VEHICLE STEERING DYNAMIC CALCULATION AND SIMULATION

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Annals of DAAAM for 2012 & Proceedings of the 23rd International DAAAM Symposium, Volume 23, No.1, ISSN 2304-1382
ISBN 978-3-901509-91-9, CDROM version, Ed. B. Katalinic, Published by DAAAM International, Vienna, Austria, EU, 2012
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Annals & Proceedings of DAAAM International 2012
VEHICLE STEERING DYNAMIC CALCULATION AND SIMULATION
VU, T[rieu] M[inh]
Abstract: This paper presents fundamental mathematical
estimations of vehicle sideslip in stationary conditions
regarding the influences of the vehicle parameters such as the
tire stiffness, the position of gravity center, the vehicle speed
and the turning radius. The vehicle dynamics on steady state
and transient responses are also investigated to see the effects
of the yaw natural frequency and yaw damping rate on the
steering system. Results from this study can be used in
designing an automatic control of tracking vehicle in the future.
Keywords: sideslip angle, yaw damping rate, steady state
response, transient response
1. INTRODUCTION
Calculation and simulation of vehicle steering
dynamic are essential for any control systems since most
of the modern vehicles are currently equipped with new
electronic stability and auto-guided systems. The
accurate determination of the sideslip angle can help to
improve the yaw and the steering stability performance.
Sideslip estimation is based on the vehicle physical
variables (mass, gravity position, tire stiffness), vehicle
speed, lateral acceleration, steering angle, and yaw rate.
Unlike yaw rate, the vehicle sideslip angle cannot be
measured directly; hence estimation methods have been
developed to calculate the sideslip angle from the
available above variables.
Among the latest research papers on this issue, Kim
H. and Ryu J. in [1] have proposed a sideslip angle
estimation method that considers severe longitudinal
velocity variation over the short period of time based on
extended Kalman filter (EKF). Hac A, el al. in [2] have
established an estimation method of vehicle roll angle,
lateral velocity and sideslip angle. Only roll rate sensor
and the sensors readily in electronic stability control
(ESC) are used in this estimation process. Mathematical
algorithms are based on kinematic relationships, and
then, avoiding dependence on vehicle and tire models,
which can minimize tuning efforts and sensitivity to
parameter variations. Lenain R., et al. in [3] introduce an
observer for dynamic sideslip angle with mixed
kinematic for accurate control of fast off-road mobile
robots. With respect to pure kinematic approaches, the
use of this dynamic representation for estimation of the
sideslip angle improves reactivity in sliding variable
adaptation and consequently in path tracking accuracy.
The content of this paper is mostly based on the
publication in [4] on handling model of advanced vehicle
dynamics where mathematical algorithms for a single
track vehicle are modeled regarding the effects of the
vehicle center of gravity, the front/rear tire stiffness and
the under-steering/over-steering conditions. Other
knowledge for yaw damping and steering control is
referred on Ackermann J. and Sienel W. in [5] where a
steering control system is studied with yaw damping rate
and yaw natural frequency to control the unexpected yaw
motions. Mathematical formulas and simulation for the
vehicle models are derived from Minh V.T and Aziz A.R
in [6] and from Minh V.T and Pumwa J. in [7]. Adaptive
EKF-based estimation for vehicle sideslip angle can be
referred to by Hrgetic M el. al. in [8].
This study is the first part of a project on automatic
control of tracking vehicles where the vehicle sideslip is
modeled and estimated. The accurate mathematical
model for vehicle sideslip is critically needed for
designing a free-error feedback control system for
automatic trajectory tracking vehicles.
The outline of this paper is as follows: Section 2
provides fundamental mathematic formulas for
calculation of vehicle sideslip; Section 3 presents the
vehicle behavior with steering in steady state condition;
Section 4 demonstrates the vehicle movement in transient
responses and section 5 analyses the vehicle dynamic
responses with frequency input; Finally, conclusion is
withdrawn in section 6.
2. VEHICLE SIDESLIP CALCULATION
When the vehicle is moving straight on a flat surface,
the direction of the center of gravity (CG) keeps the same
with the orientation of the vehicle. When the vehicle
turns, the yaw rate causes the change of the orientation.
Fig. 1. Sideslip angle

The vehicle demonstrates a velocity component
perpendicular to the orientation, known as the lateral
velocity. Then, the orientation of the vehicle and the
direction of the travel are no longer the same. The
vehicle is moving under the influence of different forces.
If a lateral force is acting on the tire, an angle is formed
between the direction of movement of the tire and the tire
- 0237 -

straight line. This angle is called the sideslip angle

(shown in Fig. 1).
Reason for this sideslip angle or the tire slip is the
elastic lateral deflection of the rolling tire in the tire
contact area under the effect of the lateral force between
tire and road.
For analyzing the motion behaviors of a single track
model, a linearization of the tire lateral force and the tire
slip angle is assumed via a tire stiffness
c

:
F
c



(1)
When the vehicle moves at low speed, the wheels roll
without a tire slip angle since the lateral cornering force,
F

, is small and can be ignored. The vehicle model can
be seen as the assumption of Rudolf Ackermann with the
elongations of all wheel center lines intersecting at one
point, the center of the turning curve (Fig. 2).
The steering angle,

, can be simply calculated as:
arctan l
r




(2)
The steering angle of the inner wheel,
i

, is a
function of the steering angle of the outer wheel,
o

:
arctan
tan
i
o
l
ls









(3)
And as a result of Ackermann condition, the angle of
the inner wheel,
i

, is greater than the steering angle at
the outer wheel,
o

.
Fig. 2. Model for sideslip (Ackermann condition)
where,
i

,
o

: steering angle of inner and outer wheel,
l
: wheel base,
s
: kingpin track width,
r
: radius of the curve
However, when the lateral force appears, the vehicle
front wheel orientation and the vehicle movement
direction is no longer the same. A simplified description
of the vehicle lateral dynamics is demonstrated in a
single track model (Fig. 3). The tire contact points are in
the center of tires. Longitudinal forces in the tire contact
points as well as wheel load fluctuations are not
considered. The height of the center of gravity is zero.
The Newton’s law equation of the motion for the
vehicle lateral direction is:
y Lf Lr
ma F F
(4)
The force of inertia acting on the vehicle center of
gravity,
y
ma
, corresponds to the centrifugal force:
2()
y
vv
ma m m vr mv
rr

    
(5)
Fig. 3. Deflection of the rolling tire by a lateral cornering force
F

where,
v
: Vehicle velocity,

: Vehicle angular velocity,
r
: radius of
curve,

: Yaw angle,

: Side slip angle,

: Steering angle,

: Tire
slip angle,
l
: Wheel base
And the gyroscopic effect on the z-axis at the vehicle
center of gravity:
Lf f Lr r
J F l F l


(6)
where
J
is the vehicle moment of inertia on z-axis.
The tire side forces can be calculated from the given
tire slip rigidity,
c

, in equation (1) for the front wheel:
Lf f f
Fc


(7)
and for the rear wheel:
Lr r r
Fc


(8)
The side slip angle for the vehicle at the center of
gravity,

, can be formulated from the front tire slip,
f

:
f
f
l
v
  

  
(9)
r
v





r

f

f

cent
F
Lr
F
Lf
F
l
r
l
f
l
arctan
tan
arctan
i
o
l
ls
l
r













0


i

r
l
s
O
- 0238 -

and from the rear tire slip,
r

:
r
r
l
v



(10)
It is noted that the tire slip rigidity or the sideslip
stiffness,
c

, is an elastic property for each rubber tires,
normally in the range of 30,000-50,000 N/rad.
3. STEERING IN STEADY STATE
In steady state condition, the vehicle speed,
v
, is a
constant, then, the yaw velocity,

, and the sideslip,

,
are also constant, i.e,
0
and
0


.
The torque balance equations can be formulated at the
rear contact point:
Lf y r
F l ma l
(11)
Replaced with the tire slip rigidity in equation (7):
fr
fy
ll
c ma
vl



  



(12)
And in equation (8):
f
r
ry
l
l
c ma
vl






(13)
Because in the steady state,
0


, then from equation (5),
v
r

. The transformation from equation (13) and (14) leads
to:
f
ry
fr
l
l
lm a
r l c c



  



(14)
From the above equation, the necessary steering
angle,

, during the steady state driving along a curve
composes of two parts. The first part,
l
r
, or Ackermann
angle, depends on the vehicle geometrical parameters.
And the second part,
f
ry
fr
l
ml a
l c c






, is characterized by
the influences of the lateral acceleration and the tire
rigidities, which can increase, if
f
r
fr
l
l
cc






, or reduce, if
f
r
fr
l
l
cc






, the steering angle.
From equation (9) and (10), the sideslip angle
difference between the front and the rear wheel is:
fr
l
v
   

    
(15)
With
vr
, then,
l
r

  
. Replace with

in
equation (14):
fry
rf
l
ml
a
l c c



  



(16)
Then, the difference of the sideslip angles depends on
the vehicle and the tire parameters. The driver has to
compensate the sideslip angle difference,


, with the
steering angle,

. This forms a basic knowledge of over-
steer and under-steer definition: Over-steer is if
0
fr
  
   
, neutral is if
0
fr
  
   
, and
under-steer is if
0
fr
  
   
(Fig. 4).
Fig. 4. Over-steer (left) and Under-steer (right)
Under-steer and over-steer are vehicle dynamic
characteristics used to demonstrate the sensitivity of a
vehicle steering system. The under-steer happens if the
vehicle turns less than the steering control of the driver.
Conversely, over-steer happens if the vehicle turns more
than the steering control of the driver.
The under-steer system is safer since it causes the
reduction of the lateral force at the rear axle and makes
the vehicle to stabilize at a smaller curve radius with less
lateral acceleration. While the over-steer vehicle is more
dangerous because it increases the lateral force and
increases the swerve tendency of the vehicle.
Fig. 5 demonstrates the relationship between the
sideslip and steering angle with the vehicle speed and the
turning radius. When maintaining the turning radius at
100rm
and varying the vehicle speeds,
0 40 /v m s
, the sideslips increase exponentially and
steering angles rise,
00
1.4 3.2


. Similarly, when
reducing the turning radius
100 10rm
, the sideslips
increase exponentially and steering angles rise
00
2 20


.
Fig. 5. Side and steering angle vs. velocity and turning radius
0 5 10 15 20 25 30 35 40
0
5
10
15
Increasing Vehicle Velocity v [m/s]
Angle [degree]
102030405060708090100
0
10
20
30
40
Reducing Turning Radius r [m]
Angle [degree]
Front Tire Angle

f
Reer Tire Angle

r
Vehicle Angle

Steering Angle

Front Tire Angle

f
Reer Tire Angle

r
Vehicle Angle

Steering Angle

- 0239 -

4. STEERING IN TRANSIENT RESPONSE
Transformation of equations (4-10) can lead to the
following expressions:
() fr
fr
ll
mv c c
vv

      


     




(17)
and
fr
Z f f r r
ll
J c l c l
vv

     


    




(18)
Equation (17) can be represented by yaw velocity,

:
()
fr
fr
rf
mv c c
ll
mv c c
vv


   

  


(19)
For the steady state,
v const
, then,

:
()
fr
f
r
rf
mv c c
l
l
mv c c
vv


   

  


(20)
Replace

and

in equation (18):
22
f r f f r r
Z
c c c l c l
mv vJ
   







2
2
r r f f f r
ZZ
c l c l c c l
J J mv
    






2
2
()
f f f r f r r f
ZZ
c l c c l l l c
J J mv mv
   



  



(21)
The characteristic polynomial of the dynamic
equation in (21) can be represented in an inhomogeneous
linear differential equation of 2nd order for the vehicle
slip angle

. The homogeneous part of this differential
equation has the form of a simple oscillating motion with
damping:
0AB

  
(22)
or in the vibrated frequency form:
22
00
20D s s

  
(23)
Thus, the differential equation for the slip angle

can be
viewed with a yaw undamped natural frequency,
0

:
2
02
r r f f f r
ZZ
c l c l c c l
J J mv
   



(24)
and with a yaw damping rate
D
:
22
0
2
f r f f r r
Z
c c c l c l
mv J v
D
   




(25)
Then, the dynamic yaw frequency,
omD

, is:
2
00
1
mD D


(26)
The yaw natural frequency and damping rate can be
represented for the movement of the vehicle around the
vertical axis (z).
5. VEHICLE DYNAMIC ANALYSIS
The linearized single track vehicle model is now
examined under the reaction of the driver to control the
vehicle movement with the input variable, the steering
angle,

. The output variables are the yaw velocity,

,
and the lateral acceleration,
y
a
.
The transfer function of the output,

, and the input,

can be derived from equation (14) with
1
rv


and
y
av


, then:
2
stat f
r
sf sr
v
l
ml
lv
l c c



 
 



(27)
where the relation
stat





is referred to as the stationary
yaw amplification factor.
For analyzing the dynamic behavior of the vehicle,
equations (17) and (18) can be converted to Laplace s-
form
2
2
00
1
() 21
1
z
stat
Ts
Fs Dss







 
(28)
with
z
T
is a time constant,
f
z
r
mvl
Tcl


.
This Laplace s-form can also be transformed for the
lateral acceleration:
2
12
2
2
00
1
() 21
1
yy
stat
aa
T s T s
Fs Dss


 



 
(29)
with the time constant
1r
l
Tv

, and
2Z
r
J
Tcl


.
Simulations for the driving control of the vehicle
movement are conducted with a step steering (sudden
step in input signal) and shown in Fig. 6.
Fig. 6. Transient response with a steering step angle
Steering Angle

Time
Sideslip Angle

Time
Sideslip Angle

Time
Lateral Acceleration ay
Time
- 0240 -

For a very fast input of a steering step angle to 200 in
0.4 second, the sideslip angle,

, and the lateral
acceleration,
y
a
, respond with a small overshooting
motion and then steadily fluctuate at the stable position;
While the sideslip angle,

, responds in an
undershooting motion at the beginning time.
For frequency response, the transfer function in
equation (28) now is transformed into the frequency,
j

, form:
2
.2
00
1
() 2
1
z
stat
Tj
Fj Dj






 
 
(30)
The amplitude,
ˆ()
ˆFj







, is thus a frequency
dependence.
The lateral acceleration in equation (29) is now
applied for the frequency response:
2
12
2
.2
00
1
() 2
1
y
stat
aT j T
Fj Dj




   
 
 
(31)
Simulation results of frequency response are shown
in Fig. 7. There is a peak of magnitude and phase shifting
in the low frequencies. The amplitude responses drop in
high frequencies. There is a phase lag in yaw velocity
and thus, the vehicle reaction on the steering angle
becomes larger in low frequencies.
Fig. 7. Frequency response with yaw velocity and lateral accereration
6. EXAMPLES
Example 1: The stationary steering behavior of a vehicle
is simplified in Fig. 4. The vehicle is moving in a circular
orbit with the radius r = 100 m at a speed of v = 22 m/s.
The following vehicle data are given:
Mass,
m
= 1,300 kg
Gravity center position,
front,
f
l
= 1.2 m
rear,
r
l
= 1.3 m
Lateral tire stiffness,
front,
f
c

= 55,000 N/rad
rear,
r
c

= 60,000 N/rad
(i). Find out the slip angles
f

and
r

, the attitude angle

, and the steering angle

.
(ii). Assess the stationary steering behavior of this
vehicle.
Solution:
(i). Moment balance in front point,
Lr cent f
F l F l
, while
Lr r r
Fc


and
2
cent
v
Fm
r

. Then,
2
r r f
v
c l m l
r



2f
r
r
mv l
rc l



or
2
1300 22 1.2 0.0503 2.88
100 60000 (1.2 1.3)
rrad


   
  
.
Similarly,
2r
f
lf
mv l
rc l


or
2
1300 22 1.3 0.0595 3.41
100 55000 (1.2 1.3)
frad


   
  
.
From equation (9) and (10),
rr
rr
ll
vr

  
   
.
Then,
1.3
0.0503 0.0373 2.14
100 rad

    
.
From equation (9),
1.2
0.0595 0.0373 0.0342 1.96
100
f
f
lrad
r
  
        
If the sideslip is ignored, apply the Ackermann condition
in Fig. 3, the steering angle then,
arctan 0.025 1.43
Ack
lrad
r


   


.
Remarks: The sideslip causes the vehicle movement,
2.14


, different with the vehicle orientation
(steering angle),
1.96


and greater than the
Ackermann condition,
1.43
Ack


.
(ii)
3.41 2.88 0

    
 the system is under-steer.
Remark:
fr
fr
ll
vv

  

   


and
fr
ll l
rr
  

    

l
r

  
,
then,
2f
r
fr
fr
l
l
mv
rl c c

  

    



or
f
ry
fr
l
l
lm a
r l c c



  



, when the curve radius
r const
, the lateral acceleration,
y
a
, increases, the
steering angle,

, must increase as well.
Example 2:
-80
-60
-40
-20
0
20
Magnitude (dB)
100102
-180
-135
-90
-45
0
Phase (deg)
Yaw Velocity Frequency Response
Frequency (rad/sec)
-40
-30
-20
-10
0
Magnitude (dB)
10-1 100101102
-90
-45
0
Phase (deg)
Lateral Acceleration Frequency Response
Frequency (rad/sec)
- 0241 -

Examine the yaw natural frequency and the yaw
damping of the following vehicle specifications:
2.5lm
,
1.3
f
lm
,
1.2
r
lm
,
1300m kg
,
2
1960
Z
J kgm
,
30000 /
f
c N rad


. The sideslip
stiffness of the rear tires,
r
c

, varies as
130000 /
r
c N rad


,
235000 /
r
c N rad


, and
340000 /
r
c N rad


.
Solution:
The sideslip varies and leads to different steering
system. For
130000 /
r
c N rad



f
r
fr
l
loversteer
cc






,
235000 /
r
c N rad



f
r
fr
l
lundersteer
cc






, and
340000 /
r
c N rad



f
r
fr
l
lundersteer
cc






as well.
The yaw natural frequency and the yaw damping
against the vehicle velocity,
v
, are drawn in Fig. 8.
For over-steer system, there is a critical speed,
, where the vehicle loses the stability
and begins to swerve.
For under-steer systems, the yaw damping decreases
with the vehicle speed.
Fig. 8. Yaw natural frequency and yaw damping rate
7. CONCLUSION
The single track vehicle model allows analyzing the
influences of fundamental parameters such as the effect
of the location of the center of gravity, the different front
and rear cornering stiffness as well as the under-
steering/over-steering systems on the vehicle dynamic
behavior and sideslip angle. Results from this study can
be applied to estimate the tracking errors for an
automatic vehicle tracking system in the next step of the
project. The analysis of the transient response for this
system provides essential knowledge of the vehicle
dynamic behaviors under the influences of nonlinear
dynamic variables.
8. REFERENCES
[1] Kim H.H and Ryu J., “Sideslip Angle Estimation Considering
Short-duration Longitudinal Velocity Variation”, International
Journal of Automotive Technology, Vol 12(4), pp. 545-553, 2011,
DOI: 10.1007/s12239-011-0064-2
[2] Hac A,, Nichols D., and Sygnarowicz D., “Estimation of Vehicle
Roll Angle and Side Slip for Crash Sensing”, SAE International
Congress, April 2010, Detroit, MI, USA, 2010, DOI:
10.4271/2010-01-0529
[3] Lenain R., Thuilot B., Cariou Ch., and Martinet P., “Mixed
Kinematic and Dynamic sideslip angle Observer for accurate
Control of Fast Off-road Mobile Robots”, Journal of Field
Robotics, Vol. 27(2), pp. 181-196, 2010. DOI: 10.1002/rob.20319
[4] Minh V.T., Advanced Vehicle Dynamics, 1st edition, Malaya
Press, Pantai Valley, 50603. Kuala Lumpur, Malaysia, 2012, pp.
127-144, ISBN: 978-983-100-544-6
[5] Ackermann J., and Sienel W., "Robust Yaw Damping of Cars
with Front and Rear Wheel Steering", IEEE Transactions on
Control Systems Technology, Vol. 1(1), pp. 15-20, 1993
[6] Minh V.T and Aziz A.R., “Real-time Control Schemes for Hybrid
Vehicle”, IEEE International Conference on Control Application
(CCA), Denver, CO, USA, September 2011, pp. 538-543, 2011,
DOI: 10.1109/CCA.2011.6044483
[7] Minh V.T and Pumwa J., “Simulation and Control of Hybrid
Electrolic Vehicle”, International Journal of Control, Automation
and Systems, Vol 10(2), pp. 308-316, 2012, DOI:
10.1007/s12555-012-0211-1
[8] Hrgetic M., Deur J and Pavkovic D, “Adaptive EKF-Based
Estimator of Sideslip Angle Using Fusion of Inertial Sensors and
GPS”, SAE International Journal of Passenger Cars – Mechanical
Systems, Vol 4(1), pp. 700-712, 2011, DOI: 10.4271/2011-01-
0953
max 37.8 /v m s
0 5 10 15 20 25 30 35 40
0
5
10
15
Vehicle Velocity v[m/s]
Natural Frequency

0[Hz]
Yaw Natural Frequency
0 5 10 15 20 25 30 35 40
0
2
4
6
8
10
Vehicle Velocity v[m/s]
Damping Rate D
Yaw Damping Rate
c

r1 = 30000 N/rad
c

r2 = 35000 N/rad
c

r3 = 40000 N/rad
D1 for c

r1
D2 for c

r2
D3 for c

r3
- 0242 -