Part.1 L t l V hi l D i Lateral Vehicle Dynamics

Part.1

L t l V hi l D i

Lateral Vehicle Dynamics

1 Vehicle Dynamic Model 1. Vehicle Dynamic Model 2. Planar Model

3. Tire Models 4. Bicycle Model

5. Understeer/oversteer

6 Dynamic model in terms of error w.r.t. road 6. Dynamic model in terms of error w.r.t. road 7. lane keeping model

8 V hi l St bilit C t l

1

8. Vehicle Stability Control

8. Vehicle Stability Control 8. Vehicle Stability Control

8 1 Bi l d l N li d Li d l

8.1 Bicycle model: Nonlinear and Linear models 8.2 Phase Plane Analysis

8.3 Phase Plane Analysis of Bicycle model 8.4 Electronic Stability Program (ESP)

8.5 Vehicle Stability Control Algorithm

8.1 2DOF Bicycle Model

• Assumption of 2DOF Bicycle Model

1) Longitudinal speed is Constant ( )

2) B d li l i ffi i l ll ( )

0, 0 ax   

/ i 0 1

  

2) Body slip angle is sufficiently small. ( )

3) Left and right slip angles are identical. ( ) / , sin 0, cos 1

y x f f

v v

     

,

f fL fR r rL rR

     

f

( ) 2 2

y y x yf yr

F  m a   m v    F  F



• y-axis Motion Dynamic Equation

2Ff f

f

2 2

z z z f yf r yr

M  H     I  l F   l F



• yaw-axis Motion Dynamic Equation

l

2Fyf

z z z  f yf r yr

 l

m a 



• Dynamic Equation of 2DOF Bicycle Model 2F  2 F

 

l

r



2 2

2 2

yf yr

x

f yf r yr

F F

m v

l F l F

 



  

   

   

        

   

 



yr 3

Top View 2F Iz

 

 

▲ Lateral Tire Force Model



Tire Slip Angle: small angle approximation

f v_x



• Linear Tire Slip Angle at Low Slip Angle

 

tan  if  1

Front Slip Angle

v l  v l 

tan1 ^{y f y f}

f f f f

x x

v l v l

v v

 

       ^{ }    ^{ }

v l ^

Front Tire Rear Slip Angle

1 vy lr  vy lr 

 ^{     }

y r

v  l 

tan1 ^{y r y r}

r

x x

v v

 

      

v



Linear Lateral Tire Model

• Linear Lateral Tire Force at Low Slip Angle Front Slip Angle

4000 5000

[N]

tyi i i

F C 

*

*

* 0( _i , )

i

i

yf ty m tzi

F  F   F

  

 

3000 4000

ire Force [

_{f f f f} ^{y f}

x

v l

C C

v

 ^ ^{ } ^

    

 

 

2000

Lateral T

Ci

Rear Slip Angle

*

*

* 0( i , )

i

i

yr ty m tzi

F  F   F

  

0 2 4 6 8 10 12

0 1000

i

y r

r r r

x

v l

C C

v



 ^{  } ^

    

 

 

Linear Lateral Tire Force

0 2 4 6 8 10 12

Slip Angle [deg]

 

Where, C_i  Cornering Stiffness

5

State Equation of 2 DOF Linear Bicycle Model

• Dynamic Equation of 2DOF Bicycle Model

• y-axis Motion Dynamic Equation

( ) 2 2

2 2

y y x yf yr

y f y r

f f r

F m a m v F F

v l v l

C C

v v

 

 



         

   

   

        

   



 

• yaw-axis Motion Dynamic Equation

x x

v v

   

2 2

2 2

z z z f yf r yr

y f y r

f f f r r

M H I l F l F

v l v l

l C l C

v v



 



        

   

   

          

   



 

x x

v v

   

• Define state = [Body Side Slip Angle, Yaw Rate]

State Equation of 2 DOF Bicycle Model

• Define state = [Body Side Slip Angle, Yaw Rate]

 

 

St t E ti f 2DOF Bi l M d l x 

  

 

• State Equation of 2DOF Bicycle Model

2

2( ) 2( ) 2

f r 1 f f r r f

f

x x x

C C l C l C C

m v  m v  mv 



        

       

     

     

 

2 2

2( ) 2( ) 2

x x x

f f r r f f r r f f

f

z z x z

x

l C l C l C l C l C

I I v I



   

     

                    

  

 

 



2

2( ) 2( )

1

2(

f r f f r r

x x

C C l C l C

m v m v

l C

   

  

 

  ^{2 2}

2

) 2( ) 2

f x

f

C mv l C

l C l C l C

 





   

     

    

 2(l_f C_f            

 ) 2( ) 2

f f

r r f f r r

z z x z

f

l C

l C l C l C

I I v I

Ax B





          

    

    

 

 

7

8. Vehicle Stability Control 8. Vehicle Stability Control

8 1 Bi l d l N li d Li d l

8.1 Bicycle model: Nonlinear and Linear models 8.2 Phase Plane Analysis

8.3 Phase Plane Analysis of Bicycle model 8.4 Electronic Stability Program (ESP)

8.5 Vehicle Stability Control Algorithm

8.2 Phase Plane Analysis

▪ a graphical method for analysis of dynamic behavior of a systems

▪ Phase plane analysis is limited to second-order systems.

▪ For second order systems solution trajectories can be represented by curves in the plane

▪ For second order systems, solution trajectories can be represented by curves in the plane, which allows for visualization of the qualitative behavior of the system.

▪ In particular, it is interesting to consider the behavior of systems around equilibrium points

▪ Consider the second-order system described by the following equations:

1 ( ,1 2) x₁  P x x_{1 2}

2 1 2

( , ) ( , )

x x x

x  Q x x

where,

and are states of the system x₁ and are states of the system.x₂

and are nolinear functions of states.

x x

P Q

▪ Slope of the phase plane trajectoryp p p j y

2 2 1 2

1 1 1 2

( , )

( )

dx dx dt Q x x dx  dt dx  P x x

9

1 1 ( ,1 2)

dx dt dx P x x

1) Equilibrium points of the system

0 P( _{1 2} )

1 2

0 ( , )

0 ( , )

e e

e e

P x x Q x x





Example:p

▪ Find the equilibrium points of the system described by the following equation:

 ^{2 3}

x  x a  x

 

▪ Put the system in the standard form by setting x  x₁, x  x₂

1 2 1 2

2

( , ) x  x  P x x

 ^{2 3}

2 1 2 ( ,1 2)

x  x a  x Q x x

▪ The slope of the phase plane trajectoryp p p j y





2 1 2

1 1 2 2

( , ) ( , )

x a x

dx Q x x

dx P x x x

 

 

2) Investigate the linear behavior about an equilibrium points

1 1  1

2 2 2

e e

x x x

x x x





 

 

▪ From above equation,

1 2

P P

x x

x x

x   a x b x a b x

   

     

   

 ₁  _{1 2}   _{1 2}    ₁

2 1 2 2

1 2

1 2

e e

e e

x x

x a x b x a b x

x Q Q c x d x c d x

x x

x x

   

   

 

    

           

             

          

  

 



▪ Characteristic equation

^{s  a} ^{s d}^{bc  s2  (a d) s ad bc 0}

▪ Eigen value

 

2 1,2

( ) ( ) 4

2

a d a d ad bc

  ^{     }

11

2

3) Possible Case

x

▼ are real and negative. ▼ are real and positive.

1 and 2

  ₁ and ₂

x

x

x

x

x



x

1



▼ ₁ and ₂ are real and opposite signs. ▼ ₁ and ₂ are complex and negative real parts.

x

x

x

x

x



x



< Saddle Point > < Stable Focus >



where   0 

13 where, ₁  0 ₂

▼ ₁ and ₂ are complex and positive real parts. ^▼ are complex and zero real parts.

1 and 2

 

x

x

x

x

x

Example: Satellite Control



▪ Consider a satellite control system as shown in the figure shown below:

( , ) T( , )

 

 

< Satellite Control System >

▪ The equation of motion can be written as follows:

I   T

▪ Rewriting the equation

T u

^

15

I u

  

1. PD Control

▪ Controller

1 2

( , )

u       k  k 

▪ Closed system

1 2

u k k

      

2 1 0

k k

  



    

 

x

▪ Controller can be designed to make eigen values to stable focus

1 1 2 x  x  x

 



x

2 2 1 1 2 2

x  ^ x      u k x k x



   



   

2. On/Off Controller

▪ The slope of the phase plane trajectory

T d d d d

u u

I dt d dt d

   

 

 

          

 

I dt d dt d

d u

d



 ^





0 d 0

u d



   

Case.1

^



17

m m

u u u d

d



 

   



 

Case.2 ^m

m

u u u d

d



 

     



 

Case.3

1 2

constant 2

m

m

d u d

u

  

 

  

  

 

 1 ²

constant 2

m

m

d u d

u

  

 

   

  

 



2 2





 

( 0) um

d   if  

Case.4 ( 0)

sgn( )

m

m

u u d

d u

elsewhere d

if 

 

 

 

 

   

 







Case.4

1 2

constantt ( 0)

2 ^m

d

u if

 

     

1 2

constant

2 u^m   elsewhere







19

Case.5 Lead Compensation/On-Off Controller

sgn(  ) u   um   a 

um

1 1

+ u  ^ 

um



1 s

1 s +

-

u   

0

s  a

< Block Diagram of Closed System >

▪ Sliding Region

s  a 

( 0), 0 ( ) 0 ^{a t}

s a

s a t

f e

i

 

    ^{ }

  

      

Case.5 Lead Compensation/On-Off Controller

▪ From Lyapunov Theory

1 2

0 and 0

V  2 s  V   s s System is stable.

▪ if (s>0)

( ) ^ ^ 0



 

0 s 

sgn( ) 0

m m

m

s u s a u a

u a

 



         

 

u

u

▪ if (s<0)

 a

 a

sgn( ) 0

m m

m

s u s a u a

u a

 



        

 

 





 0 s

2

u

 a u

 a

a

s  0

0 s 

21

0

s 

Case.6 Servo with Saturation

u

2

1

c a

e  

T

^ref 0⁰ _{a e}

I s   

B s

 

▪ The equation of motionThe equation of motion

0

( )

ref

if

c a a







  

 ( )

( )

( )

f if if

I B T u c a

c a a

   



        

   



 

Case.6 Servo with Saturation



▪ phase plane trajectory

( )

if  

^

( ) c a

0

( )

a

I B c

a if

if



  





     



 

c a B



( )

a

I B c a

c a if 

 





     

  

 

 

( )

ss B

if a

I B c a





 



 

 

a 

a

_ss

I B c a

c a B

 



    

 

 c a

B

  B

23

8. Vehicle Stability Control 8. Vehicle Stability Control

8 1 Bi l d l N li d Li d l

8.1 Bicycle model: Nonlinear and Linear models 8.2 Phase Plane Analysis

8.3 Phase Plane Analysis of Bicycle model 8.4 Electronic Stability Program (ESP)

8.5 Vehicle Stability Control Algorithm

i f 2 O i l d l

8.3 Phase Plane Analysis of Bicycle model

• Assumption of 2DOF Bicycle Model

1) Longitudinal speed is Constant ( )

2) Body slip angle is sufficiently small ( )

0, 0 ax   

/ , sin _f 0, cos _f 1 v v

     

2) Body slip angle is sufficiently small. ( )

3) Left and right slip angles are identical. ( ) / , sin 0, cos 1

y x f f

v v

  

,

f fL fR r rL rR

     

( ) 2 2

y y x yf yr

F  m a   m v    F  F



• y-axis Motion Dynamic Equation

2F

f f





• yaw-axis Motion Dynamic Equation

l

2Fyf

1

2 2

z z z f yf r yr tzi

i

M H I  l F l F M



         



 l

m a 

• Lateral Tire Force and Self Aligning Moment

l

r

y

 

g g

- Non-linear Characteristic using Pacejka Tire Model

25

r

2Fyr

fi d Sid Sli l

8.3 Phase Plane Analysis of Bicycle model

x 



   

 

- Define state = [Body Side Slip Angle, Yaw Rate]

2 F_tyf 2 F_tyr

m v 

  

 

   

 

 





- Dynamic Equation

4

1

2 2

x

f tyf r tyr tzi

i

m v

l F l F M

I







 

    

         

   

 

 





Iz

 

 

- Lateral Tire Force - Self Aligning Moment

4000

Fz = 2500 N

80 Fz = 2500 N

1000 2000 3000

Force [N] Fz = 3500 N Fz = 4500 N

20 40 60

Moment [N] Fz = 3500 N

Fz = 4500 N

▼

  

▪ v_x  70kph,



0 6 0.8 1

0 6 0.8 1

, g

x p f _x p , _f g

0.2 0.4 0.6

e [rad/s]

0.2 0.4 0.6

e [rad/s]

-0.4 -0.2 0

Yaw Rate

-0.4 -0.2 0

Yaw Rate

-1 -0 8 -0 6 -0 4 -0 2 0 0 2 0 4 0 6 0 8 1

-1 -0.8 -0.6

-1 -0 8 -0 6 -0 4 -0 2 0 0 2 0 4 0 6 0 8 1

-1 -0.8 -0.6

27

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

Body Side Slip [rad]

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

Body Side Slip [rad]

▼

   

▪ v_x  70kph,





1 1.5

s] ¹

1.5

/s]

, g

x p f _x ^p _f ^g

0.5

ide Slip [rad/s

0.5

Side Slip [rad/

-0.5 0

ity of Body Si

-0.5 0

city of Body S

▼ Phase Plane Trajectory

1 1.5

s]

   

0 0.5

Side Slip [rad/s

▪ Front Steering = 0 deg

▪ Vehicle Speed = 100kph

-0.5 0

elocity of Body

▪ Front Steering = 3 deg

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1.5 -1

Body Side Slip [rad]

V

▪ Front Steering = 6 deg

1 1.5

d/s]

1 1.5

d/s]

0 0.5

dy Side Slip [rad

0 0.5

dy Side Slip [rad

-1 -0.5

Velocity of Bod

-1 -0.5

Velocity of Bod

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1.5

Body Side Slip [rad]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1.5

Body Side Slip [rad]

▼

  

0.8 1

▪ Front Steering = 0 deg

▪ Vehicle Speed =100 kph

0.2 0.4 0.6

[rad/s]

▪ Front Steering = 3 deg

-0.4 -0.2 0

Yaw Rate

▪ Front Steering = 6 deg

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1 -0.8 -0.6

Body Side Slip [rad]

Body Side Slip [rad]

0.6 0.8 1

0.6 0.8 1

0 0.2 0.4

Rate [rad/s]

0 0.2 0.4

Rate [rad/s]

Time Domain Simulation-1: Vehicle Speed =100 kph

• Front Steering Angle [deg] • Vehicle Trajectory

eg] 10

Front Steering 1 deg

4 6 8

ering Angle [de Front Steering = 1 deg

Front Steering = 3 deg Front Steering = 6 deg

250 300

m]

Front Steering = 1 deg Front Steering = 3 deg Front Steering = 6 deg

0 5 10 15

0 2

time [sec]

Front Stee

100 150 200

oordinate [m

• Body Slip Angle [deg]

[ ]

0 50

Global Y Co

0 50

e [deg]

-350 -300 -250 -200 -150 -100 -50 0 50 100 150 -100

-50

Gl b l X C di t [ ]

-150 -100 -50

Body Slip Angle

Front Steering = 1 deg Front Steering = 3 deg

S

• Yaw Rate [deg/s]

Global X Coordinate [m]

-90

e [m] Front Steering = 1 deg

Front Steering = 3 deg

0 5 10 15

-200

time [sec]

B Front Steering = 6 deg

40

Front Steering = 1 deg

-110 -100

Y Coordinat

Front Steering = 3 deg Front Steering = 6 deg

20 30

Rate [deg/s] Front Steering = 1 deg Front Steering = 3 deg Front Steering = 6 deg

-100 -90 -80 -70 -60 -50 -40 -30 -20

Global -120

Global X Coordinate [m]

0 5 10 15

0 10

time [sec]

Yaw

g] 10

Time Domain Simulation-2: Vehicle Speed =100 kph

• Front Steering Angle [deg]

0 5 10

ring Angle [deg

Front Steering = 1 deg Front Steering = 3 deg Front Steering = 6 deg

• Vehicle Trajectory

500 Front Steering = 1 deg Front Steering = 3 deg

0 5 10 15

-10 -5

time [sec]

Front Steer

400 450

m]

Front Steering = 3 deg Front Steering = 6 deg

time [sec]

30 40

le [deg] Front Steering = 1 deg

Front Steering = 3 deg Front Steering = 6 deg

• Body Slip Angle [deg]

250 300 350

ordinate [m

0 10 20

Body Slip Angl

150 200 250

obal y Coo

0 5 10 15

-10

time [sec]

B

20 Front Steering = 1 deg

• Yaw Rate [deg/s]

50 100

Gl 150

] Front Steering = 3 deg 50

▼

   

▪ v_x  70kph, 0 deg,









1 1.5

/s]

, g,

x p f



1 1.5

d/s]

, g,

x p f



0.5

Side Slip [rad/

0.5

Slip Angle [rad

-0.5 0

ocity of Body S

-0.5 0

ocity of Body S

-1 -0 8 -0 6 -0 4 -0 2 0 0 2 0 4 0 6 0 8 1

-1.5

Velo -1

-1 -0 8 -0 6 -0 4 -0 2 0 0 2 0 4 0 6 0 8 1

-1.5

Velo -1

33

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1

Body Side Slip [rad]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Body Side Slip [rad]

▼

  

▪ v_x  70kph, 0 deg,









0 6 0.8 1

0 6 0.8 1

, g,

x p f





0.2 0.4 0.6

e [rad/s]

0.2 0.4 0.6

e [rad/s]

-0.4 -0.2 0

Yaw Rate

-0.4 -0.2 0

Yaw Rate

8. Vehicle Stability Control 8. Vehicle Stability Control

8 1 Bi l d l N li d Li d l

8.1 Bicycle model: Nonlinear and Linear models 8.2 Phase Plane Analysis

8.3 Phase Plane Analysis of Bicycle model 8.4 Electronic Stability Program (ESP)

8.5 Vehicle Stability Control Algorithm

35

8.4 Electronic Stability Program (ESP)

• Why is ESP ?

• Why is ESP ?

– An innovative safety system – Actively supporting the driver

– Enhanced driving stability in situations with critical vehicle dynamics

– Functions of ABS and TSC are integrated

• What does ESP do ?

• ESP actively enhances vehicle stability

• (staying in lane and in direction)

– Through interventions in the braking system or the motor management

– To prevent critical situations,

i.e., skidding, from leading to an accident – To minimize the risk of side crashs

 Under Steer  Over Steer

• What is so special about ESP ? (1) p

ESP watches out:

– surveys the vehicle’s behavior

(longitudinal and lateral dynamics)

– watches the vehicle-operator commands

(Steering angle, brake pressure, accelerator-pedal travel) – is continuously active

in the background

• What is so special about ESP ? (2)

• What is so special about ESP ? (2)

ESP knows:

– recognizes critical situations – in many cases before the driver does

– considers the possible ways of intervening:

• Wheel-individual brake applicationpp

• Intervention in the motor management

• What is so special about ESP ? (3)

• What is so special about ESP ? (3)

ESP acts as quick as lightning:

– without reaction time

– calculated intervention in the brakes or the motor anagement – reduces risk of skidding