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
&
&
&
&
&
&