lateral vehicle dynamics - university of...

Modelling of Automotive Systems 1

Lateral Vehicle Dynamics

y- lateral

x- longitudinal

Modelling of Automotive Systems 2

Kinematic Model of Lateral Vehicle Motion: Bicycle Model

0. are angles slip swheel' thei.e. n,orientatio wheelsofdirection in the are B andA at s velocitie:assumptionMajor

Instantaneous rolling centre

Wheelbase

Steering angle

Vehicle orientation (yaw angle)

Circular radius

Centre of gravity

Modelling of Automotive Systems 3

Bicycle Model 2 front wheels are represented by a single wheel at A 2 rear wheels are represented by a single wheel at B

centre rolling ousInstantane :O radiusCircular :R

n)orientatio vehicleanddirection motion ebetween th (angle vehicle theof angle Slip:c.g. of velocity :V

vehicleof angle heading - vehicle theofn Orientatio:

vehicle theof e Wheelbas:gravity of centre - C

0 case, steeringfront only For steered. becan srear wheel andfront both Assumed

angles Steering

β

δ

δ

δ

ΨllL rf

r

r

f

+=

=

Modelling of Automotive Systems 4

Bicycle Model

βψγ += vehicle theof angle Course

Triangle OCA

( )

Rl

orRl

or

Rl

ff

f

f

ff

f

f

f

=−=−

−

=−

ββδδδββδ

δπβδ

sincostancoscossincossin

2sinsin

Triangle OCB

( )

Rlor

Rlor

Rl

rr

r

r

rr

r

r

r

=−=−

+

=−

βδβδβδδβ

πδδβ

costansincoscossincossin

2sin

sin

Add both

( )R

ll rfrf

+=− βδδ costantan

Modelling of Automotive Systems 5

Bicycle Model

β angle Slip From

Rll

llRl rf

rfrf

f =−=− ββδββδ sincostansincostan

Rll

llRl fr

rffr

r =−=− βδββδβ costansincostansin

( ) ( )

+

+=

+

+=

=+−+

−

ff

frrf

ff

frrf

fffrrf

llll

llll

llll

δδβ

δδβ

ββδδ

tantantan

tantantan

0sincostantan

1

i.e. the slip angle is represented by steering angles and wheelbases.

Modelling of Automotive Systems 6

Kinematic Model of Lateral Vehicle

( )R

ll rfrf

+=− βδδ costantan

Angular velocity

RV

=ψ&

Therefore

( )rf

rf

llV

+

−=

βδδψ

costantan&

( )( )βψ

βψ

+=

+=

sincos

VYVX

&

&

Modelling of Automotive Systems 7

Kinematic Model of Lateral Vehicle: consider vehicle width Limitation of the bicycle model:

different. arefact in and io δδ

Modelling of Automotive Systems 8

Kinematic Model of Lateral Vehicle: consider vehicle width In the bicycle model

( )rf

rf

llV

+

−=

βδδψ

costantan&

If the slip angle β is small, 0=rδ

2

,

2

2Let

Therfore

handother on the

wi

wo

iof

f

f

rf

f

lR

LlR

LRL

RL

RV

LV

llV

−=

+=

=+

=

=

=

=+

=

δδ

δδδ

δ

ψ

δδψ

&

&

Modelling of Automotive Systems 9

Kinematic Model of Lateral Vehicle: consider vehicle width Trapezoidal tie rod arrangement to realize oi δδ >

Modelling of Automotive Systems 10

Dynamics of Bicycle Model

Modelling of Automotive Systems 11

Dynamics of Bicycle Model

rotation of centre theis which Opoint the toaxis lateral vehicle thealong measured position, lateral vehicle-

axis allongitudin vehicleyx−

( )yrryffz

yryfx

xy

yryf

FlFlIFFVymVyRya

FF

YX

−=

+=+

+=+=

+=

−

−

ψ

ψ

ψψ

ψ

&&

&&&

&&&&&& 2

y

x

maLaw sNewton' Using

velocityallongitudin vehicleVangle yaw-

scoordinate global

Experimental shows the lateral tyre force is proportional to the slip angle (when slip angle is small).

Modelling of Automotive Systems 12

Dynamics of Bicycle Model Experimental shows the lateral tyre force is proportional to the slip angle (when slipangle is small).

( ) ( )

,tan

,tan

stiffness cornering are, where

22ForcesRear tyre

angle steering front tyre where

: tyreof angle Slip

x

ryvr

x

fyvf

rf

vfryrvffyf

vrr

vff

VlV

VlV

CCCFCF

ψθ

ψθ

θθδ

θαδ

θδα

αα

αα

&

&

−=

+=

−=−=

−=

−=

Steering angle

Velocity angle

x

y

lf

ψ

fα angle Slip

Modelling of Automotive Systems 13

Dynamics of Bicycle Model

small are,When vrvf θθ

x

rvr

x

fvf

Vly

Vly

ψθ

ψθ

&&

&&

−=

+=

( )

x

rryr

x

ffyf

yrryffz

yryfx

VlyCF

Vly

CF

FlFlIFFVym

ψ

ψδ

ψ

ψ

α

α

&&

&&

&&

&&&

−−=

+−=

−=

+=+

2

2

Modelling of Automotive Systems 14

( ) ( )

( ) ( )

( ) ( )

( ) ( )ψ

δψ

ψδ

ψψδψ

ψψδψ

ααααα

ααααα

ααα

ααα

&&&&

&&&&

&&&&&&

&&&&&&&

xz

rrff

xz

rrff

z

ff

x

rrffx

x

rff

x

rr

z

r

x

fff

z

f

x

rr

x

fffx

VIlClC

yVI

lClCICl

mVlClC

VymV

CCm

Cy

eiV

lyCIl

VlyC

CIl

mVlyC

mVlyC

mC

Vy

22222

222

..

222

222

+−

−−=

−+−

+−=

−+

+−=

−−

+−=+

Modelling of Automotive Systems 15

( ) ( )

( ) ( )δ

ψψ

ψψ

ψψ

α

α

αααα

αααα

+

+−

−−

−−−

+−

=

z

ff

f

xz

rrff

xz

rrff

x

rrffx

x

rf

ICl

mC

yy

VIlClC

VIlClC

mVlClC

VmV

CCyy

dtd

yy

20

20

20

20

1000

20

20

0010

becomemotion of equations lateral The

variable,space state Introduce

22

&

&

&

&

&

&