Modelling of Automotive Systems. 1. Lateral Vehicle Dynamics y- lateral ...
Kinematic Model of Lateral Vehicle Motion: Bicycle Model. 0. are angles slip s
wheel'.
Lateral Vehicle Dynamics
y- lateral
x- longitudinal Modelling of Automotive Systems
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Kinematic Model of Lateral Vehicle Motion:
Bicycle Model
Instantaneous rolling centre
Circular radius Steering angle
Centre of gravity
Vehicle orientation (yaw angle)
Wheelbase
Major assumption : velocities at A and B are in the direction of wheels orientation, i.e. the wheel's slip angles are 0. Modelling of Automotive Systems
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Bicycle Model 2 front wheels are represented by a single wheel at A 2 rear wheels are represented by a single wheel at B Steering angles
δf δr Assumed both front and rear wheels can be steered. For only front steering case, δ r = 0 C - centre of gravity L = l f + lr : Wheelbase of the vehicle Ψ : Orientation of the vehicle - heading angle of vehicle V : velocity of c.g.
β : Slip angle of the vehicle (angle between the motion direction and vehicle orientation) R : Circular radius O : Instantaneous rolling centre
Modelling of Automotive Systems
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Bicycle Model Course angle of the vehicle
γ =ψ + β
Triangle OCA π sin − δ f sin (δ f − β ) 2 = lf R sin δ f cos β − sin β cos δ f
or
lf
=
cos δ f R
or
tan δ f cos β − sin β =
lf R
Triangle OCB π sin δ r + sin (β − δ r ) 2 = lr R sin β cos δ r − sin δ r cos β cos δ r = lr R
or
or sin β − tan δ r cos β =
lr R
Add both
(tan δ
f
− tan δ r )cos β =
l f + lr R Modelling of Automotive Systems
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Bicycle Model Slip angle β
From tan δ f cos β − sin β =
lf
R l sin β − tan δ r cos β = r R
(l
f
lr tan δ f cos β − lr sin β = l f sin β − l f tan δ r cos β =
l f lr R lr l f R
tan δ r + lr tan δ f )cos β − (l f + l f )sin β = 0
tan β =
l f tan δ r + lr tan δ f lf +lf l f tan δ r + lr tan δ f + l l f f
β = tan −1
i.e. the slip angle is represented by steering angles and wheelbases.
Modelling of Automotive Systems
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Kinematic Model of Lateral Vehicle
(tan δ
f
− tan δ r )cos β =
l f + lr R
Angular velocity ψ& =
V R
Therefore ψ& =
V (tan δ f − tan δ r )cos β l f + lr
X& = V cos(ψ + β ) Y& = V sin (ψ + β )
Modelling of Automotive Systems
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Kinematic Model of Lateral Vehicle: consider vehicle width Limitation of the bicycle model: δo and δi in fact are different.
Modelling of Automotive Systems
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Kinematic Model of Lateral Vehicle: consider vehicle width In the bicycle model ψ& =
V (tan δ f − tan δ r )cos β l f + lr
If the slip angle β is small, δ r = 0 ψ& =
Vδ f l f + lr
=
Vδ f L
on the other hand V ψ& = R Therfore
δf =
L R
Let δ f =
δo =
δo + δi 2
L R+
lw 2
,
=
L R
δi =
L R−
lw 2
Modelling of Automotive Systems
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Kinematic Model of Lateral Vehicle: consider vehicle width Trapezoidal tie rod arrangement to realize
δi > δo
Modelling of Automotive Systems
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Dynamics of Bicycle Model
Modelling of Automotive Systems
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Dynamics of Bicycle Model x − vehiclelongitudinal axis y - vehiclelateralposition,measuredalong the vehicle lateralaxis to the point O which is the centreof rotation
X − Y global coordinates ψ - yaw angle Vx − vehicle longitudinal velocity Using Newton's Law ma y = Fyf + Fyr a y = &y& +ψ& 2 R = &y& + Vxψ& m( &y& + Vxψ& ) = Fyf + Fyr I zψ&& = l f Fyf − lr Fyr Experimental shows the lateral tyre force is proportional to the slip angle (when slip angle is small). Modelling of Automotive Systems
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Dynamics of Bicycle Model Experimental shows the lateral tyre force is proportional to the slip angle (when slip angle is small). Slip angle of tyre : α f = δ − θ vf where δ front tyre steering angle α r = −θ vr Rear tyre Forces Fyf = 2Cα f (δ − θ vf
)
Fyr = 2Cα r (− θ vf
x
)
where Cα f , Cα r are cornering stiffness tan θ vf = tan θ vr =
V y + l f ψ& Vx V y − l rψ& Vx
Steering angle
,
Velocity angle
Slip angle α f
,
lf
ψ
y Modelling of Automotive Systems
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Dynamics of Bicycle Model When θ vf , θ vr are small y& + l f ψ& θ vf = Vx y& − lrψ& θ vr = Vx
m ( &y& + Vxψ& ) = Fyf + Fyr I zψ&& = l f Fyf − lr Fyr y& + l f ψ& Fyf = 2Cαf δ − Vx y& − lrψ& Fyr = −2Cαr Vx
Modelling of Automotive Systems
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&y& + Vxψ& =
2Cαf δ m
2Cαf ( y& + l f ψ& ) 2Cαr ( y& − lrψ& ) − − mV x mV x
2Cαf ( y& + l f ψ& ) lr 2Cαr ( y& − lrψ& ) lf + ψ&& = 2Cαf δ − Iz Vx Vx Iz i.e. 2Cαf δ
2(Cαf + Cαr )
(
) ψ&
(
)ψ&
2 Cαf l f − Cαr lr &y& = y& − Vx + − m mV x mV x 2 Cαf l 2f + Cαr lr2 l f 2Cαf δ 2(Cαf l f − Cαr lr ) ψ&& = y& − − Iz I zVx I zVx
Modelling of Automotive Systems
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Introduce state space variable, y y& ψ ψ& The lateral equations of motion become 1 0 y 2(Cαf + Cαr ) − y& 0 d mV x = 0 dt ψ 0 ψ& 0 − 2(Cαf l f − Cαr lr ) I zVx
0 y 2 Cαf l f − Cαr lr 2Cαf 0 − Vx − y& mV x + m 0 0 1 ψ 2 Cαf l 2f + Cαr lr2 2l f Cαf 0 − ψ& I I zVx z