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A Comparison Among Different Methods to Estimate Vehicle Sideslip Angle

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Abstract and Figures

Accurate measurement of the vehicle sideslip angle is fundamental to improve reliability of the vehicle dynamics control systems focused on stability and developed both for safety and performance optimization. Many experimental procedures to estimate the vehicle sideslip angle have been proposed in the last years, mainly based on GPS, INS and physical models. The aim of this paper is to compare different methods to estimate sideslip angle employing an instrumented vehicle, equipped with a system for data acquisition and time-synchronized storage capabilities, a stand-alone GPS, a GPS aided MEMS-based Attitude and Heading Reference System (AHRS) and specific sensors to collect data on the steering wheel angle and on the position of brake, throttle and clutch pedals. Further information is collected by capturing the available data at the OBD port of the vehicle. Data acquisitions (from all sensors) are synchronized by means of an external triggering signal. After driving sessions performed with specific manoeuvres in order to highlight the main phenomena concerned with the dynamic behaviour of the vehicle, the different estimation procedures have been applied, discussing on the advantages and the degree of reliability of each one of them.
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Abstract—Accurate measurement of the vehicle sideslip
angle is fundamental to improve reliability of the vehicle
dynamics control systems focused on stability and developed
both for safety and performance optimization. Many
experimental procedures to estimate the vehicle sideslip angle
have been proposed in the last years, mainly based on GPS,
INS and physical models.
The aim of this paper is to compare different methods to
estimate sideslip angle employing an instrumented vehicle,
equipped with a system for data acquisition and time-
synchronized storage capabilities, a stand-alone GPS, a GPS
aided MEMS-based Attitude and Heading Reference System
(AHRS) and specific sensors to collect data on the steering
wheel angle and on the position of brake, throttle and clutch
pedals. Further information is collected by capturing the
available data at the OBD port of the vehicle. Data acquisitions
(from all sensors) are synchronized by means of an external
triggering signal.
After driving sessions performed with specific manoeuvres
in order to highlight the main phenomena concerned with the
dynamic behaviour of the vehicle, the different estimation
procedures have been applied, discussing on the advantages
and the degree of reliability of each one of them.
Index Terms Differential GPS Tracking, Inertial
Navigation System, Instrumented Vehicle, Physical Modelling,
Vehicle Sideslip Angle.
I. I
NTRODUCTION
EAL
-
TIME
knowledge of the vehicle sideslip angle (ߚ)
(the angle the vehicle center of gravity velocity (V
G
)
forms with the longitudinal vehicle axis (x)) is a
fundamental issue to manage all vehicle’s control systems
[1] such as braking [2]-[24], stability [5] and many ADAS
control systems [6], as well as a very important factor
allowing to validate driving simulators [7], [8].
In the event of low-friction situations, it is advantageous
to control the vehicle sideslip angle, preventing it from
assuming unexpected values [9], [10]. However, despite the
absolute benefits deriving from the direct measurement of
Manuscript received June XX, 20XX; revised July XX, 20XX.
F. Farroni is with Dipartimento di Ingegneria Industriale, Università
degli Studi di Napoli Federico II, via Claudio n. 21 80125 Napoli Italy (e-
mail: flavio.farroni@unina.it)
E. Rocca is with Dipartimento di Ingegneria Industriale, Università degli
Studi di Napoli Federico II, via Claudio n. 21 80125 Napoli Italy (e-mail:
ernesto.rocca@unina.it)
F. Timpone is with Dipartimento di Ingegneria Industriale, Università
degli Studi di Napoli Federico II, via Claudio n. 21 80125 Napoli Italy
(corresponding author: phone: +39 081 76 83263; fax: +39 081 2394165; e-
mail: francesco.timpone@unina.it)
N. Pasquino is with Dipartimento di Ingegneria Elettrica e delle
Tecnologie dell'Informazione, Università degli Studi di Napoli Federico II,
via Claudio n. 21 80125 Napoli Italy (e-mail: nicola.pasquino@unina.it).
the sideslip angle and absolute vehicle velocity, these values
are not usually directly measured on production cars and
therefore must be estimated [11], [12].
While the yaw rate (r=dψ/dt) can be easily measured by a
gyroscope, Fig. 1 shows that the estimation of the vehicle
sideslip angle requires the knowledge of the vehicle heading
(ψ) and of the direction of the center of gravity’s velocity
vector (α) (all of them expressed with respect to a dedicated
reference system).
Because of the fundamental importance of ߚ for vehicle
dynamics, several methods have been developed over the
years for its estimation. All the different existing methods
display some strong points, but even some weaknesses, so
that a universally valid and advantageous method has not
been developed yet.
The most widely adopted methods are briefly described in
the following, together with their main advantages and
disadvantages, as well as their main fields of application.
A. Single Antenna GPS
The magnitude and the direction of the center of gravity
velocity vector (V
G
) can be easily determined once a
suitable reference system has been identified thanks to a
GPS receiver.
Depending on the specific receiver in use, the information
about the velocity vector can be obtained directly (some
models are able to provide the velocity vector’s components
expressed in the reference system the device has been
referred to) or has to be determined in an indirect way. The
latter is the case of a GPS receiver only indicating the
sequence of the taken position at any single time. In this
case, by simple kinematic considerations, being the elapsed
time between two consecutive points known, also the
velocity vector’s magnitude can be determined.
Once the velocity vector pertaining to the vehicle’s center
of gravity has been fully characterized in its components,
the ߙ angle is defined with respect to the assumed
coordinate system. If the vehicle is equipped with some
gyroscopes, then their signal can be integrated, and the
A Comparison Among Different Methods to
Estimate Vehicle Sideslip Angle
F. Farroni, N. Pasquino, E. Rocca, F. Timpone
R
Fig. 1. Vehicle Dynamics characteristic Angles.
vehicle’s heading angle (ψ) can be obtained. From the
knowledge of these two angles, the vehicle sideslip angle ߚ
is immediately determined.
One of the main limits affecting this procedure is
represented by its variable accuracy: position uncertainty in
a modern GPS is largely determined by the number of
satellites in view and can reach in some cases the
unacceptable magnitude of a few meters.
For these reasons, the single-antenna-GPS method is
often integrated with some correction algorithm, such as the
differential correction method [13]. Other improvements of
this method use a higher number of GPS like the Double
Antenna GPS method discussed below.
B. Double Antenna GPS
Also a double antenna GPS can be used for vehicle
sideslip angle measurement [12], [14]: either two separate
GPS receivers or a single receiver equipped with two
different antennas can be used, theoretically obtaining
equivalent accuracy. The purpose of using two different
antennas is to directly calculate the vehicle’s orientation
with respect to an ‘absolute’ reference coordinate frame,
with no necessity of using an additional gyroscope.
Using a double-antenna GPS two different points of the
same vehicle, which can be assumed as a completely rigid
body, can be simultaneously characterized.
The use of a double-antenna GPS is very easy to
understand since a single-antenna GPS cannot measure the
vehicle sideslip angle by itself, but always needs to be
flanked by a gyroscope.
C. Inertial Navigation System signals Direct Integration
Procedures
One of the easiest ways to determine the vehicle sideslip
angle consists in the direct integration of the signals coming
from the accelerometers and gyroscopes (or, more often,
from the Inertial Navigation System (INS) based on the
advanced application of inertial sensors (accelerometers)
and rotational speed sensors (gyroscopes)) installed on-
board the vehicle itself [15].
Thanks to this method, no vehicle model nor other
instrument is needed. The results obtained from the adoption
of this simple procedure are not always acceptable, because
they tend to be affected by a high rate of error, since they
are based on the direct integration of the accelerometer
signals.
D. Physical models
This approach for the sideslip angle estimation is based
on the use of some physical models, possibly running in
real-time, describing the vehicle’s behavior, such as the
‘Bicycle model’ or the ‘Quadricycle model’ [16].
Once the chosen physical model is known, in order to
determine the vehicle sideslip angle, it is sufficient to
strictly apply the model’s equations.
The precision of the obtained results is highly affected by
the degree of complexity of the employed model, that is by
the reliability of the physical and geometrical parameters
inserted in the model, like the vehicle’s characteristic
lengths and dimensions, its inertial features, transmission
ratios, tyres’ behavior parameters and so on [17].
II. E
XPERIMENTAL EQUIPMENT
A. Vehicle
The experimental tests have been carried out on a FIAT
Multipla 1.9 JTD, made available by the DICEA
(Department of Civil, Architectural and Environmental
Engineering) of University of Naples ‘Federico II’ [18].
This vehicle has been used extensively in past researches to
monitor the behaviour of a driver both for analytical model
formulation [19], and for testing of Driving Assistance
Systems (DAS) logics.
In order to define the considered test-vehicle as precisely
as possible, its physical and geometric parameters have been
carefully evaluated by means of the available technical
sheets.
Tyre behaviour has been modelled by means of a ‘Magic
Formula’ model. The vehicle was equipped with the
measurement instruments described in the following.
B. Description of the installed hardware
The vehicle is equipped with a system for data acquisition
and time-synchronized storage capabilities. The core of the
system is a National Instruments (NI) CompactDAQ
chassis, with up to eight measurement-specific modules. All
the sensors installed on-board are in direct communication
with the DAQ’s control software.
The DAQ receives digital and analog signals via
appropriate modules, as well as data and communications
via a CAN (Controlled Area Network) interface. In order to
process the big amount of acquired data, the CompactDAQ
is connected to a PC controller. The synchronization and
acquisition software is based on LabVIEW.
A TOPCON GPS supplies positioning and kinematic data
using a dual satellite constellation (GPS and GLONASS).
The GPS is equipped with an external antenna, installed on
the vehicle’s rooftop. The GPS dual-frequency receiver is
equipped with an internal memory allowing to store raw
satellite data for post-processing applications. An XSENS
MTi-G device embedding a GPS and an Inertial
Measurement Unit (IMU), together with an AHRS processor
have been placed on-board as near as possible to the center
of gravity (ܩ), with the aim of minimizing the centripetal
acceleration deriving from the rotations of the vehicle. The
GPS system complements the inertial system by providing a
stable position and velocity output. As a result, the errors
introduced by integrating accelerometer and gyroscope
signals (dead reckoning) are corrected by the position and
velocity given by the GPS receiver, resulting in an output
that is both stable and able to track fast changing
movements.
Three potentiometers are used to collect data on the
position of the brake, throttle and clutch pedals. The steering
wheel angle is measured by a draw-wire displacement
sensor.
Other on-board information is also collected by capturing
the available data at the OBD (On Board Diagnostic)-II port
of the vehicle. Data acquisitions (from all sensors) are
synchronized by means of an external triggering signal.
The MTiG control logic is based on inertial sensors and a
miniature GPS receiver, also including other supporting
sensors, such as a 3D magnetometer and a static pressure
transducer.
Orientation data are provided as relative orientation
between the above-defined reference system and the LTP
(Local Tangent Plane) reference system. LTP is a reference
system defined by the tangent plane to the Earth surface in
every point. According to the axes’ orientation, it is possible
to distinguish between different reference systems, such as
LTP
enu
(‘East North Up’) or LTP
nwu
(‘North West Up’). The
‘default’ LTP reference system with respect to which the
XSENS provides the results is LTP
nwu
.
III. D
ATA
A
CQUISITION AND
P
RE
-P
ROCESSING
A. Available Data Channels
A LabVIEW script has been used for the acquisition of
data from the different on-board instruments, with a
sampling frequency of 10 Hz.
The collected GPS data are the time, the latitude, the
longitude, and the altitude of the GPS antenna, hemispheres
(North or South and East or West), the speed of the GPS
antenna and the number of visible satellites.
The following data are gathered from the XSENS sensor:
the three Euler angles (roll, pitch, and yaw components
around vehicle longitudinal, lateral and vertical axes the
three angular velocities of vehicle (roll, pitch, and yaw rate)
and the three linear accelerations of vehicle (longitudinal,
lateral, and vertical acceleration).
Data from the other sensors are the steering wheel angle
(positive for clockwise rotation out of neutral position), the
longitudinal speed of both the rear wheels of vehicle
(computed by means of a rotational encoder), the
normalized excursion of the vehicle pedals (brake, clutch,
and throttle pedals), and the engine’s rotation speed).
By direct connection to the XSENS device, some other
useful information can be retrieved, the main one being the
components of the vehicle’s velocity vector in the LTP
nwu
reference system.
IV. D
RIVE
S
ESSIONS
In this section a description of the performed tests is
reported, together with a brief description of the particular
‘test protocol’ implemented.
A. Experimental Site
Some preliminary test drives have been performed close
to the location of the Engineering Faculty of the University
of Naples Federico II in order to become familiar with the
measurement equipment, configure it.
The final test session has been carried out on a closed race
track, an open-sky and abandoned parking area of a mall in
Northern Naples (coordinates: 40.896776° N; 14.312782°
E). The site fulfils the experimental requirements: no vehicle
transit, no considerable obstructions to GPS signals, and
quite planar area.
Some fixed elements in the site, such as road markings,
are used to define the reference points of each manoeuvres
performed in the test session.
Test drives were performed in dry road and good weather
conditions (cloud-free sky).
B. Test Protocol Definition
The test protocol has been defined following the criterion
of performing manoeuvres determining a considerable
sideslip angle occurrence.
In the following the different steps of the chosen test
protocol for the drive sessions are reported(Fig. 2):
Start: all the simulations start and finish at the same point,
marked with the chess flags.
Idle Period: before any drive is performed, a preliminary
acquisition with stationary vehicle has to be carried out,
with the aim of identifying the bias affecting all the
instrumental signals. The idle period usually lasts some tens
of seconds.
Straight drive: all drive sessions start with a set of straight
drives on a closed loop, separated by curves to revert the
direction. The straight drive are used to validate the initial
condition on yaw angle, computed by means of a compass:
if the compass procedure is correct, during the straight drive
the vehicle sideslip angle is supposed to be null, as the
centre of gravity’s velocity vector is perfectly aligned with
the vehicle’s heading.
Circle drive: performed with constant steering angle and
constant vehicle speed to get a constant sufficiently high
vehicle sideslip angle [20].
“8” shaped manoeuvres: after the circle drive some
manoeuvres on a “8-shaped” trajectory have been
performed. Also in this case, this manoeuvre has the purpose
of letting some higher vehicle sideslip angles occur.
Hard Braking: the final path consists in a straight drive,
with a very hard braking manoeuvre performed at sustained
speed, aiming to determine tyres’ lock up and subsequent
ABS logic intervention.
Stop: the drive session stops in correspondence of the
same point it started. Thanks to the white stripes on the road
surface, the exact point can be found, and the vehicle’s final
orientation is made to be approximately coincident with the
initial one.
The different phases of the above described test protocol
can be easily discerned on the sample vehicle sideslip angle
plot (obtained by means of 1Cbis differential GPS procedure
explained in the following), traced against measurement
time. The different steps of the data acquisition session can
be discerned (Fig. 3); in particular:
Idle Period: this phase can be easily recognized in the
plot, as the sideslip angle has been put null when the vehicle
is not moving (V
G
< 5 km/h).
Straight drive: as it is evident, during a straight drive the
vehicle sideslip angle is equal to zero, since the direction of
Fig. 2.
Vehicle’s trajectory during the performance of the defined test
protocol, superimposed to the satellite image of the selected area, and
colored according to the el
apsed time from the beginning of the drive session
[Blue=Initial position; Red=Final position]. The initial and final points
coincide, and have been marked with the chess flags.
the V
G
vector is coincident with the direction of the vehicle-
anchored x axis, indicating the vehicle’s heading.
In Fig. 3 it is possible to see that the straight drive
portions are characterized by null sideslip angle indeed: the
visible regular spikes represent the cyclic curves performed
to invert the vehicle’s direction, as depicted in previous Fig.
2. All the spikes have the same sign, since all the curves
have been performed in the same direction (clockwise).
Circle drive: as above described, the circle drive
manoeuvre has been performed with constant steering angle
and vehicle speed. This feature allows a very rapid
identification of this particular drive session’s step in the
plot of the vehicle sideslip angle, as the ߚ signals oscillates
around a well-defined value, different from zero.
“8” shaped manoeuvres: this path can be discerned in
the plot considering that during an “8” shaped manoeuvre
curves are performed in both directions, so that the steering
wheel has to pass through the neutral’ position at every
inversion. For this reason, it is easy to understand the
vehicle sideslip angle oscillates around an approximately
null value, reaching about the same magnitude in both
directions, as depicted in Fig. 3.
Hard Braking: the hard braking can be easily discerned
from the vehicle sideslip angle plot by the null portion of the
curve, corresponding to the vehicle complete stop after the
braking process.
The slight oscillation right before the null value (before
the vehicle has come to a complete stop) is due to the fact
the vehicle is rapidly decelerating, so that the lateral
component of velocity is comparable with the longitudinal
one, making sideslip angle value oscillating.
V. I
MPLEMENTED
P
ROCEDURES AND
R
ESULTS
To estimate the vehicle sideslip angle different procedures
have been employed based on GPS, on INS Signal Direct
Integration and on Bicycle Model.
A. GPS
The first procedure (1A) makes use of the XSENS device,
using both the GPS receiver and the gyroscopes.
The vehicle sideslip angle is determined as the difference
between the ߙ and ߰ angles, computed with respect to the
LTP
enu
frame (Fig. 1).
The ߙ angle is determined once the components of the
center of gravity’s velocity vector (V
G
) are known, from the
XSENS data acquisition. It is important to underline that the
XSENS platform automatically calculates the velocity
vector’s components in the LTP
nwu
reference system.
As it is evident in Fig. 4, starting from the vector’s
component it is immediate to obtain the ߙ angle as:
ߙ
=
1
ܸ
ܰ݋ݎݐ
ܸ
ܧܽݏݐ
=
1
ܸ
ܻ
ܸ
ܺ
(1)
The ߰ angle can be obtained either directly from the
XSENS platform or by the integration of the vehicle’s yaw
rate signal, acquired by the INS sensor.
Also the second procedure (1B) makes use of the XSENS
device, using both the GPS receiver and the gyroscopes, and
simply applies the vehicle sideslip angle definition,
reporting the velocity vector’s components into the moving,
vehicle-anchored {ܩݔݕݖ} reference system to know the ݑ
and ݒ components.
Once the ݑ and ݒ components have been determined, the
vehicle sideslip angle is simply determined as:
ߚ
=
1
ݒ
ݑ
(2)
The third procedure (1C) makes use of both the TOPCON
GPS and XSENS devices.
The vehicle sideslip angle is determined as in the previous
procedure 1A, but in this case the ߙ angle is obtained thanks
to a different method.
The TOPCON GPS receiver simply provides the
terrestrial coordinates of all the points belonging to the
vehicle’s trajectory (with a data rate depending on the
configured working frequency), and the magnitude of the
velocity vector needed to connect two consecutive points ܲ1
and ܲ2.
In order to compute the ߙ angle, some simple kinematic
consideration are employed (Fig. 5).
The instantaneous velocity of the vehicle in the ܲ1 point
can be obtained as:
ݐ
=
Δ
ݐ
0
Δ
ݐ
(3)
In such cases, the instantaneous velocity vector can be
assumed to be parallel to the displacement vector at any
given time, whose direction can be easily identified through
the difference of the coordinates recorded by the GPS
receiver. Once the velocity vector has been fully identified,
its inclination angle ߙ is used for the calculations.
Regarding the ߰ angle, it can be obtained either directly
from the XSENS platform or integrating the vehicle’s yaw
rate signal acquired by the gyroscope.
One of the main limitations affecting the preciseness of
the just described procedure is represented by its heavy
dependence on the quality of the GPS signal, hence from the
Fig. 3. Sample vehicle sideslip angle plot vs. measurement time.
Fig. 4. Decomposition of the vehicle’s centre of gravity’s velocity vector
along the moving vehicle-anchored {
ܩ
ݔ
ݕ
ݖ
} reference system and along the
absolute {
ܲ
ܺ
ܻ
ܼ
} reference system.
number of satellites in view of the receiver.
For this reason, some signal improvements have been
performed in order to increase the accuracy of the sideslip
angle estimation in the following procedure.
The fourth procedure (1C bis) is very similar to the 1C
one, since it is based on the same physical and analytical
considerations. The only difference between the two
procedures is represented by the GPS data acquisition: in
order to enhance the signal’s preciseness and reliability a
differential correction has been applied [13].
In the differential procedure (DGPS) a GPS receiver
tracking a satellite always experiences a number of ranging
errors, which will be translated into positioning errors. The
commonality of range errors between two receivers
simultaneously tracking a satellite at two different locations
can be exploited: common errors can be removed by
differencing. Recording simultaneous range observations
allows differential processing at a later time, or in real time,
if a suitable communication device is available.
The differential correction considered in the current paper
has been carried out off-line, referring to the ephemeris data
published online after the data acquisition session was
performed.
A highly precise GPS device(‘Trimble’ GPS: receiver and
antenna) has been positioned in a fixed place for the
differential procedure application and has been used as the
reference station, in order to differentiate the data acquired
by the TOPCON GPS (the mobile station).
In order to allow the reference station to fix its position,
the reference GPS has been placed in position about 1 hour
before the drive sessions’ beginning.
B. INS Signal Direct Integration
These procedures only use the accelerometers’ signal,
obtained from the XSENS acquisition.
The first procedure (2A) is a purely kinematic method for
the vehicle sideslip angle determination, based on the
integration of the accelerometers’ signals; for this reason,
the only required instruments are the accelerometers and the
gyroscopes.
The accelerometers provide a measurement of
acceleration referred to the vehicle-anchored moving
reference system {ܩݔݕݖ}. To express the center of gravity’s
acceleration vector
ܩ
with respect to the LTP
enu
reference
system it is necessary to implement a coordinate
transformation, so all the calculations are performed in the
absolute reference system. A crucial importance is assumed
by the Euler Angles used since these highly affect the
obtained accelerations in the {ܩݔݕݖ} reference system, and
in turn the resulting velocities and sideslip angle [21].
By integration of the acceleration components the
velocities, expressed in the ‘absolute’ LTP
enu
reference
system, are obtained.
The initial condition for the integration of the
accelerations has been set to zero as the vehicle had null
velocity during the first acquired seconds (idle period).
After integration, the velocity components have to be
reported again into the {ܩݔݕݖ} moving reference system. At
this point, the ݑ and ݒ components, namely the V
G
components in the {ܩݔݕݖ} reference frame, have been
obtained, hence it is immediate to obtain the desired vehicle
sideslip angle as (2).
The second Procedure (2B) calculates the vehicle sideslip
angle making use of the XSENS inertial station and of the
rotational encoder fitted on the rear wheels.
This procedure is based on the vehicle sideslip angle
definition, (2).
Regarding the ݑ component, this has been directly
measured through the rotational encoders installed on the
vehicle’s rear wheels (it is calculated as the mathematical
average of the values provided by the two encoders).
The ݒ component has been obtained from the Equation
reported below, where the lateral component of the ࢇܩ
vector with respect to the {ܩݔݕݖ} vehicle-anchored moving
reference system is defined:
ܽ
ݕ
=
ݒ
+
ݑ
ݎ
ݒ
=
ܽ
ݕ
ݑ
ݎ
(4)
Where the yaw rate ݎ is measured by the gyroscope.
Once the ݑ and ݒ components have been obtained, the
vehicle sideslip angle is obtained by applying its definition.
The third procedure (2C) aims to determine the vehicle
sideslip angle starting from the signals obtained from the
accelerometers and gyroscopes embedded into the XSENS
inertial station, installed on-board.
This procedure makes use of the following Equations:
ܽ
ݔ
=
ݑ
ݒ
ݎ
ܽ
ݕ
=
ݒ
+
ݑ
ݎ
ݑ
=
ܽ
ݔ
+
ݒ
ݎ
ݒ
=
ܽ
ݕ
ݑ
ݎ
(5)
Once the a
x
, a
y
and ݎ = dψ/dt quantities have been
obtained from the XSENS data, these equations can be
solved, determining the ݑ and ݒ.
Once the ݑ and ݒ components have been obtained, the
vehicle sideslip angle is obtained by applying its definition.
C. Bicycle Model
This procedure uses a vehicle dynamics’ physical model
to predict the vehicle sideslip angle, given the input
parameters.
The physical model considered for the current paper is the
so-called bicycle model [16].
Both the kinematic bicycle model (absence of drift) and
the dynamic bicycle model (with drift) have been
considered, in order to provide a comparison between the
different cases.
In the first procedure (3A) the Classic Kinematic Bicycle
is used, which is only valid in case of low vehicle’s speed
and high curvature radius of the performed trajectory, and
therefore very uncommon to be observed in actual
conditions. The input signals for this model are the vehicle’s
longitudinal speed (u) and the steering angle of the vehicle’s
Fig. 5. Determination of the instantaneous vehicle’s velocity vector
starting
from its centre of gravity’s trajectory.
front wheels (ߜ).
The vehicle sideslip angle, in case of kinematic
conditions, can be easily determined by means of simple
geometric and trigonometric considerations, Fig. 6. Being
the (ACB) ̂ angle equal to ߜ, it is immediate to obtain:
ߚ
=
1
ܽ
2
ܴ
ߜ
=
tan
1
݈
ܴ
ܴ
=
݈
ߜ
(6)
Combining the above reported Equations (6) it is possible
to obtain the vehicle sideslip angle according to the Classic
Kinematic Bicycle Model
ߚ
0
=
1
ܽ
2
݈
ߜ
(7)
In the procedure (3B) the Classic Dynamic Bicycle model
is used. Taking into consideration the presence of a certain
sideslip angle makes the model closer to reality than the
above kinematic model, since greater speeds and real
manoeuvres can be considered.
Also in this case the input signals for this subsystem are
the vehicle’s longitudinal speed (u) and the steering angle of
the vehicle’s front wheels (ߜ).
Making use of these signals, the ݑ and ݒ quantities can be
determined by means of the following Equations [16]:
ݒ
ݎ
=
ۏ
ێ
ێ
ێ
ۍ
ܥ
1
+
ܥ
2
ݑ
݉
ܥ
1
ܽ
1
ܥ
2
ܽ
2
ݑ
݉
+
ݑ
ܥ
1
∙ ܽ
1
− ܥ
2
∙ ܽ
2
ܬ
ݖ
ݑ
ܥ
1
∙ ܽ
1
2
+ ܥ
2
∙ ܽ
2
2
ܬ
ݖ
ݑ
ے
ۑ
ۑ
ۑ
ې
∙ ቊ
ݒ
ݎ
ቋ + ܥ
1
∙ ߜ∙ ቐ
1
/
݉
ܽ
1
ݖ
(8
)
After ݒ and ݎ have been determined, the vehicle’s slip
angle can be obtained (2).
VI. R
ESULTS AND
C
ONCLUSION
The vehicle slip angle estimation method that exhibits the
best fit with what is physically expected is obtained by
procedures 1Cbis (GPS with differential correction), 3A and
3B (Bycicle Model) (Fig. 7). The other GPS based
procedures still provide good results (Fig. 8). In revers, INS
based methods show results not compliant with physical
expectations probably due to errors in Euler Angles’
estimation attributable to the magnetometer, whose signal is
highly affected by the metallic masses on-board the test
vehicle. Future analysis and experimental activities will
provide a sounder base for the claim of the better
performance of procedures 1Cbis, 3A and 3B. Future
experiments will also be carried out according to a designed
experimental plan, and results from the experimental
campaign will be analyzed by means of statistical data
analysis techniques to determine the influence of some
relevant factors on the estimates provided by the different
methods [22]-[26].
Future developments will also deal with a real-time GPS
differential procedure, with a double antenna GPS
procedure, with combined methods able to fuse signals
coming out from different devices treating them by means of
statistical methods.
R
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