運用狀態觀察器技術之車輛翻覆預測系統

國 立 交 通 大 學

機械工程學系

碩 士 論 文

運用狀態觀察器技術之車輛翻覆預測系統

Vehicle Rollover Prediction System

Using States Observers

研 究 生：許齡元

指導教授：陳宗麟 博士

運用狀態觀察器技術之車輛翻覆預測系統

Vehicle Rollover Prediction System Using States Observers

研 究 生：許齡元 Student：Ling-Yuan Hsu

指導教授：陳宗麟 Advisor：Tsung-Lin Chen

國 立 交 通 大 學

機 械 工 程 學 系

碩 士 論 文

A Thesis

Submitted to Department of Mechanical Engineering College of Engineering

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master of Science

in

Mechanical Engineering July 2006

Hsinchu, Taiwan, Republic of China

運用狀態觀察器技術之車輛翻覆預測系統

學生: 許齡元 指導教授: 陳宗麟 博士

國立交通大學機械工程學系 碩士班

摘 要

在本篇論文中，我們提出了一個車輛翻覆預測系統。此一系統主要是以含有道路狀 況之＂完整車輛模型＂為出發點，藉由狀態觀察器技術來獲得車輛即時動態，並且將即 時動態傳輸至車輛模型中，來預測在未來時間內的車輛動態，進而以在未來時間內的車 輛側傾角，判斷車輛是否即將翻覆。此種預測方法能應用於各種不同動態的車型，並提 供可靠的物理根據來宣告車輛翻覆事件。 此預測系統的挑戰之一為如何建立以完整車輛模型（高階、高度非線性系統）為基 礎之狀態觀察器。我們提出一適用於非線性系統之新型觀察矩陣，藉由此觀察矩陣來簡 化完整車輛模型，並拆解成兩個低階子模型：側傾、橫擺子模型。如此一來，即可針對

兩低階子模型分別建立狀態觀察器，再經由相似於傳統ADI(Alternative Direction Implicit)

之切換式數值演算法，來進行車輛即時動態估測。由本論文中的ADI-like 切換式演算機 制之收斂穩定度分析中可知，此演算機制能成功的使兩個從複雜系統中解析出之子系 統，近似於原複雜系統之動態行為。 由模擬結果得知，當車輛在一斜坡上做快速轉彎之操作行為下，上述之觀察器演算 機制可藉由三種感測器：縱向速度感測器、側向加速度感測器以及懸掛系統位移感測 器，來正確地預測出車輛翻覆之發生。

Vehicle Rollover Prediction System

Using States Observers

Student: Ling-Yuan Hsu Advisor: Dr. Tsung-Lin Chen

Department of Mechanical Engineering

National Chiao-Tung University

Abstract

In this thesis, we present a vehicle rollover prediction method, which employs the “full-car model” accompanied with road conditions and states observer techniques, to predict vehicle dynamics and declare a rollover happening by the vehicle roll angle in future time. This prediction method presents a strong evidence for a rollover occurrence, and the methodology can be widely applied to vehicles with different dynamic characteristics.

Based on the novel observability matrix proposed in this thesis, the “full-car model” is broken down into two subsystems. Two states observers are constructed for each subsystem respectively and do the switching scheme for the vehicle states estimation, which the approach is similar to the conventional alternative direction implicit method (ADI). The proposed ADI-like computation scheme enables a states observer design for a highly nonlinear and high order dynamic system.

Simulation results indicate that, with the following three sensors: longitudinal velocity sensor, lateral accelerometer and suspension displacement sensor, we are able to predict a vehicle rollover occurrence correctly, which is initiated by a quick wheels maneuvering on a slope.

Acknowledgement

本論文得以完成，首先要感謝作者的指導教授陳宗麟老師，在老師的研究領導之下， 能夠周詳地考量本論文中各項難題，並且指引出最佳方向，讓作者能夠輕鬆地面對挑 戰。在此過程中，不但學習到相關專業知識以及技術上的應用，更學習到如何去面對各 種議題，更甚於在技術中找出議題並解決。此兩年的光陰，作者在老師身上學到很多， 最後，在此致上最誠摯的謝意。 同時感謝實驗室學長、同儕與學弟在各項課題的指導、在心靈層面的關懷以及在日 常生活的解悶，並且在課業與研究上皆能互相砥礪求進步，使得作者能夠順利成就本論 文。 順便感謝大學同窗好友，在網路上不時地給予鼓勵，能夠在充滿研究氣息的日常生 活中，增加另一份趣味。 最後僅將本論文獻給最親愛的父親許添富與母親吳沂瀛，感謝他們多年來所付出的 辛勞，從小以開明方式教導作者讀書，今時今日，作者才能成就此論文，希望能與他們 分享成就之快樂；女友雲理，亦在本論文之過程中，能給予相當多的支持，感謝她時常 舟車勞頓，並且時常鼓勵作者學習，使作者能無憂無慮之下完成此論文，在此獻上最誠 摯之謝意。

Contents

摘 要 ...i

Abstract...ii

Acknowledgement...iii

Contents...iv

List of Tables ...vi

List of Figures...vii

Mathematical Notations ...viii

Chapter 1 Introduction ...1

1.1 Motivations and Objectives...1

1.2 Previous Research Survey ...2

1.2.1 Dynamic Modeling of Full-State Vehicle ... 2

1.2.2 Prediction Method in Vehicle Rollover... 2

1.2.3 Neglect of the Vehicle Pitch Motion ... 3

1.2.4 Numerical Algorithm in Switching Scheme ... 3

1.3 Construction of this Vehicle Rollover Prediction System ...4

1.4 Outline of this Thesis...4

Chapter 2 Full-Car Model ...6

2.1 Dynamic Frames of the Vehicle...7

2.1.1 Euler Transformation ... 7

2.2 Sprung Mass System ...10

2.2.1 Vehicle Rotational Motion ... 10

2.2.2 Vehicle Translational Motion... 17

2.3 Unsprung Mass System ...19

2.3.1 Wheel Steering System... 19

2.3.2 Suspension Force... 20

2.3.3 Nonlinear Tire Model ... 22

2.3.4 Wheel Dynamics ... 24

2.4 Road Condition...25

2.5 Summary...27

2.6 Full-Car Model Validation...28

2.7 Conclusions ...28

Chapter 3 System Observability of Full-Car Model...30

3.1 Nonlinear Observability Matrix...30

3.3 Negligence of Pitch Motions ...31

3.4 Integrated Yaw-Roll Model ...32

Chapter 4 Vehicle Rollover Prediction System ...34

4.1 Separated Yaw-Roll Model...34

4.1.1 Vehicle Yaw Model ... 35

4.1.2 Vehicle Roll Model... 36

4.1.3 Separated Yaw-Roll Model Validation ... 39

4.2 Switching Observer Scheme...39

4.2.1 Error Source ... 40

4.2.2 Preliminaries for the Stability Analysis of Switching Computation Scheme 40 4.2.3 Stability Analysis for “Explicit Euler Method” Approximation ... 41

4.2.4 Stability Analysis for “Runge-Kutta Method” Approximation ... 44

4.3 Sensor Selections...48

4.3.1 Sensors for Yaw Model ... 49

4.3.2 Sensors for Roll Model ... 51

4.4 Nonlinear Observer Algorithm ...51

4.5 Block Diagram for the Prediction System...52

Chapter 5 Simulation and Results...53

5.1 Case I ...54 5.2 Case II...54 5.3 Case III ...54 5.4 Case IV ...57 5.5 Case V...57 5.6 Conclusions ...59

Chapter 6 Conclusions and Future Works ...60

6.1 Conclusions ...60

6.2 Future Works ...62

Reference ...64

Appendix ...67

A. The Separation of the Integrated Yaw-Roll Model from Euler Transformation...67

B. Parameters of the Full-Car Model ...67

B.1 Vehicle Inertial and Geometric Parameters ... 68

B.2 Suspension Coefficients ... 68

List of Tables

Table 3.1 States covariance matrix in partial part. (subscript 1: front-left side, 2: front-right

side, 3: rear-right side, and 4: rear-left side)...32

Table 4.1 The eigenvector of the observability grammian when the output is lateral acceleration...49

Table 4.2 The eigenvalue of the observability grammian when the output is lateral acceleration...49

Table 4.3 The eigenvector of the observability grammian when the output is longitudinal velocity ...50

Table 4.4 The eigenvalue of the observability grammian when the output is longitudinal velocity ...50

Table B.1 The inertial and geometric parameters of the full-car model...68

Table B.2 Coefficients of the nonlinear suspension model ...69

Table B.3 Tire geometric parameters...69

Table B.4 Nonlinear tire stiffness coefficients in the longitudinal direction...70

List of Figures

Figure 2.1 Diagram of frames about vehicle ...6

Figure 2.2 Diagram of Euler transformations from the global frame to the body frame ...9

Figure 2.3 Diagram of Euler transformations from the global frame to the wheel frame...9

Figure 2.4 Free body diagrams of the vehicle ...10

Figure 2.5 Ackerman principle ...19

Figure 2.6 Diagram of the passive suspension system ...21

Figure 2.7 Diagram of front-wheel free-body ...24

Figure 2.8 Vehicle motion in the three road conditions. a) the relationship between vehicle yaw angle (ε) and road yaw angle (εroad), b) the vehicle motion on a slop, c) the vehicle motion on a downward ...25

Figure 4.1 Comparison of the full-car model and separated yaw-roll model. a) diagram of the full-car model (in the light gray) and the yaw model (in the dark gray) b) diagram of the full-car model (in the light gray) and the roll model (in the dark gray) ...35

Figure 4.2 Dynamic responses of two models (the integrated yaw-roll model and the separated yaw-roll model) in the general case...38

Figure 4.3 Dynamic responses of two models (the integrated yaw-roll model and the separated yaw-roll model) in the rollover case...38

Figure 4.4 Block diagram of the vehicle rollover prediction system ...52

Figure 5.1 Comparison of vehicle response and rollover prediction system output in Case I, in which the vehicle does not rollover. ...55

Figure 5.2 Comparison of vehicle response and rollover prediction system output in Case II, in which the vehicle rollover. ...55

Figure 5.3 Comparison of vehicle response and rollover prediction system output in Case III, in which the vehicle rollover due to road bank angle...56

Figure 5.4 Comparison of vehicle response and rollover prediction system output in Case IV, in which the vehicle rollover but the prediction failed, due to neglecting the road bank condition. ...56

Figure 5.5 Comparison of vehicle response and rollover prediction system output in Case V.58 Figure 5.6 Comparison between full-car model, separated yaw-roll model with pitch motion and separated yaw-roll model...58

Mathematical Notations

Variable symbol

G E ： global frame W E ： wheel frame B E ： body frame road E ： road frame

Q ： rotation orthogonal tensor for Euler transformation

θ φ

ε, , ： Euler angles presented in body frame, for yaw, roll, and pitch motion

road road road φ θ

ε , , ： Euler angle presented in road frame, for yaw, roll, and pitch motion

r

ε ： relative yaw angle between vehicle yaw angle and road yaw angle

z y

x, , _{： vehicle displacement in longitudinal, lateral, and vertical directions}

ε θ φ

ω , , ： Euler angular velocity along three Euler axes , ,

x y z

ω _{： vehicle angular velocity presented in body frame}

i

ω ： tire angular velocity in wheel frame, for i = 1~4.

L ： angular momentum about the CG of the vehicle body M ： external moment about the CG of the vehicle body

i

σ ： momentum arms associated with the ith external force

body z y x

F_{, ,} ： effective forces presented in the body frame in three directions

z y x

F_{, , ： effective forces presented in the wheel frame in three directions}

road z y x

F_{, ,} ： effective forces induced by road conditions and presented in road frame in three directions

W i

Z ： length variation at the ith suspension in wheel frame

i

ei

r ： effective rolling radius of the ith tire

ρ _{： road curvature}

i

δ ： tire steering angle of the ith tire

i

λ ： slip ratio of the ith tire

i

α ： slip angle of the ith tire

Vehicle parameter symbol

vehicle

m ： total vehicle mass from sprung mass and unsprung mass

s

m ： sprung mass of the vehicle body

ui

m ： unsprung mass of the vehicle at the ith side

z y x

I , , ： moment of inertia of the vehicle along three axes 2

, 1

sb ： tread width of the vehicle at the front/rear side

2 , 1

l ： distance from the vehicle CG to the front/rear axle h ： height of the vehicle CG

Z ： distance from the vehicle CG to the road surface K ： spring stiffness coefficient

m

C ： nonlinear spring stiffness modeling, for m = 1, 2, 3

damper

D ： damper coefficient

i brake

T _{, ： braking torque acting on the ith tire}

i motor

T _{, ： motor torque acting on the ith tire}

Tire parameter symbol

i

r ： real radius of the ith tire

wheel

I ： moment of inertia of the tire along three directions

vertical

K ： tire vertical stiffness coefficient

i

κ ： roll steer coefficient of the ith tire

E D C

B, , , ： characteristic coefficients of the nonlinear tire model

Chapter 1

Introduction

A vehicle rollover prediction system proposed in this thesis is to study this topic of the vehicle safety problem. This system can help drivers know where, when and how the vehicle rollover happens, and provide the dynamic information to rebuild the happened rollover accident. Furthermore, motivations and objective will be introduced in section 1.1. Section 1.2 surveys previous researches about issues extended from this prediction system. Then, the construction and organization of this thesis will be described in section 1.3 and 1.4, respectively.

1.1 Motivations

and

Objectives

In 1999, National Highway Traffic Safety Administration (NHTSA) announced its plan to take rollover stability in the one-to-five star rating system for safety performance. One major driving force behind this initiative was the well-published rollover incidents of several Sport Utility Vehicles and consumer passenger cars. In particular, from CBS news, 62% of all SUV deaths occurred in rollover accidents. It seems fair to say that the rollover stability has become an important measure in the car safety performance.

With the increasing amount of the car, the rise of this vehicle safety problem comes as expected. The vehicle safety performance, such as the above-mentioned star rating system, safety factor, etc., becomes one of the important factors when consumers purchase a car. However, the cost of this vehicle, equipping more safety systems, will deter consumers from buying this vehicle. Therefore, a safety system with low cost and high accuracy is necessary to be planned.

Most of the research works, which study in rollover accidents, were focus on vehicles with higher center of the gravity (CG), such as trucks, trailer, etc. [1] [18] [21] [26]. However, the dynamic maneuver of these heavy vehicles is more different than that of consumer passenger cars, even the SUV. The factor of that is heavy vehicles carry more than four tires

and in some cases, a trailer. Therefore, a proper vehicle rollover prediction system with low cost and high accuracy for four-wheel vehicles is proposed in this thesis.

1.2 Previous Research Survey

In this section, several topics studied in this thesis will be reviewed. Additionally, the topics include: dynamic modeling of full-state vehicle, prediction method in vehicle rollover, neglect of the vehicle pitch motion, and numerical algorithm in switching scheme.

1.2.1 Dynamic Modeling of Full-State Vehicle

The complication of the vehicle model is decided from concerned vehicle states. Most of previous researches were developed their own vehicle models with less degree of freedom (DOF) [4] [7] [10] [23] [27]. In order to keep accurate dynamics of the vehicle, vehicle states, which are not integrated into the vehicle model, will be supplied by empirical parameters. Furthermore, the empirical parameter, verified by the experimental data, may be only valid in some operating regions. However, this method is not suitable for rollover accidents, for the reason that the vehicle rollover may happen in any possible situation.

In contrary, some research works construct a vehicle model with complex dynamics, as shown in VDANL (Vehicle Dynamics Analysis, Non Linear) [2], CarSim [29], EDVSM (Engineering Dynamics Vehicle Simulation Model) [6], ADAMS (Agencywide Documents Access and Management System) [22], etc. With substituting vehicle geometry parameters of this vehicle model, we can obtain accurate dynamics of this vehicle in any possible operating region. However, these vehicle models hiding in the unavailable codes are not suitable for observer and/or controller design.

1.2.2 Prediction Method in Vehicle Rollover

Obviously, a credible vehicle rollover prediction method can effectively lower the amount of rollover accidents. As a result, researchers proposed various prediction methods including: time-to-rollover (TTR), rollover velocity, genetic algorithm predictor (GAP), rollover index

(RI), rollover stability advisor (RSA) and etc. [4] [10] [18] [23] [27] [28] [31]. Most of these methods employ vehicle current states information along with either heuristic formulas or over-simplified vehicle models to predict rollover happenings. These approaches may be applicable to certain rollover events and specific types of vehicles. However, they generally can not be applied to different types of vehicles nor account for different types of rollover events. This is because the vehicle rollover is a consequence of multiple vehicle dynamics, vehicle maneuvering, road conditions, etc. All these factors have to be examined carefully for an accurate vehicle rollover prediction.

1.2.3 Neglect of the Vehicle Pitch Motion

Most of the research work developed their rollover stability measure based on a simplified vehicle model. Furthermore, the frequently neglected vehicle dynamics is the pitch motion [4] [10] [19] [23] [28]. The negligence of pitch motion, or any other dynamics, in the rollover stability could be practical but it needs a feasibility check. Unfortunately, we did not find much theoretical discussion from the published materials. From the dynamic viewpoint, the system observability matrix can elucidate the connections between system output and system states [5]. This technique has been successfully utilized in determining the best sensor locations [11] [24], model reduction [5], etc. Therefore, conceptually, it is possible to adopt this technique in a rollover prediction system to examine the feasibility of a simplified vehicle model and determine the suitable sensors deployment. However, challenges arise for the case that a system of interest is highly nonlinear and involves many states. In that case, the construction of observability matrix would involve intensive math derivations and thus impractical.

1.2.4 Numerical Algorithm in Switching Scheme

Alternative Direction Implicit (ADI) [14] methods have been widely utilized in numerical calculations for partial differential equations. This method breaks a partial differential equation into a set of difference equations and doing the switching scheme mainly to save for the computation time. In this thesis, an ADI-like computation scheme is proposed to break a set of highly nonlinear differential equations into two sets of nonlinear differential equations,

not for the computation time but for obtaining less complicated subsystems. It is because most of control methodologies require vast math derivation and, in practice, can only apply to systems with less complicated math models. Similar to the ADI method, the proposed method gain its advantages at the price of simulation accuracy. Therefore, the stability and accuracy of this ADI-like method need to be resolved before use.

1.3 Construction of this Vehicle Rollover Prediction System

Theoretically, the response of a dynamic system can be well described by a precise system model and associated initial conditions. Stem from this concept, we proposed a novel vehicle rollover prediction system, which composed of an observer-based estimator and a model-based predictor. The estimator uses three sensors (longitudinal velocity sensor, lateral acceleration sensor and suspension displacement sensor) to work with a simplified vehicle model for the observer construction. The predictor uses the vehicle dynamic model along with those estimated vehicle states to predict the vehicle roll angle in future time. The predicted vehicle roll angle can be utilized to declare vehicle rollover in future time and/or determining which rollover prevention measure to implement. Furthermore, the simplified model is excerpted from the vehicle full-car model with the feasibility check via system observability matrix, and it incorporates the road bank condition. This approach, based on the well-defined system model, present a strong evidence for the vehicle rollover prediction.

1.4 Outline

of

this

Thesis

The organization of the thesis is shown as follows.

In chapter 2, the nonlinear full-car model is presented. The sprung mass system and unsprung mass system of the vehicle are discussed and linked together. Furthermore, the road condition is also considered to be integrated into the vehicle modeling.

In chapter 3, the basic concept and procedure of the novel observability matrix along a trajectory are both presented. Then, simulation results provide the evidence for the neglect of

the vehicle pitch motion. After neglecting the vehicle pitch motion, the integrated yaw-roll model is excerpted from the full-car model.

In chapter 4, the vehicle rollover prediction system is described in detail. The components of this prediction system, such as the predictor, estimator, switching scheme, and nonlinear observer, are also introduced, respectively. Additionally, the stability analysis of the switching scheme is proved in the mathematical methodology.

In chapter 5, simulation results are shown to verify this prediction system.

In chapter 6, we summarize conclusions addressed in this thesis and the suggestions of future works.

Chapter 2

Full-Car Model

A full-car model with 21 states is constructed to mimic a moving vehicle on real road conditions. This model contains two parts: a vehicle body (sprung mass) and four wheel-axle assemblies (unsprung mass). In this thesis, the sprung-mass system assumes rigid body motion and its math derivation work mostly follows Hingwe’s dissertation [13]. The unsprung-mass system contains five sub-systems, which are wheel steering system, suspension system, nonlinear tire model, wheel dynamics and road bank conditions. The mathematic models of these subsystems, which often derived for different coordinates system (dynamic frame) at its own convenience, can be integrated into one system via physical principles and Euler angle transformation.

Section 2.1 introduces some dynamic frames of the vehicle system. The sprung mass system and the unsprung mass system of the vehicle are individually introduced in the section 2.2 and 2.3. The road conditions are also considered in the vehicle modeling and described in the section 2.4. The whole dynamics equations of the vehicle model, as discussed above, are summarized in section 2.5, and then, section 2.6 will introduce how to check out the model validation. Section 2.7 describes the conclusion of the vehicle modeling.

G

E

W

E

B

E

road

E

road sideline (global frame) (road frame) (body frame) (wheel frame)

2.1 Dynamic Frames of the Vehicle

Dynamic frames of a vehicle system, most moving objects as well, are often considered for two frames: global frame and body frame. However, these two frame systems are not convenient enough to describe the dynamics of a vehicle system, since a vehicle system can contain lots of subsystems. Therefore, we introduce 4 coordinate systems along with the vehicle dynamics modeling work. They are global frame ( G

E ), which is fixed to earth; road frame ( road

E ), which is set on the road and changed with the road bank angle; wheel frame ( W

E ), which is set on the tire; body frame (E ), which is set on the center of gravity (CG) of B a vehicle. These frame systems are shown in figure 2.1, and they can switch around through Euler angle transformation.

Unlike most of the vehicle modeling works, we introduce the road frame system to accommodate the change of road angle. This road frame system is particularly useful because lots of vehicle motions are performed relative to the road surface, instead of a fixed point on earth. In other words, it is more effective to describe the vehicle dynamics in terms of the road frame than of the global frame.

2.1.1 Euler

Transformation

In this section, we will introduce two Euler transformations for 4 coordinate systems. First Euler transformation is shown in figure 2.2, and three Euler’s angles (ε,θ,φ ) are used to represent the coordinate relationship between the global frame and body frame. In the Euler transformation, yaw motion (ε) rotates along the vertical axis ( ), pitch motion (z θ) rotates

along the lateral axis ( ) and roll motion (y φ ) rotates along the longitudinal axis ( x ). Then,

we can define the proper orthogonal rotation tensor (Q) such that the motion in the global

frame can be transformed to the body frame by the following equations,

(2.1) G z y x G B E Q Q Q E Q E ⋅ ⋅ ⋅ = ⋅ =

with ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = φ φ φ φ cos sin 0 cos sin 0 0 0 1 x Q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = θ θ θ θ cos 0 sin0 1 0 sin 0 cos y Q ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = 1 0 0 cos 0 sin sin 0 cos ε ε ε ε z Q

where represents the transformation matrix from the global frame to the body frame. Therefore, we can describe the vehicle motion in the global frame, wheel frame, and body frame by Euler transformation.

z y x

Q _{, ,}

Second Euler transformation is set between the global frame and the wheel frame to represent the road frame for the coordinate transformation. As shown in figure 2.3, we can describe the road curvatures by the three Euler angles (ε_road, φ_road, θ_road). Again, the proper orthogonal tensor ( ) is defined such that the motion in the global frame can be transformed to the road frame by the following equations,

road Q (2.2) G road road road G road road E Q Q Q E Q E ⋅ ⋅ ⋅ = ⋅ = θ φ ε with ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = 1 0 0 0 cos sin 0 sin cos road road road road road Q ε ε ε ε ε ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = road road road road road Q φ φ φ φ φ cos sin 0 sin cos 0 0 0 1 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = road road road road road Q θ θ θ θ θ cos 0 sin0 1 0 sin 0 cos

where road represents the transformation matrix from the global frame to the road frame. Q_ε,φ,θ

Therefore, we can take these two Euler transformations to derive the vehicle motion in the following sections.

Global frame (EG₎ G x E G y E G z E Body frame (EB₎ B x e =ξ B y e B z e φ ζ ξ W y e η= θ W z e W x e W y e ε Wheel frame (EW₎

Figure 2.2 Diagram of Euler transformations from the global frame to the body frame

Global frame (EG₎ G x E G y E G z E Wheel frame (EW₎

Road frame (Eroad₎

G y Ey Ψ = z Ψ x Ψ road θ road x e y ϒ z ϒ x x ϒ = Ψ road φ W road z z e =e W x e W y e r road ε = −ε ε road z z e = ϒ road y e road ε

2.2 Sprung Mass System

Assuming the rigid body motion, as the free body diagrams shown in figure 2.4, the sprung mass clearly has six degree-of-freedoms, which are three rotational motions and three translational motions for the center of gravity (CG). These motions of the sprung mass are briefly described as follows.

sprung mass (global frame) (body frame) 1 l 2 l 2 sb 1 sb h unsprung mass (wheel frame) G x E G y E G z E B x e B y e B z e CG 1 σ σ12 11 σ

Figure 2.4 Free body diagrams of the vehicle

2.2.1 Vehicle

Rotational

Motion

The vehicle rotation dynamics can be conveniently written in equations in the coordinate that is rotated with the car (body frame) and then transformed back to the coordinate that is fixed to the earth (global frame) by the above-mentioned Euler’s angles (ε ,θ ,φ ) transformation. Hence, the angular velocity along three axes which rotated by Euler’s angles can be expressed as follows,

(2.3) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = φ θ εε θ φ θ ε ε ω φ θ φ θ θ ω φ φ φ φ φ φ φ ω ε θ φ cos coscos sin

sin 0 0 sin cos0 0 0 0 0 0 0 cos sin 0 cos sin 0 0 0 1 0 0          x y z y z z Q Q Q Q Q Q

where ω_φ,_θ,_ε is the Euler angular velocity, and in these angular velocity, ωε rotates along the axis W,

z

e ω_θ rotates along the axis η , and ω rotates along the axis _φ . Therefore, the vehicle angular rate and the vehicle angular acceleration can be expressed in terms of the Euler angle dynamics as follows:

B x e (2.4) φ θ ε φ θ ω φ θ ε φ θ ω θ ε φ ω cos cos sin sin cos cos sin       + − = + = − = z y x (2.5) φ θ φ ε φ θ θ ε φ φ θ φ θ ε φ θ ω φ θ φ ε φ θ θ ε φ φ θ φ θ ε φ θ ω θ θ ε θ ε φ ω sin cos cos sin cos cos cos sin cos cos sin sin sin sin cos cos cos sin                             − − − + − = + − − + = − − = z y x

where ωx,y,z is the vehicle angular rate represented in body frame.

The rotational motion of the vehicle body is to rotate about the roll center (RC), instead of the CG. Hence, the external moment consists of two parts: the angular momentum about the RC measured from the global frame and the angular momentum about the CG measured from the RC frame. Furthermore, because the RC frame is fixed in and move with the body frame, the angular velocity of the RC is the same with that of the CG. Therefore, using the well-known Euler equations of motion, one can presents the vehicle rotational dynamics as follows. M L dt dL b = × + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{ω (2.6)}

where L is the angular momentum about the CG of the vehicle body, and M is the

external moment about the CG of the vehicle body ( ). Furthermore, we can expand the above equation in terms of the angular velocity in body frame.

z y x M M M , ,

(

)

(

)

(

x y

)

x y z z z x z x z y y y z y z y x x x I I I M I I I M I I I M ω ω ω ω ω ω ω ω ω − − = − − = − − =    (2.7)

where is the moment of inertial of the vehicle body. Then, from equations (2.4), (2.5), and (2.7), the vehicle rotational motion can be represented in terms of Euler’s angles as follows, z y x I , ,

(

)(

)

(

)(

)

(

φ ε θ

)(

θ φ ε θ φ

)

φ θ φ ε φ θ θ ε φ φ θ φ θ ε φ θ φ θ ε φ θ θ ε φ φ θ φ ε φ θ θ ε φ φ θ φ θ ε φ θ φ θ ε φ θ φ θ ε φ θ θ θ ε θ ε φ sin cos cos sin sin cos cos sin cos cos cos sin cos cos sin sin cos cos sin sin sin sin cos cos cos cos sin sin cos cos cos sin                                       + − − − + + + = + − + − − − − + − + = + + − + − − + = − z x y z z y z x y y x y z x x I I I I M I I I I M I I I I M (2.8)

Therefore, using the external moment discussed in the following section, we can obtain the information of Euler’s angles, and then, can also obtain the vehicle angular velocity in body frame.

2.2.1.1 External

Moment

The external moment contains the external forces and the associated moment arms. In order to match the above-mentioned Euler equations of motion, we will discuss the external moment on the CG of the vehicle body. Then, we can easily express the external moment as follows, (2.9)

(

∑

= × = 4 1 i body i i F M σ

)

where σi is the ith moment arms associated with the ith external force, and is the body

i

F ith

external force in body frame.

2.2.1.2 External

Forces

External forces of the vehicle body mainly come from tire forces and can be expressed as follows, (2.10)

∑

∑

∑

∑

= = = = ⋅ + ⋅ + ⋅ = 4 1 4 1 4 1 4 1 i W z zi i W y yi i W x xi i body i F e F e F e F

where F_x,y,zi represents effective force in the wheel frame (eW_x ,eW_y ,eW_z ) from ith wheel in three directions, for i = 1~4 to represent a 4-wheels vehicle. However, the structure of tire forces does not suit to discuss in this section, and we will go into detail about this topic in section 2.3. Using the transformation matrix ( ), we can transform tire forces from the wheel frame to the body frame.

y xQ Q (2.11) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + − + + − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ φ θ φ φ θθ φ φ θ φ θ θ φ θ φ φ θθ φ φ θ φ θ θ cos cos sin cos sin sin cos cos sin sin sin cos cos cos sin cos sin sin cos cos sin

sincos 0 sin

zi yi xi zi yi xi zi xi zi yi xi body zi body yi body xi F F F F F F F F F F F F F F

Therefore, external forces in body frame are obtained, and in the next section, momentum arms in body frame will be introduced.

2.2.1.3 Momentum

Arms

We take one of the momentum arms as the example for simplicity. As shown in figure 2.4, it will clearly find out where the momentum arm (σ₁, the blue dash line) is. This momentum arm is composed of two parts: σ₁₁ and σ₁₂ (the red dash lines), as shown in figure 2.4. We will separately discuss two momentum arms that locate in the different frames.

Firstly, the momentum arm (σ₁₁) locates in body frame. From the figure 2.4, we can intuitively set the CG of the vehicle body as the origin and write down the length of the momentum arm as vector term,

B z B y B x e h e sb e l 2 2 1 1 11 = + − σ (2.12)

Secondly, the momentum arm (σ₁₂) locates in wheel frame. Again, we can also set the edge of the vehicle body as origin and write down the length of the momentum arm in vector term,

(2.13) W z e z Z ) ( − −

where Z represents the height of the CG in the static situation, and represents the height

variation of the CG. However, the momentum arm (

z

11

motion will induce the length variation of the momentum arm (σ₁₂). Here, we assume the length variation largely depends on the z axis in wheel frame for simplicity. Using the transformation matrix (QxQy), we can inversely transform the momentum arm (σ11) from the

body frame to the wheel frame.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − 2 2 cos cos sin cos

sinsin sin cos cos sin sin 0 cos * * *** * 1 1 1 1 _h sb l ZW θ φ φ θ φ φ θ φ φ θ θ θ (2.14) W z W e h sb l Z ⎟ ⎠ ⎞ ⎜ ⎝ ⎛_{− + −} = θ θ φ cosθcosφ 2 sin cos 2 sin 1 1 1 (2.15)

where is the length variation at the front-left suspension in the wheel frame. Hence, using (2.13) and (2.15), we can obtain the partial momentum arm (

W Z₁ 12 σ ) in wheel frame,

(

)

W z e z Z h sb l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛_{− + − − −} = θ θ φ θ φ σ cos cos 2 sin cos 2 sin 1 1 12 (2.16)

Two of partial momentum arms, as shown in equation (2.12) and (2.16), are derived in the different frames. However, the external forces ( ), as shown in equation (2.11), are represented in the body frame. Therefore, we should transform the momentum arm to the body frame for calculation. Again, using the transformation matrix ( ), we can transform this momentum arm (

body z y x F_{, ,} y xQ Q 12

σ ) from the wheel frame to the body frame,

(2.17) B E ⋅ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − θ φ σσ θ θ θ σ σ φ θ φ φ θθ φ φ θ φ θ θ cos cos sin cos sin 0 0 cos cos sin cos

sinsin sin cos cos sin sin 0 cos 12 12 12 12

Then, the whole momentum arm (σ₁) in body frame can be composed of (2.12) and (2.17), shown as follows, B E l l h sb h sb _⋅ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − = θ φ σ φ θ σ θ σ θ φ σσ θ θ θ σ σ cos cos sin cos sin cos cos sin cos sin 12 12 1 12 1 12 12 12 1 1 1 2 2 2 2 (2.18)

In the same way, the other momentum arms (σ₂, σ₃, σ₄) can be calculated soon. ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − + − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − + − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − + = θ φ σ φ θ σ θ σ σ θ φ σ φ θ σ θ σ σ θ φ σ φ θ σ θ σ σ cos cos sin cos sin cos cos sin cos sin cos cos sin cos sin 42 42 2 42 2 2 32 32 2 32 2 3 22 22 1 22 1 2 2 2 2 2 2 2 h sb h sb h sb l l l (2.19) with

(

)

W z e z Z h sb l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛_{− − − − −} = θ θ φ θ φ σ cos cos 2 sin cos 2 sin 1 1 22

(

)

W z e z Z h sb l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{− − − −} = θ θ φ θ φ σ cos cos 2 sin cos 2 sin 2 2 32

(

)

W z e z Z h sb l ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{+ − − −} = θ θ φ θ φ σ cos cos 2 sin cos 2 sin 2 1 42

From the above two sections, we obtain two essential materials for the external moment, which are the external forces in body frame and the momentum arms in body frame. Therefore, we can derive the external moment in the next section.

2.2.1.4 External

Moment

Arrangement

Using equations (2.11), (2.18), and (2.19), the external moment, shown in equation (2.9), can be expanded as follows,

(

)

12 22 32 42 12 1 2 2 1 2 2 2 2 2 2 2 2 1 ₂ cos 1 1 1 cos 2 2 2 cos 3 3 3 cos 4 4 4

cos sin sin

1 1 1 1 1 sb h sb h sb h sb h h body body F_z F_y F_y body body F_z F_y F_y B ex body body F_z F_y F_y body body F_z F_y F_y body body l F_z F_x F_x F_y l F_z M θ θ θ θ φ θ φ σ σ σ σ σ + + − + + − + + + + − − − − − + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

(

)

(

)

(

)

(

)

22 32 42 12 2 2 2 1 2

cos sin sin

2 2 2 2

cos sin sin

2 3 3 3 3

cos sin sin

2 4 4 4 4

sin sin cos

1 1 1 1 1 1 2 h h h sb sb body body F_x F_x F_y B ey body body l F_z F_x F_x F_y body body l F_z F_x F_x F_y body body l F F F F y x x y body l F_y φ θ φ φ θ φ φ θ φ φ θ φ σ σ σ σ − − − − − − − − − − + + + + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

(

)

(

)

(

)

22 32 42 1 2 2 2 2 2

sin sin cos

2 2 2

sin sin cos

2 3 3 3 3

sin sin cos

2 4 4 4 4 sb sb body F_x F_x F_y B ez body body l F_y F_x F_x F_y body body l F_y F_x F_x F_y φ θ φ φ θ φ φ θ φ σ σ σ + − − + + − − − + − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ _(2.20)

Then, substituting (2.11), (2.18), and (2.19) into (2.20), the partial moments ( , , ) can be expressed as follows,

x M M_y M_z

(

)

(

)

(

)

(

)

(

)

1 2 1 2 2 2 2 2 2 2

sin cos sin cos cos

1 1 1

sin cos sin cos cos

2 2 2

sin cos sin cos cos

3 3 3

sin cos sin cos cos

4 4 4

sin sin cos cos sin

1 1 1 sin sin co 2 2 x sb sb sb sb h h F_x F_y F_z F_x F_y F_z F_x F_y F_z F_x F_y F_z F_x F_y F_z F_x F_y M θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ φ − + − − + − − + + − + + + + + + =

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 2 1 2 2 1 2 2 2 2 2 s ₂cos sin

sin sin cos cos sin

3 3 3

sin sin cos cos sin

4 4 4

sin cos sin cos cos cos

1 1

sin cos sin cos cos cos

1 2

sin cos sin cos cos

2 h h sb h sb h sb h F_z F_x F_y F_z F_x F_y F_z l Z _y l Z _y l φ θ φ θ φ φ θ φ θ φ φ θ φ z F z F θ θ φ θ φ θ θ θ φ θ φ θ θ φ θ φ + + + + + + + + − + − − − + − − − − − +

(

− − −

(

)

)

θ

(

)

(

2

)

2 2 cos 3

sin cos sin cos cos cos

2 sb h Z z F_y l Z z F_y4 θ θ θ φ θ φ θ − + + − − − (2.21)

(

)

(

)

(

)

(

)

(

) (

)

(

)

2 2 2 2

sin cos sin cos cos

1 1 1 1

sin cos sin cos cos

1 2 2 2

sin cos sin cos cos

2 3 3 3

sin cos sin cos cos

2 4 4 4

cos sin cos sin

1 1 2 2 cos sin c 3 3 4 y h h h h l F_x F_y F_z l F_x F_y F_z l F_x F_y F_z l F_x F_y F_z F_x F_z F_x F_z F F F x z x M θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ θ θ θ θ θ − − + − − + + − + + − + − − − − − − − =

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1 2 2 1 2 2 2 2 2 2 2 2 os ₄sin

sin cos sin cos cos cos sin sin

1 1

sin cos sin cos cos cos sin sin

1 2

sin cos sin cos cos cos sin sin

2 3

sin cos sin co

2 sb h sb h sb h sb h F z l Z z F_{x y} l Z z F_{x y} l Z z F_{x y} l θ θ 1 2 3 F F F θ θ φ θ φ φ θ φ θ θ φ θ φ φ θ φ θ θ φ θ φ φ θ φ θ θ φ − − − + − − − − − − − − − − − − − − − − −

−

(

+ − s cosθ φ−

(

Z−z

)

)

(

F_x4cosφ−F_y4sin sinθ φ

)

(2.22)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1 1 2 2 2 2

sin sin cos cos sin

1 1 1 1

sin sin cos cos sin

1 2 2 2

sin sin cos cos sin

2 3 3 3

sin sin cos cos sin

2 4 4 4

cos sin cos sin

1 1 2 2 cos sin 3 3 z sb sb sb s l F_x F_y F_z l F_x F_y F_z l F_x F_y F_z l F_x F_y F_z F_x F_z F_x F_z F_x F_z M θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ φ φ θ φ θ θ θ θ θ θ + + + + + − + + − + + − − + − + − − =

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 2 1 2 2 1 2 2 2 2 2 2 cos sin 4 4

sin cos sin cos cos sin sin cos

1 1

sin cos sin cos cos sin sin cos

1 2

sin cos sin cos cos sin sin cos

2 ₂ 3 sin 2 b sb h sb h sb sb F_x F_z l Z z F_{x y} l Z z F_{x y} h l Z z F_{x y} l θ θ 1 2 3 F F F θ θ φ θ φ φ θ φ θ θ φ θ φ φ θ φ θ θ φ θ φ φ θ φ θ − + − + − − − + + − − − − − + + − − − − + + + ⎛ ⎞ ⎜ ⎟ ⎝ ⎠

(

)

(

cos sin 2cos cos

)

(

4sin 4sin cos

)

h

Z z F_x F_y

θ φ− θ φ− − φ+ θ φ

(2.23)

The derivation of the vehicle rotational motion is complete here. Therefore, vehicle angular velocity presented in body frame can be obtained by Euler’s angles.

2.2.2 Vehicle

Translational

Motion

Using Newton’s equations, we can clearly express the linear motion for the CG of the vehicle body. However, we should check what terms the acceleration contains. Therefore, we can obtain the vehicle translation dynamics.

2.2.2.1 Newton’s

Equation

The vehicle translation dynamics can be conveniently written in equations by Newton’s equation as follows: (2.24)

∑

∑

∑

− = ⋅ = ⋅ = ⋅ g F a m F a m F a m zi z vehicle yi y vehicle xi x vehicle

where x ,y ,z represent longitudinal, lateral and vertical displacement of CG, respectively,

represents effective force from

i

F ith wheel in each direction, for i = 1~4 to represent a 4-wheels vehicle, mvehicle is the total vehicle mass from sprung mass and unsprung mass, and

g is the earth gravity.

2.2.2.2 Acceleration

The acceleration contains not only linear acceleration along three axes, but also inertial acceleration induced by the angular velocity and angular acceleration. Therefore, the acceleration can be written down as follows,

( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≅ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ × ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ≅ × + = × × + = z y x a a a z x y y x z y x z y x V a r a a                     ε ε ε θφ ω ω ω . 0 . 0 (2.25) where T z y x a a a

a=[ , , ] represents the acceleration in each direction, ω≅[φ,θ,ε]T

represents the angular velocity along each direction, and T

z y x

V =[, , ] represents the velocity in each direction. Here, the vehicle angular velocity (ω_x,ω_y,ω_z), shown in equation (2.4), contains many terms of Euler’s angles. However, we assume the vehicle angular velocity is close to the Euler angular velocity ( ), and two of the Euler angular velocity, ( ) are smaller than the other angular velocity, (

ε θ φ ,,  

θ

φ_,  _{ε ). Hence, we only consider the}

acceleration induced by the vehicle yaw angular velocity (ε ), as shown in equation (2.25). Therefore, substituting equation (2.25) into equation (2.24), we can write down the equations of the linear motion as follows,

(2.26)

∑

∑

∑

− = = + = − g F z m F x y m F y x m zi vehicle yi vehicle xi vehicle ) ( ) ( ) (           ε ε

The derivation of the vehicle translational motion is complete here.

2.3 Unsprung Mass System

The unsprung mass, which consisted of axles, chassis, and four tires, is crucial to a full-car modeling. Its dynamic characteristics are described in various subsystems for expression clarity, which include wheel steering system, suspension system, tire model, wheel dynamics and road bank condition. The dynamics of these subsystems are first discussed individually and put back together via physical principles and Euler angle transformation.

2.3.1 Wheel Steering System

IC inner δ δouter (body frame) 1 sb 1 2 l +l (wheel frame) eBx B y e W x e W y e

Figure 2.5 Ackerman principle

The Ackerman Steering principle is to ensure a vehicle can be smoothly cornering. As shown in figure 2.5, the vehicle, turning slickly around the instantaneous center (IC), has the

different angles at the outer and inner tire. This principle specifies the angle relations between steering wheel angle, inner tire angle and outer tire angle [9]. The equation from the simple geometry in figure 2.5 can be written as follows:

2 1 1 cot cot l l sb inner outer − δ = ₊ δ (2.27)

where δouter is the steering angle of the outer tire, δinner is the steering angle of the inner tire,

represents the front tread width, and represent the distance from the CG to the front/rear axle.

1

sb l_1,2

After each tire angle is specified, the adhesive force generated by tires [15] [16] can be transformed from the wheel frame to the body frame. These force outputs are then fed into equations (2.8) and (2.26) for further derivation.

i bi i ai yi i bi i ai xi F F F F F F δ δδ cosδ sin sin cos + = − = (2.28)

where δ_i is the steering angle of the ith tire, F_a,b is the longitudinal/lateral tire force of the ith tire, for i = 1~4 to represent 4-wheels. Additionally, this full-car model is set front-wheel steer (δ₃ =δ₄ =0).

2.3.2 Suspension

Force

Without losing much generality, a spring-damper system is considered for the vehicle suspension system. Most of the suspension-modeling works assume linear operations. However this assumption is likely to be erroneous in rollover incidence since a rollover usually accompanied with suspensions lift-off on one end and reach compression limits on the other end. The suspension at the lift-off end generates force to balance its own wheel weight and produces no net force on the vehicle body. The suspension on the other end reaches its maximum compression limit and the output force gradually saturated. For these reasons, the spring coefficient is modified to be nonlinear to handle these extreme cases. From the figure 2.6, the equations of suspension force can be written as follows:

∑

= + + = i ui i damper i zi H z g m H D KH F 4 1  (2.29) with ( ) 1 2 3 C H C _i e C K = − ⎩ ⎨ ⎧ − ≤ − ≤− = K g m H for K g m K g m H for H H ui i ui ui i i i / , / / ,

where K represents spring stiffness coefficient; Cm, m = 1, 2, 3 for nonlinear spring

stiffness modeling, represents damper coefficient, represents unsprung mass from each tire weight and represents spring compression at

damper

D mzi

i

H ith wheel. Lastly, the mean

value of the displacement from each suspension is the vertical displacement of the unsprung-mass system.

The calculation of the spring compression at each suspension mainly focuses on the height induced by the vehicle pitch and roll motion. As discussed before, this induced height has been shown in equation (2.16). However, when considering the equation (2.16), we set the origin at the edge of the vehicle body. At this moment, we should set the origin at the bottom of the suspension system, shown in figure 2.6. In this regard, we will not consider the height (Z ) of the CG, because of concerning the spring compression. Therefore, the spring

compression at each suspension can be written down as follows,

The part of Vehicle

K D_d (wheel frame) W z e W x e

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + − − − + + − − + + − + − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = z h sb l z h sb l z h sb l z h sb l H H H H H φ θ φ θ θ φ θ φ θ θ φ θ φ θ θ φ θ φ θ θ cos cos 2 sin cos 2 sin cos cos 2 sin cos 2 sin cos cos 2 sin cos 2 sin cos cos 2 sin cos 2 sin 2 2 2 2 1 1 1 1 4 3 2 1 (2.30)

Additionally, suspension forces will cause the deflection of tires and change the radius of tires. Therefore, we should consider this variation in radius for further derivation.

vertical zi i ei K F r r = − (2.31)

where rei is the effective rolling radius of ith tire, is the real radius of ri ith tire, and

is the tire vertical stiffness.

vertical

K

2.3.3 Nonlinear

Tire

Model

The study of the tire model in previous research can be classified into three approaches: empirical, physical, and hybrid model [17]. In this thesis, we use the hybrid model, which is named the magic formula tire model [15] [16], for its accuracy. The forces generated by the tire are obtained from the magic formula tire model and associated tire parameters, used in this simulation, are excerpted from Feng’s dissertation [7]. This nonlinear tire model takes the vertical loads to identify tire parameters, and uses slip ratio, slip angle, and tire parameters to get tire forces. In this section, the construction of the nonlinear tire mode will be described.

2.3.3.1 Pacejka’s Magic Formula Tire Model

From [15] [16], the magic formula tire model is shown as follows:

(

)

(

)

[

]

{

}

( )

(

[

{

α β α λ α λ y y y y y y b x x x x x x a B B E B C D F B B E B C D F 1 1 1 1 tan tan sin tan tan sin − − − − − − = − − =

)

]

}

(2.32)

where F_a,b represent the longitudinal/lateral adhesive force, λ is the slip ratio of the tire,

Additionally, the slip ratio is defined as the respective speed difference between the tire and vehicle at the each side. The slip angle is defined as the respective angle difference between the tire and vehicle at the each side. These two physical quantities are both induced by the frictional coefficient of the road, and are the best suitable parameter of the tire force.

According to the magic formula, the tire longitudinal and lateral forces are functions of slip ratio, slip angle and tire parameters. Moreover, tire parameters change with vertical loads and thus they need to be calculated in real-time. Additionally, the tire “self-alignment torque” and “longitudinal-lateral force coupling” effect, mentioned in [15] [16], are neglected in this simulation for simplicity.

However, the lateral force has the other component, mentioned in [7], are also considered in this thesis. Hence, the total lateral force can be expressed as,

( )

(

)

[

]

{

α − β − α

}

+γφ = − − y y y y y y b D C B E B B

F sin tan 1 tan 1 (2.33)

where γ is the respective coefficients from the camber thrust, and φ is the Euler roll angle.. Additionally, the camber thrust is assumed linear respect to the Euler roll angle for simplicity [7].

2.3.3.2 Slip Ratio and Slip Angle

As discussed before, the slip ratio depends on the vehicle speed and the tire speed in the longitudinal direction. Hence, the slip ratio is expressed as follows,

{

ei i i i

}

i i i ei i V r V r α ω α ω λ cos , max cos − = (2.34) with W y W x y l e e x V₁ =(−sb₂1ε) +(+ ₁ε) W y W x y l e e x V₂ =(+sb₂1ε) +(+ ₁ε) W y W x y l e e x V₃ =(+ sb₂2ε) +(− ₂ε) W y W x y l e e x V₄ =(−sb₂2ε) +(− ₂ε)

where r_ei is the effective rolling radius of the ith tire, ω_i is the ith tire angular velocity, and is the vehicle speed at the

i

the denominator, it means the vehicle does the tracking maneuver. Furthermore, when choosing the vehicle speed (V_icosα_i) as the denominator, it means the vehicle does the braking maneuver.

As discussed before, the slip angle value depends on the tire attitude and vehicle attitude. The tire attitude contains steering angle, sideslip angle, roll steer, kingpin inclination etc. However, in this thesis, the slip angle only considers the steering angle, sideslip angle, and roll steer. Therefore, the slip angle is expressed as follows,

(2.35) φ κ β δ αi = i − i − i −1 tan with 1 1 1 1 ( )( ₂ ) − − + = ε ε β y l  x sb  1 1 1 2 =( + ε)( + ₂ ε)− β y l  x sb  1 2 2 3 ( )( ₂ ) − + − = ε ε β y l  x sb  1 2 2 4 ( )( ₂ ) − − − = ε ε β y l  x sb 

where βi is the side slip angle at ith tire, and κi is the roll steer coefficient. Additionally,

we assume the roll steer angle, induced by the vehicle roll motion, is also linear respect to the Euler roll angle for simplicity [7].

2.3.4 Wheel

Dynamics

F_a1 T_m1 T_b1 r₁ (wheel frame) W z e W x e

As shown in figure 2.7, the spinning wheels are accelerated by motor torque and decelerated by both braking torque and adhesive tire forces.

i motor i brake ai i i wheel rF T T I ω =− − _, + _, (2.36)

where Iwheel represents moment of inertia of the tire, ωi represents the angular rate of the

each tire, represents the effective rolling radius of the each tire, represents the braking torque acting on the each tire, and represents the motor torque acting on the each tire. Additionally, this full-car model is set front-wheel drive (

i r T_brake,i i motor T _, 0 4 , 3 , = motor = motor T T ).

Thanks to the equation obtained for wheel dynamics, we are able to link wheel angular velocity with the power train system, braking system and adhesive tire force all together.

2.4 Road

Condition

As shown in figure 2.8, the road bank condition, which include curves and slopes of roads, are introduced into the vehicle dynamics modeling by inserting a “road frame” in between the conventional “global frame” and “body frame”.

road φ road θ road ε ε r ε a)

(body frame) (body frame)

(global frame) (global frame) (road frame) (body frame) b) c) O road sideline (road frame) (road frame) (global frame) G x E G y E G z E B x e B y e ezB G z E G y E G x E B y e B x e B z e road x e road z e road y e road z e road x e road y e vehicle m g mvehicleg

Figure 2.8 Vehicle motion in the three road conditions. a) the relationship between vehicle yaw angle (ε) and road yaw angle (εroad), b) the vehicle motion on a slop, c) the vehicle motion on a downward

These road curvatures can be intuitively described by the three Euler angles (εroad, ,

road

φ θroad) for the coordinate transformation. Since the rollover incidence is declared by the