Where, Fixed Frame R t ti F b t OXYZ O OXYZ

3D D i

3D Dynamics

2010 vds-3D

Professor Kyongsu Yi

©2010 VDCL

©2010 VDCL

Vehicle Dynamics and Control Laboratory

1

• Kinetics of Rigid Bodies in Three Dimensions

m a 

 F

• Translational Dynamic Equation of Rigid Body (Newton Equation)

F   m a



• Rotational Dynamic Equation of Rigid Body (Euler Equation)

 ^M

^{ H }

 ^

• Relation between Positions in the Two Frame:

Fixed Frame and Rotating Frame Fixed Frame and Rotating Frame

• unit vectors,

i j ˆ ˆ ,

 

ˆ( ) cos(

) sin(

) i t     t   t 

 

ˆ( ) j t   sin( 

 t ) cos( 

 t )

z

t

 

ˆj ˆi

 

ˆ _zsin( _z ) _z cos( _z ) _z ˆ

di t t j

dt  

 

 



Where,

Fixed Frame

R t ti F b t

OXYZ

O OXYZ



 

 

ˆ cos( _z ) sin( _z ) _z ˆ

dt

dj t t i

dt  







Rotating Frame about

Angular Veocity of w.r.t.

Oxyz OXYZ

Oxyz OXYZ



  • time derivative of unit vectors

(General Form)

ˆ ˆ 0 ˆ ˆ

ˆ ˆ 0 ˆ ˆ

ˆ 0 ˆ ˆ ˆ

z z

i i i i

d j j j j

dt

k k k k







          

           

         

          

       

• Relation between Positions in the Two Frame

ˆ cos( ) sin( )

ˆ sin( ) cos( )

z t z t X

xi

t t Y

yj

 

 

 

     

 

           

 

3

0 ^z

k _k ^{ }_{ } k k

       

sin(

) cos(

)

        

 

• Rate of Change of a particle in the Two Frame:

Fixed Frame and Rotating Frame Fixed Frame and Rotating Frame

• Position Vector of a particle,

Q ˆ ˆ ˆ

Q  Q i   Q   j Q k 



^Q Oxyz

x y z

Q Q i  Q j  Q k

ˆi



y

  ^{Q }

^{ Q i }

^{  ˆ Q }

^{  ˆ j Q k }

^{ ˆ}

k

ˆ

^Q OXYZ

ˆ ˆ ˆ

di dj dk

 

 

 

ˆ ˆ ˆ

Oxyz

x y z x y z

OXYZ

Q Q

di dj dk

Q Q i Q j Q k Q Q Q

dt dt dt



           



   

 

Where,

Fixed Frame

Rotating Frame about OXYZ

Oxyz OXYZ





 

Q

Q

   

g

Angular Veocity of w.r.t.

y

Oxyz OXYZ

 

• Three-dimensional Motion of a Particle Relative to a Rotating Frame.

V

/

V

• Velocity Vector of Particle P w.r.t.

OXYZ

   

P OXYZ Oxyz

V  r   r    r

'

V

/ '

OXYZ Oxyz

P F P

V V

 

VP = absolute velocity vector of particle P Where,

r

'

VP

V

= velocity vector of point P' of moving frame coinciding with P

l i f P l i i f

r

F

VP_/ = velocity vector of P relative to moving frame



Where,

Fixed Frame

Rotating Frame about OXYZ

Oxyz OXYZ





5

g

Angular Veocity of w.r.t.

y

Oxyz OXYZ

 

• Three-dimensional Motion of a Particle Relative to a Rotating Frame. Coriolis Acceleration

• Absolute Acceleration of Particle P w.r.t.

OXYZ

   

a

 V 

 d    r 

  r 

a V r r

dt  

Where, _dt^d

  



^{ }

^{ }

/

 

P F Oxyz

V  r

   

   

OXYZ

d r r r

dt

r r r

    

     

 

 

 

c

2

a    r

   

   r r _Oxyz  r

Therefore,

r

       

     

2

P Oxyz Oxyz Oxyz

Oxyz Oxyz

a r r r r r

r r r r

        

       

   

  

 



/ '

/ '

P F aP c

a a

P F P c

a a a

  

  

   

coriolis acceleration Where a



= coriolis acceleration

ac

Where,

• General Motion in Rigid Body.



• Absolute Position Vector of Position P

/

P A P A

r   r r





• Absolute Velocity Vector of Position P

/

P A P A

/

P P A P A

A P A

V r r r

V V

  

 

  



/ /

0

V

r

r



     constant

 r 

• Absolute Acceleration Vector of Position P

  

/

constant r

 

/

/

(

)

P P A P A

A P A P A

a V V V

a r r

  

    

Where, Fixed Frame

Rotating Frame about OXYZ

Oxyz OXYZ





7

g

Angular Veocity of w.r.t.

Angular Acceleration of w.r.t.

y

Oxyz OXYZ Oxyz OXYZ

 

 

• Rigid Body Translational Dynamics

F m a



• Translational Dynamic Equation of Rigid Body

(Newton Equation)



F   m a



• Acceleration Vector of the Rigid Body

in the Global Frame

 

( )

x x x x

G A

a v w v

a a V V v w v

       

       

             



 

in the Global Frame

 

( )

( )

G y Axyz Axyz y y y

z z z z

x z y y z

a a V V v w v

a v w v

v v w v w

  

       

       

       

    

   



 ( )

( )

( )

x z y y z

y x z z x

z y x x y

v v w v w v v w v w

   

          

 

    

   



 (

x z y y

v v w v   



 )

( )

z

y x z z x

w v v w v w

  

     

 

 



( v v w v w

)

     

  

• Rigid Body Rotational Dynamics

i l i i f i id d

H

M  



• Rotational Dynamic Equation of Rigid Body (Euler Equation)



: Angular momentum of the Rigid body

Where,

   

1 n

G i i

i

H r v m r r dm



        

i i

v   r

with respect to the frame of fixed orientation.

• Angular momentum about x-axis

  ^{ }



 ^{   }

x x y z x

H y w y w x z w x w z dm

w y z dm w xy dm w zx dm

         

    



  

• Angular momentum about x-axis

  ^{   }

x y z

x x y xy z xz

w y z dm w xy dm w zx dm w I w I w I

      

     

  

x

G y yx x y y yz z

H I w I w I w

H H I w I w I w

   

 

 

 

        

 

There,

z zx x zy y z z

H I w I w I w

 

 

    

   

9

• Rigid Body Rotational Dynamics

H

M 



• Rotational Dynamic Equation of Rigid Body (Euler Equation)



H

M 



• Angular Momentum (Symmetric moment of inertia)

x x x

G y y y

H I w

H H I w

   

   

     

   

xy xz yz

0

I I I

   

• Rigid Body Rotational Dynamics

z z z

H I w

   

   

 

( )

( )

G G xyz G

H H H

I  I I w w

 

   

     

 

• Rigid Body Rotational Dynamics

 

 

 

( )

( )

( )

x x z y y z

x x x x x

y y y y y y y x z z x

I I I w w

I w I w

I w I w I I I w w

I w I w I I I

 

 



    

       

       

    ^{  I}

^

            ^{       }  ^w

 ^{I w}

     ^{  }  _  ^I

^

 ^  ^{( I}

^{ }   ^I

^{) w w}

   ^ _

Major Course Contents

Part 1: Lateral Vehicle Dynamics

1 1 Vehicle Dynamic Model

j

1.1 Vehicle Dynamic Model 1.2 Planar Model

1.3 Tire Models 1.4 Bicycle Model y

Bank angle/crosswind 1.5 Understeer/oversteer

1.6 Dynamic model interms of error wrt road 1 7 lane keeping model

1.7 lane keeping model 1.8 Lateral stability Control

Part 2: Longitudinal Vehicle Dynamics Part 2: Longitudinal Vehicle Dynamics

1 Longitudinal Dynamic Model 2 Engine model

3 Transmission 3 Transmission 4 Brake

Part 3: Vehicle Control Systems

Part 3: Vehicle Control Systems

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

8. Vehicle Stability Control

1. Vehicle Dynamic Model

• Vehicle State

- Roll:

- Pitch:



 ^{Rear View}

Left Vie - Yaw:

- X: x

 ^{Left View}

X:

- Y:

- Z:

x

z

y

l

l

Z: z

Top View

1. Vehicle Dynamic Model

F   m a



• Translational Dynamic Equation of Vehicle (Newton Equation)

• Rotational Dynamic Equation of Vehicle (Euler Equation)

G G

M  H

 ^

obal YGlo



1. Vehicle Dynamic Model

F   m a



• Translational Dynamic Equation of Vehicle (Newton Equation)

. . . .

( )

( ) ( )

( )

x x x x z y

G y C G C G y y y x z

a v v v v v

a a V V v v v v v

  

  

 

       

     

   

     

                      

   

     

     

  

 

 

   

 

  (  )

z z z z y x

a v ^{ }  v ^ v v  v  ^

         

            

G G

M  H

 ^

• Rotational Dynamic Equation of Vehicle (Euler Equation)

 

 

( )

( ) ( )

x z y

x x x x

G y y G xyz G y y y x z

I I I

I I I

H I H H I I I I I

  

   

      

      

     

 

         

   

                             

  

  

    

    

 ^{( )} 

z z z z

z y x

I I I

I I I

      

     

         

     

                    

15

2. 3DOF Planar Motion Model

2. 3DOF Planar Motion Model

2. 3DOF Planar Motion Model

• Assumption of 3DOF Vehicle Planar Motion Model

1) Ignore Roll, Pitch Motion ( )^{  }_

^{  }



^



2) Ignore Suspension Dynamics ( )F_tzi  constant  v_z  0

• Translational Dynamic Equation of Vehicle (Newton Equation)

F   m a



0 ( )

x x x x y

a v v v   v 

         

            

y q q

. . . .

( ) 0 ( )

0 0 0

G y C G C G y y y x

z

a a V V v v v v

a





         

                    

         

         

   



• Rotational Dynamic Equation of Vehicle (Euler Equation)

G G

M  H

 ^

0 0 0 0

( ) 0 0 0 0

x x

G y y G xyz G

I

H I H H

I I I I





    

          

         

                 

         

         

 

   

17

z z z z z

I  I   I  I 

            

         

i i i i

2. 3DOF Planar Motion Model

  ^{ }

2

( )

x x x y

F  m a  m v  v 





 

- x-axis Motion Dynamic Equation

F

f

  ^

^

1

_txi cos( _f ) _tyi sin( _f ) _{tx tx}

i

F  F  F F







- y-axis Motion Dynamic Equation

ty1

F

F

   

2

3 4

( )

sin( ) cos( )

y y y x

txi f tyi f ty ty

F m a m v v

F F F F



 

     

     





 

y y q

v

F

F

l

  



1

( ) ( )

txi f tyi f ty ty



M H I



- yaw-axis Motion Dynamic Equation

v

 ^

2 4

1 3

[ sin( ) cos( )]

z z z

f txi f tyi f r tyi

i i

M H I

l F F l F



 

 

  

       

 

 



 



l

   

 

1 2 3 4

1 2

cos( )

sin( )

w tx tx f w tx tx

w ty ty f

t F F t F F

t F F





 

         

 

   

 

 

 

F

4

F

Fty 3

Fty

 

  ^tx4

2  t

3. Tire Model

3.1 Pacejka Tire Model 3 1 1

3.1.1 Slip Angle

3.1.2 Lateral Tire Model 3.1.3 Slip Ratio

3.1.4 Longitudinal Tire Model 3.1.5 Combined Tire Model 3 1 6 Self Aligning Moment 3.1.6 Self Aligning Moment 3.2 Dugoff’s Tire Model

19

3. Tire Model

Longitudinal Tire Force F



Lateral Tire Force Self Aligning Moment

tyi tzi

F M



 M

Slip Angle Wheel Angular Speed

i i







   _i



F

F

3. Tire Model

Tire Deformation

• Tire Deformation

 x

 

• Longitudinal Tire Force

 y

 

• Lateral Tire Force

txi

tzi

F x

F

  

  

 

F y

F

  

  

 

F

^F

 x  y

L it di ll ti d f^tzi ti L t ll ti d f ti

F

 x

F

 y

< Longitudinally tire deformation > < Laterally tire deformation >

REF: Reza N. Jazar, “ Vehicle Dynamics: Theory and Application”, pp101 ~ 105, Springer, 2008

3. Tire Model

• Longitudinally Tire Deformation

• Longitudinal Tire Force

V

r

 

txi

tzi

F x

F

  

  

 

 x r



V V

   

x r  V

   

• Longitudinally Tire Deformation

3. Tire Model

• Laterally Tire Deformation Laterally Tire Deformation • Shear Stress Distribution• Shear Stress Distribution

< Bottom view of a laterally deflected and turning tire >

• Lateral Tire Force and Self Aligning Moment



ction of l Travel

x

• Lateral Tire Force



Direc Wheel

tyi y

F    dA ^F

a

F

y M

tzi tyi x

M  F  a

a

tyi

Pneumatic Trail

3. Tire Model

3.1 Pacejka Tire Model 3 1 1

3.1.1 Slip Angle

3.1.2 Lateral Tire Model 3.1.3 Slip Ratio

3.1.4 Longitudinal Tire Model

3.1.5 Combined Tire Model

3 1 6 Self Aligning Moment

3.1.6 Self Aligning Moment

3.2 Dugoff’s Tire Model

3.1.1 Slip Angle

• The angle between the orientation of the tire and the orientation of the Wheel



_tan

txi

V

 

V

V



V

tyi

i i i

    

Tire Slip Angle at - i th Wheel

 

Where,

V

Tire Slip Angle at Wheel Steering Angle at - Wheel

Angle between and at i-th Wheel

i i

i txi tyi

i th i th

V V













Where,

i

g



y f y f

v   l   v   l  

1 2

tan( ) tan( )

tan( ) tan( )

y f y f

x w x w

y r y r

v t v t

v l v l

 

 

 

 

 

 

   

   

 

 

 

3 4

tan( ) tan( )

x w x w

v t v t

 

 

 

     

25

3.1.2 Lateral Tire Model

sin( tan (

))

tyi y y y y vy

F  D C

B   S

• Lateral Tire Force at the i-th Wheel

(1 )( )

tan (

( )

y y i hy y i hy

y

E S E B S





     

Where,

F

y

M

2

5200 5200

0.22 1.26

40000 32750

0.00003 1.0096 22.73

tzi tzi

y y

y tzi tzi

F F

B C

D F F

 

   

     

F

1.6 0 0

y

y hy vy

E

 





• Normal Tire Force at the i-th Wheel

< Lateral Tire Force >

    ^{ }  

1 1 1 1 2 2 2 2

2 2

r r

tz t o tz t o

f r f r

m l m l

F K r r g F K r r g

l l l l

m l m l

 

         

 

    ^{ }  

3 3 3 3 4 4 4 4

2 2

f f

tz t o tz t o

f r f r

m l m l

F K r r g F K r r g

l l l l

 

         

 

Original Radius of the Tire Effective Rolling Radius of the - Wheel

io i

r  r  i th

Where,

3.1.2 Lateral Tire Model

• Slip Angle versus Lateral Tire Force Curve

27

3.1.3 Slip Ratio

• During Braking

cos( )

i i ti i

r  V 

  ^{  } 

r

• During Traction

cos( )

i

ti i

 V

 



cos( )

r  V 



cos( )

ti i

V 



cos( )

i i ti i

i

i i

r V

r

 

 

  

  

V

Side View

Where,

Angular Velocity of the - Wheel Tire Radius of the - Wheel

Tire Slip Angle at Wheel

i i

i th

r i th

i th







 

V

V

Tire Slip Angle at - Wheel

i

i th

 

2 2

1

( ) ( )

t y f x w

V  v   l    v   t  

V

1

2 2

2

2 2

( ) ( )

( ) ( )

( ) ( )

t y f x w

t y f x w

V v l v t

V v l v t

 

 

 

     

     

 

 

3

( ) ( )

t y r x w

V  v   l   v   t 

3.1.4 Longitudinal Tire Model

• Longitudinal Tire Force at the i-th Wheel

sin( tan (

))

txi x x x x vx

F  D C

B   S

(1 )( ) ^x tan (1 ( ))

x x i hx x i hx

x

E S E B S

 B ^ 

     

Where,

1940 1940 1940

22 1.35 2000

645 16125 0 956

tzi tzi tzi

x x x

F F F

B    C    D   

• During Traction ( )



^ 0

645 16125 0.956

3.6 0 0

x hx vx

E   S  S 

1940 1940 1940

22 F

1 35 F

1750 F

B    C    D   

• During Braking ( )



 0

22 1.35 1750

430 16125 0.956

0.1 0 0

x x x

x hx vx

B C D

E S S

 

  

29

3.1.4 Longitudinal Tire Model

• Slip Ratio versus Longitudinal Tire Force Curve

3.1.5 Combined Tire Model

jk i d l

• Pacejka Tire Model

1) Longitudinal Tire Model:

2) L l Ti M d l

0

( , )

txi tx i tzi

F





( )

F F F

2) Lateral Tire Model:

F

 F

( 

, F

)

▲ There is no correction between Longitudinal and Lateral Tire Model

• Normalized Slip

• Normalized Slip

1) Normalized Slip Ratio: ^{* i}

i

m

 

 

^{0.058 ( 0)}

m

0.1

if

elsewhere

 _{ } ^  ^

 

2) Normalized Slip Angle: _i^{* i}

m

 

 

*



6.3 650

3500

tzi m

   F ^



• Correction Factor:

• Combined Tire Model based on Pacejka Tire Model

* * 2 * 2

( ) ( )

i i i

    

j

*

*

* 0

*

(

, )

i

txi tx m tzi

i

F  F   F

  

*

*

* 0

(

, )

i

i

tyi ty m tzi

F  F   F

  

3.1.5 Combined Tire Model

• Combined Tire Force

Longitudinal Tire Force g Lateral Tire Force a e a e o ce

* *

* *

0 0

*

(

, )

(

, )

i

i i

txi tx m tzi tyi ty m tzi

i

F  F   F F  F   F

 

   

4000

l h 0[d ]

3000 4000

0 2000

alpha=20[deg]

alpha=15[deg]

alpha=10[deg]

alpha=5[deg]

alpha=0[deg]

nal force [N]

0 1000 2000

Slip ratio=0.1 Slip ratio=0.05 Slip ratio=0.01 Slip ratio=0.005 Slip ratio=0

orce [N]

-2000

Longitudin

-2000 -1000 0

Lateral fo

-1.0 -0.5 0.0 0.5 1.0

-4000

-30 -20 -10 0 10 20 30

-4000 -3000

Slip ratio Slip angle [deg]

3.1.6 Self Aligning Moment

sin( tan (

))

tzi z z z z vz

M  D C

B   S

• Self Aligning Moment at the i-th Wheel

x

(1 )( )

tan (

( )

z z i hz z i hz

E S E B S





     

Where,

Direction of Wheel Travel

Bz

 ^{1.86 10}



 ^{2.73 10}



B C D

      



D Wh

 

   

exp 0.11 10

2.40

z z z

tzi

z

B C D

F C

  

   

 

 F

y M

 ^{-2.72 10}



 ^{-2.28 10}



z tzi tzi

z z z

z

D F F

B C D

B C D

     

 

 

< Self Aligning Moment >

 ^{-0.0 10}



 ^{-0.643 10}

 ^4.04

0

z z

z tzi tzi

hz

C D

E F F

S



      



 0

g g

hz vz

3.1.6 Self Aligning Moment

• Slip Angle versus Self Aligning Moment Curve

80

40 60

]

Fz = 2500 N Fz = 3500 N Fz = 4500 N

20 40

oment [Nm

-20 0

Aligning Mo

-60

Self A -40

-15 -10 -5 0 5 10 15

-80 -60

Slip Angle [deg]

Slip Angle [deg]

3.2 Dugoff’s Tire Model

Longitudinal Tire Force

1 ( )

i

txi x i

F C  f 

   





Lateral Tire Force

1  

tan( )

i

( )

F C  f 



1 ( )

i

tyi y i

i

F C f 

   



Where,

 F

 ¹ 



 ^{  }

Ftxi

 

 

 

2 tan( )

i

x i y i

C C

 ^     

  ^{(2 ) ( 1)}

1 ( 1)

i i i

i

i

f if

if

  

 

  

    

Longitudinal Tire Stiffness Lateral Tire Stiffness

Ti /R d F i ti C ffi i t

x y

C C





35

Tire/Road Friction Coefficient

 

3.2 Dugoff’s Tire Model

Longitudinal Tire Force Lateral Tire Force

1 ( )

i

txi x i

F C  f 

   



tan( ) 1 ( )

i

tyi y i

F C  f 

   



1  

1  

Where,



 0



 0

4000 4000

2000 3000 4000

[N]

Fz = 2500 N Fz = 3500 N Fz = 4500 N

2000 3000 4000

Fz = 2500 N Fz = 3500 N Fz = 4500 N

0 1000

nal Tire Force

0 1000

Tire Force [N]

-2000 -1000

Longitudin

-2000 -1000

Lateral T

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

-4000 -3000

Slip Ratio

-20 -15 -10 -5 0 5 10 15 20

-4000 -3000

Slip Angle [deg]

3.2 Dugoff’s Tire Model

Longitudinal Tire Force Lateral Tire Force

1 ( )

i

txi x i

F C  f 

   



tan( ) 1 ( )

i

tyi y i

F C  f 

   



4000

Sli A l 0 d 4000

Sli R ti 0

1  

1  

Where,

F

 4500 N

F

 4500 N

2000 3000

e [N]

Slip Angle = 0 deg Slip Angle = 3 deg Slip Angle = 6 deg Slip Angle = 9 deg

2000 3000

N]

Slip Ratio = 0 Slip Ratio = 0.04 Slip Ratio = 0.08 Slip Ratio = 0.12

0 1000

nal Tire Force

0 1000

l Tire Force [N

3000 -2000 -1000

Longitudi

3000 -2000 -1000

Lateral

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

-4000 -3000

Slip Ratio

-20 -15 -10 -5 0 5 10 15 20

-4000 -3000

Slip Angle [deg]

37