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.

E

G

E

W

E

B

E

road

road sideline

(global frame)

(road frame) (body frame)

(wheel frame)

Figure 2.1 Diagram of frames about vehicle

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 (E ), which is fixed to earth; road ^G frame (E^road), which is set on the road and changed with the road bank angle; wheel frame (E ), which is set on the tire; body frame (^W 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

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

Qx_{, ,}

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,

Qroad

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

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

Global frame (E^G)

G

Ex G

Ey G

Ez

Body frame (E^B)

B

ex =ξ

B

ey

B

ez

φ

ζ

ξ

W

ey

η= θ

W

ez

W

ex W

ey

ε

Wheel frame (E^W)

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

Global frame (E^G)

G

Ex G

Ey G

Ez

Wheel frame (E^W)

Road frame (E^road)

G y Ey

Ψ = Ψz

Ψx

θroad

road

ex

ϒy

ϒz

x x

ϒ = Ψ

φroad

W road

z z

e =e

W

ex W

ey

r road

ε = −ε ε

road

z z

e = ϒ

road

ey

εroad

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

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

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, coscossinsin 00

where ω_φ,θ,ε is the Euler angular velocity, and in these angular velocity, ω_ε rotates along the axis e^W_z , 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

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. 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

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,

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, 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,

where F_x,y,zi represents effective force in the wheel frame (e^W_x ,e^W_y ,e^W_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

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,

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)

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

σ11) with the vehicle pitch and roll

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 (Q_xQ_y), we can inversely transform the momentum arm (σ₁₁) from the body frame to the wheel frame.

⎥⎥

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 (

Z₁W

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

In the same way, the other momentum arms (σ₂, σ₃, σ₄) can be calculated soon.

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,

( )

cos sin sin

1 1 1 1

body body ex

Fz Fy Fy

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

body body ey

l Fz Fx Fx Fy

sin sin cos

2 2 2

sin sin cos

2 3 3 3 3

sin sin cos

2 4 4 4 4

sb sb

Fxbody Fx Fy B

body body ez

l Fy Fx Fx Fy

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

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 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

sin cos sin cos cos cos

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

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 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

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

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)

where x ,y ,z represent longitudinal, lateral and vertical displacement of CG, respectively, represents effective force from

Fi ith wheel in each direction, for i = 1~4 to represent a

4-wheels vehicle, m_vehicle 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,

( )

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 δ^outer

(body frame)

sb1

1 2

l +l

(wheel frame) e^Bx

B

ey W

ex W

ey

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

cot 1

cot l l

sb

inner

outer − δ = +

δ (2.27)

where δ_outer is the steering angle of the outer tire, δ_inner 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.

sb1 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 δδ δδ

cos 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

14



(2.29)

with K =C₁e^C2(Hi−C3)

⎩⎨

⎧− ≤≤−−

= m g K for H m g K

K g m H for H H

ui i

ui

ui i

i

i / , /

/ ,

where K represents spring stiffness coefficient; C_m, 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 m_zi

Hi 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

D_d K

(wheel frame)

W

ez W

ex

Figure 2.6 Diagram of the passive suspension system

⎥⎥

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 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:

( )

where F_a,b represent the longitudinal/lateral adhesive force, λ is the slip ratio of the tire, α is the slip angle of the tire, and B,C,D,E represent the associated tire parameters.

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,

{

_eiⁱ _i ⁱ _i ⁱ _i

}

ei

i r V

V r

α ω

α λ ω

cos , max

− cos

= (2.34)

with V₁ =(x−^sb₂¹ε)e^W_x +(y+l₁ε)e^W_y

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

Vi ith side. Additionally, when choosing the tire speed (r_eiω_i) as

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