Integrated application of network traffic and intelligent driver models in the test laboratory analysis of autonomous vehicles and electric vehicles

Abstract

Abstract: The aim of the research is to develop a laboratory model-based diagnostic procedure that performs tests of the motion processes of autonomous electric vehicles in a particular city, on a transport network or track. The test consists of a laboratory based generation of the corresponding speed and steering angle signals, being in accordance with real driving and traffic conditions, which are also used in the test procedure. The procedure takes into account the real trajectory tracking process as well (Péter and Lakatos, 2017).

Full text

I
nt. J. Heavy Vehicle Systems, Vol. 2
7
, Nos. 1/2, 2020 227
Copyright © 2020 Inderscience Enterprises Ltd.
Integrated application of network traffic and
intelligent driver models in the test laboratory
analysis of autonomous vehicles and electric
vehicles
Tamás Péter*
Department of Control for Transportation and Vehicle Systems,
Budapest University of Technology and Economics (BME),
Stoczek utca 2. 1111, Budapest, Hungary
Email: peter.tamas@mail.bme.hu
*Corresponding author
Ferenc Szauter
Széchenyi István University,
SZE KVJT and JKK,
Egyetem tér 1. 9026, Gyor, Hungary
Email: szauter@ga.sze.hu
Zoltán Rózsás
Automotive Proving Ground Zala Ltd..
Jozsef nador ter 2-4. H-1051 Budapest, Hungary
Project office: Feszek u. 4. H – 8900 Zalaegerszeg,
Email:zoltan.rozsas@apz.hu
Website: https://zalazone.hu/en/
István Lakatos
Széchenyi István University,
SZE KVJT and JKK,
Egyetem tér 1. 9026, Gyor, Hungary
Email: lakatos@sze.hu
Abstract: The aim of the research is to develop a laboratory model-based
diagnostic procedure that performs tests of the motion processes of autonomous
electric vehicles in a particular city, on a transport network or track. The test
consists of a laboratory based generation of the corresponding speed and
steering angle signals, being in accordance with real driving and traffic
conditions, which are also used in the test procedure. The procedure takes into
account the real trajectory tracking process as well (Péter and Lakatos, 2017).
Keywords: autonomous electric vehicles; laboratory model-based diagnostic
procedure; corresponding speed and steering angle signals; laboratory based
generation; real traffic environment.

228 T. Péte
r
et al.
Reference to this paper should be made as follows: Péter, T., Szauter, F.,
Rózsás, Z. and Lakatos, I. (2020) ‘Integrated application of network traffic
and intelligent driver models in the test laboratory analysis of autonomous
vehicles and electric vehicles’, Int. J. Heavy Vehicle Systems, Vol. 27, Nos. 1/2,
pp.227–245.
Biographical notes: Tamás Péter is a Research Professor, he obtained his MSc
degree in Mechanical Engineering from the BME, Hungary in 1972,
Mathematics Doctorate in 1978 and a Technology Doctor-PhD in 1998,
respectively. He has been Habilitated Doctor in Transport Engineering since
2012. His research field includes non-linear stochastic dynamical systems,
mathematical modelling, analysis and optimisation, computer-mathematics,
equivalence classes of vehicle vibration systems, stochastic vehicle dynamics,
road traffic and logistic models and their applications, mathematical analysis
and optimal control of large-scale road traffic networks. He is a Lecturer of the
following courses: computing, applied mathematics for engineers,
mathematical methods for PhD students.
Ferenc Szauter is an Assistant Professor, he obtained his MSc degree in
Mechanical Engineering from the Széchenyi István University (SZE), Hungary
in 2005, and a Technology Doctor-PhD in 2017. His research field covers the
followings: integrated analysis of processes concerning traffic and vehicle
dynamics. Analysis of the Complex Environmental Impact on Urban
Trajectories. Vibration analysis of a suspension system subject to high level of
measurement noise. AR and ARMA spectral analysis of suspension system of
city bus. Questions regarding vehicle safety and the mathematical analysis. An
in-depth analysis of cycling and pedestrian accidents. Two Operating States-
Based Low Energy Consumption Vehicle Control. Robust Reconfigurable
Control for In-wheel Electric Vehicles. Examinations of complex traffic
dynamic systems and new analysis, modelling and simulation of electric
vehicular systems. Maximising Battery Life and Usable Capacity with Battery
Management System in Electric Vehicles. Investigating the application of
aluminium as a winding material in high efficiency electric motor.
Development of laboratory applying real traffic scenarios.
Zoltán Rózsás is Education and R&D Coordinator at Automotive Proving
Ground Zala Ltd, obtained his BSc degree in Mechanical Engineering from
University of Miskolc, Hungary, Specialisation manufacturing automation in
2008. He has professional engineering management experience and strong
industrial background with 10+ years of experience that has been gained in
multinational environment, different technical related leadership roles.
Automotive Proving Ground Zala is the test, research and development venue
of autonomous vehicles. He is leading the proving ground related activities
with the education institution, additional study groups and optional lessons.
Supports the education and R&D related programs, events. He is leading the
mentoring program of 100+ project-related students, thesis, part time students,
summer practice, dual students. Takes part in establishing the research and
development vision of the company, preparation of funding. Provides support
for preparation of different kind of cross-border projects like Horizon 2020
proposal as a consortium partner and takes part in the specification of the ICT
and 5G related technical background of the proving ground.
István Lakatos is a Professor, he obtained his MSc in Mechanical Engineering
of Vehicles from the BME, Hungary in 1989, his Technology Doctor-PhD in
2003, and Habilitated Doctorate in 2013, respectively. His technical expert
activities are mobility and vehicle industry, energy and environmental
researches, development of hybrid-electric vehicles, analysis of truck accidents,

I
ntegrated application of network traffic and intelligent driver models 229
on-board diagnostics of environmental systems and development of a new
method for the analysis of diesel injection pumps. His expert activity is in
connection with higher education vocational training and standardisation of
official trainings as a Professional Leader. He is Head of Department of Road
and Rail Vehicles, SZE University, Hungary.
1 Introduction
The vehicle movement on the designated trajectory is executed in such a way that the
autopilot or driver can even commit limited arbitrary errors in trajectory tracking during
driving. When generating the speed of each vehicle, account must be taken of the
particular traffic conditions, i.e., the composition and movement of vehicles in the
environment of the network section (Csiszár and Földes, 2018; Koryagin, 2018; Lakatos
and Mándoki, 2017), as well as the complex transport environment typical (Farooq et al.,
2018; Ghadi et al., 2018; Iordanopoulos et al., 2018) of the time of day and seasonality
determined by processes of the large-scale transport network (Péter et al., 2015, 2016).
The model-based method subject to this study is therefore a theoretical foundation that
takes into account and examines the drivers’ (Szabó et al., 2004, 2009) characteristics
during the movement along the trajectory as well as the relationship between the
microscopic vehicle environment and the macroscopic traffic environment (Busznyák
and Lakatos, 2017; Lakatos et al., 2016).
2 Traffic related application of the intelligent driver model (IDM)
Adaptive cruise control (ACC) is a vehicle system that allows the vehicle to adjust its
speed to the environment. The IDM is an ACC model that is widely used in
transportation research to model longitudinal movement. Treiber, Hennecke and Helbing
developed the IDM, which is being used by the car company BMW, in 2000 at the
transport laboratory of Dresden Technical University.
The IDM model is used for modelling continuous traffic flows in simulations of
highway and city traffic (Kovács et al., 2016). As a vehicle tracking model, IDM
describes the dynamics of the position and speed of each vehicle. In the case of multi-
model open source traffic simulator (Treiber and Helbing, 2002) use IDM to simulate the
longitudinal movement of the vehicle and this simulator also introduces a lane change
strategy. Model-based single-lane traffic inhomogeneity is studied by Treiber et al.
(2004).
Treiber et al. (2006) study vehicle stability and IDM parameter sensitivity.
Kesting et al. (2008) propose to extend the driver parameters of the IDM model. They
study the impact of vehicles equipped with IDM on traffic flow and travel times as
bottlenecks. Jerath (2010) also uses the IDM model and examines the impact of ACC on
traffic flows. The results of the above work show that increasing the proportion of ACC
vehicles will result in increased traffic efficiency by reducing travel times. Treiber and
Kesting (2011) used IDM to study instability in congested traffic.

230 T. Péte
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et al.
The IDM has many advantages over other ACC models from calibration and intuitive
parameters point of view, and the also modelling requires simple simulation. However,
there are also disadvantages in respect of assuring the proper features of the vehicle and
the driver. The IDM is a collision-free model, therefore, in critical accident situations, the
desired minimum distance is no longer sufficient to guarantee driver safety and, in the
event of an emergency braking, it tends to overshoot the actual deceleration of the
vehicle.
Derbel et al. (2012, 2013) developed a proposal for a more accurate operation of the
IDM and studied possible modifications to IDM, taking into account the driver’s safety
and the real capabilities of the vehicle. As a result of this amendment, the driver has to
take into account the behaviour of the following vehicles, and thus a modified IDM has
been developed and tested with a microscopic simulator considering string stabilisation.
This modified IDM already highlights the proper vehicle capabilities.
In our present work we rely on the model joint-developed with French researchers to
overcome the disadvantages. Based on this, the IDM is already providing greater
performance in driver security by following real reactions in near-collision critical
situations. The paper shows the modification and the state-of-the-art operation of the
intelligent driver model in connection with the proper capabilities of the vehicle.
Modelling and research work encompasses a complex area and includes approaches of
both microscopic and macroscopic modelling (Derbel et al., 2017).
The complex macroscopic traffic environment is generated by the large-scale network
model, in which the microscopic traffic simulation model provides the individual vehicle
movement in traffic on the sections of the defined trajectories. However, this microscopic
model must properly reproduce dynamic traffic processes and must also be validated.
Accordingly, at this stage of our work we rely on the IDM research and development of
Treiber et al. (2000a, 2000b) and Derbel et al. (2017).
The features of the classical IDM are the following: a single system of differential
equations that analyses the case of n vehicles travelling on a single lane; the microscopic
model describes a chain model-like longitudinal dynamics; each driver looks only
forward and aims to keep an appropriate distance; there is no overtaking, the vehicles
keep their order and the first vehicle has a dominant role, as do the slow-moving vehicles
in the group.
The classical IDM is written with separate differential equations member by member.
In our study this is summarised in the following system of differential equations (1),
where the current position of the ith vehicle is described by function xi(t). The parameters
and functions used in the model are as follows:
ai is the maximal acceleration of the ith vehicle
vi is the desired speed of the ith vehicle
si is the required distance between the ith and the preceding vehicle (i = 1,2, ..., n).
1
1
12
() (()) (()) 1Axt V fxt Sfxt
−
−
++=
  (1)
The notation used in differential equation system (1) is the following
1
12
11 1
, ,...,
n
Aaa a
−
=;
1
44 4
12
11 1
, ,...,
n
Vvv v
−
=;12
, ,..., n
Ssss=

I
ntegrated application of network traffic and intelligent driver models 231
0.,
ii
s
s const== or 1
(,)
iiii
s
sx x
−
=
(i = 1,2,…,n).
4
1
4
2
1
4
(()) ...
n
x
x
fxt
x
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦




,
2
01
2
12
2
2
1
1
()
1
()
(())
...
1
()
nn
xx
xx
fxt
xx
−
⎡
⎤
⎢
⎥
−
⎢
⎥
⎢
⎥
⎢
⎥
−
=
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
−
⎣
⎦
,
1
1
1...
1
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
The modified IDM applied in this work is represented by formula (2). Detailed
description of the above is provided by Derbel et al. (2012, 2013). This model, using a
function h(t) also takes into account the fact that drivers monitor the movement of the
following vehicles as well, Figure 1.
1
12
() (()) (()) 1() ()
A
xt Vf xt Sf xt t ht
−
++ =+
  (2)
Figure 1 Setting of the relative distances at a stabilised speed state after a vehicle group
consisting of five elements is started (see online version for colours)
The notation used in differential equation system (2) is as follows:
1
12
11 1
, ,...,
n
Aaa a
−
=;
1
44
12
2
44
23
44
1
4
1
1
1
1
i
ii
n
h
vv
h
vv
Vh
vv
v
+
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
−−−−−
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
;

232 T. Péte
r
et al.
22
112
22
223
22
1
2
iii
n
shs
shs
Sshs
s
+
⎡⎤
⎢⎥
⎢⎥
⎢⎥
=⎢⎥
−− − − −
⎢⎥
⎢⎥
⎣⎦
1
2
()
()
() ...
()
n
ht
ht
ht
ht
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
⎢⎥
⎣⎦
; 1
() () i
ii
i
a
ht hft a
+
=⋅; (i = 1,2, …, n–1); () 0
n
ht=.
0.,
ii
s
s const== or 1
(,)
iiii
s
sx x
−
=
(i = 1,2,…,n).
4
1
4
2
1
4
(()) ...
n
x
x
fxt
x
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦




,
22
101
22
212
2
22
1
1
()
1
()
(())
...
1
()
nn n
xx
xx
fxt
xx
ε
ε
ε
−
⎡
⎤
⎢
⎥
+−
⎢
⎥
⎢
⎥
⎢
⎥
+−
=
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
+−
⎣
⎦
,
1( )
1( )
1( ) ...
1( )
t
t
t
t
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
3 Relationship between the IDM and the large-scale network
The speed of a given vehicle and the distance kept are determined by the driver. Their
decision, however, depends on their own perceptions, on signals that are transmitted by
the physical environment and received by the vehicle and on the local and general effects
of network traffic (Lakatos et al., 2016; Pokorádi, 2018).
Physical impacts resulting from road quality, meteorological and visibility conditions
at a given vehicle density determine a selectable speed range. The modified IDM
discussed in the previous section can be used to describe the dynamic traffic connections
originating from forward-moving vehicle-vehicle effects in a given section.
Figure 2 The n-element vehicle group and the environment determining their movement
(see online version for colours)

I
ntegrated application of network traffic and intelligent driver models 233
At the same time, the dynamics of the movement of the IDM group is not arbitrary.
It is determined by control speeds formed in the large-scale network or network sections.
The vehicles slow down if a congestion occurs, and stop when the traffic light switches to
red, but after the reaction delay time, they will accelerate to the maximum permitted
speed if the road section ahead is free. This is indicated in Figure 2 by the control speed
function x0(t) defined by the large-scale macroscopic network processes for each
trajectory.
4 Application of the network traffic model
For this research we apply the reduced network traffic model (Péter and Bokor, 2011,
2010a, 2010b; Péter, 2012; Dömötörfi et al., 2016) which contains internal network
elements of n sections located in a domain characterised by state vector x. The model
incorporates m external sections that are directly related to an internal section or sections.
The state vector s of the latter is considered known by measurement. In this model, out of
the matrices that form the link hyper-matrix, only matrices K11 and K12 play a role,
because they represent each transfer that applies to the internal sections.
1
11 12
[(,) (,)]
x
L K xsx K xss
−
=+
 (3)
where x ∈ ℜn, n
x∈ℜ
, s ∈ ℜm, L = diag{l1, ..., ln}, li is the length of the internal sectors
in the main diagonal (∀li > 0, i = 1,2, …, n), K11 ∈ ℜnxn, K12 ∈ ℜnxm.
Taking into account the delays in practice, which largely originate from the time lags
that can be derived from the reaction time (perception, decision, action: 0.6 ... 0.7 s) and
the time from the actuation to its effect (0.15–0.3 s), results in a mathematical model
describing reality more precisely. In this case, we assume that internal automatisms S(x)
and E(x) are functions continuously differentiable with x, while traffic control light
functions ui,j(t) are continuously differentiable with t. This can be accomplished in the
model without special restrictions.
Similarly, the traffic light signal ui,j(t) can be made continuously differentiable in its
domain if the above method is applied within every
ε
t-radius of each t0 breakpoint where
values change either from 1 to 0 or from 0 to 1. In this way we use a continuous dynamic
traffic model considering also the actual deceleration and reaction delay time
phenomenon in it. In case of S a deceleration phenomenon occurs, as drivers become
cautious when they realise that the section they want to drive over is already heavily
loaded. There is no delay effect in case of function E, however, since when a section
becomes empty (which takes place at the time the last vehicle leaves it in a given
interval) it determines a continuous vehicle density function in this section, thus there is
no contradiction to the above model paradigm applied to E. In the case of traffic lights,
the reaction delay phenomenon occurs in two ways.
On the one hand, the vehicles do not start immediately when the light changes to
green, and on the other hand, there could be irregular vehicle crossings at road
intersections when the light changes to yellow (or to red). These real-life phenomena can
be taken into consideration in the transfer processes with the application of a
continuously differentiable light signal.

234 T. Péte
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et al.
4.1 Validation of the model
In Budapest, the validation was performed on the Grand Boulevard, in the section
northwards from the Pest-side bridgehead of Petőfi Bridge to the Nyugati Square, at the
traffic light junctions on the basis of the actual traffic light program data provided to the
BME (Budapest University of Technology and Economics) by the Traffic Engineering
Directorate of FKF ZRt. (Metropolitan Public Domain Maintenance Ltd.) and the traffic
counts performed on-site. The test section was also surveyed at each simulation time with
vehicles equipped with a GPS device. The actual speed profiles were recorded during the
vehicle measurements. This Boulevard model is a typical line model where the typical
speed process is determined by the traffic light programs. We used the PannonTraffic
software for the validation. The simulation took into account the actual traffic light
programs, so it followed the timely formation of measured speed processes well. In the
validation, the best approximation of speed limits was achieved by the software by
adjusting the vehicle densities most appropriate to reality in each section of the
Boulevard. The applied speed-density law was the Greenshields (linear) function in all
cases. Comparison of the speed profiles obtained from the simulation and vehicle
measurements naturally showed that speed-time functions should be considered as single
realisations of a stochastic process and should be analysed accordingly with probability
theory and statistical analysis methods. The analysis was carried out in urban
environment as described above, in heavy traffic passing through and crossing several
traffic light junctions. A large number of non-parametric statistical probes for speed
profiles and engine performances were used for homogeneity testing. The subject of the
study was that if the sample set consisting of two independent probability variables (the
one measured by the GPS device and the other simulated by the traffic model) came from
a population of equal distributions, so in practice can the two samples be considered as
having the same distribution. In the analysis it was found that in both cases the two
samples could be considered homogeneous at 95%. The results of our study reassuringly
demonstrated that the model allows the extraction of custom speed processes and derived
engine performance processes that are true to reality (Peter et al., 2011).
In Győr the validated model included the Szent István Road (Highway no. 1) and its
surroundings, with one of the busiest traffic regions in the city centre. We validated the
model based on the cross-sectional traffic counts conducted by the City in 2012.
The main features of the network are 228 road sections, nine traffic light junctions
and 38 other junctions. The network domain had 18 input sections and 15 output sections.
The phase-plans of traffic lights were provided by the Győr Directorate of Hungarian
Public Road Non-profit Plc and the Municipality of Győr. In the model 63 cross-sectional
data were available for the validation. The simulation ran for a 24-h real-time period with
a computer run-time of 2 min and 14 s. The software can be restarted from any point in
time by taking into account the state parameters that are valid at that time as initial
values. During the validation the software reviewed the actual distributions and factors
influencing the transfers quarter-hourly to approximate the measured cross-sectional
traffic data as good as possible. During this progress correlation analysis was performed
hourly by taking into account the 63 measurement points. At the 63 examined cross-
sections the correlation coefficient of values between the measured and modelled hourly
cross-sectional traffic data was very close to 1, e.g., rxy = 0.993 was obtained for peak
hours between 7am and 8am, which in practice is 100% correlation (Péter and Fazekas,
2014).

I
ntegrated application of network traffic and intelligent driver models 235
4.2 Analysis of speed processes
A model assumption is that the speed value vi ≥ 0 can also be assigned to the state
parameter ∀xi, (xi ∈ [0,1], i = 1, 2, …, n) using function fi which is continuously
differentiable with xi:
(())
iii
vfxt= (4)
By getting the individual speed processes from the macroscopic network model and by
the application of a driver-vehicle model engine performance demands and emissions of
individual vehicles can be analysed as well (Csonka and Csiszár, 2016). Speed processes
are also suitable for model validation. In addition, this macroscopic model can be used to
integrate the control signal x0(t) and to produce speed processes generated by the
modified IDM along the designated trajectories.
4.3 Methods and instruments used in the measurement in Gyor
The advantage of the instruments with a GPS receiver is that it helps the connection to
the database system and the automatic processing of data in a manner consistent with the
additional tasks (Figures 3 and 4). Thus, not only the speed values but also the GPS
location and time coordinates are recorded during the measurement. (Data of at least
three working satellites is required for the measurements) (Szauter et al., 2014; Istenes
et al., 2017). The data files of the GPS measurements were stored according to the GPS
coordinates. The above method assigned the results of the speed measurements in the
domain to the corresponding road sections, so we obtained a unique speed function for all
the road sections involved in the measurement, Figures 5 and 6.
Figure 3 Vehicle measurement system equipped with GPS receiver (see online version
for colours)

236 T. Péte
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Figure 4 Routes travelled with GPS measurements in Győr (see online version for colours)
Figure 5 Speed values measured on route 004 as a function of distance (see online version
for colours)

I
ntegrated application of network traffic and intelligent driver models 237
Figure 6 Speed values measured on route 004 as a function of time (see online version
for colours)
5 Trajectory tracking
The algorithm followed by the driver or the autopilot is modelled in a fixed Descartes
coordinate system as follows: The driver or the autopilot follows trajectory G. At starting
time t0 they are on trajectory G, so at that time P0 = G0. Henceforth, trajectory tracking
decisions are generated at times t0,t1,t2, ..., tn. At time ti they want to reach the selected
point Gi+1 on the trajectory from their position Pi with the chosen angle αi and speed Vi.
However, at time ti+1 Pi+1 is reached due to errors εα and εs made in choosing the angle
and the speed, respectively. This is how the geometric stability continues and the
trajectory G itself provides.
5.1 The geometric stability of the trajectory tracking
Taking into account simple equidistant intervals, the motion process can be written with
the following assignments, Figure 7.
Figure 7 The algorithm followed by the autopilot

238 T. Péte
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et al.
Let Pi denote the points affected by the driver or the autopilot. Gi+1 the targeted trajectory
points, ti+1 discrete times, tg(αi) direction tangents and si the distance between Pi and Gi+1.
[, ]
iii
Pxy=→111
[( ),( )]
iii
GXtYt
+++
=; 1(1)
i
tit
+=+Δ; ()
ii
tg m
α
=; 1
ii
i
s
PG +
=;
(i = 0, 1, 2,…)
Taking into account the direction vector, the section lengths and speed are:
[1, ]
ii
rm=, → 0
22
1,
11
i
i
ii
m
rmm
⎡⎤
⎢⎥
=⎢⎥
++
⎣⎦
(5)
i
(( 1) )
m(( 1) )
i
i
Yi t y
X
itx
⎛⎞
+Δ−
=
⎜⎟
+Δ−
⎝⎠
22
(((1) ) (((1) )
iii
s
Xi t x Yi t y= + Δ− + + Δ− (6)
i
i
s
vt
⎛⎞
=
⎜⎟
Δ
⎝⎠
Taking into account the above the discrete points of trajectory tracking can be easily
described with the following equation (7) recursive formula:
12
12
1
1
i
ii
i
ii
ii
i
s
xx m
s
m
yy m
+
+
=+
+
⋅
=+
+
(7)
The mechanism of error-free trajectory tracking is shown in Figure 8.
Figure 8 X(t) and y(t) in the case of error-free trajectory tracking
h denotes the driver or the autopilot and V denotes the vehicle.
In the event that trajectory tracking is subject to errors, the following algorithm
describes and Figure 9 illustrates the mechanism of the process:
()
real
iisis
real
iivis
s s Error s
v v Error v
ε
ε
⎯⎯⎯→+ =⋅
⎯
⎯⎯→+ =⋅ (8)
real
ii i
mmErrorm
αα
ε
⎯
⎯⎯→+ =⋅ (9)
0.7 1.3
s
ε
≤≤ 0.7 1.3
α
ε
≤≤

I
ntegrated application of network traffic and intelligent driver models 239
In case of small angles:
tg
αα
≅ m
α
≅
Discrete points of trajectory tracking with error:
122
122
:1
:1
is
ii
i
iis
ii
i
s
xx m
ms
yy m
α
α
α
ε
ε
εε
ε
+
+
=+
+
=+
+
(10)
where
ε
s is the distance (or speed) estimation error,
ε
α is the angle estimation error.
These are errors that are re-generated during every step in interval [ti, ti+1]. Their
distribution largely depends on driver skills and visibility.
Figure 9 X(t) and y(t) in case of trajectory tracking with error
The geometric stability of the trajectory tracking method is well underpinned
by the preliminary simulation tests, as illustrated in Figure 10(a) when the angle and
distance estimations are accurate, but stability is provided even if the direction
and distance estimation is subject to different rates (5–30%) of random errors,
see Figure 10(a)–(d).
Further studying the trajectory tracking at discrete points, we move towards
the continuous time tracking. In this case, starting from an initial value that is not
necessarily a trajectory point, continuous error functions εα = εα (t) and εs = εs (t) are
taken into account in the construction of the differential equations describing trajectory
tracking.
5.2 The differential equation system for trajectory tracking
When applying formula (10), in order to move to a continuous case, we use the following
relations and notation:
122
122
:1
:1
is
ii
i
iis
ii
i
s
xx m
ms
yy m
α
α
α
ε
ε
εε
ε
+
+
=+
+
=+
+
In order to move to a continuous case, we use the following relations and notation:
0tΔ→

240 T. Péte
r
et al.
1ii
x
xx
+
Δ= − ; 1ii
yy y
+
Δ= − ; 22
i
s
sxy=Δ = Δ +Δ ; i
y
m
x
Δ
=Δ
22
2
2
1
s
xy
x
y
x
α
ε
ε
Δ+Δ⋅
Δ=
Δ
⎛⎞
+⎜⎟
Δ
⎝⎠
22
2
2
1
s
yxy
x
y
y
x
α
α
εε
ε
ΔΔ+Δ⋅⋅
Δ
Δ=
Δ
⎛⎞
+⎜⎟
Δ
⎝⎠
(11)
At equation (11) we use the difference quotient in discreet time
22
2
2
1
s
xy
tt
x
ty
t
x
t
α
ε
ε
⎛Δ⎞Δ
⎛⎞ ⎛⎞
+⋅
⎜⎟ ⎜⎟
⎜⎟
ΔΔ
Δ⎝⎠ ⎝⎠
⎝⎠
=
Δ⎛Δ⎞
⎛⎞
⎜⎟
⎜⎟
Δ
⎝⎠
⎜⎟
+Δ
⎛⎞
⎜⎟
⎜⎟
⎜⎟
Δ
⎝⎠
⎝⎠
22
2
2
1
s
y
xy
t
xtt
t
y
ty
t
x
t
α
α
εε
ε
⎛Δ⎞
⎛⎞
⎜⎟
⎜⎟
⎛Δ⎞Δ
Δ⎛⎞ ⎛⎞
⎝⎠
⎜⎟ +⋅
⎜⎟ ⎜⎟
⎜⎟
ΔΔΔ
⎛⎞
⎜⎟
⎝⎠ ⎝⎠
⎝⎠
⎜⎟
⎜⎟
Δ
Δ⎝⎠
⎝⎠
=
Δ⎛Δ⎞
⎛⎞
⎜⎟
⎜⎟
Δ
⎝⎠
⎜⎟
+Δ
⎛⎞
⎜⎟
⎜⎟
⎜⎟
Δ
⎝⎠
⎝⎠
(12)
By generating boundary transition and assuming that time derivatives of track trajectories
exist:
If 0
lim ( ), ( )txtytΔ⎯⎯→∃
Formulas (13) and (14) are the following:
22
222
22
222
()
()
GG Gs
GG
GG G s
GG
xx y
xt xy
yx y
yt xy
α
α
α
ε
ε
εε
ε
+⋅
=
+
+⋅
=
+
 


 


(13)

I
ntegrated application of network traffic and intelligent driver models 241
Since equation (13) can be converted to the following equation (14) form as well
()
()
G
G
y
yt
xt x
α
ε
=


22 2 2
GGs
xy x y
ε
+= +
   (14)
If, for any arbitrary initial value and with zero error (
ε
s:=1
ε
α
:=1), we apply the result
obtained, the trajectory curve provides an orthogonal and directional mapping, i.e., the
trajectory mapping performs a congruent transformation, in Figure 11 a mapping of two
adjacent circular routes can be seen for an arbitrary initial value: [{A∪ B }→ {A*∪B*}].
Figure 10 (a) During a 12 s period a bend is correctly tracked by the error-free algorithm at
discrete times; (b) during a 12 s period a bend is well tracked by the algorithm with an
error of 5% at discrete times; (c) during a 12 s period the algorithm is stable with
extreme errors of 20% and (d) during a 12 s period the algorithm is stable with extreme
errors of 30% (see online version for colours)

242 T. Péte
r
et al.
Figure 11 A mapping of two adjacent circular routes can be seen for an arbitrary initial value
(see online version for colours)
A
B
A
*
B
*
A
B
A*
B*
A
B
A*
B*
6 Conclusions
Based on the system of differential equations derived from the continuous trajectory
tracking it can be established that the physical considerations used for trajectory tracking
can be utilised well during laboratory tests. In that case a suitable GPS-based set of points
must be provided at the selected trajectory points to describe the geometry. The point to
be tracked runs on this trajectory, taken into account the macroscopic environment
influenced by complex traffic conditions and the microscopic environment according to
the joint dynamics of vehicles, during the simulation.
This process provides the steering angle in the given conditions, which can be
calculated from the simulation angle, the vehicle and steering geometry. In parallel with
this, the speed of movement on the trajectory is directly provided by the simulator in a
manner that is adequate for the real disordered traffic conditions. Further measurements
and validations will of course be required in both urban traffic conditions and on the
Zalaegerszeg test track (Szalay, 2016; Asaithambi et al., 2017; Pagliara et al., 2017;
Szalay et al., 2017, 2018; Török et al., 2017; Mihály et al., 2018; Takács et al., 2018).
Acknowledgement
The research presented in this paper was carried out as part of the EFOP-3.6.2-16-2017-
00002 project in the framework of the New Széchenyi Plan. The completion of this
project is funded by the European Union and co-financed by the European Social Fund.
References
Asaithambi, G., Sekhar Mourie, H. and Sivanandan, R. (2017) ‘Passenger car unit estimation
at signalized intersection for non-lane based mixed traffic using microscopic simulation
model’, Periodica Polytechnica Transportation Engineering, Vol. 45, No. 1, pp.12–20,
doi: https://doi.org/10.3311/PPtr.8986

I
ntegrated application of network traffic and intelligent driver models 243
Busznyák, T. and Lakatos, I. (2017) ‘Automotive engineering possibilities in combining global
positioning and vehicle diagnostics’, 5th International Scientific Conference on Advances in
Mechanical Engineering (ISCAME 2017, 12–13 October, Debrecen, Hungary, pp.1–6.
Csiszár, Cs. and Földes, D. (2018) ‘System model for autonomous road freight transportation’,
Promet-Traffic and Transportation, Vol. 30, No. 1, February, pp.93–103, https://doi.org/
10.7307/ptt.v30i1.2566
Csonka, B. and Csiszár, Cs (2016) ‘Service quality analysis and assessment method for European
carsharing systems’, Periodica Polytechnica Transportation Engineering, [S.l.], Vol. 44,
No. 2, pp.80–88, ISSN 1587-3811, https://pp.bme.hu/tr/article/view/8559, doi: https://doi.org/
10.3311/PPtr.8559
Derbel, O., Péter, T., Mourllion, B. and Basset, M. (2017) ‘Generalized velocity–density model
based on microscopic traffic simulation’, Transport, doi: 10.3846/16484142.2017.1292950 To
link to this article: http://dx.doi.org/10.3846/16484142.2017.1292950, ISSN: 1648-4142
(Print) 1648-3480 (Online) Journal homepage: http://www.tandfonline.com/loi/tran20
Derbel, O., Peter, T., Zebiri, H., Mourllion, B. and Basset, M. (2012) ‘Modified intelligent driver
model’, Periodica Polytechnica Transportation Engineering, Vol. 40, No. 2, pp.53–60,
https://doi.org/10.3311/pp.tr.2012-2.02
Derbel, O., Peter, T., Zebiri, H., Mourllion, B. and Basset, M. (2013) ‘Modified intelligent driver
model for driver safety and traffic stability improvement’, IFAC Proceedings Volumes,
Vol. 46, No. 21, pp.744–749, https://doi.org/10.3182/20130904-4-JP-2042.00132
Dömötörfi, Á., Péter, T. and Szabó, K. (2016) ‘Mathematical modeling of automotive supply chain
networks’, Period. Polytech. Transp. Eng., Vol. 44, No. 3, pp.181–186, doi: 10.3311/
PPtr.9544.
Farooq, A., Xie, M., Williams, E., Gahlot, V., Yan, D. and Yi, Z. (2018) ‘Downsizing strategy for
cars, Beijing for people not for cars: planning for people’, Periodica Polytechnica
Transportation Engineering, Vol. 46, No. 1, pp.50–57, doi: https://doi.org/10.3311/PPtr.10851
Ghadi, M., Török, Á. and Tánczos, K. (2018) ‘Study of the economic cost of road accidents in
Jordan’, Periodica Polytechnica Transportation Engineering, Vol. 46, No. 3, pp.129–134, doi:
https://doi.org/10.3311/PPtr.10392
Iordanopoulos, P., Mitsakis, E. and Chalkiadakis, C. (2018) ‘Prerequisites for further deploying
ITS systems: the case of Greece’, Periodica Polytechnica Transportation Engineering,
Vol. 46, No. 2, pp.108–115, doi: https://doi.org/10.3311/PPtr.11174
Istenes, G., Szauter, F. and Rödönyi, G. (2017) ‘Vibration analysis of a suspension system subject
to high level of measurement noise’, 2017 4th International Conference on Control, Decision
and Information Technologies (CoDIT), 5–7 April, Barcelona, Spain, doi: 10.1109/
CoDIT.2017.8102707.
Jerath, K. (2010) Impact of Adaptive Cruise Control on the Formation of Self-Organized Traffic
Jams on Highway, Master’s Thesis, The Pennsylvania State University The Graduate school.
Department of Mechanical and Nuclear Engineering.
Kesting, A., Treiber, M. and Helbing, D. (2008) ‘Agents for traffic simulation’, Physics and
Society, Vol. 11, pp.325–356.
Koryagin, M. (2018) ‘Urban planning: a game theory application for the travel demand
management’, Periodica Polytechnica Transportation Engineering, Vol. 46, No. 4,
pp.171–178, doi: https://doi.org/10.3311/PPtr.9410
Kovács, T., Bolla, K., Gil, R.A., Csizmás, E., Fábián, Cs., Kovács, L., Medgyes, K., Osztényi, J.
and Végh, A. (2016) ‘Parameters of the intelligent driver model in signalized intersections’,
Technical Gazette, Vol. 23, No. 5, October, pp.1469–1474, ISSN 1330-3651 (Print), ISSN
1848-6339 (Online) doi: 10.17559/TV-20140702174255.
Lakatos, I. and Őri, P. (2017) ‘Diagnostic measurement and data evaluation for the effective
performance of motor vehicles with free acceleration’, IEEE 15th International Symposium on
Intelligent Systems and Informatics: SISY 2017, 14-16 September, Szabadka, Szerbia,
New York, IEEE, pp.217–222 (ISBN: 978-1-5386-3855-2).

244 T. Péte
r
et al.
Lakatos, A. and Mándoki, P. (2017) ‘Quality evaluation of the long-distance bus and train
transportation in Hungary’, Transportation Research Procedia, Vol. 27, pp.365–372,
ISSN: 2352-1465.
Lakatos, I., Peter, T., Oberling, J., Pup, D. and Szauter, F. (2016) ‘An in-depth analysis of cycling
and pedestrian accidents in Hungary’, ASME/IEEE International Conference on Mechatronic
and Embedded Systems and Applications (MESA), 29–31 August, Auckland New Zealand
pp.1–6.
Mihály, A., Németh, B. and Gáspár, P. (2018) ‘Real-time look-ahead cruise control simulator’,
Periodica Polytechnica Transportation Engineering, Vol. 46, No. 1, pp.11–16, doi:
https://doi.org/10.3311/PPtr.9896
Pagliara, F., Esposito, M. and Troisi, C. (2017) ‘Planning and designing transport infrastructures
for sustainability: the great project NAPLEST’, Periodica Polytechnica Transportation
Engineering, [S.l.], Vol. 45, No. 2, pp.53–57, ISSN 1587-3811, https://pp.bme.hu/tr/article/
view/9812, doi: https://doi.org/10.3311/PPtr.9812
Péter, T. and Bokor, J. (2011) ‘New road traffic networks models for control’, GSTF International
Journal on Computing, Vol. 1, No. 2, February, pp.227–232, doi: 10.5176_2010-2283_1.2.65.
Péter, T. and Bokor, J. (2010a) ‘Research for the modelling and control of traffic’, Scientific
Society for Mechanical Engineering, 33rd Fisita-World Automotive Congress: Proceedings,
Budapest, Magyarország, 30 May–4 June, Budapest: GTE. 2010, pp.66–73 (ISBN: 978-963-
9058-28-6).
Péter, T. and Bokor, J. (2010b) ‘Modeling road traffic networks for control’, Annual International
Conference on Network Technologies and Communications: NTC. (2010) 30 November,
Thaiföld, pp.18–22, Paper 21 (ISBN: 978-981-08-7654-8).
Péter, T. and Fazekas, S. (2014) ‘Determination of vehicle density of inputs and outputs and model
validation for the analysis of network traffic processes’, Periodica Polytechnica
Transportation Engineering, Vol. 42, No. 1, pp.53–61 (Budapest University of Technology
and Economics).
Peter, T., Fülep, T. and Bede, Z. (2011) ‘The application of a new principled optimal control for the
dynamic change of the road network graph structure and the analysis of risk factors’, 13th
EAEC European Automotive Congress, 13–16 June, Valencia – SPAIN Society of Automotive
Engineers (STA) (2011), pp.26–36 (ISBN: 978-84-615-1794-7).
Péter, T. (2012) ‘Modeling nonlinear road traffic networks for junction control’, International
Journal of Applied Mathematics and Computer Science (AMCS) (2012), Vol. 22, No. 3,
pp.723–732, doi: 10.2478/v1006-012-0054-1.
Péter, T. and Lakatos, I. (2017) ‘Hybrid model of vehicle and traffic for combined dynamic
analysis’, Int. J. Heavy Vehicle Systems, Vol. 24, No. 2, pp.1–14, ISSN: 1744-232X.
Péter, T., Lakatos, I. and Szauter, F. (2015) ‘Analysis of the complex environmental impact on
urban trajectories’, ASME 2015 International Design Engineering Technical Conferences and
Computers and Information in Engineering Conference: Mechatronics for Electrical
Vehicular Systems, 2–5 August, Boston, Massachusetts, USA, Paper issue DETC2015-47077,
pp.V009T07A071; p.7, doi: 10.1115/DETC2015-47077, ISBN: 978-0-7918-5719-9.
Peter, T., Lakatos, I., Szauter, F. and Pup, D. (2016) ‘Complex analysis of vehicle and environment
dynamics’, 12th IEEE/ASME International Conference on Mechatronic and Embedded
Systems and Applications (MESA), 29–31 August, Auckland New Zealand, pp.1–7, Paper 34
(ISBN: 978-1-5090-6190-7).
Pokorádi, L. (2018) ‘Graph model-based analysis of technical systems’, IOP Conf, Series:
Materials Science and Engineering, Vol. 393, No. 012007, pp.1–9, doi: 10.1088/1757-
899X/393/1/012007.
Szabó, K., Szabó, G. and Renner, P. (2009) ‘Emberi hibamodellezés alkalmazása a légiközlekedési
kockázatelemzésekben’, Közlekedéstudományi Szemle, Vol. 59, No. 5, pp.29–35.
Szabó, G., Szabó, K. and Zerényi, R. (2004) ‘Safety management systems in transportation:
aims and solutions’, Periodica Polytechnica Transportation Engineering, Vol. 32, No. 1,
pp.123–134.

I
ntegrated application of network traffic and intelligent driver models 245
Szalay Zs (2016) ‘Structure and architecture problems of autonomous road vehicle testing and
validation’, Proceedings of the 15th Mini Conference on Vehicle System Dynamics,
Identification and Anomalies – VSDIA. (2016), pp.229–236, ISBN: 978-963-313-266-1.
Szalay, Z., Nyerges, A., Hamar, Z. and Hesz, M. (2017) ‘Technical specification methodology for
an automotive proving ground dedicated to connected and automated vehicles’, Periodica
Polytechnica Transportation Engineering, Vol. 45, No. 3, pp.168–174, doi: https://doi.org/
10.3311/PPtr.10708.
Szalay, Z., Tettamanti, T., Esztergár-Kiss, D., Varga, I. and Bartolini, C. (2018) ‘Development of a
test track for driverless cars: vehicle design, track configuration, and liability considerations’,
Periodica Polytechnica Transportation Engineering, Vol. 46, No. 1, pp.29–35, doi:
https://doi.org/10.3311/PPtr.10753
Szauter, F., Péter, T. and Lakatos, I. (2014) ‘Examinations of complex traffic dynamic systems and
new analysis, modeling and simulation of electrical vehicular systems’, 10th IEEE/ASME
International Conference on Mechatronic and Embedded Systems and Applications, 10–12
September, Senigallia, INSPEC Accession Number: 14701078, pp.1–5, doi: 10.1109/Mesa.
2014.6935613, Print ISBN: 978-1-4799-2772-2.
Takács, Á., Drexler, D.A., Galambos, P., Rudas, I.J. and Haidegger, T. (2018) ‘Assessment and
standardization of autonomous vehicles’, 2018 IEEE 22nd International Conference on
Intelligent Engineering Systems (INES), 21–23 June., Las Palmas de Gran Canaria, Spain,
pp.185–192., ISSN: 1543-9259, doi: 10.1109/INES.2018.8523899.
Török, Á., Fedorko, G., Molnar, V., Husaková, N. and Csiszár, C. (2017) ‘How to choose and
when to start best ITS projects that enhance logistic performance?’, Periodica Polytechnica
Transportation Engineering, Vol. 45, No. 1, pp.8–11, doi: https://doi.org/10.3311/PPtr.9222
Treiber, M. and Helbing, D. (2002) ‘Realistische mikrosimulation von straenverkehr mit einem
einfachen modell’, Symposium ‘Simulationstechnik’, ASIM. 2002.
Treiber, M., Hennecke, A. and Helbing, D. (2000a) ‘Congested traffic states in empirical
observations and microscopic simulations’, Physical Review E, Vol. 62, No. 2, pp.1805–1824,
https://doi.org/10.1103/PhysRevE.62.1805
Treiber, M., Hennecke, A. and Helbing, D. (2000b) ‘Microscopic simulation of congested traffic’,
in Helbing, D., Herrmann, H.J., Schreckenberg, M. and Wolf, D.E. (Eds.): Traffic and
Granular Flow’99: Social, Traffic, and Granular Dynamics, pp.365–376. https://doi.org/
10.1007/978-3-642-59751-0_36
Treiber, M., Hennecke, A. and Helbing, D. (2004) ‘Microscopic simulation of congested traffic’,
Physical Review, Vol. 62, pp.1805–1824.
Treiber, M., Hennecke, A. and Helbing, D. (2006) ‘Delays, inaccuracies and anticipation in
microscopic traffic models’, Physica A, Vol. 360, pp.71–88.
Treiber, M. and Kesting, A. (2011) ‘Evidence of convective instability in congested traffic flow: a
systematic empirical and theoretical investigation’, Procedia Social and Behavioral Sciences,
Vol. 17, pp.698–716.