Vehicle Dynamics Approach to Driver Warning

Vehicle Dynamics Approach to Driver Warning
Hindawi Publishing Corporation
International Journal of Vehicular Technology
Volume 2013, Article ID 109650, 18 pages
http://dx.doi.org/10.1155/2013/109650
Research Article
Vehicle Dynamics Approach to Driver Warning
Youssef A. Ghoneim
Research and Development Center, General Motors Corporation, 30500 Mound Road, Warren, MI 48090, USA
Correspondence should be addressed to Youssef A. Ghoneim; [email protected]
Received 12 November 2012; Accepted 4 January 2013
Academic Editor: Martin Reisslein
Copyright © 2013 Youssef A. Ghoneim. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper discusses a concept for enhanced active safety by introducing a driver warning system based on vehicle dynamics
that predicts a potential loss of control condition prior to stability control activation. This real-time warning algorithm builds
on available technologies such as the Electronic Stability Control (ESC). The driver warning system computes several indices based
on yaw rate, side-slip velocity, and vehicle understeer using ESC sensor suite. An arbitrator block arbitrates between the different
indices and determines the status index of the driving vehicle. The status index is compared to predetermined stability levels which
correspond to high and low stability levels. If the index exceeds the high stability level, a warning signal (haptic, acoustic, or visual)
is issued to alert the driver of a potential loss of control and ESC activation. This alert will remain in effect until the index is less
than the low stability level at which time the warning signal will be terminated. A vehicle speed advisory algorithm is integrated
with the warning algorithm to provide a desired vehicle speed of a vehicle traveling on a curve. Simulation results and vehicle tests
were conducted to illustrate the effectiveness of the warning algorithm.
1. Introduction
Freeway entrance and exit ramp interchanges are the sites
of far more crashes per mile driven than other segments of
interstate highways. Crashes most common on exit ramps—
run-off-road crashes—frequently occurred when vehicles
were exiting interstates at night, in bad weather, or on curved
portions of ramps. When the vehicle is driving under these
conditions at a higher speed than the surface can allow, the
understeer gradient of the vehicle can increase causing the
vehicle to plow or decrease and becomes negative causing the
vehicle to spinout.
In recent years, electronic stability control systems for
motor vehicles have become increasingly popular [1–12].
Conventionally, such systems monitor vehicle stabilityrelated quantities such as a yaw rate error, that is, a deviation
between yaw rates expected based on vehicle speed and
steering wheel angle and an observed yaw rate, and selectively
brake the wheels at one side of the vehicle in order to assist
cornering, that is, to decrease the deviation between expected
and observed yaw rates.
Electronic Stability Control (ESC) helps keep the vehicle
on its steered path during a turn, to avoid sliding or skidding.
It uses a computer linked to a series of sensors—detecting
wheel speed, steering angle, yaw rate and lateral acceleration
of the vehicle. During normal driving, ESC works in the
background and continuously monitors steering and vehicle
direction. It compares the driver’s intended direction (determined through the measured steering wheel angle) to the
vehicle’s actual direction (determined through measured lateral acceleration, vehicle rotation (yaw), and individual road
wheel speeds). ESC intervenes only when it detects a probable
loss of steering control, that is, when the vehicle is not going
where the driver is steering. This may happen, for example,
when skidding during emergency evasive swerves, understeer
or oversteer during poorly judged turns on slippery roads,
or hydroplaning. If the vehicle starts to drift, the system
momentarily brakes one or more wheels and, depending on
the system, reduces engine power to keep the car on the
steered course. However, Electronic Stability Control cannot
override the laws of physics. If a driver exceeds the friction
capabilities of the road surface, ESC cannot prevent a crash.
It is a tool to help the driver maintain control.
In this paper, we introduce a driver warning algorithm
integrated with the Electronic Stability Control system. The
warning system is designed to further assist a driver by
warning of an impending ESC activation so that the driver
will reduce the vehicle’s speed prior to the need for ESC
2
International Journal of Vehicular Technology
Vehicle and driver block
Driver
inputs
Steering
Acc. pedal
Brake pedal
ESC Block
Warning and speed advisory block
Driver
command
interpreter
Warning algorithm
∼
..
.
∼
Lat. acc.
Steering angle
Yaw rate
Vehicle Long. accel.
sensors Throttle
MCP
Vehicle speed
Wheel speeds
Speed advisory
Vehicle
speed
advisory
Calculate non
linear SS
understeer index
Determine stability
index
Arbitration
logic
Driving states
Surface
capability
index
∼
..
.
∼
Driver warning index
(i) Haptic warning
(ii) Text message
(iii) Chime
(iv) · · ·
Vehicle speed advisory
(i) Text message
Figure 1: Schematic diagram of the warning algorithm.
intervention. It is hoped that the warning system will assist
the driver in recognizing and avoiding instances of potential
loss of vehicle control and also reduce usage of ESC.
The body of the paper begins first with system architecture including a brief description of the different blocks used
in the warning algorithm. Second, we develop the warning
signal command for both the transient and the steady-state
modes of the vehicle. Third, we calculate a vehicle advisory
speed in a curve based on vehicle understeer gradient. Finally,
we present simulation results and vehicle tests.
2. Warning Algorithm Architecture
Figure 1 illustrates the schematic diagram of the warning
algorithm which consists of three major parts:
(1) the vehicle and driver block which contains the
following:
(i) driver inputs (steering, accelerator pedal, and
brake pedal),
(ii) vehicle sensors (lateral acceleration, yaw rate,
wheel speeds to estimate the vehicle longitudinal speed, throttle position, and master cylinder
pressure sensor);
(2) the ESC block:
(i) command Interpreter block;
(3) the warning and speed advisory block:
(i) the driving states,
(ii) the Lateral Surface Capability Index,
(iii) the warning Algorithm:
(a) the understeer index,
(b) the stability index,
(c) the arbitration logic,
(iv) the vehicle speed advisory.
2.1. The ESC Block
2.1.1. The Command Interpreter Block. Vehicle Yaw-Plane
Dynamics. In this section, we describe the equations of
dynamics for the yaw-plane motion of a vehicle. While
a vehicle is undergoing handling maneuvers, it not only
incurs a yaw motion, it also experiences a side-slip motion
at the same time. The yaw-plane dynamics determine the
performance of vehicle yaw motion characterized by vehicle
yaw rate, as well as the lateral motion characterized by sideslip velocity.
The vehicle yaw-plane dynamics can be described by a
second-order state [7]
.
𝑣
𝑣
𝑏
𝑎 𝑎
[ ..𝑦 ] = [ 11 12 ] [ .𝑦 ] + [ 1 ] 𝛿,
𝑎21 𝑎22 𝜓
𝑏2
𝜓
.
(1)
where 𝑣𝑦 and 𝜓 are the vehicle side-slip velocity (defined
as the component of the vehicle velocity vector in the 𝑦
direction [13]) and yaw rate (defined as vehicle’s angular
velocity around its vertical axis), respectively, and the system
coefficients, 𝑎𝑖𝑗 ’s and 𝑏𝑖 ’s, are functions of vehicle mass, vehicle
speed, vehicle inertia, and the front and rear cornering
International Journal of Vehicular Technology
3
stiffness, and inevitably, the location of vehicle center of
gravity are described by the parameters 𝑎 and 𝑏:
𝑎11 = −
𝑎21 =
𝐶𝑓 + 𝐶𝑟
𝑎12 =
𝑀𝑣 𝑣𝑥
−𝑎𝐶𝑓 + 𝑏𝐶𝑟
𝐼𝑧 𝑣𝑥
𝑏1 =
𝐶𝑓
𝑀𝑣
−𝑎𝐶𝑓 + 𝑏𝐶𝑟
𝑀𝑣 𝑣𝑥
𝑎22 = −
𝑏2 =
𝑎𝐶𝑓
𝐼𝑧
𝑣𝑦𝑑 (𝑠)
− 𝑣𝑥 ,
𝑎2 𝐶𝑓 + 𝑏2 𝐶𝑟
𝐼𝑧 𝑣𝑥
substituting the system coefficients with the nominal values
forms the first-stage command:
,
𝜓𝑑 (𝑠)
𝑏2 𝑠 + (𝑎21 𝑏1 − 𝑎11 𝑏2 )
= 2
.
𝛿 (𝑠)
𝑠 − (𝑎11 + 𝑎22 ) 𝑠 + (𝑎11 𝑎22 − 𝑎12 𝑎21 )
(2)
.
𝛿 (𝑠)
𝑏1 𝑠 + (𝑎12 𝑏2 − 𝑎22 𝑏1 )
.
2
𝑠 − (𝑎11 + 𝑎22 ) 𝑠 + (𝑎11 𝑎22 − 𝑎12 𝑎21 )
𝑣𝑦𝑑 (𝑠)
(3)
𝛿 (𝑠)
Desired Vehicle Response. The desired yaw rate and the sideslip velocity commands are determined by the desired vehicle
response to the driver’s steering input. There are mainly two
approaches of implementation: (a) using the state in (1) to
perform real-time integration or (b) using transfer functions
such as (4) and (5) which consist of the steady-state value of
the desired yaw rate and side-slip velocity and a dynamic filter
representing the desired vehicle dynamics.
When the state approach is employed to generate the
commands, the desired vehicle side-slip velocity and desired
yaw rate are computed based on the system differentials using
nominal values of system parameters defined in (1)–(5).
When the state equation approach is employed to generate the commands, the desired vehicle side-slip velocity and
desired yaw rate are computed based on the system differential equations using nominal values of system parameters:
=
(𝑠/𝑧𝑣 + 1) 𝜔𝑛2
𝑉
,
𝑠2 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛2 𝑦 dss gain
.
(𝑠/𝑧𝜓. + 1) 𝜔𝑛2 .
𝜓𝑑 (𝑠)
𝜓
,
= 2
𝛿 (𝑠)
𝑠 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛2 dss gain
(4)
Equations (4) and (5) can be used to design a closedloop control system for vehicle stability enhancement by
regulating the measured vehicle performance to its desired
value determined in the following section.
(5)
In this reference model, the 𝐶𝑓 and 𝐶𝑟 are replaced with
constants 𝐶𝑓0 and 𝐶𝑟0 representing the values of a highcoefficient condition on dry surface. Integration of (1) results
in a desired time trace of vehicle side-slip velocity and yaw
rate.
The transfer-function approach for obtaining the desired
vehicle response is based on the structure of the system
input-output transfer function derived in (4) and (5), but
(7)
where
𝑉𝑦 dss gain =
(𝑎12 𝑏2 − 𝑣𝑥 ) − 𝑎22 𝑏1
,
𝑎12 𝑎22 − 𝑎12 𝑎21
.
𝜓dss gain =
𝑎21 𝑏1 − 𝑎11 𝑏2
𝑎12 𝑎22 − 𝑎12 𝑎21
(8)
(9)
are the steady-state gains of desired steady-state side-slip
velocity and yaw rate. The damping ratio and natural frequency of the desired vehicle performance can be expressed
in terms of system parameters
𝜔𝑛 = √𝑎11 𝑎22 − 𝑎12 𝑎21 ,
𝜁=−
𝑎11 + 𝑎22
2𝜔𝑛
(10)
and tabulated as functions of vehicle speed in control software. The variables 𝑧𝑣 and 𝑧𝜓. are the negative of the system
zeroes for both side slip velocity and yaw rate and can be
represented by
.
𝑣
𝑣
𝑎 𝑎
𝑏
[ ..𝑦𝑑 ] = [ 11 12 ] [ .𝑦𝑑 ] + [ 1 ] 𝛿.
𝑎
𝑎
𝑏2
𝜓𝑑
𝜓𝑑
21
22
(6)
Equations (6) are mathematically equivalent to (5). The
practical difference is that the desired natural frequency and
damping ratio determined by (6) can be specified without
regard to their original formation derived from the vehicle
parameters.
Therefore, rewriting (6) in terms of system natural frequency and damping ratio yields
.
𝑏2 𝑠 + (𝑎21 𝑏1 − 𝑎11 𝑏2 )
𝜓 (𝑠)
=
,
𝛿 (𝑠) 𝑠2 − (𝑎11 + 𝑎22 ) 𝑠 + (𝑎11 𝑎22 − 𝑎12 𝑎21 )
=
𝑏1 𝑠 + (𝑎12 𝑏2 − 𝑎22 𝑏1 )
,
𝑠2 − (𝑎11 + 𝑎22 ) 𝑠 + (𝑎11 𝑎22 − 𝑎12 𝑎21 )
.
The transfer functions from the steering input to the
vehicle yaw rate and side-slip velocity can be derived from
the state
𝑣𝑦 (𝑠)
=
𝛿 (𝑠)
𝑧𝑣 = (𝑎12 − 𝑣𝑥 )
𝑧𝜓. = 𝑎21
𝑏2
− 𝑎22 ,
𝑏1
𝑏2
− 𝑎22 .
𝑏1
(11)
(12)
The steady-state value of the vehicle side-slip velocity can
be obtained by multiplying the gain with the steering angle;
that is,
𝑉𝑦 dss = 𝑉𝑦 dss gain 𝛿.
(13)
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International Journal of Vehicular Technology
Substituting expressions in (3) and (8) into (13) yields
𝑉𝑦 dss =
𝑣𝑥 ((𝑎 + 𝑏) 𝑏𝐶𝑓 𝐶𝑟 − 𝑎𝐶𝑓 𝑀𝑣 𝑣𝑥2 )
(𝑎 + 𝑏)2 𝐶𝑓 𝐶𝑟 + 𝑀𝑣 𝑣𝑥2 (−𝑎𝐶𝑓 + 𝑏𝐶𝑟 )
𝛿
2
𝑣𝑥 𝛿
𝑎 𝑀𝑣 𝑣𝑥
= (
)
(𝑏
−
),
𝑎 + 𝑏 𝐶𝑟
(𝑎 + 𝑏) + 𝐾𝑢 𝑣𝑥2
(14)
where
𝑀𝑣
𝑎
𝑏
− ).
(
𝑎 + 𝑏 𝐶𝑓 𝐶𝑟
𝐾𝑢 =
(15)
Equation (14) is composed of a product of two terms
where the first term represents the steady-state of desired yaw
rate. From this fact, the steady-state value of desired side-slip
velocity can be implemented in terms of steady-state desired
yaw rate
.
𝑉𝑦 dss = 𝜓dss (𝑏 −
2
𝑎 𝑀𝑣 𝑣𝑥
).
𝑎 + 𝑏 𝐶𝑟
(16)
Equations (18) and (19), which take separate desired
steady-state values as the inputs, can be combined into
one set of dynamic filters with a common input. Dynamics
are imparted to steering wheel angle through a secondorder filter with two states. Desired yaw rate and lateral
velocity are computed based on linear combinations of these
states. Steady-state yaw gain, as mentioned before, can be
implemented in a table look-up to account for variation of
understeer coefficient when the vehicle is operated at the limit
of tire adhesion. Steady state lateral gain is computed from the
steady-state yaw gain through kinematics relationships. Both
filter natural frequency and damping ratio are functions of
vehicle velocity and are implemented as look-up tables.
The dynamic relationship between the steering angle and
desired side-slip velocity can be expressed into the following
form:
𝑣𝑦𝑑 (𝑠)
𝛿 (𝑠)
.
.
𝑣𝑥 𝛿
,
(𝑎 + 𝑏) + 𝐾𝑢 𝑣𝑥2
(17)
where 𝐾𝑢 is obtained from computation of (15).
Therefore, (14), (15), and (16) can be used to obtain the
value of 𝑉𝑦 dss .
Given the steady-state side-slip velocity, the next step is
to process such value through a dynamic filter with desired
damping ratio and natural frequency representative of the
system dynamics described in (7) and (8). But one question
remains regarding to the significance of the non-minimumphase zero in the implementation of the side slip command.
The dynamic filter with desired damping ratio, natural
frequency, and zero can be implemented using a set of two
first-order differentials
.
𝑥1 = 𝑉𝑦 dss − 2𝜁𝜔𝑛 𝑥1 − 𝜔𝑛2 𝑥2 ,
𝑣𝑦𝑑 = 𝜔𝑛2 (𝑥2 +
(18)
𝑥1
),
𝑧𝑣
.
.
𝑥1 = 𝜓𝑦 dss − 2𝜁𝜔𝑛 𝑥1 − 𝜔𝑛2 𝑥2 ,
.
𝑥2 = 𝑥1 ,
𝑣𝑦𝑑 = 𝜔𝑛2 (𝑥2 +
(19)
𝑥1
).
𝑧𝜓.
.
𝑥2 = 𝑥1 ,
(21)
where 𝛿 = steering wheel angle, 𝜁 = damping ratio, function of
vehicle speed, and 𝜔𝑛 = natural frequency, function of vehicle
speed.
Defining
𝑉𝑦 gain = Ωgain 𝑔𝑟 (𝑏 −
𝑎𝑀𝑣 𝑉𝑥 2
),
𝑎 + 𝑏 𝐶𝑟
Ωgain = the yaw gain from 3-d table,
(22)
𝑔𝑟 = steering ratio,
we can compute for both the dynamic desired side-slip
velocity and yaw rate using the two states 𝑥1 and 𝑥2 of a single
dynamic filter described in (21):
𝜓des = [
𝑎
] 𝐶𝑓 𝑥1 + Ωgain 𝑔𝑟 𝜔𝑛2 𝑥2 ,
𝐼𝑧
(23)
1
] 𝐶𝑓 𝑥1 + 𝑉𝑦gain 𝜔𝑛2 𝑥2 .
𝑀𝑣
(24)
𝑉𝑦 des = [
where 𝑧𝑣 is expressed in (11) and implemented in a lookup
table as a function of vehicle speed. A similar second-order
dynamic filter can be implemented in control computation to
provide the desired yaw-rate command. Consider
(20)
.
𝑥1 = 𝛿 − 2𝜁𝜔𝑛 𝑥1 − 𝜔𝑛2 𝑥2 ,
.
.
𝑥2 = 𝑥1 ,
𝑏1 𝑠 + 𝑎12 𝑏2 − 𝑎22 𝑏1
.
𝑠2 + 2𝜁𝜔𝑛 𝑠 + 𝜔𝑛2
Using this relationship, a dynamic filter can be established
with steering angle as input:
The steady-state desired yaw rate 𝜓dss during control
computation can be obtained using the following:
𝜓dss =
=
A block diagram of the command interpreter block is
illustrated in Figure 2.
2.2. The Warning and Speed Advisory Block
2.2.1. The Driving States. This block reads all available sensor
signals, namely, the lateral and longitudinal accelerations,
steering wheel angle, and yaw rate to detect the current
driving situation. This block distinguishes 11 different driving
modes such as cruising, braking, cornering, transient, low
speed maneuvering, and reversing.
International Journal of Vehicular Technology
5
Figure 2: Block diagram of the command interpreter algorithm.
Figure 3: Block diagram of the driving states algorithm.
For the purpose of this study, it is sufficient to distinguish between a transient mode and a steady turn for
both linearized and nonlinear relations between lateral tire
forces and slip angles. We define four intermediate flags 𝐹𝑖tr ,
𝑖 = 1, 4.
The transient mode is determined as follows:
󵄨󵄨 .. 󵄨󵄨 ..
󵄨󵄨𝜓󵄨󵄨 > 𝜓trans 2 th (𝑉𝑥 ) 󳨀→ 𝐹1 tr = true,
󵄨 󵄨
󵄨󵄨 .. 󵄨󵄨 ..
󵄨󵄨󵄨𝜓󵄨󵄨󵄨 ≤ 𝜓trans th min (𝑉𝑥 ) 󳨀→ 𝐹1 tr = false,
󵄨󵄨 . 󵄨󵄨 .
󵄨 󵄨
. {
{ 󵄨󵄨󵄨𝑉𝑦 󵄨󵄨󵄨 > 𝑉𝑦trans th (𝑉𝑥 ) 󳨀→ 𝐹2 tr = true,
𝑉𝑦 { 󵄨 . 󵄨 .
{ 󵄨󵄨 󵄨󵄨
{ 󵄨󵄨󵄨𝑉𝑦 󵄨󵄨󵄨 ≤ 𝑉𝑦trans th min (𝑉𝑥 ) 󳨀→ 𝐹1 tr = false,
.
󵄨󵄨 . 󵄨󵄨
󵄨 󵄨
. {
{ 󵄨󵄨󵄨𝛿󵄨󵄨󵄨 ∗ 𝑓 (𝑉𝑥 ) > 𝛿trans th (𝑉𝑥 ) 󳨀→ 𝐹3 tr = true,
𝛿{ .
.
{ 󵄨󵄨󵄨 󵄨󵄨󵄨
{ 󵄨󵄨󵄨𝛿󵄨󵄨󵄨 ∗ 𝑓 (𝑉𝑥 ) ≤ 𝛿trans th min (𝑉𝑥 ) 󳨀→ 𝐹3 tr = false,
{
𝜓{
{
..
󵄨󵄨 . 󵄨󵄨 .
{ 󵄨󵄨󵄨𝜓󵄨󵄨󵄨 > 𝜓cr th 󳨀→ 𝐹1 cr = true,
𝜓{ 󵄨 . 󵄨 .
󵄨 󵄨
{ 󵄨󵄨󵄨𝜓󵄨󵄨󵄨 ≤ 𝜓cr th min 󳨀→ 𝐹1 cr = false,
.
(25)
∨ 𝐹2
tr
∧ 𝐹3
tr
∧ 𝐹4 tr ) , 𝑛} .
|𝛿| > 𝛿cr th 󳨀→ 𝐹2 cr = true,
|𝛿| ≤ 𝛿cr th min 󳨀→ 𝐹2 cr = false,
(27)
󵄨󵄨 󵄨󵄨
󵄨󵄨𝑎𝑦 󵄨󵄨 > 𝑎𝑦
󳨀→ 𝐹3 cr = true,
󵄨 󵄨
cr th
󵄨󵄨 󵄨󵄨
󵄨󵄨󵄨𝑎𝑦 󵄨󵄨󵄨 ≤ 𝑎𝑦 crth min 󳨀→ 𝐹3 cr = false,
Steady turn = {𝐹1 cr ∨ 𝐹2 cr ∧ 𝐹3 cr } .
Transient mode
tr
𝛿{
{
𝑎𝑦 {
{
{ [𝛿] > 𝛿trans th (𝑉𝑥 ) 󳨀→ 𝐹4 tr = true,
𝛿{
{ |𝛿| ≤ 𝛿trans th min (𝑉𝑥 ) 󳨀→ 𝐹4 tr = false.
=delay sample {(𝐹1
The delay mode starts when the input signal becomes
false. The variable count is incremented by one as long as the
old value of the variable is less than 𝑛, where 𝑛 is the number
of samples. The output in this case is only true if the old value
is less than 𝑛. Thus, the falling edge is delayed by 𝑛 samples.
The reason for introducing the delay sample function is to
avoid transient mode switching during signals’ zero crossing.
We define three intermediate flags 𝐹𝑖 cr , 𝑖 = 1, 3. The
steady turn is determined as follows:
(26)
Figure 3 illustrates the driving states algorithm described
earlier.
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International Journal of Vehicular Technology
2.2.2. The Lateral Surface Capability Index. The lateral surface
capability index is determined based on the comparison
between actual vehicle motion obtained from sensor inputs
and vehicle behavior obtained from a linear vehicle motion
model. When the vehicle is in the linear range of operation,
the vehicle motion model is close to the actual vehicle motion
obtained from the sensor inputs. In this case, the lateral
surface index is set to the maximum lateral acceleration that
the vehicle can sustain on dry surface. When the vehicle
lateral motion approaches the limit of adhesion, the vehicle
motion model is substantially different from the actual
vehicle motion, and it can be concluded that the lateral
surface index must at least equal 𝑎𝑦 /𝑔, wherein 𝑔 denotes the
gravity acceleration.
The first step in this block is to detect whether the vehicle
is in a linear mode of motion or not. To this effect, the linear
mode detection evaluates the following three conditions:
We define four intermediate flags 𝑓𝑖 lin 𝑖 = 1, 4. The linear
mode detection is determined as follows:
. 󵄨 󵄨 󵄨
󵄨.
if 󵄨󵄨󵄨󵄨𝜓𝑑 𝑣𝑥 + 𝑣𝑦𝑑 󵄨󵄨󵄨󵄨 − 󵄨󵄨󵄨󵄨𝑎𝑦 󵄨󵄨󵄨󵄨 < 𝑎𝑦 Thr1 󳨀→ 𝑓1 lin = true,
else 𝑓1 lin = false,
.
.
if (𝜓𝑑 𝑣𝑥 + 𝑣𝑦𝑑 ) 𝑎𝑦 > −𝑎𝑦 Thr2 󳨀→ 𝑓2 lin = true,
else 𝑓2 lin = false,
.󵄨
.
󵄨.
if 󵄨󵄨󵄨󵄨𝜓𝑑 − 𝜓󵄨󵄨󵄨󵄨 ≤ 𝜓𝑒 max 󳨀→ 𝑓3 lin = true,
else 𝑓3 lin = false,
.
.󵄨
󵄨.
if 𝜓𝑒 min ≤ 󵄨󵄨󵄨󵄨𝜓𝑑 − 𝜓󵄨󵄨󵄨󵄨 󳨀→ 𝑓4 lin = true,
else 𝑓4 lin = false.
The linear flag is given by
𝑓lin (𝑡) = 𝑓lin (𝑡 − 1) ∨ 𝑓4 lin ∧ {𝑓1 lin ∧ 𝑓2 lin ∧ 𝑓3 lin } .
. 󵄨 󵄨 󵄨
󵄨󵄨 .
󵄨󵄨𝜓𝑑 𝑣𝑥 + 𝑣𝑦𝑑 󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑎𝑦 󵄨󵄨󵄨 < 𝑎𝑦Thr1 ,
󵄨 󵄨 󵄨
󵄨
.
(28)
.
(𝜓𝑑 𝑣𝑥 + 𝑣𝑦𝑑 ) 𝑎𝑦 > −𝑎𝑦 Thr2 ,
.
𝜓𝑒
min
.󵄨
.
󵄨.
≤ 󵄨󵄨󵄨󵄨𝜓𝑑 − 𝜓󵄨󵄨󵄨󵄨 ≤ 𝜓𝑒
(29)
max .
.
(30)
.
The desired yaw rate 𝜓𝑑 is obtained from (23). 𝑣𝑦𝑑 is
obtained by differentiating (18) as follows:
.
𝑣𝑦𝑑 =
𝜔𝑛2
1
(𝑥1 + [𝑉𝑦dss − 2𝜉𝜔𝑛 𝑥1 − 𝜔𝑛2 𝑥2 ]) .
𝑧𝑣
(31)
(32)
(33)
The second step in the lateral surface capability index block is
the straight driving mode detection algorithm. We define the
following intermediate flags 𝑓𝜓. max , 𝑓𝜓. min , 𝑓𝛿 max , and 𝑓𝛿 min .
The straight line flag is determined as follows:
󵄨.󵄨 .
if 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨 ≥ Ωth max 󳨀→ 𝑓𝜓. max = false,
else 𝑓𝜓. max = true,
if |𝛿| ≥ 𝛿th max 󳨀→ 𝑓𝛿 max = false,
else 𝑓𝛿 max = true,
󵄨.󵄨 .
if 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨 ≤ Ωth min 󳨀→ 𝑓𝜓. min = true,
else 𝑓𝜓. min = false,
(34)
if |𝛿| ≤ 𝛿th min 󳨀→ 𝑓𝛿 min = true,
The difference between measured and expected lateral
accelerations in (28) is due to the fact that the assumed linear
relationship between lateral force and sideslip angle does not
hold exactly. In fact, the direct proportionality between lateral
force and sideslip angle is a good approximation as long as
the lateral forces are moderate. When the difference between
measured and desired lateral accelerations exceeds a certain
threshold, the sideslip angle increases much more strongly,
indicating that the tire is moving towards its lateral limit of
adhesion.
Equation (29) compares the product of signed desired and
observed lateral accelerations, to a small negative number
−𝑎𝑦 Thr2 . An excessive negative value is symptomatic of a
situation where the state of motion of the vehicle cannot
follow the driver intended steering. Equation (30) compares
the difference between a desired and observed yaw rate to
upper and lower thresholds. Obviously, if this difference is
.
above the upper threshold 𝜓𝑒 max , control of the vehicle is not
exact indicating that the vehicle has departed from its linear
range of operation. On the other hand, if it is below the lower
.
threshold 𝜓𝑒 min , it is likely that the vehicle is going straight,
and that no information about the lateral friction properties
of the road surface can be inferred from the data of the various
sensors.
else 𝑓𝛿 min = false.
The straight driving flag is given by
𝑓sl (𝑡) = 𝑓sl (𝑡 − 1) ∨ {𝑓𝜓. min ∧ 𝑓𝜓. max ∧ 𝑓𝛿 min ∧ 𝑓𝛿 max } .
(35)
The lateral surface capability index is determined based
on input data from the sensors and the flags 𝑓sl (𝑡) and
𝑓lin (𝑡). The operation is described referring to the flowchart
of Figure 4. During the initialization step, the index 𝜇 is
set to a predetermined default value 𝜇0 , which may be
a typical friction coefficient of a dry, solid road surface.
The lateral acceleration 𝑎𝑦 is read from lateral acceleration
sensor. The straight line flag 𝑓sl (𝑡) is then verified. If it is
“false,” that is, if the vehicle is going through curves and is
subject to a substantial lateral acceleration, a timer is reset
to zero. Next, if the linear flag 𝑓lin (𝑡) is “true,” it can be
concluded that the lateral surface capability index must at
least equal 𝑎𝑦 /𝑔, wherein 𝑔 denotes the gravity acceleration.
The lateral surface capability index is therefore updated to be
𝜇(𝑡) = max(𝜇(𝑡 − 1), 𝑎𝑦 (𝑡)/𝑔) the maximum of 𝑎𝑦 /𝑔 and an
estimate obtained from a previous iteration 𝜇(𝑡 − 1). In this
way, if this step is executed repeatedly in subsequent iterations
International Journal of Vehicular Technology
7
𝜇 := 𝜇0
#𝑎𝑦
No
𝑓sl (𝑡) = true?
Timer reset
Yes
𝜇(𝑡) = max(𝜇(𝑡 − 1),
𝑓lin (𝑡) = true?
𝑎𝑦
𝑔
)
Yes
Timer enabled
Timer out?
No
Yes
𝜇(𝑡) =
𝑎𝑦
No
𝑔
Figure 5: Schematic diagram of stability index implementation.
Figure 4: Schematic diagram of the lateral surface capability index.
of the procedure of Figure 4, 𝜇(𝑡) will grow and converge
towards the true friction coefficient of the road surface.
On the other hand, if the linear flag is found to be “false,”
this may be due to the fact that the quality of the road surface
has deteriorated and its friction coefficient has decreased, or
that the vehicle is going at the stability limit. In that case, 𝑎𝑦 /𝑔
is set as the new the lateral surface capability index.
If the straight driving flag is found to be “true,” no
estimation of lateral surface capability index is possible.
In this case, the timer mentioned is enabled; that is, the
timer starts to run if the straight flag has just switched to
“true,” or it simply continues to run if the straight flag was
“true” already in the previous iteration of the procedure.
The value of the timer is thus representative of the time
in which the vehicle has been going straight. If the timer
has exceeded a predetermined limit, the algorithm resets the
friction coefficient to 𝜇0 . In this way, if the vehicle has been
going straight for such a long time such that the previously
acquired estimate of the lateral surface capability index is no
longer reliable, the estimate is reset to 𝜇0 , and the process
of iteratively approximating its true value restarts when the
straight driving flag 𝑓sl (𝑡) becomes “false.”
2.2.3. The Warning Algorithm. The warning algorithm computes indices based on yaw-rate error, yaw rate dead-band,
side slip velocity error, and side slip velocity dead-band, an
understeer error, and an understeer dead-band, using various
sensors. It consists of 3 major blocks:
(i) stability index block;
(ii) steady state linear/nonlinear understeer index block;
(iii) arbitration block.
Stability Index Block. The stability index block is mainly
used during transient driving situations, in which a driver
has to turn the steering wheel of the vehicle quickly and/or
in alternating directions, so that a reliable estimation of
the vehicle understeer is difficult. Figure 5 illustrates the
schematic diagram of the stability index implementation.
.
The stability index block requires the yaw rate error 𝜓error ,
..
the rate of the yaw error 𝜓error , and the lateral velocity error
.
rate 𝑉𝑦 error for the computation of the stability index. These
quantities are defined as follows:
.
.
.
𝜓error = 𝜓𝑑 − 𝜓,
.
.
..
𝜓error =
.
.
.
(𝜓error (𝑡) − 𝜓error (𝑡 − 𝑇))
𝑇
.
,
.
𝑉𝑦error = 𝑣𝑦𝑑 − 𝑣𝑦 = 𝑣𝑦𝑑 − (𝜓𝑣𝑥 − 𝑎𝑦 ) .
(36)
The yaw rate and the lateral velocity rate dead-band
lookup tables store a yaw rate dead-band DB𝜓. and a lateral
velocity rate dead band DB𝑉. , as a function of longitudinal
𝑦
velocity 𝑣𝑥 and steering wheel angle 𝛿. These dead bands
represent (sometimes called a neutral zone) an area of a signal
range or band where no action occurs (the system is dead),
The purpose is of the dead band to prevent oscillation or
repeated activation-deactivation cycles (called “hunting” in
control systems). a predetermined percentage of a yaw rate or
a lateral velocity derivative above which, for a given vehicle
speed and steering wheel angle, control over the vehicle is
lost. The warning algorithm is used in combination with an
electronic stability control (ESC system), and the ESC system
may use associated dead bands of the yaw rate and the lateral
velocity derivative for deciding whether to intervene or not.
If, for example, the dead bands of the ESC system are at 70% of
a value at which control of the vehicle is lost for a given vehicle
speed and steering wheel angle, the dead bands used in the
stability index block may be at 50% of such a value, ensuring
that the warning algorithm will issue a warning signal prior
to any intervention of the ESC.
8
International Journal of Vehicular Technology
This allows solving for the forces
𝐹𝑦𝑓 =
𝑎𝑀𝑣2𝑥
,
𝐿𝑅
(40)
𝑏𝑀𝑣2𝑥
.
𝐹𝑦𝑟 =
𝐿𝑅
.
Using the relation 𝜓 = 𝑣𝑥 /𝑅, the steer angle equation can be
written as
𝐿
.
𝑅
𝛿 = −𝛼𝑓 + 𝛼𝑟 +
(41)
The linearized lateral forces are expressed in terms of the
tire slip angles
𝐹𝑦𝑓 = 𝐶𝑓 𝛼𝑓 ,
(42)
𝐹𝑦𝑟 = 𝐶𝑟 𝛼𝑟 .
Figure 6: Vehicle in steady turn.
.
From (40) and (42), the steer angle relation becomes
From the dead bands DB𝜓. , DB𝑉. and the error signals
..
𝛿=(
𝑦
.
𝜓error , 𝜓error , and 𝑉𝑦 error , the yaw rate and lateral velocity
rate stability indices SI𝜓. , SI𝑉. as follows:
𝑦
..
󵄨󵄨 .
󵄨
󵄨󵄨𝐺𝑝 𝜓error + 𝐺𝑑 𝜓error 󵄨󵄨󵄨
󵄨
󵄨,
.
SI𝜓 =
DB𝜓.
SI𝑉.
𝑦
󵄨󵄨 .
󵄨
󵄨󵄨𝑉𝑦 error 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
=
.
.
DB 𝑉
𝑀 (𝑏𝐶𝑟 − 𝑎𝐶𝑓 )
𝐿𝐶𝑓 𝐶𝑟
𝛿 = 𝐾und
(37)
(38)
Steady State Linear/Nonlinear Understeer Index Block. In this
section, we will discuss the structure and operation of the
understeer index. For estimating the understeer of a vehicle,
it is important to know whether the vehicle is moving in
a linear regime, in which the sideslip angle of the vehicle
is approximately directly proportional to the lateral forces.
Figure 6 illustrates the vehicle in a steady turn; to keep the
slip angle small, we assume that the turn radius is large. This
is normally the case for high-speed turns, in which the vehicle
is not skidding.
For steady state mode, the dynamic equation of motion in
the body-centered coordinate system is given by
0 = 𝑎𝐹𝑦𝑓 + 𝑏𝐹𝑦𝑟 .
(43)
𝑣𝑥2 𝐿
+ ,
𝑅 𝑅
(44)
where
The decision whether the vehicle is in a situation
approaching ESC activation or not during a transient maneuver can be based on the yaw rate and lateral velocity derivative
indices defined in (37), and a general stability index SI is set
equal to
𝑀𝑣2𝑥
= 𝐹𝑦𝑓 + 𝐹𝑦𝑟 ,
𝑅
𝑣𝑥2 𝐿
+ .
𝑅 𝑅
Equation (43) relates the steer angle to the speed via the
understeer coefficient 𝐾und
𝑦
SI = max (SI𝜓. , SI𝑣. 𝑦 ) .
)
(39)
𝐾und = (
𝑀 (𝑏𝐶𝑟 − 𝑎𝐶𝑓 )
𝐿𝐶𝑓 𝐶𝑟
)
(45)
𝑎
𝑀 𝑏
−
).
=
(
𝐿 𝐶𝑓 𝐶𝑅
From (44), we can calculate the change in the steer angle
when the 𝐿/𝑅 is changed by differentiating (44).
In the linear case
2
𝑉
𝜕𝛿
= 𝐾und linear 𝑥 + 1,
𝜕 (𝐿/𝑅)
𝐿
𝐾und linear
𝐿
𝜕𝛿
− 1) .
= 2(
𝑉𝑥 𝜕 (𝐿/𝑅)
(46)
Figure 7 shows the steering wheel angle 𝛿 versus𝐿/𝑅 for
constant speed. For the linear tire force assumption, the
slopes of the plot are constants since the understeer and the
vehicle speed are constant.
For the nonlinear tire characteristics, in which the tire
force-sideslip angle becomes significantly nonlinear, the
slopes are not constant and vary with the turn radius when
the vehicle speed is constant as illustrated in Figure 8. In this
case, the understeer relates to the local slope of the steer angle
curve.
International Journal of Vehicular Technology
9
In this section, an identification algorithm using the Kalman
filter is developed to estimate the linear understeer and
nonlinear understeer variable. The Kalman filter is a set
of mathematical equations that provide an efficient computational (recursive) solution of the least-squares method.
The filter is very powerful in several aspects; it supports
estimations of past, present, and even future states, and it can
do so even when the precise nature of the modeled system is
unknown.
The Kalman filter addresses the general problem of trying
to estimate the state of a discrete-time controlled process that
is governed by the linear stochastic difference equation
𝐾und 𝑉𝑥2 > 0,
Understeer stable
𝛿
𝐾und 𝑉𝑥2 = 0,
Neutral stable
𝐾und 𝑉𝑥2
< 0,
𝐿
Oversteer stable
−1 <
𝐿
𝑅
𝑥 (𝑡 + 1) = 𝑥 (𝑡) + 𝑣 (𝑡)
𝐾und 𝑉𝑥2
< −1,
𝐿
Oversteer unstable
with a measurement 𝑦 ∈ R that is
𝑦 (𝑡) = 𝐻 (𝑡) 𝑥 (𝑡) + 𝑛 (𝑡) .
Figure 7: Steer angle versus 𝐿/𝑅 for linear case [14].
𝛿
Stable
Unstable
Oversteer
𝐾undnl
Understeer
𝑥 (𝑡) = 𝐾und (𝑡) ,
𝐿
𝑅
Figure 8: Steer angle versus 𝐿/𝑅 for nonlinear case [14].
=
𝑑 (𝛼𝑓 − 𝛼𝑟 )
𝑑 (𝑎𝑦 )
,
(47)
where
(𝛼𝑓 − 𝛼𝑟 ) = 𝛿 −
𝑑𝑡
=
𝑣𝑥2
,
𝑑 (𝛼𝑓 − 𝛼𝑟 )
𝑑𝑡
(51)
.
𝐿𝑎𝑦
𝑎+𝑏 .
𝜓=𝛿− 2 .
𝑣𝑥
𝑣𝑥
𝑦 (𝑡) = 𝛿 − 𝐿
𝑎𝑦
𝑣𝑥2
,
𝐻 (𝑡) = 𝑎𝑦 .
(52)
In the nonlinear range we define
Since in the nonlinear case there is no constant understeer
coefficient, we define a variable coefficient that expresses how
the slip angle difference changes as the lateral acceleration
changes
𝑑 (𝑉𝑥2 /𝑅𝑔)
𝑑𝑎𝑦
𝑎𝑦
In the linear range we define
𝐿
𝛿=
𝑅
Neutral steer
𝑑 (𝛼𝑓 − 𝛼𝑟 )
(50)
We will be using two Kalman filters for the linear and
nonlinear operations. Equations (44) and (47) can be recast
as
𝐾und 𝑎𝑦 = 𝛿 − 𝐿
𝐾und nl =
(49)
𝑥 (𝑡) = 𝐾und nl (𝑡) ,
𝑦 (𝑡) =
𝑑 (𝛼𝑓 − 𝛼𝑟 )
𝑑𝑡
By applying Kalman filter technique, it is possible to
estimate the linear understeer and nonlinear understeer
variable.
Estimation of the Linear and Nonlinear Understeer. In the last
section, we developed linear and nonlinear expression for the
understeer tendency of the vehicle under steady cornering.
𝐻 (𝑡) =
𝑑𝑎𝑦
.
𝑑𝑡
(53)
The random variables 𝑣(𝑡) and 𝑛(𝑡) represent the process
and measurement noise, respectively.
They are assumed to be independent (of each other),
white, and with normal probability distributions
𝑝 (𝑣) ∼ 𝑁 (0, 𝑄) ,
(48)
,
𝑝 (𝑛) ∼ 𝑁 (0, Γ) .
(54)
𝑄 is the process noise covariance and, Γ is measurement
noise covariance. 𝐻(𝑡) in the measurement equations (52)(53) relates the state to the measurement 𝑦(𝑡). In practice, it
will change with each time step or measurement.
Consider the signal model of (49)–(54), and assume that
the initial state and noise sequences are jointly Gaussian.
̂ + 1) denote the conditional mean of 𝑥(𝑡 + 1) given
Let 𝑥(𝑡
10
International Journal of Vehicular Technology
Figure 9: Schematic diagram of the understeer index implementation.
̂ + 1)
observation {𝑦(𝑡)} up to and including time 𝑡; then, 𝑥(𝑡
satisfies the following recursion:
𝑥̂ (𝑡 + 1) = 𝑥̂ (𝑡) + 𝐿 (𝑡) (𝑦 (𝑡) − 𝐻 (𝑡) 𝑥̂ (𝑡)) ,
𝑥̂ (𝑡0 ) = 𝑥0 ,
𝑆 (𝑡 + 1) = 𝑃 (𝑡) + 𝑄 (𝑡) ,
−1
𝐿 (𝑡) = 𝑆 (𝑡) 𝐻𝑇 (𝑡) (𝐻 (𝑡) 𝑆 (𝑡) 𝐻𝑇 (𝑡) + Γ (𝑡)) ,
(55)
𝑃 (𝑡 + 1) = 𝑆 (𝑡 + 1) − 𝐿 (𝑡) 𝐻 (𝑡) 𝑆 (𝑡 + 1) ,
𝑃 (𝑡0 ) = 𝑃0 ,
where 𝐿(𝑡) is the filter gain and 𝑃(𝑡) is the state error
covariance.
The linear flag defined in (35) controls the operation of the
two Kalman filters (52) and (53) for estimating the understeer
of the vehicle in linear and nonlinear regimes, respectively.
When the vehicle is determined to be in non-linear
regime, the linear flag becomes zero, filter (52) stops, and
filter (53) starts to operate, initialized with the most recent
understeer value from filter (52). Similarly, when the linear
flag changes back to 1, filter (52) becomes operative again and
is initialized with a nominal understeer based on the front
and rear lateral tire stiffness and the front and rear vehicle
weight distribution and with initial covariance values.
In principle, the two filters might be regarded as a single
Kalman filter which swaps 𝑦(𝑡) and 𝐻(𝑡) according to the
value of the linear flag.
In analogy to what was described earlier for the yaw rate
and lateral velocity rate stability indices SI𝜓. , SI𝑉. , lookup
𝑦
tables provided values of a desired understeer 𝐾𝑢 des and a
dead band of the understeer DBund . As in case of the yaw
rate and the lateral velocity derivative, desired understeer
values can be predetermined empirically by measuring the
understeer of a test vehicle at given speeds and steering wheel
angles. Alternatively, they may be calculated in advance or in
real time, for example, using the following formula:
𝑊𝑓 𝑊𝑟
1 𝑣 𝛿
−
).
𝐾𝑢 des = max ( 2 ( . 𝑥 − 𝐿) ,
(56)
𝑣𝑥 𝜓des
𝐶𝑓 𝐶𝑟
Since 𝐾𝑢 des may take impractically high values according
to (56), at high speeds and steering wheel angles, it is
preferred to define an upper limit of the desired understeer
𝐾𝑢 des as follows:
𝐾𝑢 des
= min (max (
𝑊𝑓 𝑊𝑟
1 𝑣𝑥 𝛿
(.
− 𝐿) ,
−
) , 𝐾und
𝑣𝑥2 𝜓des
𝐶𝑓 𝐶𝑟
max ) .
(57)
∘
The upper limit 𝐾und max may be set to, for example, 8 /g or
5∘ /g, g denoting the gravity acceleration.
Similar to the yaw rate dead band DB𝜓. and a lateral
velocity rate dead band DB𝑉. , the understeer dead band
𝑦
DBund gives values of the understeer which can be regarded
as safe as function of longitudinal velocity 𝑣𝑥 and steering
wheel angle 𝛿. The understeer index UI based on the effective
vehicle understeer 𝐾und or 𝐾und nl , defined as 𝐾𝑢 , estimated
by filter (52) or (53) the desired understeer 𝐾𝑢 des from (57)
and the understeer dead band DBund :
󵄨
󵄨󵄨
− 𝐾𝑢 󵄨󵄨󵄨
󵄨𝐾
(58)
UI = 󵄨 𝑢 des
.
DBund
The structure and operation of the understeer index are
depicted in Figure 9.
International Journal of Vehicular Technology
11
Figure 10: Flow chart of the arbitrator logic.
The Arbitrator Block. Referring again to Figures 5 and 9, three
indices SI𝜓. , SI𝑣. 𝑦 , and UI are supplied to the arbitrator. Albeit
of different origin, the three indices are comparable in that
they are dimensionless and that a value above 1 indicates a
critical driving situation.
The operation of the arbitrator is explained referring to
the flowchart of Figure 10. The first step is that we distinguish
between a transient mode, and a steady state mode. In the
transient mode the direction of the vehicle is changing so
rapidly that the understeer cannot be relied upon. Therefore,
if the driving mode is found to be the transient mode, the
decision whether the vehicle is approaching ESC activation
or not can only be based on the yaw rate and lateral velocity
derivative indices SI𝜓. and SI𝑣. 𝑦 .
A general stability index is set equal to the max(SI𝜓. , SI𝑣. 𝑦 ).
If the vehicle is not in the transient mode, the steering wheel
angle 𝛿 is compared to a predetermined upper threshold 𝛿HI .
If this threshold is exceeded, there is a considerable risk of
the vehicle being unstable, and the system should be rather
liberal in issuing a warning. In that case, the general stability
index is set equal to the max(SI𝜓. , UI). Otherwise, a weighted
sum(𝑘SI𝜓. + (1 − 𝑘)UI) of the yaw rate stability index and
understeer index is calculated. The weighting factor 𝑘 is tuned
to a value between 0 and 1. This weighting factor may be set
dependent on the vehicle speed 𝑣𝑥 and decreases with the
vehicle speed, giving increasing importance to the understeer
at high speeds.
If the vehicle is in a straight line driving mode, the general
stability index determined prior to this step is multiplied by
positive factor 𝛼SL , which is smaller than 1, reflecting the fact
that the vehicle is least susceptible to a loss of control if it is
driving a longer straight line.
Finally, the general stability index is compared to an
upper threshold Stabth hi , for example, 0.8. If it exceeds this
12
International Journal of Vehicular Technology
2.2.4. The Speed Advisory Block. Very often the road signs
indicate the safe speed in a curve. However, on low 𝜇 surfaces,
the suggested posted speed might not be adequate for the
road condition.
When the vehicle is driving in a curve at a higher speed
than the surface can allow, the understeer gradient of the
vehicle increases causing the vehicle to plow or decreases and
becomes negative causing the vehicle to spinout. The warning
algorithm described earlier will issue a warning to alert the
driver that he/she is traveling faster than the road surface can
allow. In this section, we develop an advisory speed algorithm
in a curve based on vehicle dynamics in conjunction with
the driver warning algorithm. The advisory speed algorithm
computes the advisory speed which allows a vehicle to travel
around the turn or curve in its travel lane without causing
an uncomfortable “side force” to its driver or passengers and
helps maintain control of the vehicle. The advisory speed is
based on the maximum lateral capability of the surface, the
driver steering input, the actual vehicle speed, and the actual
understeer of the vehicle. A visual advisory speed can be
displayed, for example, in the DCI or a HUD display when the
warning signal is issued. The visual advisory HMI is outside
the scope of this paper.
Based on the steering input 𝛿 and vehicle speed 𝑣𝑥 , if we
assume that the coefficient of the surface would allow the
vehicle (theoretically) to stay in the linear range, the desired
understeer is determined from (57). If the surface coefficient
is much lower, we can compute a vehicle speed (less than the
actual speed of the vehicle) such that the understeer gradient
is kept within a small deviation from the desired understeer
of the vehicle. Assume that Δ𝐾und is the understeer deviation
from the linear performance. Thus, the stable vehicle speed
limit can be determined from (44) as follows:
𝑣lim
=
√
𝐿𝜇𝑔
𝑊𝑓 𝑊𝑟
𝑣𝑥 𝛿
1
− 𝐿) ,
−
) , 𝐾und
|𝛿| − [min (max ( 2 ( .
𝐶𝑓
𝐶𝑟
𝑣𝑥 𝜓des
,
max )
+ Δ𝐾und ] 𝜇𝑔
(59)
where 𝜇 is the maximum capability of the surface and is
determined as described in Section 2.2.2.
Finally, the advisory speed is calculated as follows:
𝑉adv
={
min (𝑣lim , 𝑣𝑥 ) if the warning flag is set
speed limit if available, when warning flag is not set.
(60)
Under straight line condition 𝑣lim is set to ∞.
0.6
0.55
0.5
Friction coefficient
upper threshold, a warning flag is set to TRUE. Since the
dead bands for warning algorithms are set lower than those
of an ESC system, the warning flag will become TRUE prior
to any intervention of the ESC system. If the stability index
was found not to exceed the upper threshold, the index is
compared to a lower threshold Stabth lo , for example, 0.2. If it
is below this lower threshold, the warning flag is set to FALSE;
if not, the warning flag is left as it is until the procedure is
repeated.
0.45
0.4
0.35
0.3
0.25
0.2
0
5
10
15
20
Time (s)
25
30
Figure 11: Time trace of the road friction coefficient.
The advisory speed is communicated to the driver to
allow the driver to make a choice on what action should
be taken, or through an intervention system where the
engine and/or braking systems are controlled automatically
to reduce the vehicle’s speed.
3. Simulation Results
In this section, we present some typical simulation results
showing the performance of the driver warning algorithm
described in Section 2. The vehicle was traveling at 80 kph on
a 40 m radius loop with 200 m straight section with variable
friction coefficient to simulate the exit ramp of a freeway.
A driver model with a driver preview time of 1 second and
driver lag of 0.12 second was used in the simulation. Figure 11
shows the time trace of the road friction coefficient as the
vehicle travels. In this simulation, the driver did not react to
the warning at the time when the warning was issued.
Figure 12 illustrates the time trace of the warning signal
and the ESC activation flag. The warning was issued at around
5 seconds when the driver entered the curved section of the
road at 80 kph. In order to make ESC system as unintrusive
as possible, its activation threshold will have to be set rather
high. If the vehicle starts to drift, the ESC system is activated
to keep the vehicle back on course. Figure 13 shows the vehicle
actual path as compared to the target path. It is noticed
that although the ESC intervened to maintain the vehicle
stability, the vehicle had started sliding towards the edge of
the road and did not follow closely the target path. This is
also confirmed in Figure 14, where the vehicle trajectory is
illustrated as the vehicle entered the curved section of the
road.
To illustrate the performance of the speed advisory
algorithm, the simulation was repeated this time with the
International Journal of Vehicular Technology
13
1
Vehicle trajectory
0.8
0.6
0.4
0.2
0
0
10
20
30
Figure 14: Vehicle trajectory in the curved section of the road, driver
not reacting to the warning at the time the warning is set.
Time (s)
Warning flag
ESC flag
50
Figure 12: Warning and ESC flags time trace, driver not reacting to
the warning at the time the warning is set.
40
Vehicle speed (kph)
80
Global 𝑌 coordinate (m)
60
40
30
20
20
10
0
0
−100
100
200
0
5
10
15
20
25
30
Time (s)
−20
Global 𝑋 coordinate (m)
Vehicle
Design path
Target path
Figure 13: Vehicle path compared to the target path, driver not
reacting to the warning at the time the warning is set.
driver reacting to the warning signal by reducing the vehicle
speed based on the speed advisory algorithm.
Figure 15 shows the vehicle advisory speed and the vehicle
speed as function of time. The advisory speed is initially set
equal to the speed of the vehicle before the activation of
the warning signal. This speed will not be displayed until
Vehicle total CG
Advisory speed
Figure 15: Vehicle advisory speed.
the warning is activated. At the end of the warning signal,
the advisory speed will no longer be displayed to the driver.
Instead if the vehicle is equipped with GPS receiver and a
conventional navigation system, the GPS receiver enables the
speed limit detection unit to find out the exact geographic
location of the vehicle, to identify, based on map data of the
navigation system, a road on which the vehicle is currently
moving and to retrieve from the navigation system data on an
eventual speed limit on this road. Figure 16 shows the vehicle
trajectory; in this case, the vehicle is following more precisely
the target path as opposed to the first case where the driver
14
International Journal of Vehicular Technology
1
80
70
0.8
50
Warning flag
Global 𝑌 coordinate (m)
60
40
30
0.6
0.4
20
0.2
10
−50
0
0
50
100
0
10
Global 𝑋 coordinate (m)
20
30
Time (s)
Vehicle
Design path
Target path
Figure 18: Warning and ESC flags time trace, driver reacting to the
warning at the time the warning is set.
Figure 16: Vehicle path compared to the target path, driver reacting
to the warning at the time the warning is set.
Vehicle trajectory
1
0.8
0.6
0.4
0.2
0
0
Figure 17: Vehicle trajectory in the curved section of the road, driver
reacting to the warning at the time the warning is set.
did not react at the time the warning was set. Figure 17 shows
a stable trajectory of the vehicle. It is also noticed that the
duration of the warning signal has been reduced, and the ESC
did not activate as shown in Figure 18.
To study the effect of the vehicle being on a banked
road on the warning algorithm, the previous simulation was
repeated with the vehicle traveling at 80 kph on a 40 m radius
loop with 200 m straight section of dry surface and variable
bank angle. When the vehicle operator is driving on a banked
road, the measured lateral acceleration will include the effect
of the banked road and therefore affecting the straight driving
5
10
15
20
25
30
Time (s)
Warning flag
ESC flag
Figure 19: False warning issued on a banked road with uncompensated lateral acceleration.
detection algorithm. In addition, the operator introduces a
correction to the steering angle to maintain the vehicle on
the road, and therefore the desired yaw rate and understeer
commands will indicate that the driver wishes to travel on the
bank and not across it. In this case, the warning signal might
be triggered unnecessarily especially when the vehicle is in
the linear range of operation. Figure 19 shows that algorithm
issued a warning signal even though the vehicle was stable
International Journal of Vehicular Technology
15
70
When the vehicle is driven on a banked road, the
measured lateral acceleration is corrupted by the bank angle
of the road given by the following equation:
60
𝑎𝑦𝑚 = 𝑎𝑦 + 𝑔 sin 𝜙.
Global 𝑌 coordinate (m)
80
(61)
The kinematics relationship between the lateral acceleration
.
𝑎𝑦 and the yaw rate of the vehicle 𝜓 is given by the following
equation:
50
40
.
.
𝑣𝑦 = 𝑎𝑦 − 𝜓𝑣𝑥 .
30
(62)
.
20
Under steady state equation, 𝑣𝑦 = 0, and therefore (62)
becomes
10
𝑎𝑦 = 𝜓𝑣𝑥 .
.
(63)
Define 𝜀(𝑘) as
0
100
150
200
Global 𝑋 coordinate (m)
250
𝜀 (𝑘) = 𝑎𝑦𝑚 (𝑘) − 𝑎𝑦 (𝑘) = 𝑔 sin 𝜙
.
= 𝑎𝑦𝑚 (𝑘) − 𝜓𝑣𝑥 (𝑘) .
Vehicle
Design path
Target path
Figure 20: Vehicle path compared to the target path on a banked
road.
Next, we will develop a Kalman filter to estimate 𝜀(k).
Define the following state vector
1
𝑥 (𝑘) = [
];
𝜀 (𝑘)
0.5
Lateral acceleration (g)
(64)
(65)
0.4
the state vector is governed by the linear stochastic difference
equation
0.3
𝑥 (𝑘 + 1) = 𝑥 (𝑘) + 𝜔 (𝑘)
with a measurement 𝑦 ∈ R that is
0.2
.
𝑦 (𝑘) = 𝜑𝑣𝑥 (𝑘) = ⌊𝑎𝑦𝑚 (𝑘) − 1⌋ 𝑥 (𝑘) + 𝜂 (𝑘) .
0.1
0
−0.1
(66)
100
20
Tim
me (s)
Time
30
−0.2
The random variables 𝜔(𝑘) and 𝜂(𝑘) represent the process
and measurement noise, respectively.
They are assumed to be independent (of each other),
white, and with normal probability distributions
𝑝 (𝜔) ∼ 𝑁 (0, 𝑄) ,
−0.3
𝑝 (𝜂) ∼ 𝑁 (0, Γ) .
True 𝑎𝑦
Measured 𝑎𝑦
Figure 21: True and measured uncompensated lateral acceleration
on a banked road.
(68)
The Kalman filter developed in (55) is applied to obtain
an estimate 𝜀̂(𝑘) for the difference between the measured and
the actual lateral acceleration.
The previous simulation was repeated using the compensated lateral acceleration
𝑎𝑦 comp = 𝑎𝑦 + 𝜀̂ (𝑘) .
and followed the target path as shown in Figure 20. Figure 21
compares the measured acceleration including the effect of
the bank component to the true lateral acceleration of the
vehicle.
Therefore, it is important to compensate for the effect of
the bank in the lateral acceleration measurement.
(67)
(69)
In Figure 22, the compensated measured lateral acceleration shows a good agreement with the true lateral acceleration
of the vehicle. Referring to Figure 23, the simulation result
represents an example of a correct warning (no warning
issued) on a banked curve in the linear range of the vehicle
when the lateral acceleration is compensated.
16
International Journal of Vehicular Technology
0.5
Warning results—low 𝜇 handling course
1
0.4
0
50
51
52
53
54
55
56
57
58
59
60
130
80
30
−20
Lateral acceleration (g)
0.3
ESC active
Warning flag
SWA (deg)
0.2
(a)
0.1
0
10
20
30
Time
Tim
me (s)
−0.1
Warning results—low 𝜇 handling course
1.5
1
0.5
0
50
51
52
53
54
56
57
58
59
60
Lateral 𝜇 capability
Understeer (deg/g)
−0.2
−0.3
(b)
Bank compensation
True 𝑎𝑦
Measured 𝑎𝑦
Warning results—low 𝜇 handling course
60
55
50
45
40
50
Figure 22: True lateral acceleration and measured uncompensated
and compensated lateral acceleration on a banked road.
51
52
53
54
55
56
57
58
59
60
Vehicle speed (kph)
(c)
Figure 24: Vehicle simulating a freeway exit on low 𝜇 surface.
1
0.5
Warning flag
55
50
30
10
−10
0
10
20
30
Time (s)
−0.5
−1
Figure 23: Warning signal on a banked road with compensated
lateral acceleration.
4. Vehicle Test Results
To evaluate the warning algorithm performance, the following results are based on experimental data obtained using
an Opel Omega vehicle equipped with ESC sensors. The
tests were conducted on low-mu handling track with straight
and curved sections to simulate a freeway exit. Time slices
from different driving sessions are zoomed in to illustrate the
performance of the warning algorithm.
A time slice of 10 seconds (50–60 seconds) is shown in
Figure 24. The graphs show the performance of the warning
system as the vehicle simulates a free way exit. In the first 4
seconds, the driver enters a curved section of the course.
At time 𝑡 = 54 seconds, the driver maintains an 80 deg
steering wheel angle at speed of 58 kph. The vehicle starts to
understeer and reaches an understeer of 20 deg/g. The lateral
mu capability algorithm identifies a maximum lateral surface
capability of 0.3. At this moment, the warning signal was
issued, and the driver released his accelerator pedal but did
not apply his brakes. The ESC was activated 3.5 seconds after
the warning signal and slowed down the car. The sudden
change in the understeer at 58 seconds is due to the fact that
vehicle was not in a steady state mode and the understeer is
not defined outside the steady state behavior of the vehicle.
Figure 25 represents a series of transient maneuvers on
the handling course. As seen from the warning signal and the
ESC active flag, the warning signal was issued 3 to 4 seconds
prior to the ESC activation which allows the driver to react to
the warning signal and reduce the vehicle speed.
Figure 26 shows a large increasing steering angle maneuver at relatively low speed. It is noticed that under this
scenario the warming is issued almost at the same time as the
ESC activation. The warning signal is not effective in this case
International Journal of Vehicular Technology
17
Warning results—low 𝜇 handling course
1
50
−50
0
30
35
40
45
50
55
60
−150
Warning results—low 𝜇 handling course
1
0
100
102
Warning results—low 𝜇 handling course
102
104
106
108
110
112
(a)
Figure 25: Warning and ESC signals under transient maneuvers.
0
100
106
ESC active
Warning flag
SWA (deg)
ESC active
Warning flag
SWA (deg)
1
104
50
−50
−150
114
108
110
112
114
116
118
500
400
200
0
−200
−400
120
ESC active
Warning flag
SWA
Figure 26: Warning and ESC signals with increasing steering angle
at low speed.
since the driver does not have time to react to the warning
signal before the ESC activation.
Figure 27 shows the vehicle going into a steady state
maneuver. In this case, the speed advisory algorithm was
triggered as the driver enters the curved section of the road.
The vehicle starts to understeer and the warning signal is
set. The driver releases the accelerator pedal and reduces
the steering input. The warning signal was then terminated
and the speed advisory ended. The ESC did not activate in
this case since the driver reduced his steering input and the
understeer gradient of the vehicle is reduced. The vehicle did
not a have an automatic brake control to control the vehicle
speed to the advisory speed. The warning was set until the
driver reduced the steering input.
5. Conclusions
(1) In some of the conditions evaluated, the warning
signal is issued 3 to 4 seconds prior to ESC activation
to allow the driver to react to the warning.
(2) If the driver reacts to the warning signal by reducing
the vehicle speed in the curve, the vehicle will follow
more precisely the road curves with minimum or no
ESC intervention.
(3) For large and increasing steering angle maneuver at
relatively low speed, the warning is issued almost at
the same time as the ESC activation. The warning
signal is not effective in this case since the driver does
not have time to react to the warning signal before the
ESC activation. However, at these low speeds, the ESC
is very effective and can stabilize the vehicle without
heavy brake intervention.
1.5
1
0.5
0
100
Warning results—low 𝜇 handling course
102
104
106
108
110
112
30
20
10
0
−10
114
Lateral 𝜇 capability
Understeer (deg/g)
(b)
80
60
40
20
0
100
Warning results—low 𝜇 handling course
102
104
106
108
110
112
80
60
40
20
0
114
Vehicle speed (kph)
Advisory speed (kph)
(c)
Figure 27: Vehicle simulating a freeway exit on low 𝜇 surface with
speed advisory.
(4) When the vehicle is driven on a banked road, the
uncompensated lateral acceleration measurement can
false trigger the warning algorithm. Therefore, it is
important to compensate for the effect of the bank in
the lateral acceleration measurement.
(5) This warning system should be evaluated in a wider
range of road and vehicle conditions to more fully
evaluate its usefulness.
Nomenclature
𝑎 : Distance from the center of gravity of
vehicle to the front axle (m)
𝑏: Distance from the center of gravity of
vehicle to the rear axle (m)
𝐶𝑓 : Cornering stiffness of both tires of front
axle (N/rad)
𝐶𝑟 : Cornering stiffness of both tires of rear
axle (N/rad)
𝑔: Acceleration of gravity (m/s2 )
𝑔𝑟 : Steering gear ratio
𝐼𝑧 : Moment of inertia of entire vehicle
about the yaw axis (kgm2 )
𝑀𝑣 : Total vehicle mass (kg)
𝑊𝑓 : Vehicle weight on the front axle
18
𝑊𝑟 :
𝐾𝑢 :
𝑇𝑤 :
𝑣𝑦 :
Vehicle weight on the rear axle
Desired understeer coefficient (deg/g)
Vehicle track width (m)
Lateral velocity of vehicle’s center of
gravity (m/s)
Lateral acceleration of vehicle’s center of
𝑎𝑦 :
gravity (m/s2 )
Measured lateral acceleration at the
𝑎𝑦𝑚 :
vehicle’s center of gravity (m/s2 )
Desired lateral velocity of vehicle’s
𝑣𝑦𝑑 :
center of gravity (m/s)
.
𝑣𝑦𝑑 :
Desired lateral velocity rate of vehicle’s
center of gravity (m/s2 )
Lateral velocity gain (m/s/rad)
𝑉𝑦 gain :
Steady sate desired lateral velocity of
𝑉𝑦 dss :
vehicle’s center of gravity (m/s)
Longitudinal velocity of vehicle’s center
𝑣𝑥 :
of gravity (m/s)
𝑥𝑖 , 𝑖 = 1, 2: State variables of the second-order filter
Negative of system zero for desired side
𝑧𝑣 :
slip velocity
Negative of system zero for desired yaw
𝑧𝜓. :
rate
𝛿:
Steering angle of the front wheels (rad)
.
𝜓:
Yaw rate of vehicle (rad/s)
Φ:
Bank angle (rad)
.
𝜓𝑑 :
Desired yaw rate of vehicle (rad/s)
.
𝜓dss :
Desired steady state yaw rate of vehicle
(rad/s)
Yaw velocity gain (rad/s/rad)
Ωgain :
𝜁:
Damping ratio of desired vehicle
performance
Natural frequency of desired vehicle
𝜔𝑛 :
performance (rad/s).
References
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Shock and Vibration
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Acoustics and Vibration
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
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