lateral vehicle dynamics - university of...
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![Page 1: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/1.jpg)
Modelling of Automotive Systems 1
Lateral Vehicle Dynamics
y- lateral
x- longitudinal
![Page 2: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/2.jpg)
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
![Page 3: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/3.jpg)
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
+=
=
![Page 4: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/4.jpg)
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
![Page 5: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/5.jpg)
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.
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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
&
&
![Page 7: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/7.jpg)
Modelling of Automotive Systems 7
Kinematic Model of Lateral Vehicle: consider vehicle width Limitation of the bicycle model:
different. arefact in and io δδ
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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
&
&
![Page 9: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/9.jpg)
Modelling of Automotive Systems 9
Kinematic Model of Lateral Vehicle: consider vehicle width Trapezoidal tie rod arrangement to realize oi δδ >
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Modelling of Automotive Systems 10
Dynamics of Bicycle Model
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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).
![Page 12: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/12.jpg)
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
![Page 13: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/13.jpg)
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
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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
+−
−−=
−+−
+−=
−+
+−=
−−
+−=+
![Page 15: Lateral Vehicle Dynamics - University of Sussexusers.sussex.ac.uk/~tafb8/mas/MAS_03a_LateralVehicleDynamics.pdf · Modelling of Automotive Systems 2 Kinematic Model of Lateral Vehicle](https://reader031.vdocuments.net/reader031/viewer/2022030413/5a9f21537f8b9a84178c6797/html5/thumbnails/15.jpg)
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
&
&
&
&
&
&