Lehigh University
Lehigh Preserve
Theses and Dissertations
1993
Simulation and analysis of a braking and turning
with an antilock braking system
Taesoo Chi
Lehigh University
Follow this and additional works at: http://preserve.lehigh.edu/etd
Recommended Citation
Chi, Taesoo, "Simulation and analysis of a braking and turning with an antilock braking system" (1993). Theses and Dissertations. Paper
239.
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' 'T··'l.···
U
,
'
Ii)
..
m
aes
,
';
;
'. m
I•"· T'·,'····· l....i E"····.·
1:3
imulati nand
a
an
nalysis f
rakin and Turning wit
ntilock Braking System
: January 16, 1994
Simulation and Analysis
of a Braking and Turning with an Antilock Braking System
by
Taesoo Chi
A Thesis
Presented to the Graduate Committee
of Lehigh University
in Candidacy for the Degree of
Master of Science
in
Department of Mechanical
Engineering and Mechanics
Lehigh University
Bethlehem, Pennsylvania
December, 1993
t.
To my Parents and Wife with all my Love
iii
Acknowledgements
I would like to express sincere thanks to my advisor, Prof. Stanley H.
Johnson, for his invaluable guidance and support throughout this research. Thanks
also go to Hyundai Motor Company which has supported me for this opportunity.
I am deeply indebted to Korean Students in Lehigh University, especially to Mr.
Sang-Koo Lee, a candidate for Ph. D. in Mechanical Engineering and Mechanics, for
helpful discussions.
To my wife, Eunhee, I offer sincere thanks for her continuous understanding
and encouragement.
iv
Table of Contents
i\bstract --------------------------------------------------------------------------------------------1
1. Introduction -------------------------------------------------------------------------------------1
1.1 Introduction -------------------------------------------------------------------------1
1.2 Background ------------------------------------------------------------------------ 3
1.3 Motivation and Outline of Thesis ----------------------------------------------- 4
2. Mathematical Modeling
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Equations of Motion of the Vehicle ------------------------------------------- 6
Wheel Rotational Equation ----------------------------------------------------- 14
Wheel Velocities ----------------------------------------------------------------- 18
Vehicle Position ----------------------------------------------------------------- 19
Tire Force -:----------------------------------------------------------------------- 20
Brake System Model
----------------------------------------------------------- 25
i\ntilock Brake System(i\BS) ------------------------------------------------- 30
3. Simulation and i\nalysis
3.1
3.2
3.3
3.4
3.5
--------------------------------------------------------------------- 6
----------------------------------------------------------------35
Introduction to Simulation ----------------------------------------------------- 35
Braking in a Turn --------------------------------------------------------------- 37
Braking in a Turn with i\BS
------------------------------------------------- 38
i\nalysis of the Vehicle while Braking in a Turn -------------------------- 43
Improved i\lgorithm for i\BS -------------------------------------------------- 58
4. Conclusion
-----------------------------------------------------------------------------------61
5. References
-----------------------------------------------------------------------------------63
6. i\ ppendix
-------------------------------------------------------------------------------------66
Tire Modeling
vita
----------------------------------------------------------------------------------------------70
v
Abstract
1
Chapter 1
Introduction
1.1 Introduction
An Antilock Brake System(ABS) can enhance braking performance as well as
handling performance, i.e., stability and steerability, during combined braking and
steering maneuvers. These improvements are accomplished by controlling and
maintaining the wheel slip within a proper range. It is no doubt that a computer
"
simulation is a useful tool to analyze the behavior of a braked vehicle and to develop
control strategies for ABS because the simulation has the ability to describe the
complex interaction caused by nonlinear variables.
Directional vehicle dynamics involve yawing, rolling, pitching, lateral and
directional body motion, including interactions, that are responsible for a stability
problem. The stability of the braked vehicle depends on whether or not the forces
required to maintain or change course as desired by the driver can be transmitted at
the contact point between tire and road surface. Therefore, the analysis of stability is
quite interesting in the study on the braked vehicle equipped with a conventional
brake and ABS.
The objectives of this research- are an investigation of the response of the
braked vehicle(or equipped ABS) while braking in a tum, an analysis on stability and
an application of a control method of ABS. Therefore, a nonlinear seven-degree-of2
freedom computer simulation model is developed, and a nonlinear tire model and
brake model are included for this research. To make inquiry into the effects of ABS,
a control method is introduced and improved. The stability and steerability is analyzed
through several methods for a nonlinear system.
1.2 Background
Mathematical modeling and computer simulation is a valuable research tool for
studying, understanding, and improving the handling performance of vehicles. Vehicle
dynamics modeling starts with a consideration of forces and moment equations, where
thedominant physical force-producing device is the tire. Therefore, a realistic tire
model is essential in evaluating the performance of the vehicle. The behavior of a
rigid body of the vehicle is derived from the equation of motion of a classical rigid
body dynamics [1,2,3]. Early researchers used the friction circle concept to account
for the simultaneous effects of lateral tire forces [1]. However, the experimental
research helped several tire models based on an empirical formula to be formulated
[4,6,7,8]. The tire model by [6], which is adopted in this research, was based on the
data frem a tire test machine and the tire friction theory [5]. In this m9del the basic
tire input variables are tire normal load, lfteral slip angle, longitudinal slip ratio and
tire camber angle, along with the resulting response variables of lateral and
longitudinal force and aligning torque. The tire model by [7,8] was formulated by a
combined trigonometric function through experimental data.
3
j'
The simulations on combined braking and steering maneuvers have been
accomplished with the nonlinear model [6,9,10,11]. In general, a yaw rate and a final
position of the vehicle was evaluated in the time-domain as a index of stability [6,9].
Directional stability was noted to be strongly influenced by lateral load transfer
between the front and rear axles [12]. The yawing motion of the vehicle while
braking in a tum was characterized by the yaw acceleration as a function of
longitudinal acceleration [10].
Antilock Brake System has been developed and produced by several companies
worldwide. The development process of these systems relies heavily on testing [13].
Several control algorithms for ABS was introduced [14,15] and are based on
experimental data. A new systematic design has been tried using sliding-mode control
[16].
1.3 Motivation and Outline of Thesis
Both analytical and numerical approaches are necessary for a system
development. This research begins on a preliminary step for a new slip control
system. Therefore, the characteristics of the braked vehicle while braking in a tum
are required to be analyzed, and this research is concerned with the methods of the
analysis of stability properties. In order to further this research, a control method is
introduced and improved.
The body of this thesis is divided into two chapters, and the contents of each
4
chapter are summarized in this section.
In Chapter 2, a set of equations of motion for a braked vehicle, a tire model
and a brake model are derived. A three-degree-of-freedom model for expressing -the
motion of the vehicle body is derived from classical rigid body dynamics, and fourdegree-of-freedom is allocated in four wheels of the vehicle. Four nonlinear wheel
equations are solved by analytical method through linearlization. A nonlinear tire
model [6] is adopted for combined braking and steering maneuvers. In addition, the
concepts of ABS are depicted, and a control method is introduced.
In Chapter 3, simulation and analysis are performed. The simulation is set up
so that the ramp step input for the steering angle and the first order input fot the
-brake force input are applied as an open loop input. The response of the braked
vehicle over a slippery and high friction road is presented in various figures. The
yawing motion is broadly investigated and evaluate using a variety of methods: timedomain analysis, phase-plane,
ro~t-Iocus
plot and Liapunov's indirect method.
Finally, a new control idea is suggested; this idea improves
steerability of the vehicle.
5
t~e
stability and
Chapter 2
Mathematical Modeling
Computer simulation is a valuable research tool for studying the dynamic
characteristics of an automobile. This type of simulation is desired to develop a
simple but powerful model which is proper for its purpose. The objective of the
modeling approach presented in this research is to investigate the response of the
vehicle while braking in a tum.
A nonlinear seven-degree-of-freedom model - the longitudinal, lateral and
yawing motion of the body and the rotational motion of four wheels - is developed to
analyze the behavior of the braked vehicle. (Fig. 2-1) A nonlinear tire model for
calculation of the longitudinal and lateral tfre forces at the tire-road contact point and
a static brake model are included. DASSL, a solver of differential / algebraic
equation, solves the nonlinear first order differential equations for the body of the
vehicle, and the linear ordinary differential equations for each wheel.
2.1 Equations of Motion of a Rigid Body Model of the Vehicle
. The body centered axis concept is well known and in the present circumstances
has the major advantage that the moments and products of inertia of the body remain
constant and independent of the position of the body in space [1-3]. It may be
convenient to solve the translational as well as the rotational motions of a rigid body
6
Fp
lIT
Driver's
Input
Brake
System
r-----·-·-·-·-·-·-·---·-·-·-·~
!
!
ABS
Modulator
Steering
System
ABS
Controller
•
!
j
i
r
Wheel
Spin
Mode
w·1
Tire
Model
(4 DOF)
Fx
I
t
Fx,Fy
Vehicle
Dynamics
(3 DOF)
Fz, vx ,vy , Wz
!
l_._._._.
._._._._._. ._._._._._._._.__._._._._._._._._._._._. ._._.j
Fig. 2-1 Block Diagram for the Dynamic System
7
I
Vx,vy,W Z
in terms of a coordinate system fixed in the body, particularly if the applied forces
are most easily specified in the body-axis system.
The axis system for a vehicle is represented by Fig. 2-2 in which the axes x,y
and z are mutually perpendicular. It consists of a moving axis system fixed at the
vehicle center of mass, and with the x-axis forward, the y-axis to the right, and the
z-axis down.
The equations of motion of the rigid vehicle are derived from general rigid
body dynamics.
2.1.1 General Body-axis Translational Equations
Let us write the translational equation in the form
F
=
mv
(2-1)
where F is the total external force acting on the rigid body, and v is the absolute
velocity of the center of mass. Expressing these vectors in terms of their
instantaneous body-axis components, we can write
F=Fi+Fi+Fk
x
Y'
z
and
v = vi+vi+vk
x
Y'
z
8
(2-2)
x
3
z
y
Fig. 2-2 The Axis System for a Rigid Vehicle
9
where i, j, and k are an orthogonal triad of unit vectors which is fixed in the body.
The absolute acceleration
v in terms of the body-axis coordinate system can
be expressed as follows:
where
and w is the absolute angular velocity of the body-axis coordinate system.
The cross product w x v is evaluated by using a determinant.
Then, the following equations of motion can be written:
Fx
=
m(vx +vz wy -vy w)
z
Fy
=
m(vy +vxw z -vzw)
Fz
=
m(vz +vyw x -vxw)
(2-3)
2.1.2 General Body-axis Rotational Equation
The rotational motion of a rigid body is described by the general relationship
10
between the external moment M and the angular momentum H.
Using this body-axis coordinate system, the absolute rate of change of H can
be expressed as follows:
(2-4)
where (ffl r is the rate of change of the absolute angular momentum with respect to
a body-axis origin. If Hx ' Hy and Hz are
t~~
instantaneous values of the projections of
H onto the x, y, and z axes, then
where the unit vectors i, j and k rotate with xyz system. Expressions for Hx , Hyand
Hz for the body-axis coordinate system are as follows:
Hx = I n wx +1 wy +1 wz
~
~
Hy = I ~ wx +1Wwy +1~ wz
Hz
=
I zxx
w +1zyy
uj +1 w
zzz
Also,
iIx = I
iIy
=
iu +1 iu +1 iu
nx~y~z
I ~ iu x +1Wiu y +1~ iu z
iIz = I zxx
iu +1zyy
iu +1zzz
iu
11
Now, the cross product wxH is evaluated by using a determinant.
Ca)xH
=
(Hzy
<.> -Hyz
w)i+(Hxz
w -Hzx
w)j+(Hyx
w -HxCa»)k
y
Corresponding components of M and
if
must be equal, so the following equations
are obtained as the general rotation equations in terms of a body-axis coordinate
system:
Mx = 1xi»x x+1x y
(wy -w xz
<.»+1xzz
(w +w xwy1 +(1z z
-iy1<'>
+1y z
(w2_W~
y Jw
yz
z
y
My = 1xy(w x +<.> y wz)+1yy Wy +1yz (w z -U> x wy1+(1xx -izz)w x wz +1,xz(w2-w~
z
x
(2-5)
M z = 1xzx
(w -w yz
<.»+1yzy
(w +w xz
<.»+1zzz
W +(1y-i)<.>
<.> +1 (<.>2_w~
yzzxyxyx
Y
2.1.3 Equation of Motion of 3 DOF
The model used for this research is simplified into three motions of a rigid
body: longitudinal, lateral and yawing motion. Rolling, pitching, and vertical motions
are ignored( wx=wy=vz=O ) because it is a simple model which does not include the
suspension of the vehicle. The products of inertia of the rigid body
~,
Iyz are equal to
zero for a body symmetrical about the xz plane. Then a simplified set of equations is
derived:
Fx
=
m(vx -vyw z)
Fy = m(vy +vx w)
z
Mz
=
1zz Wz
12
(2-6)
Fci
x
y
Fig. 2-3 Resolution of Tire Force
13
2.1.4 External Forces and Moments
Resolution of the resulting forces acting on each tire into components along the
vehicle-fixed axis system is depicted in Fig. 2-3. From these figures, the summation
of the tire forces acting in the directions of their respective vehicular axes are:
Fxi = Fci cos rJri - Fsj sin rJri
FYi = Fci sinrJrj + Fsj COSrJrj
(2-7)
The resultant forces that act on the vehicle in x- and y-directions are determined from
the summations. From Fig. 2-2 the sum of the yaw moments about the car
e.G is
evaluated:
4
Fx = ,EFxi
;=1
4
(2-8)
Fy=,EFyi
;=1
2.2 Wheel Rotational Equation
The analytical representation of wheel rotation involves four degrees of
freedom associated with the rotational velocities of each wheel about its spin axis.
The wheel rotational degrees of freedom are incorporated in order to enable the
calculation of rotational slip due to braking and to investigate its effect on tire forces.
14
A free body diagram of a rolling tire with applied brake torque is given in Fig.
2-4(a). The equation of motion yields:
(2-9)
This equation is a nonlinear first order equation because of tire torque and brake
torque, but it can be solved by an analytical method through linearization over a small
time steps in the computer simulation. This approach [8] is depicted as follows:
Longitudinal slip ratio is defined as the change in distance traveled per
revolution due to driving or braking conditions divieded by the distance traveled
under the free-rolling condition. Thus, the slip ratio is computed:
s
=
1
(2-10)
From the definition of the slip, the following equations can be expressed:
vG(l-s)
<.)=---
(2-11)
r
Let So be the value of s at t=to for a wslip curve in the form shown in Fig. 24(b). Expanding the Wslip relationship in a Taylor series about S=So,
pes) = Po+ apl (s-s~ + Higher order terms
as =80
15
(2-12)
(2-13)
F ;;; jJ(s)N
Neglecting the high order terms and arranging the equations for s and the
derivative of s, a differential equation may be written:
(2-14)
The solution is
where
Eq.(2-15) may easily be solved for s(slip) by updating
So
at the end of
appropriate intervals, thus avoiding virtually all the integration costs inherent in
the original form of Eq. (2-9).
16
~Tqi
\~
Fci
(a) Free Diagram of a Rolling Tire
/I.l. 0
_._.-
!
f
slip
(b) I.l. - slip Curve
Fig. 2-4 Wheel Dynamics
17
2.3 Wheel Velocities
To enable the calculation of tire forces at the wheels of the vehicle, it is
necessary to evaluate velocities at the four wheels. Referring to Fig. 2-2, the position
of the individual wheel centers are obtained as follows:
T
T
2
2
T,
r =-bi--j
4
2
r ""ai+...1J•
1
1':
2'
T,
r. ""-bi+-j
3
2'
""ai-...1j
(2-17)
The velocities of the tire contact points are first evaluated along the vehicle axes in a
moving frame of reference using the cross product of angular velocity and the
distances to each wheel.
(2-18)
VG "";""6>Xr
Therefore, the forward velocities of the wheel centers along the vehicle x-axis are
T
uGZ =Vx +...16)
2 z
(2-19)
T,
u =V +-w
G4 x 2 z
The lateral velocities of the tire contact points are obtained as
VGZ=Vy-awl. '
VG3 ""vy-bwz. '
18
VGZ=Vy+awz.
vG4""vy+bw 1.
(2-20)
2.4 Vehicle Position
Since the equations of motion are written in terms of the moving coordinate
system, a transformation is required to relate the vehicle's position to the inertial
coordinate system.The moving coordinate system (x, y, z) fixed at the center of mass
of the vehicle with respect to the space-fixed axis system (X, Y, Z) is given by single
angular rotation. The velocity components in the space-fixed axis system are obtained
in the followipg form:
dX
.
='VxCos If!-vysm If!
dt
-
dY
.
-='V ~mlf!-v coslf!
dt:C
Y
(2-21)
d\f1 =6)
dt
Z
The position and orientation of the vehicle with respect to the inertial frame are
obtained by integration as follows:
X= J(t(vxcoslf!-vysinlf!)dt+X(O)
o
Y= J(t(vxsinlfr-vycoslf!)dt+Y(O)
o
Tf = (\.u dt + If(O)
Jo
z
19
(2-22)
2.5 Tire Force
The handling behavior of a vehicle is dependent to a great extent on the
mechanical characteristics of the tires. The forces and moments(the self-aligning
moment, the overturning moment and the rolling resistance moment of the tire are
ignored in this work) generated by the tires at the tire-road contact patch include the
normal force, the side force arising from the slip angle and the camber angle(which is
ignored) and the circumferential force arising from applied torque.
Several mathematical models have been developed which take into account the
dependence of the longitudinal and lateral forces on both the tire slip angle and the
longitudinal slip. The tire model used in this work is based on the comprehensive
work described in [6]. It is developed in a convenient computational form which can
be easily applied in a vehicle dynamics simulation. The procedure which calculates
the tire forces is shown in appendix A, and the results of this method are shown in
Fig. 2-6.
2.5.1 Slip Angle
The most important component of the tire forces which influences vehicle
handling is the side force because the control of a vehicle is dependent on the side
force generated by the two pairs of wheels. When a tire is steered across the path of
20
i
,i
11,
,,
Direction of
Wheel Heading
/
,,
/
i
i
i
,i
i
i
Direction of
,i
__._._._.
.
.
._._._._.,ii ._. ._. .
. .
,,
i
i
,i
,i
._._. . ._._._. .__
Wheel Travel
,/
,
,,
,
i
i
,,
Fig. 2-5 Tire Kinematics in Ground Plane
21
motion, a deformation and displacement of the contact patch occurs which gives rise
to a lateral force and a moment which attempts to realign the wheel in the rolling
direction. As a definition the slip angle is the angle formed between the direction of
the wheel heading and the direction of travel of the center of tire contact. The
corresponding kinematics are depicted in Fig. 2-5. Then, the slip angle is calculated
as
p. = tan-I (VUGi ) - rfr.
I
where the forward velocity
UGi
Gi
(2-23)
I
and the lateral velocity vGi of the tire contact point on
the ground are given by Eq. (2-19).
2.5.2 Nonnal Force of the Tire
The forces predicted by the tire model depend on the instantaneous value of
the normal force of the road on the tire. The normal forces change due to the
. longitudinal acceleration. The effect of the suspension system is neglected. Thus, the
normal forces on each front and rear tire are obtained by summing the moments about
the tire contact points.
mgb -maxh
Fz3,4
a+b
mga +maxh
=
a+b
22
(2-24)
where ax is the instantaneous longitudinal acceleration(ax < 0 for braking).
2.5.3 Wheel Slip
As mentioned before,the longitudinal slip ratio is defined as the change in
distance traveled per revolution due to driving or braking conditions divided by the
distance traveled under the free-rolling condition.
slip = 1 -
W;'j
------:.-~--
(2-25)
uGicoslJr j + vGjsinlJrj
It can be seen that the denominator represents the tire velocity along the direction of
wheel heading. Thus, the wheel longitudinal slip is constrained to the value of
-1.0 ~ slip ~ 1.0.
2.5.4 Tire Characteristics
Tire forces are affected by various parameters, such as slip angle, longitudinal
slip, normal load, velocity, inflation, tread contour and so on. Thus, the proper
mathematical tire model is required for simulation purposes. In this work the most
important characteristics of the tire for braking during steering is the relationship
between longitudinal and lateral force with respect to the slip angle and the wheel
slip. These forces are functions of the slip angle, wheel slip and normal force:
23
1.00
Q)
u
L
0.80
0
l...t...
...
Q)
1=
-0
c
'6
0.60
0.40
0.20
~
0.00 - j - - - - - - - - - - - - - , { - - - - - - - - - - - - - j
en
§ -0.20
....J
-g
-0.40
N
'§ -0.60
~
z
-0.80
~~~~~~;;;~~
-1 .00 +----,-----r-----,-----,--+----,---,-----,------r----I
-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0
Longitudinal Slip
(a) Longitudinal Tire Force
1.00 . . - - - - - - - - - - - - , - - - - - - - - - - - - - - - . ,
Q)
u
0.90
o0.80
l...t...
~
1=
o
Slip Angle 20°
0.70
0.60
L
10°
.2 0.50
o
....J 0 40
<J
•
Q)
.~
o
0.30
§ 0.20
o
z 0.10
0°
0.00 -F§~;:::::=__.,.-__,_-_,_-_l_-J,--.,____,-=:::::;:~~
-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8
1.0
Longitudinal Slip
(b) Lateral Tire Force
Fig. 2-6 Tire Characteristics
24
Fe = f(s, {J,FJ
(2-26)
Fs = f(s, {J,Fz)
Longitudinal force is divided into two regions based on the peak. The one
region corresponding to values of the wheel slip less than that corresponding to the
peak point is called the stable region. The other region for larger wheel slip values is
called the unstable region. Longitudinal force increases as the wheel slip increases in
the stable region, but it decreases as the wheel slip increases in the unstable region.
On the other hand, as the slip angle increases, the peak decreases and moves to the
right.(See Fig. 2-6(a))
The side force, as may be seen from Fig. 2-6(b), is maximum at zero
longitudinal slip and decreases with increasing longitudinal slip.
2.6 Brake System Model
In order to study vehicle response in this research, it is necessary to consider
the brake pedal force by a driver as a primary input with a steering angle. The
dynamics of the braking system is approximated by a first-order system which has an
exponential response curve with time constant T. This response approximates the
effects of all the components of the braking system from the master cylinder to the
brake wheel cylinder.
25
Power Booser
Io
Brake Pedal
/
Tire~
(
Proportioning Valve
')
I
Brake Wheel Cylinder
Brake Pad
Brake Disc
\.
)
Fig. 2-7 Brake System
26
2.6.1 Calculation of Brake Torque
The brake system consists of a brake pedal, a power booster, a master
cylinder, a proportioning valve and a brake wheel cylinder as shown Fig. 2-7. The
mechanical linkages and the hydraulic system which comprise the brake system
transform the pedal force into the brake torque.
When the pedal force Fp is amplified by the pedal linkage, the force Fpi which
is used as the input to the power booster, is calculated in Eq (2-27).
(2-27)
where lp is the ratio of the pedal linkage and TJ p is the mechanical efficiency of the
pedal linkage. The force F bo ,the output of the power booster, which pushes the
connecting rod of the master cylinder, is amplified by the power booster as shown in
Fig. 2-8(a) in which the characteristic of a general vacuum power booster is
expressed. The equations to calculate the force F ba can be expressed as follows:
Fbo
=
0
Fbi < Fbd
= ab(Fbi-Fbd)+Fbj
Fbd<Fbi<Fbr
= Fbi+Fbu -Fbr
Fb?Fbr
(2-28)
According to Fig. 2-8(a), the output force F ba is zero until the input force Fbi is
greater than the drag force of the booster Fhd • As soon as the input force exceeds the
drag force, the output force suddenly increases into force F bj and is multiplied by the
ratio C¥b with respect to the input force. After the input force reaches a force F br at
27
last, the amplification is stopped.
The pressure of master cylinder Pme is calculated with its area Arne and the
output force of the booster. As the distribution of brake pressure is accomplished by
the proportioning valve(Fig. 2-8(b», the front and rear line pressures Plf, Pir are
depicted as follows:
Fha
Pme = -TJ
A
m
me
(2-29)
Plf=Pme
P"
=
«p(Plj.-Pk)+Pk
where TJm is the efficiency of the master cylinder and am and Pk are the ratio and
knee pressure of the proportioning valve. The brake torques Tqf and T qr are calculated
from the line pressure of the brake system and the radius of each brake disk rdf and
Tqf
= Awc/Plf-Pof) (BFf ) r df
Tqr
= Awe, (P" -PorHBF,) rdr
(2-30)
where Pof and Por are the offset pressures of each brake wheel cylinder, and BFf and
BFr are the braking factors corresponding to the brake pad of each brake.
28
F bu ._._._._._._._._._._.-._._._._._._._._._._._._._.
Booster Input Force, Fbi
(a) Characteristics of Power Booster
Front Line Pressure, PIf
(b) Characteristics of ProportioniIm
Fig. 2-8 Power Booster and Proportioning Valve
29
2.7 Antilock Brake System(ABS)
The Antilock Braking System(ABS) has been developed to reduce the tendency
for wheel lock and improve vehicle control during sudden braking on slippery and
high-friction road surfaces. The concept of ABS has long been addressed by the
automobile industry. Development of ABS started as early as 1932. In 1969, the
Ford Motor Company introduced the Sure-Track ABS which controlled only the rear
wheels with a special vacuum modulator. The Chrysler Corporation introduced the
first four-wheel ABS system in 1971. In 1978, Bosch developed a modern computercontrolled hydraulic four-wheel ABS [13], and these days many companies
manufacture ABS for various types of vehicles.
The general concepts for ABS control are introduced, and the simulation
model of ABS is developed in order to analyze the stability and steerability of the
braked vehicle equipped with ABS.
2.7.1 General Concepts for ABS
Basically, an ABS is a device installed in the brake line between the upper
brake system (brake pedal linkage, master cylinder, power booster) and the lower
brake system (caliper, pad, disc) which acts so as to prevent the braked wheel from
locking up while the vehicle is in motion. The ABS does this by automatically
modulating the applied brake pressure so that the longitudinal slip of the braked wheel
30
is maintained within a narrow slip range as shown in Fig. 2-9. By doing this, the tires
are able to retain most of their lateral force capability, allowing the vehicle to remain
steerable and the driver/vehicle system to remain stable. Also, in most cases the
stopping distance is shortened in comparison to a locked-wheel stop.
As shown schematically in Fig. 2-1, the ABS consists of several key
components. These are the modulator, the controller, and the wheel sensor. Together
these elements, which can be implemented and integrated in a variety of ways,
provide the ABS function. Referring to Fig. 2-1, the driver provides the force
command to the system. This, acting through the brake system, causes the vehicle and
the braked wheel to decelerate. Some kind of sensed variable - usually wheel speed
and sometimes vehicle acceleration - is fed back to the controller. When a certain
threshold is exceeded, the controller unit generates a signal which causes hydraulic
pressure to be reduced. As the wheel begins to recover speed, a second threshold is
crossed and the modulator begins to reapply the full brake effort. The system
continues cycling until the vehicle comes to rest, or the brake pedal force is released.
As mentioned above, the objective of the ABS is to prevent wheel lock and
thus to improve the performance of the vehicle: i.e., short stopping distance, lateral
stability and steerability.
2.7.2 Control Method
In order to accomplish good ABS performance, the slip of the wheels must be
31
maintained in a proper region (8 - 30 %). The first function of ABS is to predict
incipient wheel lock up and to decrease the brake pressure without exceeding a proper
slip region. Another function of ABS is to reapply the brake pressure when the danger
of wheel locking is averted so that the maximum braking is accomplished. This cycle
is repeated as long as the brake pedal is depressed until the vehicle comes to rest.
The ABS controller consists of several parts: the calculation of the wheel
velocity and acceleration, the calculation of slip thresholds, the determination of the
control phase and the control of the modulator. The wheel is controlled with the
control loop, presented in Fig. 2-10.
The wheel velocities of each wheel and the wheel accelerations, which are
calculated by the derivative of velocity with respect to time, are two control variables.
In the case of initial braking, the wheel lock up is predicted and the brake pressure
begins to be decreased when wheel acceleration exceeds dI(-l.O G) and wheel
velocity exceeds the first slip threshold(about 5 %). This is called Phase-3 which the
wheel is decelerated and the brake pressure is decreased. Phase-4 occurs when the
wheel begins to be accelerated but the recovery of the braked wheel is not sufficient
while the brake pressure is continuously decreased. The brake pressure begins to be
increased when the wheel acceleration exceeds al(about +0.4 G) in order to prevent
the efficiency of braking from deteriorating. this is called Phase-5. In Phase-2, which
follows phase-5, the wheel is decelerated again with increasing brake pressure. Until
'---.-now, the first control loop is completed and the second control loop will be started if
the wheel is overbraked again.
32
g
1.00
~
0.90
~
0.80
4:l
S
~
-~
~
0.70
0.50
0.40
~
0.30
"g
Slip Angle 20°
0.60
a
'5h
Slip Angle 0°
Flong.
0.20
j
0.10
~
0.00
a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 2-9 Slip Range for ABS Control
33
0.8
0.9
Vehicle, Vx
._._._._._._._._._
Slip Threshold
..
wheel, Wi R
time
g
'.0
"8t
al
.(
0
~
dl
f------~--~------"~--_____,.,L------'--
time
Fig. 2-10 Control Loop for ABS Control
34
Chapter 3
Simulation and Analysis
3.1 Introduction to Simulation
The block diagram of the nonlinear vehicle system is expressed in Fig 2-1.
When the brake and/or steering input is commanded, the longitudinal slip and slip
angles of each wheel are calculated based on the instantaneous values of the vehicle
longitudinal and lateral velocities, yaw rates and angular velocities of the wheels as
discussed in the previous section. The nonlinear tire produces the tire forces that drive
the motion of the vehicle; this motion is then fed back into the tire model.
The computer program implementation of this model basically carries out numerical
integration of a nonlinear set of first order differential equations. The equations of
motion of the vehicle are integrated by DASSL. The wheel spin mode for each wheel
is calculated algebraically in order to compute longitudinal slip.
The simulation program is set up so that the ramp step input for the steering
wheel and the first order input for the brake force input are applied as an open loop
input. The vehicle is initially travelling at a constant speed in a straight line over an
even road before the brake and/or steering are applied. A constant speed is required
to start the steady state turn before brake torque is exerted. A driving torque should
be applied into the driven wheels for a constant speed because the tire drag makes the
vehicle decelerate. The vehicle handling characteristics during braking in a turn are
35
evaluated using the ISO procedure [27]. Under this procedure, braking is carried out
while the vehicle is executing a steady state turn with a fixed steering angle.
Vehicle and brake parameters used in this simulation model are taken from the
front wheel drive Hyundai Elantra and [26] as listed in Table 3-1.
Table 3-1 Vehicle Parameters
Parameter
I
I
Car weight, mg
Values
2.75 KN
Distance to front axle from car
1.0 m
e.g, a
Distance to rear axle
1.5 m
from car e.g, b
1.8 m I 1.8 m
Wheel track, Tfl Tr
Yaw inertia of the body, Iz
1627 Kg m2
Wheel inertia, Iw
4.07 Kg m2
36
I
3.2 Braking in a Turn
The vehicle is initially travelling at 50 kph (31.25 mph) on a dry road (jt=0.9)
or a slippery road (jt=0.3) when the ramp step input for steering is commanded. The
maximum steering angle is reached within a specific time (tstr =0.5 sec). The steering
angle is determined in order that the vehicle has a steady-state turn within a time (2.5
sec). The driver's pedal force is applied with a first order input (an exponential
function with a time constant, Tc =O.13 sec) when the vehicle reaches steady state.
The brake torque with respect to the driver's pedal force and the brake torque
distribution are expressed in Fig. 3-1. This brake system is designed to give
significantly less brake torque to the rear wheels than the front wheels. This is done
in order to prevent rear wheel lock-up under most braking maneuvers.
A simulation of braking in a turn is presented in Fig. 3-2, where the
coefficient of friction of a road is 0.3 and the driver's pedal force is 10 kg. The
velocities of the e.g. of the vehicle and the front and rear wheels are shown in Fig. 32(a). The front wheel is locked within 1 sec after the brake input is commanded and
also loses lateral tire force. But the rear wheel maintains the lateral tire force due to a
small slip. The results cause the yawing motion of the vehicle. The vehicle loses its
steerability as the yaw rate is severely decreased as shown Fig. 3-3(a). The time
history of the longitudinal acceleration of the vehicle clearly demonstrates the
characteristic of the longitudinal tire force when the wheel is locked. The maximum
37
deceleration is found just before the wheel is locked because the longitudinal tire
friction does not have maximum value at full skid (slip = 1.0). Instead, the
longitudinal tire force has a maximum value at a critical slip (approximately 0.2).
As the result of this simulation, the various characteristics of braking in the
vehicle are able to be evaluated, such as stability, steerability and braking
performance. On the other hand, these characteristics can be controlled by ABS which
. controls the wheel slip.
3.3 Braking in a Turn with ABS
A simulation with ABS is shown in Fig 3-4. The front and rear wheels are
controlled by ABS within a proper slip range (0.8-0.3) without wheel lock-up during
panic braking (pedal force: 50 kg) on a slippery road (",=0.3), while the front and
rear wheels without ABS are locked within 0.3 and 2.0 sec after braking.
The average vehicle deceleration is 2.48 m/sec 2 when ABS is operating but
2.75 m/sec2 when ABS is not operating, that is to say, the average deceleration is
increased by 10 % with ABS. The stability and steerability would be improved
because each wheel does not lose its lateral force. The detailed analysis of stability
and steerability is performed in the following section.
38
350.00 . - - - - - - - - - - - - - - - - - - - - - - - - ,
300.00
E 250.00
.*....(J)
~
L-J
Q}
200.00
Front
:J
()
....
~
150.00
Q}
.::::t.
Cl
di 100.00
50.00
o.00 -jL----,----,-------.---,---,---,------,---,--------,------1
o 102030405060708090100
Pedal Foree [Kgf]
(a) Brake Toque with Respect to Driver's PedaJ Force
50.00
45.00
40.00
,--,
E 35.00
.*....
{l30.00
Q}
~
25.00
....
0
t-
20.00
....
Cl
~
15.00
10.00
5.00
0.00
0
50
100
150
200
250
Front Torque [Kgf*m]
(b) Brake Torque Distribution
Fig. 3-1 Brake Torque
39
300
350
14.000
12.000
10.000
"....,.
U
QJ
Ul
"""'- 8.000
E
.......,
>-
0+-
u
6.000
0
Qj
>
4.000
~
Front Wheel
2.000
0.000 + - - - , - - - - - . - - - , - - ' - - - - . - - - , - - - , - - - - - - , - - - - - 1
3.0
4.0
5.0
6.0
7.0
2.0
0.0
1.0
8.0
Time (sec)
(a) Velocities of the Vehicle and Wheels while Braking in a tum.
1.000
0.900
0.800
0.700
J
Front
0.600
Q.
iii 0.500
DADO
0.300
)
0.200
0.100
0.000
0.0
1.0
2.0
3.0
I
Rear
4.0
5.0
Time (sec)
6.0
7.0
8.0
(b) Wheel Slip while Braking in a turn
Fig. 3-2 The Behavior of Vehicle and Wheels while Braking in a Turn
40
0.200 . . . , . . . - - - - - - - - - - - - - - - - - - - - - - - - - ,
0.150
'"
OJ
()
VJ
"'0.100
1J
0
L
"-"
OJ
0
0::
0.050
~
0
>0.000
-0.050 - t - - - , - - - - - , - - - - , . - - - - , - - - . , - - - - - - , - - - - - , - - - - j
3.0
4.0
5.0
6.0
7.0
2.0
0.0
8.0
1.0
Time (sec)
(a) Yaw Rate in Response to Braking in a Turn (l-L=O.3)
3.000
I
2.000
()
'"
OJ
VJ
cy
Lateral Acceleration
1.000
VJ
"'-
E
~ 0.000
"-----
.Q
o
L
"*u -1.000
I
()
<{
Longitudinal Acceleration
-2.000
-3.000
0.0
1.0
2.0
3.0
4.0
5.0
Time (sec)
6.0
7.0
8.0
(b) Longitudinal and Lateral Acceleration while Braking in a Turn (l-L=O.3 Fp=lO)
Fig. 3-3 Lateral and Yawing Motion while Braking in a Turn
41
14.000 l""'"'~~~~~~-----------~
12.000
,.......
10.000
Vehicle w/
0
ASS
U
QJ
Front WHL w/ASS
(/)
/vehicle w/ASS
""- 8.000
E
"--'
>-
=E
6.000
o
Qj
>
4.000
Front WHL w/
0
ASS
2.000
Rear WHL w/
0
ASS
0.000 + - - - , - - - - - - - . . . , - - - - - - ' - r - - - - - - - , - - - ' - - - , - - - , - - - . - - - - - - j
0.0
2.0
3.0
4.0
5.0
1.0
7.0
8.0
6.0
Time (sec)
(a) Velocities
wI and w/o ABS
1.000
0.900
0.800
0.700
0.600
Q..
(/) 0.500
0.400
0.300
0.200
0.100
0.000
0.0
1.0
2.0
3.0
4.0
Time (5)
(b) Wheel Slip
5.0
6.0
7.0
wI ABS
Fig. 3-4 The Behavior of Vehicle and Wheels wi ABS
42
8.0
3.4 Analysis of the Vehicle While Braking in a Turn
3.4.1 Yawing Motion
The lateral and directional motions of a vehicle involve a yawing motion which
depends on the variation of tire force with slip angle and slip ratio. When a driver
turns the steering angle to change the course during driving, the lateral tire force Fyi
is due to a slip angle. The sum of the lateral forces Fy at the point of contact between
tire and road surface must be equal to the centrifugal force resulting from the driving
speed
Vx
and the instantaneous curvature p of the course selected by the driver as
shown Fig. 3-5. The sum of the longitudinal forces Fx determines the acceleration or
deceleration of the vehicle. The sum of the yawing moments of the tire forces caused
by braking and/or steering about the vertical axis of the vehicle is expected to cause
the change in angular momentum.
The undesired behavior of yawing motion while braking in a turn can be
divided into two parts: spin and drift-out (Fig. 3-5). The difference of the lateral force
between the front and rear wheels causes the vehicle to understeer or oversteer. The
spin occurs when the lateral force of the front wheels is much larger than that of the
rear wheels, in other words, when the rear wheels lose the lateral force, the vehicle
become unstable. The drift-out occurs contrary to the spin. It occurs when the front
wheels lose the lateral force, therefore, the vehicle is apt to lose its steerability.
43
(a) Force Distribution on Tum
~
(Q
Drift out
\\\\
Reference circle
~'"
;'
"
".'".
spU
\J~
(b) Spin and Drift-out
Fig. 3-5 Braking in a Turn
44
(a) The Vehicle with Yaw Freedom Only
In order to understand the characteristics of the yawing motion of the vehicle,
a simple elementary system is introduced [1]. In the body-centered-axis model as
.
shown Fig. 2-2, the front and rear wheels are replaced by single wheels at the center
of the vehicle, and tire-cornering stiffness has a linear property with respect to the
small slip angle. The following equations can be rearranged from Eq. (2-6):
(3-1)
where slip angles are expressed by definition:
awz
v
fJ = - - 0
I
x
(3-2)
Yawing response to the steering angle with non-lateral motion (vy =0) and constant
velocity (vx = constant) is essentially a problem similar to the torsional oscillation of a
body fixed at the center of the gravity and oscillation under the action of two springs
located at distances a and b on either side of the point of fixation:
(3-3)
where
45
(3-4)
Nr =
Nf is the restoring torque produced from the tires by a change in the angular position
of the vehicle, hence
(3-5)
but if the vehicle is assumed initially to have no angular heading, Eq. (3-3) can be
rewritten
Iu~,r, - Nr r{r - N ~~.Ir = N d 0
(3-6)
Therefore, yawing motion can be expressed in second order form, such as Eq. (3-6)
which has the spring constant N f and the damping coefficient Nr•
The character of the natural response of a second order system, such as this
one, is determined by the roots of the characteristic equation:
(3-7)
If Nf is positive, one of the roots will be positive and the motion of the vehicle will
be divergent or unstable. When N f is zero, the system is neutrally stable. If Nf is
~.
4,6
negative but 14Nflu. I is less than N,2, the system will decay to a position of
equilibrium (over damped); if 14Nflu. I is greater than N/, the system will decay in an
oscillating manner (underdamped).
As mentioned above, the yawing motion is typical of a second-order dynamic
system, therefore this motion is a very important component in analyzing the
characteristic behavior of a braked vehicle in a tum.
(b) Yawing Motion While Braking in a Turn
Yawing motion is not quite as simple during braking in a tum because of the
complex interaction and nonlinearity of the variables. But the results of the numerical
simulation has the ability to portray the complex interactions caused by load transfer
and tire characteristics under combined braking in a tum maneuver. Fig. 3-6 and 3-7
represent the yawing motion while braking in a tum on both a high friction and a
slippery road. The position of the vehicle corresponding to Fig 3-6 is presented in
Fig. 3-8.
Braking forces break the steady state of the vehicle in a tum and cause the
yaw rate to decrease or increase as shown Fig. 3-6(a) and 3-7(a). The yaw rate is
investigated in terms of the increase of the driver's pedal force; the yaw rate has the
tendency of increasing for a while as soon as the braking force is initiated. After this
tendency disappears, the greater the pedal force, the faster the yaw rate is decreased.
As shown in Fig. 3-6(b) and 3-7(b), this behavior can be explained by the phase-plane
47
in which the yaw rate
Wz
and the yaw acceleration Wz describe the history of the
yawing motion. The tendency of increasing the yaw rate indicates 1) a positive yawacceleration in the phase-plane, and 2) a tendency of spinning toward unstable state.
However, this tendency disappears and another unexpected tendency follows which is
called drift-out defined in the preceding section. The drift-out makes the vehicle
deviate from the expected course and has negative yaw-acceleration. Even though this
characteristic guarantees a stable state, the severe drift-out deteriorates the steerability
because of the large deviation from the expected course. (This is called strong
understeer in the handling analysis.) These tendencies can be changed according to the
parameters of a vehicle, but can not be prevented perfectly even with a slip control
device [28].
As shown in Fig 3-8, the position of the vehicle on braking in a turn is
affected by the yawing motion. The instantaneous curvature
V
x
p --
p
can be expressed as:
(3-8)
The curvature is constant while the steady state turn is maintained; at the same time,
the rate of change between the longitudinal velocity and the yaw rate is the same.
However, the curvature becomes smaller when the yaw rate is increased during
braking, Le. this motion tends toward spinning. Also, the curvature becomes larger
when the yaw rate decreases so quickly, Le. the motion tends to result in a drift-out.
This almost occurs during heavy braking in a turn without an external disturbance.
48
,
0.50
Steady Stole
0.40
.--...
u
Q)
(f)
0.30
"<J
0
Podd Forel 10 kg
I.........
>-
0.20
0
Qj
0.10
-'u
I
>
~
0
>-
0.00
-0.10
-0.20
0
2
3
4
Time (sec)
5
6
7
(a) Time Domain
0.80
..,..--------r------==-~==__-----_...,
0.60
~
0.40
Pecci Force 10 kg
0-
~
0.20
<J
-=-o -0.00
c
o
:g
lI])
Qj
u
u
+--------l-----+I----4\------=~*_-__j
-0.20
-0.40
1 -0.60
3
~ -0.80
/
Fp=50 kg
Fp=50 kg w/ABS
-1.00
-1 .20 +---~----+---~--~--~--~-----0.1
o
0.1
0.2
0.3
0.4
0.5
-0.2
Yaw Velocity (rad/s)
(b) Phase Plane
Fig. 3-6 Braking in a Turn (p=O.8)
49
0.20
0.15
,.......
Vl
..........
lJ
0.10
0
L
'--"
....>-
·0
0.05
0
Q)
>
~
0
0.00
>-0.05
rp~30
-0.10
0
2
3
4
Time (sec)
I
kg
w/o
5
ABS
6
7
8
(a) Yaw Response on Time-domain
0.40 - , - - - - - - - - , - - - - - - - - - - - - - - - - ,
0.30
,.......
Vl
0Vl
..........
lJ
0
L
0.20
0.10
0.00
'--"
6 -0.10
....
~
rp=30 kg w/ABS
-0.20
/
QJ
Q)
u
u
-0.30
I
-0040
«
~
o
>- -0.50
-0.60
rF30 kg
I
w/o
ASS
-0.70 + - - - - - - - - - , - - - - - 1 - - - - - - , - - - - - , - - - - . , - - - - - - - 1
-0.1
-0.05
o
0.05
0.1
0.15
0.2
Yaw Velocity (rad/s)
(b) Yaw Response on Phase-plane
Fig. 3-7 Braking in a Turn (JL=O.3)
50
40.00
.--------------------:~
P'cJ<jForcel~
35.00
30.00
E 25.00
'-"
c
~ 20.00
'iii
o
0..
I 15.00
>10.00
Fp=50 kg
5.00
0.0 0 -1-----==1:;::0=------2'0- - - - - 3 ' 0- - - - - - - - 1
40
0
X-Position (m)
Fig. 3-8 Vehicle Trajectories (JL=O.8)
51
3.4.2 Analysis for Stability
In the previous section, the stability is determined by the roots of the
characteristic equation(Le. eigenvalues) for the model with yaw degree of freedom
only as a linear system. However, we have three state variables in the nonlinear
system with a nonlinear tire model, and this model can be expressed by the following:
(3-9)
Currently, we are interested in how to determine the stability for the nonlinear
system. The approach [29,30,31] to determine the stability for the nonlinear system is
following:
According to the basic definitions, stability properties depend only on the
nature of the system near the equilibrium point. Therefore, to conduct an analysis of
stability, it is often theoretically legitimate and mathematically convenient to replace
the full nonlinear description by a simpler description that approximates the true
system near the equilibrium point. Often a linear approximation is sufficient to reveal
the stability properties. This idea of checking stability by examination of a linearized
version of the system is referred to as Liapunov's first method, or sometimes as
Liapunov's indirect method. It is a simple and powerful technique and is usually the
52
first step in the analysis of any equilibrium point.
The linearization of a nonlinear system is based on linearization of the
nonlinear function f in its description. An n-th order system is defined by n functions,
each of which depends on n variables. In this case, each function is approximated by
the relation:
The linear approximation for the vector f(x) is made up of the n separate
approximations for each component function. The complete result is compactly
expressed in vector notation as
j(x+y)
In this expression F is the
I(x) +Fy
(3-11)
n x n matrix
Of!
ax!
Of
- z
ax!
-
'{
e
Of!
axz
-
Of
- z
axz
Of,.
aXn
Ofz
aXn
-
(3-12)
F=
53
The matrix F is referred to as the Jacobian matrix of f.
Next, suppose that
x is an equilibrium
point of the system
,
(3-13)
i(t) = f(x(t»
x
Setting x(t) = +y(t) and using the approximation Eq. (3-11) leads in a similar way
to the linear approximation
j(t)
=
(3-14)
Fy(t)
Thus, the linear approximation of a nonlinear system has F as its system matrix. The
state vector of the approximation is the deviation of the original state from the
equilibrium point. The stability properties of a linear system are determined by the
location (in the complex plane) of the eigenvalues of the system matrix, and the
stability properties of the linearized version of a nonlinear system can be determined
that way. Then, stability properties of the original system can be inferred from the
linearized system using the following general results:
(1) If all eigenvalues of F are strictly in the left half-plane, then
x is asymptotically
stable for the nonlinear system.
(2) If at least one eigenvalue of F has a positive real part, then
nonlinear system.
54
x is unstable for the
(3) If the eigenvalues of F are all in the left half-plane, but at least one has a zero real
part, then
x
may be either stable, asymptotically stable, or unstable for the
nonlinear system.
The essence of these rules is that the eigenvalues of the linearized system
completely reveals the stability properties of an equilibrium point of a nonlinear
system. The reason is that, for small deviations from the equilibrium point, the
performance of the system is approximately governed by the linear terms.
The 3 x 3 Jacobian matrix is formed at each calculation step in this
simulation, and three eigenvalues of the Jacobian matrix are calculated using the
software package LAPACK. In the case of Fig. 3-9, in which the eigenvalues for the
same condition with that of Fig. 3-6 (jL=O.9/hard braking) are presented, two
eigenvalues are complex conjugates and the other has a negative real value. According
to the above approach, the system is for a while unstable as soon as the braking force
is initiated, because the eigenvalue has a positive real part. This analysis accords with
the previous analysis for the yaw rate.
On the other hand, the eigenvalues can be presented on the complex (s-o)
plane. This plot, called the root-locus plot, indicated the positions of the roots of the
characteristic equation of the system, so that the evaluation for the system can be
accomplished using a classical control theory [31]. First of all, the damping ratio can
be evaluated from the location of the roots, that is, the damping ratio
r is equal to
cosine of the angle formed between the negative s-axis and the line formed between
55
10.000
mog;nory-3
0.000
E -1 0.000
~
Q)
0::
........,
-20.000
Real-I
R."-2
1Il
Q)
:J
a
-30.000
>
c
Q)
OJ
W -40.000
-50.000
-60.000
0.0
0.5
1.0
1.5
2.0
2.5
Time (sec)
3.0
3.5
Fig. 3-9 Time History of Eigenvalues (w/o ABS)
56
4.0
5.000..,-------------------.------,----,
4.000
3.000
(= .5
2.000
>La
.~
a
1.000
0.00 0 rz!E!i!!iilm------~======:m::lllEHIIE!lI3:~>__-1
1; -1.000
-2.000
-3.000
-4.000
-5.000 +--.---,---,--------,----,----,---r----.~-+--1
-45.0-40.0 -35.0 -30.0-25.0 -20.0-15.0 -10.0 -5.0 0.0 5.0
Real
(a)
~=O.3
Fp=30 kf w/o ABS
5.000 . . , - - - - - - - - - - - - - - - - - - - , - - , - - - - - - ,
4.000
3.000
2.000
>L-
a
.~
1.000
0.00 0 -t-----'l:8!'E23!iE:---,::,;;::m-"'E*--iiE-~
iHHllllIE--$;>__----j
o
1; -1.000
-2.000
-3.000
-4.000
-5.000 +--.---,---,---------,----,----,------,-----,-'---j--------j
-45.0-40.0-35.0 -30.0-25.0-20.0-15.0-10.0 -5.0 0.0 5.0
Real
(b)
~=O.3
Fp=30 kg wI ABS
Fig. 3-10 Eigenvalues on s-u Plane
57
~
the point of the root and the origin as shown Fig. 3-10.
(3-18)
On the
S-(J"
plane in Fig 3-1O(a) and (b), these trajectories of the roots and the
damping ratio are found; one real eigenvalue is closed to zero; the other two remain
complex while in the initial transient range, in the steady state, and during braking.
However, these two complex pairs change to real values during the final phase of
braking, e.g. slow speed. A damping ratio (t=O.5) is indicated in this plane. The
damping is relatively larger with ABS than without ABS as mentioned in the
following discussion.
3.5 Improved Algorithm for ABS
The control algorithm introduced in Section 2.7.2 uses two kinds of fixed
control thresholds to enable the wheels to be controlled within the proper slip range:
the fixed slip threshold and the fixed wheel acceleration threshold. When this method
is applied, the steerability is more or less poor on the slippery road as shown in Fig.
3-7 even though the ABS controls the wheels within the expected slip range. The
reason is that the lateral force of the rear wheel is comparatively larger as compared
to that of the front wheel. Therefore, the system has a small damping ratio, and as a
result, the yaw rate decreases quite quickly, and the vehicle has a tendency toward
58
drift-out.
Here is a suggestion for improving the stability and steerability of the braked
vehicle. The idea is that the threshold may be variable for the characteristics of the
vehicle in order to enable the vehicle to easily trace the driver's designated course
without deteriorating braking performance. That is, when the system has a small
~
damping ratio, which is evaluated by eigenvalues, the damping ratio can be increased
by decreasing the lateral force of the rear wheels. This strategy is achieved using the
variable slip threshold which varies corresponding to eigenvalues as shown in Fig 311 (a). In addition, this concept prevents the vehicle from having the tendency of
spinning using the variable slip threshold; when the vehicle has the tendency of
spinning, the eigenvalues approach zero, so that the slip threshold can be varied to
avoid instability.
In Fig. 3-11(b), the three kinds of slip thresholds are applied: Case (a): front
fixed 0.08/rear fixed 0.08, Case (b): front 0.08/rear variable 0.02-0.05, and Case (c):
front fixed 0.05/rear variable 0.02-0.03. In the case of (a), the yaw velocity decreases
fast in incipient braking so that drift-out occurs. In the case of (b), the yawing
motion is fluctuating and tends to spin so that the driver can feel uncomfortable and
the system is unstable for a long time. In the case of (c), the system is stable and yaw
, -
velocity smoothly decreases so that this combination is the best for the stability and
the steerability while braking in tum.
59
Modulator <1f----
ASS
1==
<.l (WHL Velocities)
Algorithm .- . ; r - - s (Eigenvalue)
,
f------·_-------_·_---------------------~---·_·_---_·----_._---------_._--._._-------------_._._._----._._------·_·_·-1
i
j
ij
Ii
i
i
iL_._._._.
ij
Variable Slip Threshold
THSUP~. ~a~ ~.
I!
s
i
j
.
.
._.
.
._.
._._._" ._._.
.
._._. . .
._.
!
.J
(a) Concept for Variable Slip Threshold
0.180
0.160
0.140
CJl
'"'
0.120
"-...
"'0
0
L
'-'
>-
0+-
()
0.100
0.080
0
Q)
> 0.060
~
0
>-
0.040
0.020
0.000
-0.020
0.0
1.0
2.0
3.0
4.0
Time (sec)
5.0
6.0
(b) Yaw Velocity for Each Variable Slip Threshold
Fig. 3-11 Improved ABS
60
7.0
Chapter 4
Conclusion
A nonlinear seven-degree-of-freedom vehicle model with a realistic brake
model has been developed, and a nonlinear tire model has been adopted in order to
simulate and analyze the behavior of the vehicle equipped with Antilock Brake
System(ABS) while braking in a turn. A control method for ABS was employed and
improved to maximize its stability and steerability.
The simulation was performed with a brake pedal input and a steering input on
a slippery and high friction road. The results of the numerical simulation presented
the ability to portray the complex interactions by load transfer and tire characteristics
under combined braking in a turn maneuver.
The characteristics of the braked vehicle on a steady-state-turn was basically
investigated and closely analyzed using several methods such as a time-domain
analysis, a phase-plane, a root-locus plot and Liapunov's indirect method, in other
words, a yawing motion can be directly presented in the time domain and phaseplane, therefore, this motion can be evaluated through the analyses of a distribution
of the braking forces and a relationship between its curvature and a velocity rate. At
the same time, the roots of the characteristic matrix directly reveal the stability
properties and give us a system which can be evaluated in the root-locus plot.
Finally, a new idea was suggested to improve the stability and steerability of
61
the braked vehicle in ABS. The characteristic roots were adopted in a control
algorithm, that is, the motion of the braked vehicle was fed back into the controller so
that the improvement of the performance was accomplished with this research.
The following recommendations are suggested for future research:
(1) The vehicle model should be expanded to investigate the effects of the lateral
weight transfer which affects the longitudinal and lateral tire forces. This model will
include the pitching and rolling motion with a realistic suspension model. In addition,
the effects of the inertia of a power train(engine, transmission) mightbe considered
due its importance to the controlled wheels.
(2) A vehicle test should be conducted on the development process of ABS. After a
real time simulation, combining the simulation model with the realistic actuator in
order to validate the control algorithm and the hardware equipment, various vehicle
tests should be conducted on various road conditions.
62
Chapter 5
References
1. Ellis, J.R. Vehicle Dynamics. London Business Book Limited,1964
2. "Computer Simulation of Vehicle Handling", The Bendix Corporation Research
Laboratories, Contact No. FH-1l-7563, September 1972
3. Jindra, F. "Mathematical Model of Four-Wheeled Vehicle for Hybrid Computer
Vehicle Handling Program", NHTSA, DOT HS-801 800, Jan. 1976
4. Dugoff, H., et al, " An Analysis of Tire Traction Properties and their Influence on
Vehicle Dynamic Performance", SAE 700377
5. Sakai, H., "Theoretical and Experimental Studies on the Dynamic Properties of
Tire, Part 1: Review of Theories of Rubber Friction", International Journal of Vehicle
Design, Vol. 2, No.1, 1981
6. Allen, R.W. and Rosenthal, T.J. et al. "Steady State and Transient Analysis of
Ground Vehicle Handling", SAE 870495, 1987
7. Bakker, E, Pacejka, H.B., et al, "Tyre Modeling for Use in Vehicle Dynamics
studies", SAE 870421
8. Bakker, E, Pacejka,H.B, et al, "A new Tire Model with an Application in Vehicle
Dynamics Studies", SAE 890087
9. Xia,X., et al, "Response for Four-Wheel-steering vehicle to Combined Steering
and Braking Input", ASME 1989 Winter Annual Meeting, DSC-Vol. 13, 1989
10. Nakazato, H. et al, "A New System for Independently Controlling Braking Force
63
Between Inner and Outer Rear Wheels", SAE 890835, 1989
11. Yamamoto, M., "Active Control Strategy for Improved Handling and Stability",
SAE 911902, 1992
12. Allen R. W., et al, "Characteristics Influencing Ground Vehicle
Lateral/Directional Dynamics Stability", SAE 910234, 1991
13. Zellner, J.W., "An Analytical Approach to Antilock Brake System Design", SAE
840249, 1984
14. Hussain, S.F., "Digital Algorithm Design for Wheel Lock Control System", SAE
860509, 1986
15. Watanabe, M., et al, "A New Algorithm for ABS to Compensate for RoadDisturbance", SAE 900205, 1990
16. Tan H.S., et aI, "Vehicle Traction Control: Variable-Structure Control
Approach", Journal of Dynamic Systems, Measurement, and Control, Trans. of the
ASME Vol. 113, Jun 1991
17. Greenwood, D.T. Principle of Dynamics. 2nd ed. Prentice-Hall, Inc., 1988
18. Petzold, L.R.
subroutine DASSL - differential/algebraic system solver,
Computer and Mathematics Research Division, Lawrence Livermore National
Laboratory, 1983
19. Bernard J.E, "Digital computer method for the prediction of braking performance
of trucks and tractor-trailers", Tran vol 82, SAE 730181, 1973
20. Bleckmann, H.W. "The new four-wheel anti-lock generation: a compact antilock
and booster aggregation and advanced electronic safety concept", Proc Instn Mech
64
Engrs, Vol. 200 No. D4, 64/86, 1986
21. Allen, R.W, et al,_IIField Testing and Computer Simulation Analysis of Ground
Vehicle Dynamic Stability", SAE 900127, 1990
22. Leiber, H. and Czinczel, A., IIFour Years of Experience with 4-Wheel Antiskid
Brake System(ABS) II, SAE 830481, 1983
23. Satoh, M., et al, "Performance of Antilock Brakes with Simplified Control
Technique ll , SAE 830484
24. Hasida, K., et al, "Compact 4Ch-ABS Hydraulic
Unit'~,
SAE 910697, 1991
25. Allen, R.W., et al, IITest Methods and Computer Modeling for Analysis of
Ground Vehicle Handling", SAE 861115, 1986
26. "Service Manual for Hyundai Elantra", Hyundai Motor Company, 1992
27. Road Vehicle - Braking in a tum: Open loop test procedure, ISO TC22/SC9
N200, Jan. 1980
28. Dreyer, A and Heitzer, H-D IIControl Strategies for Active Chassis Systems with
Respect to Road Friction." SAE 910660, 1991
29. Luenberger, D.G, Introduction to Dynamic Systems, John Wiley & Sons, Inc.
1979
30. Takahashi, Yet al, Control and Dynamics Systems, Addison-Wesley Publish Co.
1970
31. Ogata, K, Modem Control Engineering, Prentice-Hall, Inc., 1970
65
Chapter 6
Appendix
A. Tire Modeling
..
This model development was motivated by three primary objectives, 1) to
account for measured interactive force properties over the complete maneuvering
range from pure adhesion to pure sliding, 2) to use standard variable definitions and
commonly available parameters, and 3) to achieve a computational efficient analytical
form for computer simulation application.
The basic tire input variables are tire normal load, lateral slip angle,
longitudinal slip ratio, and camber angle, along with the resulting response variables
of lateral and longitudinal force and aligning torque. Table A-I presents the parameter
variations with load. Lateral and longitudinal forces are derived in Table A-2. Model
coefficients are provided as inputs, and the program generates user specified force and
moment response plots. Some typical plots are shown in Table A-3.
66
A-l. Parameter Variation with Load
K = ~(AO+AIF _Al ~
s
2
z A2 z
1. Lateral Stiffness Coefficient
apO
-
2. Longitudinal stiffness Goefficient
Kc =
4Fz(CS/FZ)
apo
Aa 2
3. Camber Thrust Stiffness
Yyo = AaF'z -AFz
4
Km= KF
t z
4. Aligning Torque
5. Peak Tire/Road Coefficient of
Friction
J.l o
~SNo
= (BtFz+B3+B4Fz SN
T
where SNr = 85 (Test Skid No.)
a
p
apo =
6. Tire Contact Patch Length
F
x
= apo(l-K-)
aF
z
O.0768JFzFzT
Tw(Tp+5)
where
F zr = tire design load at operating
pressure( lbs)
Tw = tire width (inches)
Tp = tire pressure (psi)
67
A-2. Summary of Basic Equations
1. Composite
2. Force Saturation Function
2 4
3
C 0 +C 0 +-o
2
1
1r
.f{ 0)
Fy
F
J.l z
3. Normalized Side Force
2
3
C 0 +C3 0 +C 0
1
4
+1
= _;:::f{=O=)=K=stan==a~ +Y y
(l.
Y
.Ik 2tJm2 a +K c S2
Ys
4. Normalized Longitudinal
Force
5. Aligning Torque
1
Jsin a +s2cos2 a
J.l =~o[ l-K Vsin a+s cos a]
2
K c = K c +(Ks -Kc)
6. Slip to Slide Transition
Il
68
2
2
2
Table A-3. Tire Parameter for Main Text Tire/Vehicle Test Cases
Tire /Vehicle
Standard Cross
Section
Radial/RWD
Bias
Ply/RWD
Wide Section
Low Profile
Radial/FWD
155 SR 13
P155/80 D 13
P185170 R 13
Tire Width
6
6
7.3
Tire Pressure
24
24
24
Tire Design Load
810
900
980
AO
914.02
817
1068
Al
12.9
7.48
11.3
A2
2028.24
2455
2442.73
A3
1.19
1.857
0.31
A4
-1019.2
3643
-1877
Ka
0.05
0.2
0.05
KIJ.
0.234
0.234
0.234
Bl
0.0003396
-0.000257
-0.000169
B3
1.19
1.19
1.04
Parameter
Tire Designation
B4
4.98
x 10-8
2.64
X
10-8
1.69
X
10-8
CS/PZ
18.7
15.22
17.91
IJ.nom
0.85
0.85
0.85
Kl
-1.22
x
10-4
-1.95
69
X
10-4
-0.8
X
10-4
Vita
July 2, 1964
Feb. 1986
Born in Danyang, Korea
B.S., Korea University
Seoul, Korea
Research Engineer,
Hyundai Motor Company
Ulsan, Korea
Graduate Student,
Lehigh University
Supported by Hyundai Motor Co.
1986 - 1992
Jan. 1993 - Dec. 1993
Will be a Research Engineer
with the Hyundai Motor Co.
1994 -
Future Address: Research and Development Dept.
Hyundai Motor Company
700 Yangjungdong Jungku
Ulsan, Korea, 681-380
Tel) 522-80-2993, Fax) 522-80-5784
70