limit value of the corneringstiffness, which can be seen in equation 4.4.

C34 >C12mv

2xf

L2C12 + bmv2x

(4.4)

The values tested in the simulation for different cornering stiffness can be seen in Table 9.

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Table 9: Cornering stiffness values

C12 kN/rad C34 kN/rad45 5160 6075 7590 90105 105120 120135 135

To analyse the individual effects of front/rear cornering stiffness, separate simulations have been made.In Figure 4.4 and Figure 4.5, the ranges were set to be the same. Despite the line with a corneringstiffness of 51 kN/rad, the other results show similarity when changing the same amount of front orrear cornering stiffness. For instance, the result of 90 kN/rad for front with 60 kN/rad for rear issimilar to the one of 90 kN/rad for rear and 60 kN/rad for front. Since centre of gravity is set atcentre of vehicle length, the front and rear effects are expected to be the same.

Figure 4.4: Force as function of radius with changing rear cornering stiffness

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Figure 4.5: Force as function of radius with changing front cornering stiffness

For comparison, results from different cornering stiffness have been put in a single plot, see Figure 4.6.The figure indicates that the results show similarity within the same amount of cornering stiffness.For example, the results of C12 = 90 kN/rad and C34 = 60 kN/rad, C12 = 60 kN/rad and C34 =90 kN/rad are equal. The lowest traction force can be obtained from the highest cornering stiffnesscombination. As discussed earlier, at the same lateral acceleration the cornering stiffness determinesthe slip angle variation. A lower traction force can be thus obtained from higher cornering stiffness.Too high cornering stiffness, however, might also cause some side-effects like wearing out the tyres andball-joint faster, which may lead to high cost. A compromise therefore needs to be made to create awell-balanced vehicle.

Figure 4.6: Force as function of radius showing the results of Figures 4.5 and 4.4 together for com-parison

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4.3 Vehicle behaviour simulation with λ as a factor of relevance

The last setup to check if the model works correctly is to verify the behaviour given by the centre ofgravity as a factor of importance. Two different simulation setups were made to analyse the model.The first setup was simply to have all other parameters constant with increasing radius, as shown inTable 10.

Table 10: Constant parameters for the simulation

C12 90 kN/radC34 90 kN/radv 20 m/sR 40-100 m

The changing values for the centre of gravity coefficient, λ, are shown in Table 11.

Table 11: Simulated values of λ

λ 0.40 0.45 0.50 0.55 0.60

Figure 4.7 shows a comparison between cornering resistance force and changing radius where plottedlines are various settings of the position of CoG. As Figure 4.7 suggests, the difference in force asa function of radius when λ varies is small and the lines overlap each other. There are mainly twodifferent force changes, when λ is over 0.5 or when λ is lower than 0.5 which makes the graph looklike only two lines. The rest is overlapped due to similar values given by the simulation. This can alsobe seen more clearly in Figure 4.8.

Figure 4.8 shows the percentage of the force change for different λ values in comparison to λ = 0.5.The values for the force λ = 0.4 are almost identical to λ = 0.6. The behaviour of vehicles withλ = 0.45 and λ = 0.55 also shows the same trend. This is due to equal front/rear cornering stiffness.For instance, the sum of the total force at front and rear is the same for λ = 0.4 and λ = 0.6.

Figure 4.7: Force as function of radius with changing λ

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Figure 4.8: Force ratio as function of radius with changing λ

The second setup that the model had to run through was to analyse the difference between the forceswhen the cornering stiffness is proportional to the weight distribution (equations 4.5, 4.6 and 4.7)given by both the front and the rear wheel.

Ctot = C12 + C34 (4.5)

C12,new = Ctot(1− λ) (4.6)

C34,new = Ctotλ (4.7)

Equations 4.5 to 4.7 give the values for the second setup to run, as shown in Table 12. The conditionfor these values is that the total cornering stiffness is calculated from when λ = 0.5 and C12 = C34 = 90kN/rad.

Table 12: Cornering stiffness for different λ

λ C12 kN/rad C34 kN/rad0.40 108 720.45 81 990.55 99 810.60 72 108

Figure 4.9 shows the difference between the forces when the cornering stiffness is proportional to theweight distribution, e.g. if λ is 0.4 then C12 is multiplied by 0.6 and C34 by 0.4 which gives thecorresponding values of 108 kN/rad and 72 kN/rad. Since cornering stiffness and lambda both havelinear relationship with slip angle, the corresponding cornering stiffness change can have the sameeffect as a change in lambda.

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Figure 4.9: Force as function of radius with changing λ and corresponding cornering stiffness

5 Vehicle behaviour with PID controller for yaw moment

By having a velocity controller that worked on the model, a yaw controller was also inserted (Figure5.1). The controller is built inside the magenta coloured subsystem where the vehicle parameter block,velocity reference, radius reference, lateral acceleration feedback, radius feedback and velocity feedbackare inserted in the block. This controller is needed because when torque is added to the vehicle modelthe radius becomes unsettled, which is not the behaviour that the vehicle should have. The tractionforce in equation 2.13 is calculated in the ’vehicle dynamic’ block (which contains equation 2.13). Forthe first iteration an initial value is then needed; the initial value of the traction force is set as 0. Dueto this process, the vehicle states (especially radius) after the first iteration do not correspond to thereference value. A yaw controller is therefore applied.

Figure 5.1: Main view of Simulink model with PID controller

As in section 4, the controller is built in a similar way but controlling yaw rate (see Figure 5.2). Theerror inserted to the PID controller is the difference between the reference yaw rate and the actual yaw

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rate. The reference value for yaw rate, ψref , is calculated using the reference values of the velocityand radius as in equation 5.1. The actual yaw rate ψout, is the output of bicycle model calculation.

ψref = Vx,refRref

(5.1)

The PID controller feeds out δPID which is the actual steering angle. δPID is the output of thiscontroller block and will be used in the vehicle cornering calculation. The effect of this PID controlleris to minimise the variation between previous and next iteration.

Figure 5.2: Yaw rate controller block

Knowing that the model behaves according to expectations from previous chapters, the behaviour ofthe model when adding yaw moment to the vehicle has to be analysed. An important aspect thathas to be studied and analysed for this chapter is how the traction force varies when yaw moment isapplied. It is also vital to analyse the influence of the centre of gravity on vehicle behaviour with ayaw rate controller and applied yaw moment. Since the lateral forces are of major importance withregard to vehicle behaviour, it is crucial to study lateral acceleration and understeer gradient.

5.1 Traction force analysis

When the model was completed after adding a yaw rate controller, an observation and analysis hasto be made concerning the behaviour of the vehicle when a yaw moment is applied (see Figure 5.3)and to study the torque required to obtain the lowest traction force considering the limit of momentgenerated from the traction force. The first step was to make an analysis of the traction force.

The total traction force in the model is calculated as follows consider the geometry of vehicle forces:

Ft = Ftf + Ftr = max + F12sinδ (5.2)

Meanwhile, the force distribution has been analysed for three conditions, front wheel drive (FWD),four wheel drive (4WD) and rear wheel drive (RWD) as well. The velocity = 20 m/s, ay = 4 m/s2,λ = 0.4 have been applied to simulations. As Table 13 indicates, the differences in cornering resistanceresults among drive modes are within 1%. The torque vectoring influence is the same for all threeconditions. As in the previous chapter, for the following analysis, four wheel drive has therefore beenchosen for force distribution and the front and rear traction forces are proportional to the corneringresistance forces.

Table 13: Cornering resistance force distribution analysis

Drive mode YawMoment (Nm)

Force withoutmoment control (N)

Force withmoment control (N)

ForceReduction(%)

FWD 3000 190.5 161.5 154WD 3000 190.7 161.7 15RWD 3000 191.1 161.6 15

However, as discussed in the previous chapter, the expected traction force is then considered to bethe combination of the added moment and the cornering force contribution (equation 2.29). Thisequation is used as verification.

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The expectations concerning this equation are that at some point the applied yaw moment will changethe characteristics of the force reduction and start increasing the force. For this graph, the simulationparameters were set as shown in Table 14 with moment variation from 0 to 4000 Nm. To understandthe contribution better, the plots were first made with considering the total force and second with anoverview of the different contributions plus the total force contribution.

Table 14: Traction force analysis settings

λ 0.5ay 6 m/s2

C12 90 kN/radC34 90 kN/radltw 1.5 m

As can be seen from Figure 5.3, the limit of moment generated from the traction force is calculatedwith half track width, ltw, which is 0.75 m. Multiplying 0.5ltw by the traction force generates themaximum yaw moment when traction forces are applied on one side of the vehicle.

Figure 5.3: Torque and traction force illustration

5.2 Results of traction force analysis

A typical force moment relationship curve of the base line vehicle (λ = 0.5) at a velocity of 20 m/sand with a radius of 66.66 m is shown in Figure 5.4. The figure indicates that the force decreasesas the moment increase before the turning point, while after the turning point the force increases asthe moment increases. To decrease traction force, the yaw moment should therefore be applied as theamount of turning point (2250 Nm).

The moment needed to obtain the lowest traction force, however is rather high compared to the trac-

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tion force decrease. The red dashed line in Figure 5.4 shows the limit of moment generated fromtraction force due to cornering resistance. In this case (Track width = 1.5m), a yaw moment of up to310 Nm can thus be generated by traction force distribution. Considering the aerodynamic resistanceand rolling resistance, the yellow dashed line shows that the limit of moment generated from the totalresistance force is around 675 Nm. Meanwhile, to obtain a larger yaw moment than 675 Nm, thetraction force on the outer wheel has to be increased, at the same time as a braking force has to be ap-plied on the inner wheel to keep velocity constant. The total energy efficiency will thereby be reduced.

Figure 5.4: Total traction force vs yaw moment, ay = 6 m/s2, V = 20 m/s

The turning point however might vary with parameter changes and further analysis is therefore nec-essary. As discussed in the previous chapter, the traction force is contributed by two major parts asshown in equation 2.29.

One part is cornering resistance force contribution and another is yaw moment contribution. These twoparts are calculated and illustrated separately in Figure 5.5 (vehicle with λ = 0.5). The turning pointis when increasing cornering resistance force becomes larger than the negative moment contribution.As the figure indicates, compared to the cornering resistance force contribution the yaw moment has asmaller absolute value, which leads to less contribution to traction force. Meanwhile, the yaw momentcontribution is negative while the cornering resistance force increases the traction force. The yawmoment is therefore vital when analysing the reduction of the traction force.

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Figure 5.5: Analysis of Traction Force Contribution, λ = 0.5, ay = 6 m/s2, V = 20 m/s

In order to further analyse the cornering resistance contribution, a similar traction force contributionanalysis was performed for a vehicle with λ = 0.4. When yaw moment increases, the front slip angledecreases while the rear slip angle increases. For a neutral steer vehicle (see Figure 5.5), the corneringresistance contribution therefore increases as moment increases. In an understeer vehicle (see Figure5.6), however, the cornering resistance contribution decrease as moment increases in the low momentrange and then increases after a certain amount of moment.

Figure 5.6: Analysis of Traction Force Contribution, λ = 0.4, ay = 6 m/s2, V = 20 m/s

5.3 Parameter analysis with λ

From the simulations made in the previous chapter, it has been clarified that λ (centre of gravityposition) is an important variable with yaw moment applied to the model. The simulations are madeto analyse the vehicle’s performance when lambda and cornering stiffness vary, which means certainsets of vehicles have the same understeer gradient.

Table 15 shows the different parameter settings for the simulation at a velocity of 20 m/s, with a

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lateral acceleration of 4 m/s2 and with a turning radius of 100 m.

Table 15: Traction force analysis settings for changing λ

λ C12 kN/rad C34 kN/rad0.5 72 1080.5 90 900.5 108 720.4 108 720.4 90 900.6 90 90

During these simulations, the velocity and lateral acceleration are constant. The results are shown inFigure 5.7.

Figure 5.7: Parameter analysis with λ

The result suggests several trends of λ and cornering stiffness influence on traction force.

First, with the same cornering stiffness of 90 kN/rad, at zero yaw moment, the traction forces withλ = 0.5 are the lowest while with λ equal to 0.6 and 0.4 are the same. With increasing yaw moment,the vehicle with λ = 0.4 decreased the most to a lowest point around a yaw moment of 3000 Nm.This yaw moment is much higher than for the vehicle with λ = 0.5. The vehicle with λ = 0.6 showsincreasing traction force in the positive yaw moment range, which is reasonable due to its oversteercharacteristic. A vehicle with λ = 0.6 would need a negative moment to achieve lowest traction forcedue to its oversteer characteristics. Second, with λ = 0.5, the centre of gravity is located in thelongitudinal centre and the changing cornering stiffness makes the vehicle unbalanced. In this case, avehicle with lower front stiffness shows a similar result as corresponding equal cornering stiffness withlower λ. For instance, a vehicle with λ = 0.5 and front/rear cornering stiffness 72/108 kN/rad hassimilar characteristics to a vehicle with λ = 0.4 and equal front/rear cornering stiffness of 90 kN/rad.The result fits the expectation that the characteristic of traction force is similar to the corneringresistance force.

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5.4 Parameter analysis with lateral acceleration, ay

The lateral force has the greatest contribution to the traction force, meaning that the lateral ac-celeration changes the traction force dramatically. The simulations were conducted by inserting theparameters from Table 16 and plotting all different parameters into one plot to illustrate the overallinfluence on the vehicle’s behaviour when the yaw moment is increased.

Table 16: Parameter settings for changing ay, V and R

λ ay (m/s2) V (m/s) R (m)0.5 4 20 1000.5 6 20 66.660.5 8 20 500.5 6 24.5 1000.5 8 28.3 100

Meanwhile, the methods of obtaining different acceleration values affect the results in different way.Three values have therefore been simulated: 4, 6 and 8 m/s2. Two approaches for increasing lateralacceleration have been applied: increasing velocity and decreasing cornering radius. The results areshown in Figure 5.8.

Figure 5.8: Parameter analysis with lateral acceleration, velocity and radius

Figure 5.8 indicates that the traction force increases dramatically with increasing lateral acceleration.However, within the same lateral acceleration different combinations of velocity and radius showdifferent traction forces when the yaw moment is increased. The force reduction rate and correspondingyaw moment are shown in Table 17. With lower velocity and smaller radius, the traction forcedecreases more. The smaller radius would result in a higher yaw moment contribution while thecornering resistance force contribution remains the same with constant lateral acceleration.

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Table 17: Parameter analysis with lateral acceleration, velocity and radius

ay(m/s2) V (m/s) R (m) Moment(Nm) Force withoutmoment control (N)

Force withmoment control (N)

ForceReduction(%)

4 20 100 1500 184.6 177 46 20 66.66 2000 415 398.4 46 24.5 100 1500 415 408 28 15.5 30 5000 737.0 656.4 118 20 50 3000 737.0 707.6 48 28.3 100 3000 737.0 730 1

Velocity is also considered to affect the vehicle’s behaviour, which makes it important to compare thechange in force when yaw moment is added to the system. To analyse the influence on the vehiclewhen yaw moment is added with changing velocity, the simulation was carried out using parametersshown in Table 18.

Table 18: Parameter settings for changing ay, V and R

λ ay (m/s2) V (m/s) R0.5 6 15 37.50.5 6 20 66.70.5 6 24.5 100

The velocity analysis at constant lateral acceleration is shown in Figure 5.9. With decreasing velocityat constant acceleration, the turning radius decreases. The figure indicates that decreasing radiusreduces the traction force significantly. This is mainly due to the yaw moment contribution.

Figure 5.9: Parameter analysis with yaw moment, ay = 6 m/s2

5.5 Handling characteristics with constant velocity

To understand the vehicle’s behaviour, it is essential to analyse the handling characteristics. Oneimportant aspect is to study how different settings will influence the vehicle’s steering behaviourduring cornering. There are different ways to study the handling characteristics of the vehicle. Forthe purpose of this thesis, the analysis has been focused on the linear part of how the steering angle,

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δ, reacts to increasing the lateral acceleration, ay. The vehicle’s cornering performance is illustratedby characteristics of steering wheel angle as a function of lateral acceleration and is shown in Figure5.10. During yaw moment control, to obtain the lowest traction force certain yaw moments have beenchosen for corresponding lateral accelerations. For comparison, the characteristics of a vehicle withvarious λ (with or without yaw moment control, equal cornering stiffness 90 kN/rad) are shown inthe figure. The data are obtained at the same simulation condition, i.e. the same vehicle parametersand a velocity of 20 m/s.

Figure 5.10: Vehicle steering behaviour

As Figure 5.10 indicates, the relationship between lateral acceleration and steering wheel angle islinear within a certain range of lateral acceleration. The neutralsteer vehicle is marked in the figureas a reference for oversteered and understeered vehicles. It can be seen from the graph that whenthe vehicle has a yaw moment control, the understeer gradient will be lowered, causing the vehiclecharacteristic to be oversteered. Since the front slip angle decreases and the rear slip angle increaseswhen a positive moment is applied, the result is reasonable. With a reduced understeer gradient thevehicle tends to be difficult to control, which makes it less stable than if it were understeered.

However, the moment-controlled steering behaviour is constant regardless of λ settings. This indicatesthat with yaw moment control the vehicle tends to oversteer rather than understeer. I should beobserved that the yaw-moment-controlled vehicle shows a similar steering trend to the vehicle with λequal to 0.6.

Yaw moment control can also be used to extend the linear region of the understeer characteristic [2].

5.6 Summary of baseline vehicle

To summarise the results from previous discussion, the reduction (%) in total force has been calculatedcorrespondingly and is shown in Table 19. The vehicle’s cornering stiffness remained constant at 90kN/rad for both front and rear.As Table 19 shows, the vehicle with lower λ shows a higher force reduction rate. On the other hand,a higher yaw moment is required to achieve this force reduction.

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Table 19: λ analysis

ay(m/s2) V (m/s) λ Moment(Nm) Force without

moment control (N)Force withmoment control (N)

ForceReduction(%)

4 20 0.4 3000 190.7 162.4 154 20 0.5 1500 183.5 176.4 4

Table 20 shows that at the same velocity with varying lateral acceleration, the percentages of corneringresistance force reduction are the same, although the accuracy of the reduction rates is limited bythe force calculation accuracy. This phenomenon indicates that the velocity is the major parameteras regards force reduction rate rather than lateral acceleration. Meanwhile, despite the similar forcereduction rate, the yaw moment required increases as lateral accelerations increases. The powerdifference, however, increases dramatically with higher lateral acceleration. A higher power differenceindicates more power gain when the yaw moment control is applied.

Table 20: Lateral acceleration analysis

ay(m/s2)

V(m/s)

R(m)

Moment(Nm)

Force withoutmoment control (N)

Force withmoment control (N)

ForceReduction(%)

PowerDifference(W )

1 20 400 400 11.5 11.0 4 102 20 200 750 45.9 44.1 4 363 20 133.3 1100 103.3 99.3 4 804 20 100 1500 184.6 177.0 4 1526 20 66.7 2000 415.0 398.4 4 3328 20 50 3000 737.0 707.6 4 588

The force reduction rate and moment required for various velocities as well as lateral accelerationsare shown in Table 21. At the same velocity, the force reduction are similar regardless of lateralacceleration and the moment required increases as lateral acceleration increases. Meanwhile, at thesame lateral acceleration the force reduction and moment required decrease as velocity increases.Lower velocity would lead to a higher force reduction rate while a high yaw moment is required.

Table 21: Lateral acceleration and velocity analysis

ay(m/s2) V (m/s) R(m) Moment

(Nm)Force withoutmoment control (N)

Force withmoment control (N)

ForceReduction(%)

2 15 112.5 1300 45.9 40.1 136 15 37.5 4000 414.4 362.1 132 20 200 750 45.9 44.1 46 20 66.7 2000 415.0 398.4 42 25 312.5 450 45.9 45.2 26 25 104.2 1500 415.5 408 2

6 Comparison with other vehicles

To understand how the model works, it is important to analyse it with different kinds of vehicle, forinstance, a heavier vehicle. The simulations so far have only been evaluated with characteristics andparameters of a conventional car. The model could also be used to simulate other types of vehicle, inthis case a bus. The vehicle parameters of the bus are shown in Table 22.

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Table 22: Parameter settings for the bus

λ 0.63m 14000 kgJz 200000 kgmC12 250 kN/radC34 450 kN/rad

6.1 Moment control of the bus

First, it is crucial to analyse if the bus has a similar behaviour to the conventional car tested in theprevious chapter. The simulation is done by adding a yaw moment in the range from 0 to 35 kNm.The settings for this simulation are shown in Table 23. The simulations are run with a constant lateralacceleration, ay, of 2 m/s2.

Table 23: Simulated velocities for moment control of a bus

vx (m/s) 15 20 25R (m) 112.5 200 312.5

As Figure 6.1 indicates that the moment control shows a similar trend to the previous vehicle. Thetotal traction force is reduced more as the moment increases until the turning point, where it can beseen in equation 2.29 that the (−M/R) contribution to the resistance force takes over up to a certainradius, after which cornering resistance force become larger than the reduction due to the yaw moment.

Figure 6.1: Yaw moment control on bus

6.2 Comparison of baseline vehicle with the bus for different velocities

After analysing the plot made during the simulation, a comparison with the conventional car wouldbe of interest as regards how much the force is reduced. This analysis is made for the three differentspeed tests in Table 23 with the same constant acceleration for the conventional car as the bus. The

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conventional car that is compared was chosen to be the neutral/baseline vehicle that was tested inthe previous chapter. The simulation settings are the same as in Table 23.

To compare the two vehicles, the percentages of traction force reduction have been calculated cor-respondingly and are shown in Table 24. The table indicates that at the same velocity the bus andbaseline vehicle show a similar force reduction rate regardless of the variety in their vehicle parameters.The rate decreases dramatically as velocity increases. However, the moments needed to achieve thisare quite different, mainly due to differences in vehicle parameters such as mass, cornering stiffnessand λ.

Table 24: Comparison between baseline vehicle (BV) and bus at a lateral acceleration of 2 m/s2

VehicleType

ay(m/s2)

V(m/s) R(m) Moment

(Nm)Force withoutmoment control (N)

Force withmoment control(N)

ForceReduction(%)

BV 2 15 112.5 1300 45.9 40.1 13Bus 2 15 112.5 27000 1121.8 992.5 12BV 2 20 200 750 45.9 44.1 4Bus 2 20 200 18000 1122.4 1077.5 4BV 2 25 312.5 450 45.9 45.2 2Bus 2 25 312.5 11000 1121.9 1101.5 2

6.3 Comparison of baseline vehicle with the bus for different lateral accel-erations

The vehicles are also simulated and compared with different lateral accelerations, ay, and constantvelocity. For this simulation, the settings shown in Table 25 were used.

Table 25: Simulation settings for BV and bus comparison

V ehicle type ay m/s2 V m/s

BV 1 20Bus 1 20BV 2 20Bus 2 20BV 3 20Bus 3 20

Table 26 indicates that the vehicles show similar percentages of total force reduction at the samelateral acceleration. Regardless of the fact that this lateral acceleration varies, which varies due tothe radius, the reduction rates are the same.

Table 26: Comparison between baseline vehicle and bus at velocity of 20 m/s

VehicleType

ay(m/s2)

V(m/s) R(m) Moment

(Nm)Force withoutmoment control (N)

Force withmoment control(N)

ForceReduction(%)

BV 1 20 400 400 11.5 11.0 4Bus 1 20 400 8000 280.6 269.0 4BV 2 20 200 750 45.9 44.1 4Bus 2 20 200 18000 1122.4 1077.5 4BV 3 20 133.3 1100 103.3 99.3 4Bus 3 20 133.3 24000 2523.0 2420.7 4

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7 Validation of the model with experimental data

In this chapter the vehicle model is validated with real-life testing data. The importance of this isto know the difference between the linear model simulation and reality. The validation is conductedwith three different types of test and the vehicle settings used are shown in Table 27.

The settings for the cornering stiffness at front and rear were collected by testing the model andcomparing it with the experimental data. The testing began with equal cornering stiffness at frontand rear of 90 kN/rad, which gave the opportunity to study and compare the lateral acceleration,ay, and the yaw rate, ψ. If the lateral acceleration were to be too low for the simulation comparedto the experimental data, the total cornering stiffness would increase and if the yaw rate were to belower than the experimental results, then the front cornering stiffness, C12, would increase. From thismethod the settings for the vehicle were fulfilled and ready for the validation.

Table 27: Vehicle parameters for validation of the model

Mass of the vehicle, m 1684 kgWheel base position, L 2.647 mCentre of gravity, λ 0.4Front cornering stiffness, C12 85 kN/radRear cornering stiffness, C34 120 kN/radSteering wheel angle ratio, Is 15

Because of the noise that the measurement data has, it can be difficult to study the behaviour of theresult in the graphs clearly. The data was therefore filtered using a low pass filter in MATLAB onthe yaw rate. The MATLAB command is called ”filtfilt (B,A, tData.Wzrps)” (see the appendix),where tData.Wzrps is the measured wheel speed in radians per second. The data also goes throughanother filter before being inserted in the simulation model, which can be seen in Figure 7.1. This isto filter out the unstable peaks of the signal.

Figure 7.1: Experimental data passed through a filter and then inserted into the model

The different tests that have been compared with the results from the model are shown in Table 28,which also shows the manoeuvres of the vehicle, the velocity and the radius used for the tests.

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Table 28: Experimental tests

Test no: Manoeuvre R (m) V (km/h)1 Cruise controlled with constant radius 20 30, 40, 45, 502 Straight line driving - 403 Slalom - 404 Slalom - 555 Slalom - 656 Cruise controlled with decreasing radius Constantly decreasing 407 Cruise controlled straight line driving - 40

7.1 Velocity analysis

Given from the experimental data, the speed can be collected from each of the four wheels in thevehicle. There is also a speedometer signal based on the wheel speed that gives stable data for thevelocity (see Figure 7.2). This data is collected from the different speeds in test 1 (see Table 28), wherethe test was conducted by having a cruise controller driving in a constant radius for the velocities 30,40, 45 and 50 km/h.

Figure 7.2: Velocity analysis

Figure 7.2 shows the different velocities for each wheel and the average value between them whenthe vehicle makes a turn. When the vehicle makes a turn, the inner and outer wheels have differentvelocities due to the difference in the cornering radius of the inner and outer wheels. During corneringit can be seen that the speed oscillates. One reason for this is the cruise controller that constantly triesto regulate the vehicle’s speed to maintain a constant speed during the cornering manoeuvre. Anotherreason is the inclination angle of the track that gives the vehicle a deceleration and acceleration speeddepending on which side of the circle it is driving when following the path of constant radius.

7.2 Cornering with decreasing radius using cruise control

Test 6 in Table 28 has been analysed. The test was done with a cruise control. The vehicle ranat a constant velocity of 40 km/h and in the circle test the vehicle turned constantly inwards toreduce the radius in the circle test. To validate the model it is important to insert the experimental

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data in the model and study how well it fits with the data. The velocity and steering angle fromexperimental data are used as input for the simulation model. In Figure 7.3 several crucial vehiclebehaviour parameters have been plotted to analyse the difference between the simulation model andthe experimental behaviour, such as steering angle, velocity, lateral acceleration and yaw rate.

Figure 7.3: Comparison for constant velocity test with decreasing radius (test 6)

There is a small difference in yaw rate, while the lateral acceleration is the one that fits less thanthe others and the deviation between the experimental and the simulated plot at certain points isapproximately 0.4 m/s2. It can be seen that the experimental values follows a trend; it has smalloscillating peaks higher than the simulation. This fluctuation is partly because of the slope in the testtrack, which makes the vehicle drive half of a revolution downhill and the other uphill.

7.3 Validation of the model using a slalom manoeuvre

This validation is based on slalom test 3 (see Table 28). The test starts with driving in a straight lineuntil it reaches a constant speed of 40 km/h and trying to keep it as constant as possible while goinginto the slalom manoeuvre. Figure 7.4 shows the comparison between the simulation model and theexperimental behaviour, such as steering angle, velocity, lateral acceleration and yaw rate.

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Figure 7.4: Vehicle performance during slalom manoeuvre (test 3)

As previously, the biggest difference for the comparison is in the lateral acceleration plot. Whenstudied it was seen that there was an offset of 0.12 m/s2 due to road banking that was subtractedfrom the plot to have a better fit. There are some factors that affect the experimental data, for examplethe road banking causing the vehicle to drive downhill and a small slope in the lateral direction. Itcan be seen from Figure 7.4 that the experimental values have an overshoot of around 0.14 m/s2 anda time delay of 240 ms. The yaw rate shows a good agreement but has a small time delay of 70 ms,which might be caused by the tyres and the vehicles suspension elasticity.

7.4 Cruise controlled with constant radius

The purpose of the validation is to analyse the vehicle behaviour at different velocities with constantradius. The test is conducted by using a cruise controller. As can be seen in Figure 7.5, comparison ofthe experimental and simulation model is plotted for steering wheel angle, velocity, lateral accelerationand yaw rate. From the velocity graph in the figure, the behaviour stabilizes around a constant velocityfor the time period where the tests are made. The stable part in lateral acceleration and yaw rate isthus where the constant radius around 20 m at certain speed is conducted.

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Figure 7.5: Vehicle performance at constant radius with cruise controller

Because several tests at different velocities are shown in the same graph in Figure 7.5, a certain rangeof data has been chosen to be shown in Figure 7.6. It can be seen that for the lateral acceleration,experimental data is slightly higher than in the simulation. The sensor is placed around the CoG, thedistance between sensor and COG might lead to higher lateral acceleration or be due to the elasticityof the suspension and tyres.

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Figure 7.6: Zoom in plot for constant radius with cruise controller

7.5 Resistance force assumption

In the MATLAB/SIMULINK model, the resistance force calculation does not include the aerodynamicand rolling resistance parts. In order to compare the simulation results with experimental data, thetotal resistance has to be analysed. The data has been extracted from straight-line test 7. The totalresistance force and velocity are shown in Figure 7.7. The resistance force has been calculated fromexperimental data for the torque and the vehicle’s wheel radius according to equation 7.1. The meanvalues for force and velocity are taken from the same time range where velocity is relatively constantat around 12 to 32 seconds (see Figure 7.7).

Fres,straight = Texprw

(7.1)

Considering that several tests have been performed on the test track with a small slope of around 1%,which gave a gradient resistance force, a comparison has to be made between these tests and testsperformed on a flat surface. It is therefore important to validate the force assumption with more thanone straight-line driving test.

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Figure 7.7: Resistance force and velocity compared with time

Since test 7 (straight-line test) is performed on a flat surface, the resistance force should mainly containaerodynamic and rolling resistance force. The assumption has been made that the resistance forcecan be expressed with a second degree equation containing velocity, x, and two constant values, Cand B, which can be seen in equation 7.2. Meanwhile, slalom tests are performed driving downhill ona test track with a slope. The gradient force D is subtracted from the resistance force as in equation7.3. Different straight lines at the start of the slalom tests have therefore been compared and plotted(see the experimental line in Figure 7.8).

F = Cx2 +B (7.2)

F = Cx2 +B −D (7.3)

The D indicates the gradient resistance force which can be calculated from equation 7.4. The totalmass is 1684 kg plus two passengers weighing 70 kg each. Equation 7.6 shows the 180 N subtractedto compensate for the slope in the track.

Fgradient = mgsinα (7.4)

Based on the aerodynamic force equation, the constants in Equation 7.2 can then be identified as anaerodynamic coefficient, C, as in equation 7.5.

Fres = 12ρCDAf x

2 +B (7.5)

The aerodynamic coefficient is expressed with a drag coefficient (CD), a frontal area, Af and the airdensity, ρ. An assumption has been made based on straight-lines data from experimental tests as inEquation 7.6. In the assumption, CD = 0.42, Af = 2 m2, and ρ = 1.25 kg/m3.

The comparison between the different speeds is shown in Figure 7.8 together with the values calcu-lated from the experimental data. The figure indicates the difference between experimental data andsimulation results.

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Figure 7.8: Straight line part comparison for the slalom manoeuvre

The estimation of coefficients C and B had to be higher than expected because of unexpected resistancegiven by the weather conditions. The test track was wet and a little cold during the tests, whichincreased the air density and there might also have been unnoticed winds giving a higher aerodynamiccoefficient, C, estimated to be around 0.53, which provides an assumption close enough as it can becompared in Figure 7.8.

Fres = 0.53x2 + 435− 180 (7.6)

7.6 Slalom manoeuvre

The test was performed with a slalom manoeuvre at a constant velocity of 40, 50 and 55 km/h. Themotion resistance forces are calculated for both experimental data as well as simulations. Consideringthe test was performed driving downhill on a test track with a slope, the slope’s resistance is alsocalculated. The average forces have been calculated by taking the mean value of certain time rangesduring which velocity is more or less constant.The resistance forces in the slalom test at 40 km/h are shown in Figure 7.9. As the figure indicates,the experimental result is generally higher than the simulation results with a certain phase delay withthe oscillation.

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Figure 7.9: Slalom manoeuvre resistance force experiment vs simulation at 40 km/h

The average value of cornering resistance force has been calculated to compare simulation and ex-perimental results. The aerodynamic resistance, rolling resistance and gradient resistance have beensubtracted from the experimental data. Considering the oscillations, the time range has been cho-sen to cover as much of the time period as possible. The results are shown in Figure 7.10. Bothexperimental and simulated cornering resistance force increase with increasing velocity.

Figure 7.10: Slalom manoeuvre resistance force comparison experiment vs simulation

Force difference ratios between experimental data and simulation are shown in Table 29. The differ-ence between experimental and simulated forces increases as velocity increases. Considering that theestimated aerodynamic resistance is proportional to velocity squared and rolling resistance might belower than the actual rolling resistance, the result is reasonable.

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Table 29: Force comparison for slalom

Test nr: Manoeuvre Velocity km/h Force difference N Force difference ratio %3 Slalom 40 2 54 Slalom 55 18 125 Slalom 65 50 14

7.7 Cruise control with different speeds and constant radius

This test was conducted with a constant radius of 21.7 m and various velocities. The velocities areshown in Table 28. The velocity is controlled by a cruise controller. The experimental data hasbeen input into the model and simulated for the purpose of comparison. The simulated force andexperimental data are shown in Figure 7.11. The aerodynamic force and rolling resistance have beenremoved from the experimental data since the simulation results do not contain these two parts.

Figure 7.11 illustrates forces varying at different velocities. As the figure indicates, the experimentalvalues are higher than the simulation data at a certain level of oscillation. The oscillation might beexplained by the slope of the test track while doing circle test. Considering that the vehicle is runningin a constant radius of 21.7 m, the time per vehicle path cycle can be calculated as:

T = S/V = 2πR/V (7.7)

Correspondingly, at a velocity of 30, 40, 45 and 50 km/h, the revolution time is around 17, 12.5,11 and 10 s. As Figure 7.12 indicates, the test at 30 km/h has an oscillation cycle of around 18 s,which matches the revolution time caused by the slope. However, the range between peak and lowestvalues is around 500 N . As discussed earlier, the force variation caused by the slope is around 180 N(meaning +/−180 for up-/downhill driving), which is lower than the actual range. Meanwhile, at 40,45 and 50 km/h, the ranges during the test are respectively 250, 150 and 75 N. The results of the rangebetween peak and lowest values indicate that the slope is not the only the influencing variable; thevalues seem to be proportional to velocity. Considering the resistance force estimation, there mightbe a velocity-related force term that is ignored during the estimation. The slope is therefore possiblythe cause of the oscillation but only the partial cause of force variation.

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Figure 7.11: Cornering resistance force comparison between experimental data and simulation result

Since the experimental data oscillates rather much while the simulation values are not entirely stable,the average values of certain amount of time for both are calculated and are shown in Figure 7.12. Bothresults increases as velocity increases, while close to a quadratic function of velocity. The differencebetween the two results is shown in Table 30.

Figure 7.12: Average cornering resistance force comparison between experimental data and simulationresult

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The force difference is the experimental data minus the simulation result and the force difference ratiois the difference divided by the experimental data. As shown in Table 30, the force difference varieswith velocity change. It is possible that the estimation is not accurate enough, which leads to certainerrors. Considering that there are other forces acting on the vehicle, and the simulation model is asimple linear model, this difference is acceptable and the model is therefore validated.

Table 30: Force comparison for constant radius

Test nr: Manoeuvre Velocity km/h Force difference N Force difference ratio %1 Constant radius circle 30 35 221 Constant radius circle 40 38 111 Constant radius circle 45 42 71 Constant radius circle 50 61 8

8 Simulation based on experimental results

Based on the previous discussion, the simulation results can be used to estimate real-life vehicleperformance. The yaw moment effect on reducing resistance force has therefore been analysed usingsimulation. To be able to compare with the experimental data, the model is simulated at a constantvelocity of 12.5 m/s and a constant radius of 21.7 m. The results are shown in Figure 8.1.

Figure 8.1: Yaw moment effect on traction force

As figure 8.1 indicates, the resistance force decreases as the yaw moment increases, correspondingto the results discussed in previous chapters. To indicate the possible yaw moment generated fromtraction forces, two lines have been plotted in the figure along with resistance force results. The blueline represents the maximum yaw moment created from cornering resistance force only. The green linerepresents the maximum yaw moment created from the sum of cornering resistance force, aerodynamicresistance and rolling resistance. Several velocities from test no.1 with constant radius 21.7 m havealso been simulated and the results are shown in Table 31.

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Table 31: Force comparison for different velocities

V (m/s) Moment generated fromtotal resistance limit (Nm)

Force withoutmoment control (N)

Force withmoment control (N)

ForceReduction (%)

12.5 1005 813.5 735 1011.4 812 564.4 505.2 1010 626 335.4 293.6 127.8 448 125 98.8 21

9 Conclusion

This thesis has been focused on the effect of yaw moment control on vehicle cornering resistance force,while analysing the energy efficiency level.

9.1 Modelling

The analysis of the bicycle model confirms the accuracy of predicting vehicle steady-state behaviourduring constant radius cornering. To calculate traction force, a velocity controller has been appliedto the model.

The cornering stiffness and λ have been analysed for several combinations to analyse their influenceon traction force. The total force is highly depending on the sum of the cornering stiffness. Thehigher the cornering stiffness sum is, the lower the traction force. Meanwhile, with constant totalcornering stiffness, the forces are the same for different combinations of front/rear cornering stiffness.The influence of λ change only is relatively smaller.

To fulfil the objective of this study, the influence of yaw moment has been analysed. The yaw controllerhas been applied to the model. The results indicate that traction force decrease as yaw momentincreases, while after a certain amount of yaw moment the force increases as yaw moment increases.The traction force can be considered to be contributed from two parts: one from the yaw moment andthe other from cornering resistance force. Generally, to obtain the lowest traction force, a relativelyhigh yaw moment is required. However, the yaw moment generated directly from resistance forcedistribution is limited and braking force or additional yaw moment such as torque from the engine orwheel motor must therefore be applied to the vehicle if the lowest traction force is desired.

The influence of yaw moment on traction force is also influenced by vehicle parameters. For instance,with the same total amount of cornering resistance, a more understeered vehicle has higher forcereduction rate when yaw moment is applied. Meanwhile, lateral acceleration, velocity and corneringradius also have significant influence on forces. Low traction forces can be obtained when these valuesare low due to low cornering force. The percentage of force reduction has been analysed; the forcereduction rates are similar with the same velocity. Meanwhile, at certain accelerations, with lowervelocity (smaller cornering radius), the ratio is higher.

Understeer characteristic has been analysed as an important factor concerning cornering behaviour.The vehicle is more oversteered with yaw moment control. A comparison has been made betweena bus and a passenger car and the results show similar trends of force reduction with yaw momentcontrol.

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9.2 Experimental data validation

Furthermore, the model has been validated with experimental data applied. Several tests have beensimulated and the model provides satisfactory results for lateral acceleration and yaw rate estimation.The rolling, aerodynamic and gradient resistance have been taken into account when analysing thetraction force. The estimated force has a certain error compared to the experimental results. Sincethe error is acceptable, the model has been used to simulate the yaw moment effect based on real-lifecases. The result suggests that the traction force could decrease by around 10% using torque vectoringof the traction force due to rolling, aerodynamic and cornering resistance.

10 Recommendations for future work

As described in the previous chapters, this thesis has focused on using the bicycle model to estimate thevehicle’s cornering behaviour under different driving conditions and for different vehicle configurations.The major behaviour of the model predicts the experimental data satisfactorily. However, to obtaina more accurate estimation, other factors have to be taken into account,e.g. the non-linear regions ofthe vehicle’s behaviour, tyre model and resistance calculation.

A four-wheel model would therefore be the next step in further studies, to consider wheel load trans-fer.The controller in the model can also be modified to become a driver model to predict the trackcondition and estimate the steering angle and yaw moment as control input.

Further, the yaw moment can be generated from torque distribution using for example individualmotors or braking force on one side of the vehicle. It is therefore recommended to also conduct ananalysis of how the yaw moment is created and its influence on for example longitudinal slip losses.

11 References

[1] T. Miura, Y. Ushiroda, K. Sawase, ”Development of Integrated Vehicle Dynamics Control Sys-tem”, Mitsubishi motors technical review, 2008.

[2] L. De Novellis, A. Sorniotti, P. Gruber, L. Shead, V. Ivanov, K. Hoepping, ”Torque Vectoringfor Electric Vehicles with Individually Controlled Motors: State-of-the-Art and Future Devel-opments”, 26th Electric Vehicle Symposium, Los Angeles, California, 2012.

[3] M.Mirzaei, ”A new strategy for minimum usage of external yaw moment in vehicle dynamiccontrol system”, Transportation Research Part C: Emerging Technologies, 2010, 18(2): 213-224.

[4] C. Kim, P. I.Ro, ”An Accurate Simple Model for Vehicle Handling using Reduced order Modeltechniques”, SAE Technical Paper, 2001.

[5] W.F. Milliken , D.L. Milliken, ”Race car vehicle dynamics”. Warrendale: Society of AutomotiveEngineers, 1995.

[6] S. Talukdar,S. Kulkarni, ”A Comparative Analysis of a Rigid Bicycle Model with an ElasticBicycle Model for Small Trucks”, SAE Technical Paper, 2011.

[7] Smith C, ”Tune to win, The art and science of race car development and tuning”, Aero Pub-lishers, Fallbrook, 1978.

[8] Ikushima Y, Sawase K. ”A study on the effects of the active yaw moment control”, SAE TechnicalPaper, 1995.

[9] http : //www.rallycars.com/Cars/MitsubishiEvoV /LancerSpecs.html

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12 Appendix

12.1 Radius, wheel angle and understeer gradient calculations for the sim-ple model

This code shows how the understeer gradient and the steering angle is calculated depending on, if thewheel angle is constant input or not.

1 f unc t i on [ paramOut , VxRef , mass , R1 ,SWA, Kus ] = fcn (param , Vx,R)2 % STEERING ANGLE AND UNDERSTEER GRADIENT CALCULATIONS3 Kus = mass . ∗ ( b .∗C34−f .∗C12) . / ( L .∗C34 .∗C12) ;4 SWA=0.08 %OR L/R+Kus∗Vxˆ2/R;5 %6 end

12.2 Model codes for vehicle dynamics part

This code is from a block in the simulink model that calculates the vehicle dynamics of the car.

1 f unc t i on [ YawRate , Vy, VxOut , Rout , Vtotal , ay , Frf , Frr , Ftot , Fcal , Mout ,X,Y,yawangle , alpha12 , alpha34 , Fyytotal , Fyycompare , Fyyf , Fyyr , F12 , F34 , Ftes t ]=

fcn (param , Vx,R,SWA, Ftf , Ftr , Fx ,M, x , y , Yawangle )2 %3 %4 %5 % EQUATION OF MOTION MATRIX FORMATION6 CV = [ C12∗SWA∗ cos (SWA)+Ftf ∗ s i n (SWA) ; . . .7 f .∗C12∗SWA∗ cos (SWA)+Ftf ∗ s i n (SWA) ∗ f+M] ;8

9 % CALCULATION OF Vy & YAW RATE10 V YawR = steady s ta t e mat r i c e A \(CV) ;11 V YawRn = [V YawR;Vx ] ; %New matrix [Vy Yaw rate Vx ]12

13 % SLIP ANGLE CALCULATION FOR EVERY ANGLE14 alpha12 = ( ( (V YawRn ( 1 , : ) )+(V YawRn ( 2 , : ) ) .∗ f ) . /Vx)−SWA;15 alpha34 = ( ( (V YawRn ( 1 , : ) )−(V YawRn ( 2 , : ) ) .∗b) . /Vx) ;16

17 % FORCE CALCULATION18 F12 = −alpha12 .∗C12 ;19 F34 = −alpha34 .∗C34 ;20

21 Vy=V YawRn ( 1 , : ) ;22 YawRate = V YawRn ( 2 , : ) ;23

24 % VX CHANGE CALCULATION25 VxRate=(((C12 .∗ atan ( (Vy+YawRate .∗ f ) . /Vx) .∗ s i n (SWA) ) . . .26 −C12∗SWA∗ s i n (SWA)+Ftf .∗ cos (SWA)+Ftr ) . / mass )+YawRate .∗Vy;27 VxChange = VxRate∗dt ;28 ax=VxRate−YawRate∗Vy;29

30 % VX CALCULATION PLUS THE CHANGE PART31 VxOut = Vx+VxChange ;32

33 % RADIUS CALCULATION

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34 Rout = VxOut . /V YawRn ( 2 , : ) ;%rad iu s35

36 %TOTAL VELOCITY37 Vtotal=s q r t (VxOutˆ2+Vyˆ2) ;38

39 %LATERAL ACC40 ay=VxOut∗YawRate ;

This part shows how the the force distribution is made in the vehicle dynamics block.

1 % FORCE DISTRIBUTION2 Frf=(Fx+F12∗ s i n (SWA) ) ∗( alpha12 ˆ2) .∗C12 /( ( alpha12 ˆ2) .∗C12∗ cos (SWA)+(

alpha34 ˆ2) .∗C34) ;3 Frr=(Fx+F12∗ s i n (SWA) ) ∗( alpha34 ˆ2) .∗C34 /( ( alpha12 ˆ2) .∗C12∗ cos (SWA)+(

alpha34 ˆ2) .∗C34) ;4 Ftot=Fx+F12∗ s i n (SWA)5 %6 %7 end

12.3 Model codes for calculation of kus and the Yaw controller block

The first code is from the block that calculates the understeer gradient and wheel angle. The secondpart of this code is the blocks that are inserted in the yaw controller block.

1 f unc t i on [ YawVelocity ] = fcn (param , VxDes , RDes)2 % YAW RATE REFERENCE CALCULATION3 % param ARE THE VEHICLES INSERTED PARAMETERS4 % Des = DESIRED5 YawVelocity = VxDes/RDes ;6 %7 end

12.4 Code to fit a smooth graph

This code is to have a smooth graph by spline.

1 %add d i s c r e t e po in t s2 lengthX=length (m) ;3 sampl ingRateIncrease = 10 ;4 newXSamplePoints = l i n s p a c e (yaw m (1) , yaw m( end ) , lengthX ∗

sampl ingRateIncrease ) ;5 YAWM =newXSamplePoints ;6 %using s p l i n e f o r smooth curve7 F = s p l i n e (yaw m , f ,YAWM) ;8 FC = s p l i n e (yaw m , fc ,YAWM) ;9 M = s p l i n e (yaw m ,m,YAWM) ;

12.5 Filter code for experimental data

This code is to filter some irregularities from the experimental data to have a more clear graph tostudy.

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1 % FILTERING DATA2 Ts=0.01; tau =0.3 ;3 axf=f i l t e r (Ts/tau , [ 1 (Ts/tau−1) ] , tData . Ax mps2 ) ;4 ayf=f i l t e r (Ts/tau , [ 1 (Ts/tau−1) ] , tData . Ay mps2 ) ;5

6 [B,A] = butte r ( 4 , 0 . 1 , ’ low ’ )7

8 Yawra t e f i l t = f i l t f i l t (B, A, tData . Wz rps )

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TRITA - SCI - GRU 2018:026

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