# A Methodology of Cooperative Driving based on Microscopic Traffic Prediction

Boris S. Kerner1, Sergey L. Klenov2, Vincent Wiering1, and Michael
Schreckenberg1 1 Physics of Transport and Traffic, University of Duisburg-
Essen, 47048 Duisburg, Germany  2 Moscow Institute of Physics and Technology,
Department of Physics, 141700 Dolgoprudny, Moscow Region, Russia

###### Abstract

We present a methodology of cooperative driving in vehicular traffic, in which
for short-time traffic prediction rather than one of the statistical
approaches of artificial intelligence (AI), we follow a qualitative different
microscopic traffic prediction approach developed recently [Phys. Rev. E 106
(2022) 044307]. In the microscopic traffic prediction approach used for the
planning of the subject vehicle trajectory, no learning algorithms of AI are
applied; instead, microscopic traffic modeling based on the physics of vehicle
motion is used. The presented methodology of cooperative driving is devoted to
application cases in which microscopic traffic prediction without cooperative
driving cannot lead to a successful vehicle control and trajectory planning.
For the understanding of the physical features of the methodology of
cooperative driving, a traffic city scenario has been numerically studied, in
which a subject vehicle, which requires cooperative driving, is an automated
vehicle. Based on microscopic traffic prediction, in the methodology first a
cooperating vehicle(s) is found; then, motion requirements for the cooperating
vehicle(s) and characteristics of automated vehicle control are predicted and
used for vehicle motion; to update predicted characteristics of vehicle
motion, calculations of the predictions of motion requirements for the
cooperating vehicle and automated vehicle control are repeated for each next
time instant at which new measured data for current microscopic traffic
situation are available. With the use of microscopic traffic simulations, the
evaluation of the applicability of this methodology is illustrated for a
simple case of unsignalized city intersection, when the automated vehicle
wants to turn right from a secondary road onto the priority road.

###### pacs:

89.40.-a, 47.54.-r, 64.60.Cn, 05.65.+b

##  I Introduction

###  I.1 Brief Overview of Approaches to Cooperative Driving

Cooperative driving is currently one of the most developing fields in physics
of vehicular traffic. For the realization of cooperative driving, a vehicular
networking is usually needed, i.e., a vehicular network in which vehicles can
communicate to each other through vehicle-to-vehicle (V2V) communication
or/and through vehicle-to-infrastructure communication; both types of
vehicular communications are called V2X-communication (see, e.g., Schmitz2004
; Maurer2006 ; Chen2006 ; Wang2020 ; Chen2015 ). Through V2X-communication
vehicles can have additional information about the behaviors of other
vehicles.

In particular, it is assumed that cooperative driving should support the
motion of automated-driving vehicles (automated vehicles for short) (see,
e.g., Ioannou1997 ; Ioannou1996 ; Ioannou1993 ; Meyer2014 ; Liang2000 ;
Davis2014 ; Xin_Chang2020A ; Jiang-Wang2021A ; Coop_AV_Papageorgiou-
Mountakis_2021A ; Yao-Gu2022A ; Jiang-Sun2023A ; Ma-Wang2021A ; Luo-Gu2022A ;
Ma-Qu2023A ; Monteiro-Ioannou2023A ; Weijie-Yu2023A ; Li-Xiao2023A ; Gu-
Wang2023A ; Coop_AV_P-Hang_2023A ; Lane-Changing_AV_2024A ; Coop_AV_J-
Shi_2023A ; Coop_Cont_AV_2024A ; Coop_AV_K-Hou_2023A ;
Coop_Cont_AV_Huang_2024B ; Coop_AV_Ying-Shang_2024A ; AV_KKSW_Liu_2023A ;
Coop_AV_Jiahua-Qiu_2023A ; Mixed_Maiti_2023A ; Coop_AV_L-An_2023A ;
Coop_AV_Zhang_2023A ; Mixed_Wang_2023A ; Mixed_Lihua-Luo_2024A ; Mixed_Lin-
Hou_2023A ; Mixed_Ziyu-Cui_2023A ; Coop_AV_Futao-Zhang_2023A ; Coop_AV_Shi-
Teng-Zheng_2023A ; Coop_AV_Leyi-Duan_2023A ; Coop_AV_Yangsheng-Jiang_2023A ;
Coop_AV_Ruijie-Li_2023A ; Coop_AV_C-Liu_2023A ; Coop_AV_Dian-Jing_2023A ;
Coop_AV_X-Huang_2023A ; Lane-Changing_AV_Xia-Li_2023A ; Coop_AV_Jiawei-
Lu_2023A ; Coop_AV_Yu-Du_2023A ; Coop_AV_Yunjie-Liu_2023A ; Coop_AV_I-
Gokasar_2023A ; Coop_AV_Jiawei-Wang_2023A ; Coop_AV_Jie-Wang_2023A ;
Coop_AV_Z-Huang_2023A ; Coop_AV_Mengting-Guo_2023A ; Coop_AV_Qishen-Zhou_2023A
; Coop_AV_Yan-Tao-Zhang_2023A ; Coop_AV_W-Yue_2023A ; Coop_AV_Chenguang-
Zhao_2023A ; Coop_AV_Minghao-Fu_2023A ; Coop_AV_Cong-Zhai_2023A ;
Coop_AV_Yanyan-Qin_2023A ; Lane-Ch_X-Duan-C_2023A ; Z-Huang-H_2023A ;
Coop_AV_S-S-Mousavi_2023A ; Coop_AV_Weijie-Yu_2023A ; Coop_AV_J-Wen-S_2023A ;
Coop_AV_Matin-Dia_2023A ; Coop_AV_Milad-Malekzadeh_2024A ; Coop_AV_Ying-
Feng_2024A ). The information obtained through V2X-communication can also be
used for a short-time prediction of a traffic situation, for example, based on
a so-called partially observable Markov decision process (see, e.g.,
Brechtel2014 ; Lin2019 ; Schorner2019 ; Klimenko2014 ; Hubmann2019 ;
A-Lombard-A_2023A ; J-Luo-T_2023A ; Markov_Shi_2024A ), learning algorithms
(see, e.g., Isele2018 ; Qiao2018 ; Sama2020 ; D-Chen_2023A ; X-Jiang-J_2023A ;
Learn_Pred_AV_2024A ; Zhou-Z-Cao_2023A ; Zhang-S-Li_2023A ; Shupei-Wang_2024A
; Renteng-Yuan_2023A ; Du-Y-Zou_2023A ; K-Guo-M-Wu_2023A ; Chunyu-Liu_2023A ;
J-Guo_2023A ; Bharti-Redhu_2023A ; Wensong-Zhang_2023A ; Shun-Wang_2023A ;
Siyuan-Feng_2023A ; Li-Z-Huang_2023A ; Jiaxin-Liu_2023A ; Zelin-Wang_2024A ;
C-J-Hoel_2023A ; S-Fang-C_2023A ) as well as many other artificial
intelligence (AI) and model approaches (see, e.g., Orzechowski2018 ;
Althoff2016 ; Naumann2019 ; Tas2018 ; Akagi2015 ; Morales2017 ; Yoshihara2017
; Takeuchi2015 ; Yu2019 ; Hoermann2017 ; Guojing-Hu_2023A ; Changxi-Ma_2023A ;
S-Akhtar_2023A ; Xiaoxue-Yang_2023A ; Gao-X-Li_2023A ; Z-Gu_2023A ;
Y-Zhang_2023A ; Xiaoyong-Sun_2023A ; Weibin-Zhang_2023A ; H-Chen-Y-Liu_2023A ;
W-Shao_2023A ; T-Qie-W_2023A ). The short-time prediction of vehicle variables
(vehicle locations and speeds) is required for the vehicle trajectory
planning.

Thus, the short-time prediction of vehicle variables (vehicle locations and
speeds), which is required for the vehicle trajectory planning, is usually
made based on a diverse variety of statistical approaches of AI. Some of these
statistical approaches have been mentioned above Brechtel2014 ; Lin2019 ;
Schorner2019 ; Klimenko2014 ; Hubmann2019 ; A-Lombard-A_2023A ; J-Luo-T_2023A
; Markov_Shi_2024A ; Isele2018 ; Qiao2018 ; Sama2020 ; D-Chen_2023A ; X-Jiang-
J_2023A ; Learn_Pred_AV_2024A ; Zhou-Z-Cao_2023A ; Zhang-S-Li_2023A ; Shupei-
Wang_2024A ; Renteng-Yuan_2023A ; Du-Y-Zou_2023A ; K-Guo-M-Wu_2023A ; Chunyu-
Liu_2023A ; J-Guo_2023A ; Bharti-Redhu_2023A ; Wensong-Zhang_2023A ; Shun-
Wang_2023A ; Siyuan-Feng_2023A ; Li-Z-Huang_2023A ; Jiaxin-Liu_2023A ; Zelin-
Wang_2024A ; C-J-Hoel_2023A ; S-Fang-C_2023A ; Orzechowski2018 ; Althoff2016 ;
Naumann2019 ; Tas2018 ; Akagi2015 ; Morales2017 ; Yoshihara2017 ; Takeuchi2015
; Yu2019 ; Hoermann2017 ; Guojing-Hu_2023A ; Changxi-Ma_2023A ; S-Akhtar_2023A
; Xiaoxue-Yang_2023A ; Gao-X-Li_2023A ; Z-Gu_2023A ; Y-Zhang_2023A ; Xiaoyong-
Sun_2023A ; Weibin-Zhang_2023A ; H-Chen-Y-Liu_2023A ; W-Shao_2023A ; T-Qie-
W_2023A .

For reasons that will be explained in Sec. VI.2, rather than one of the
statistical approaches of AI, in this paper we follow a qualitative different
approach, which is based on the physics of vehicle motion in vehicular
traffic. In this approach introduced in KKl2022_Pred , no statistical analysis
of a historical traffic database is used for traffic prediction. Instead of
statistical analysis of empirical traffic data, microscopic traffic modeling
with a traffic flow model that can reproduce spatiotemporal features of real
traffic is applied. Based on this microscopic traffic modeling, the
microscopic short-time prediction of vehicle variables (vehicle locations and
speeds) is made. This prediction is used for the planning of the vehicle
trajectory. Because the microscopic traffic prediction of KKl2022_Pred  is the
scientific basis of the methodology of cooperative driving presented in this
paper, for the paper understanding we present a brief summary of this
microscopic traffic prediction approach below.

###  I.2 Brief Summary of Microscopic Traffic Prediction

General features of the microscopic traffic prediction approach of
KKl2022_Pred  are as follows:

  1. (i)

We assume that through vehicular networking (in particular, V2X-communication)
at some time instants

$t=t_{p},\ p=1,2,3,\ldots$ (1)

locations (including the correspondence to road lanes) and speeds of all
vehicles moving around of a subject vehicle are known. We call the multitude
of the vehicle locations $x$ and vehicle speeds $v$ as a microscopic traffic
situation at time instant $t_{p}$. In general, a time interval $\delta
t=t_{p+1}-\ t_{p},\ p=1,2,3,\ldots$ that can be very short (e.g., between 0.1
s and 1 s) should not be necessarily a constant value; this time interval is
determined by the measurement technology of the microscopic traffic situation.
However, for simulations made in KKl2022_Pred  as well as in this paper we
apply a constant time interval

$\delta t=t_{p+1}-t_{p}=\tau,\ p=1,2,3,\ldots$ (2)

where $\tau$ is a time-independent time step of the model used for microscopic
simulations (in all simulations we use $\tau=$ 1 s).

  2. (ii)

A microscopic traffic situation measured at time instant $t=t_{p}$, i.e.,
values for locations and speeds of vehicles related to $t=t_{p}$ are set into
the prediction model. These locations and speeds of vehicles are used as an
initial condition for a microscopic traffic flow model of vehicular traffic.
For this time instant $t=t_{p}$, the microscopic traffic flow model calculates
future (predicted) microscopic traffic situation that the model predicts for
prediction horizon $\Delta T_{p}$.

  3. (iii)

Based on this microscopic prediction, parameters of control of the subject
vehicle applied at time instant $t=t_{p}$ are calculated. The automated
vehicle control application should ensure a safety planning of the subject
vehicle trajectory based on calculated control parameters.

  4. (iv)

It should be emphasized that the real application of the calculated control
parameters for the safety planning of the subject vehicle trajectory at time
instant $t=t_{p}$ (1) is only possible if a microscopic traffic model used for
the microscopic prediction during the prediction horizon $\Delta T_{p},\
p=1,2,3,\ldots$ can make the related calculation during a negligible short
time interval in comparison with time interval (2). This is because only in
this case the microscopic traffic prediction remains to be a current one for
the calculations of subject vehicle control used for the safety planning of
the subject vehicle trajectory at time instant $t=t_{p}$. Note that the time-
discrete traffic flow model for traffic of KKl2022_Pred  used in the paper
(see Sec. I.3) is able to perform such calculations during a negligible short
time interval in comparison with time interval (2).

  5. (v)

The subject vehicle moves in accordance with calculated control parameters
only during the time interval

$t_{p}\leq t<t_{p+1},\ p=1,2,3,\ldots.$ (3)

Indeed in accordance with item (i), at $t=t_{p+1}$ a new measured microscopic
traffic situation is available. Therefore, both the microscopic traffic
prediction and resulting parameters of subject vehicle control made for time
instant $t=t_{p}$ become obsolete.

  6. (vi)

For this reason, at $t=t_{p+1}$ the measured microscopic traffic situation is
used as a new initial condition for microscopic traffic prediction during
prediction horizon $\Delta T_{p+1}$. The prediction is used for the
calculation of new parameters of subject traffic control used at $t=t_{p+1}$.
The parameters of subject traffic control are applied only during time
interval $t_{p+1}\leq t<t_{p+2}$. Then, at $t=t_{p+2}$ a new measured
microscopic traffic situation is used as a new initial condition for
microscopic traffic prediction during prediction horizon $\Delta T_{p+2}$; the
microscopic prediction is used for the calculation of new parameters of
subject traffic control used at $t=t_{p+2}$, which are applied only during
time interval $t_{p+2}\leq t<t_{p+3}$, and so on.

  7. (vii)

Thus in the microscopic prediction approach, both the microscopic traffic
prediction and the resulting calculation of parameters for subject vehicle
control are repeated for each new time instant (1), whereas the control
parameters are used only during the related time interval (3) before a new
measured microscopic traffic situation is available.

  8. (viii)

The microscopic traffic prediction approach introduced in KKl2022_Pred  (see
footnote [84] of KKl2022_Pred ) is a general approach because it is applicable
for both the control of automated-driving and human-driving vehicles. For this
reason, the vehicle for which the microscopic traffic prediction approach is
applied has been called above as the subject vehicle. In KKl2022_Pred , we
have illustrated this general microscopic traffic prediction approach for the
trajectory planning of the subject vehicle that is an automated vehicle.
Indeed, the trajectory planning of automated vehicles is currently one of the
priority aims in traffic science.

With the use of Fig. 1, we make a brief explanation of one of the applications
of the general features of microscopic traffic prediction approach (items
(i)–(viii) above). We assume that an automated vehicle moving on a secondary
road wants to turn right on the priority road at a unsignalized city
intersection (Fig. 1(a)). The basic objective of the microscopic traffic
prediction approach is to predict time dependencies of locations and speeds of
vehicles in a vicinity of the interaction that allow us to control the motion
of automated vehicle. The objective of this automated vehicle control is to
make possible the turning right of the automated vehicle without the stopping
at the intersection. In this case, the parameter of automated vehicle control
for the planning of automated vehicle trajectory is deceleration/acceleration
of the automated vehicle while the vehicle approaches the intersection.

Figure 1: Qualitative schema of microscopic traffic prediction for an
unsignalized intersection: (a) Microscopic traffic situation measured at
$t=t_{p}$ (1); only red colored vehicles that are in a red colored dashed road
region can affect on the motion of the automated vehicle (black colored
vehicle). (b) The measured characteristics of red colored vehicles (vehicle
speeds, vehicle locations, vehicle lengths, and so on) in the red colored
dashed road region shown in (a) are used in the microscopic prediction model;
before, all vehicles that were at $t<t_{p}$ in this dashed road region of the
microscopic prediction model are removed. (c) Through the use of the
microscopic traffic prediction, a predicted time instant $t_{\rm E}$ is found
at which the vehicle should turn right onto the priority road. Adapted from
KKl2022_Pred .

For the application scenario under consideration (Fig. 1), the calculation of
the microscopic traffic prediction begins at some $t=t_{p}=t_{1}$ (i.e., at
$p=$ 1) at which a vehicle that is the preceding vehicle for the automated
vehicle has just left the secondary road while turning right on the priority
road. In this case, at $t=t_{1}$ control of automated vehicle can be made
based on prediction of locations and speeds of vehicles moving only on the
priority road (this case is shown in Fig. 1(a)).

For each of the given time instant $t_{p},\ p=1,2,3,\ldots$, in addition to
the microscopic traffic prediction, the prediction model calculates two
predicted time instants: $t=t_{\rm min}$ and $t=t_{\rm max}$. The physical
sense of these predicted time instants is as follows: $t_{\rm min}$ and
$t_{\rm max}$ are predicted time instants at which the automated vehicle can
reach the intersection moving, respectively, with maximum acceleration (or/and
maximum speed) and with a safe speed coming to a stop at $t=t_{\rm max}$.

With the use of $t=t_{\rm min}$ and $t=t_{\rm max}$, the prediction model
searches whether there is a pair of vehicles moving on the priority road
between which the automated vehicle can merge without stopping at the
intersection while satisfying some given safety conditions. In this case

$t_{\rm min}\leq t_{\rm E}<t_{\rm max},$ (4)

where $t=t_{\rm E}$ is a predicted time instant at which the automated vehicle
can safety turn right at the intersection between the above-mentioned pair of
vehicles moving on the priority road; $t_{\rm min},t_{\rm E},t_{\rm
max}>t_{\rm 1}$.

In microscopic traffic prediction made at each time instant $t_{p}$, the
safety conditions at which a pair of vehicles moving on the priority road
between which the automated vehicle can merge without a stop at the
intersection are as follows:

$\tau^{-}\geq\tau_{1},\ \tau^{+}\geq\tau_{2},$ (5)

in which

$\tau^{-}=(s^{-}-d)/v^{-},\ \ \tau^{+}=(s^{+}-d)/v^{\rm(AV)},$ (6)

where $s^{+}$ is a gross space gap between the automated vehicle and the
preceding vehicle; $s^{-}$ is a gross space gap between the automated vehicle
and the following vehicle; $v^{-}$ is the speed of the following vehicle;
$v^{\rm(AV)}$ is the speed of the automated vehicle; all values in (6) are
related to a predicted time instant $t=t_{\rm E}$ at which the automated
vehicle could turn right at the intersection. In (5), $\tau^{-}$ is time
headway of a vehicle following the automated vehicle on the priority road just
after the merging, $\tau^{+}$ is time headway of the automated vehicle to the
preceding vehicle on the priority road just after the merging, $\tau_{1}$ and
$\tau_{2}$ are model parameters. The predicted time instant $t=t_{\rm E}$, at
which the automated vehicle can merge without a stop at the intersection, in
addition to safety conditions (5), should satisfy conditions (4).

Such a case, when safety conditions (5) are satisfied, is shown in Fig. 1(c).
Only in this case, predicted time instant $t=t_{\rm E}$ is used for the
calculation of automated vehicle deceleration. Otherwise, when at least one of
the safety conditions (5) is not satisfied, the automated vehicle stops at the
intersection.

The predicted time instant $t=t_{\rm E}$ is used for the calculation of
automated vehicle deceleration that is applied at $t=t_{\rm 1}$. In accordance
with item (v) of the general approach to microscopic traffic prediction, the
automated vehicle moves in accordance with calculated deceleration only during
the time interval $t_{1}\leq t<t_{2}$. At time instant $t=t_{2}$, a new
measured microscopic traffic situation is available. Therefore, a new
microscopic traffic prediction is calculated at which new predicted time
instants $t_{\rm min}$, $t_{\rm max}$, and $t_{\rm E}$ as well as a new
vehicle deceleration are found. The automated vehicle moves in accordance with
deceleration calculated at time instant $t=t_{2}$ only during the time
interval $t_{2}\leq t<t_{3}$. At time instant $t=t_{3}$, a new measured
microscopic traffic situation is available, new predicted time instants
$t_{\rm min}$, $t_{\rm max}$, and $t_{\rm E}$ as well as a new vehicle
deceleration are found, and so on. After the automated vehicle has turned
right, the automated vehicle moves on the priority road in accordance with
given rules of automated vehicle motion.

###  I.3 About Mathematical Model used for Simulations of Cooperative Driving

Mathematical formulations of KKl2022_Pred  that are used for simulations of
cooperative driving in the paper are as follows: We use the same microscopic
model of mixed traffic flow as that considered in Secs. II C of KKl2022_Pred .
Model for simulations of microscopic traffic situation has been considered in
Secs. II D of KKl2022_Pred . Model of motion of human-driving vehicles has
been given in Appendix A of KKl2022_Pred . Models of the vehicle merging onto
the priority road for human-driving vehicles and for automated vehicles (when
the microscopic traffic prediction is not used) have been given in Appendix B
of KKl2022_Pred . Model for simulations of microscopic traffic prediction has
been taken from Sec. III and Appendix C of KKl2022_Pred .

It should be noted that there are a huge number of other different microscopic
traffic flow models for mixed traffic (see, e.g., Dharba1999 ; Bose2003 ;
Kesting2007 ; Kesting2008 ; Kesting2010 ; Kukuchi2003 ; Li2002 ; Marsden2001 ;
Martinez2007 ; Shladover1995 ; Shladover2012 ; Talebpour2016 ; Kerner2021 ;
Kerner2021B )111It should be noted that in many publications devoted to a
study of mixed traffic the analysis of the effect of automated driving on
vehicular traffic is made (see, e.g., Dharba1999 ; Bose2003 ; Kesting2007 ;
Kesting2008 ; Kesting2010 ; Kukuchi2003 ; Li2002 ; Marsden2001 ; Martinez2007
; Shladover1995 ; Shladover2012 ; Talebpour2016 ; Kerner2021 ; Kerner2021B ).
Contrary to such publications, in our paper no effect of automated vehicles on
vehicular traffic is discussed. Instead, we apply the model of mixed traffic
of Refs. KKl2022_Pred ; Kerner2021 ; Kerner2021B  for simulations of
cooperative driving based on the microscopic traffic prediction. We should
also emphasize that in Refs. KKl2022_Pred ; Kerner2021 ; Kerner2021B , in
which the the model of mixed traffic used in the paper has been developed, the
influence of automated vehicles and human-driving vehicle on mixed traffic as
well as characteristics of automated vehicles have already been considered in
details.. A basic requirement to a model for simulations of the microscopic
traffic prediction is that the model of human-driving vehicles should simulate
microscopic behavior of human drivers in different traffic situations as close
as possible to real measured traffic data. For motion of human-driving
vehicles in mixed traffic, we use a version of a microscopic stochastic time-
discrete model of Kerner and Klenov in the framework of three-phase traffic
theory (see Appendix A in KernerBook3 ); in the model, in addition to
hypotheses of the three-phase traffic theory, well-known slow-to-start rule
TT1993 ; Schadschneider , and Gipps’ formulation for safe speed Gipps ;
Kra_PhD ; Kra  have been used; contrary to Appendix A in KernerBook3 , the
successive number of time steps of the delay in acceleration of a human-
driving vehicle has been limited as made in KKl2022_Pred  (see formulas (A17)
and (A18) of Appendix A of KKl2022_Pred ). The use of this model for human-
driving vehicles is explained as follows: as proven in KernerBook3 , the
Kerner-Klenov model satisfies the above-mentioned basic requirement following
from empirical features of real traffic. In the traffic model, update rules of
motion of automated driving vehicles are simulated with a time-discrete
version of the classical ACC model (e.g., Ioannou1997 ; Ioannou1996 ;
Ioannou1993 ; Meyer2014 ; Liang2000 ; Davis2014 ) that is equivalent to
Helly’s model Helly_1959 .

The time-discrete traffic flow model of KKl2022_Pred  used in this paper for
simulations calculates a new microscopic traffic prediction (see items (ii)
and (vi) of Sec. I.2) for the motion of about 30 vehicles during a time
interval $\theta<$ 0.005 s, which is negligible short in comparison with time
interval $\delta t=t_{p+1}-t_{p}=\tau=$ 1 s (2).

###  I.4 Objective, Focus, and Organization

The objective of this paper is the presentation of a methodology for
cooperative driving vehicular traffic based on short-time microscopic traffic
prediction of KKl2022_Pred . It must be emphasized that contrary to the
analysis of applications of microscopic traffic prediction made in
KKl2022_Pred , the methodology of cooperative driving presented in this paper
is devoted to application cases in which microscopic traffic prediction
without the use of cooperative driving cannot lead to a successful vehicle
control and trajectory planning. In other words, the objective of the paper
under consideration makes sense only then, if subject vehicle motion control
based on the microscopic traffic prediction without cooperative driving does
not lead to the merging of the subject vehicle onto the priority road without
stopping at the intersection. For this reason, the methodology of cooperative
driving based on the microscopic traffic prediction (Sec. II), the physics and
the procedure of calculations of cooperative driving (Sec. IV) and results of
simulations of this cooperative driving (Sec. V) are qualitatively different
from results of KKl2022_Pred  in which no cooperative driving has been
assumed. We limit our analysis of this methodology of cooperative driving
based on microscopic traffic prediction by a case vehicular traffic in which
100$\%$ of vehicles can participate and are willing to participate in
cooperative driving.

The main focus of this paper is the presentation of a general methodology for
cooperative driving based on microscopic traffic prediction. To study the
physical features of this general methodology for cooperative driving, we
limit simulations by the same simple case of unsignalized city intersection
discussed in KKl2022_Pred  (Fig. 1) in which a subject vehicle is the
automated vehicle that tries to turn right from the secondary road onto the
priority road without stopping at the intersection. Contrary to the case shown
qualitatively in Fig. 1 and explained above in Sec. I.2, we simulate a traffic
scenario at which microscopic traffic prediction cannot be used for the safety
vehicle merging without the vehicle stopping at the intersection.

The paper is organized as follows: The general methodology of cooperative
driving based on the microscopic traffic prediction is presented in Sec. II. A
simple scenario of cooperative driving for simulations of the general
methodology is discussed in Sec. III. The physics of calculations of
cooperative driving has been studied in Sec. IV. Results of simulations of
this cooperative driving are presented in Sec. V. In discussion section (Sec.
VI), we consider limitations of cooperative driving (Sec. VI.1), briefly
compare applications of leaning algorithms for traffic prediction versus
microscopic traffic prediction based on the physics of vehicle motion used in
the paper (Sec. VI.2), discuss other possible applications of the general
methodology of cooperative driving (Sec. VI.3), consider the effect of data
uncertainty on prediction reliability (Sec. VI.4) as well as formulate
conclusions of the paper (Sec. VI.5).

##  II Methodology of Physical Modeling of Cooperative Driving in Vehicular
Traffic

The general methodology for cooperative driving presented here can be applied
for both automated-driving and human-driving vehicles. In this case, as
mentioned above (see item (viii) of Sec. I.2), a vehicle, which uses the
cooperative driving, can be called as a subject vehicle, to distinguish the
vehicle from cooperating vehicles. However, the trajectory planning of
automated vehicles is one of the priority aims in traffic science. For this
reason, in the paper we have limited the study of the physics of the general
methodology of cooperative driving for a case in which the subject vehicle is
the automated vehicle.

The general methodology of physical modeling of cooperative driving is as
follows.

  * (i)

First, microscopic traffic prediction KKl2022_Pred  for the subject vehicle
control without the application of cooperative driving is made, as briefly
explained in Sec. I.2.

Contrary to KKl2022_Pred , in this paper we consider microscopic traffic
situations for which already at

$t_{p}=t_{1}$ (7)

the calculated microscopic traffic prediction shows that no effective subject
vehicle control is possible. Therefore, to organize the effective subject
vehicle control, a cooperative driving based on microscopic traffic prediction
is applied for subject vehicle control. At $t_{p}=t_{1}$ (7), the following
stages (ii) and (iii) of cooperative driving based on microscopic traffic
prediction are applied.

  * (ii)

Based on the calculated microscopic traffic prediction considered in Sec. I.2,
between vehicles that are willing to cooperate vehicles that are suitable ones
for the realization of cooperative driving are found. These vehicle are called
below as $``$cooperating vehicles”.

  * (iii)

Microscopic traffic prediction found at $t_{p}=t_{1}$ (Sec. I.2) is further
used to find both motion requirements for the cooperating vehicles and related
characteristics of subject vehicle control. The prediction of the vehicle
locations and vehicle speeds are used for the calculation of the future
trajectory of the subject vehicle as well as all other vehicles involved in
cooperative driving.

The microscopic traffic prediction together with the determination of the
cooperating vehicles as well as the associated calculation of motion
requirements for the cooperating vehicles and characteristics of subject
vehicle control are only possible when the microscopic model is able to
calculate all these characteristics during negligible short time interval in
comparison with time interval $\delta t=t_{p+1}-t_{p}$. It should be
emphasized that as explained in Sec. I.2 the time-discrete traffic flow model
for mixed traffic of KKl2022_Pred  used in this paper is able to perform these
calculations during a negligible short time interval.

  * (iv)

Stages (ii) and (iii) explained above are repeated at the next time instant
$t_{p+1},\ p=1,2,3,\ldots$, when a next microscopic traffic situation is
known. The new microscopic traffic situation is used as an initial condition
for the microscopic traffic flow model. Then, the model makes the microscopic
traffic prediction for vehicle locations and vehicle speeds during prediction
horizon $\Delta T_{p+1}$ (prediction horizon $\Delta T_{p}$ can be different
for different time instants $t_{p}$). Stages (ii) and (iii) are iterated for
each new time instants, and so on, once data for new microscopic traffic
situations are available.

Thus, as in KKl2022_Pred  (Sec. I.2), in the methodology for cooperative
driving there is the repetition of the microscopic traffic prediction together
with the calculation of motion requirements for the cooperating vehicle for
any current time instant $t_{p}$ (1) at which microscopic traffic situations
are available. This allows us to update the calculations of the trajectories
of both the cooperating vehicles and the subject vehicle during the vehicle
motion in traffic flow.

##  III Application Scenario of Cooperative Driving for Unsignalized
Intersection in City Traffic

In the paper, we study physical features of the above methodology of
cooperative driving through numerical simulations of a common case in city
traffic in which the subject vehicle is an automated vehicle. The automated
vehicle (black colored vehicle $``$automated vehicle” in Fig. 2) moves
initially on a secondary road and it would like to turn right onto a priority
road. There is no traffic signal on the intersection of these two roads. The
objective of cooperative driving is to find whether there is a possibility for
the merging of the automated vehicle onto the priority road without stopping
at the intersection.

As mentioned, the use of cooperative driving makes sense, if automated vehicle
motion control based on the microscopic traffic prediction without cooperative
driving does not lead to the merging of the automated vehicle onto the
priority road without stopping at the intersection. Such a case is
qualitatively shown in Fig. 2. Indeed, when the microscopic traffic prediction
is applied for automated vehicle control, we can see in Fig. 2(b) that
$\tau^{-}<\tau_{1}$, i.e., the first of safety conditions (5) is satisfied for
none of the pairs of vehicles moving on the priority road during time interval
$t_{\rm min}\leq t<t_{\rm max}$. This means that the automated vehicle must
stop at the intersection.

To simulate the scenario qualitatively shown in Fig. 2, in accordance with
item (i) of the general methodology of cooperative driving (Sec. II), the
microscopic traffic prediction, in which no cooperative driving has been used,
is made (Fig. 3). Simulations of the predicted time instants $t_{\rm min}$ and
$t_{\rm max}$, whose physical sense has been explained in Sec. I.2, is shown
in Fig. 4. In the simulations, in safety conditions (5) we have used
$\tau_{1}=$ 2.0 s and $\tau_{2}=$ 0.5 s. It turns out that indeed already at
the beginning of traffic prediction at $t_{p}=t_{1}=$ 148 s condition
$\tau^{-}\geq\tau_{1}$ is satisfied for none of the pairs of vehicles moving
on the priority road during time interval $t_{\rm min}\leq t<t_{\rm max}$. For
this reason, the automated vehicle stops at the intersection (Fig. 3(b)). Only
at some time instant $t>t_{\rm max}$, safety conditions (5) are satisfied and,
therefore, the automated vehicle can merge onto the priority road (Fig. 3(a)).

Figure 2: Qualitative explanation of the methodology for cooperative driving
based on microscopic traffic prediction. Figure 3: Simulations of microscopic
traffic prediction of automated vehicle motion with approach of KKl2022_Pred
without the use of cooperative driving at $t_{p}=t_{1}=$ 148 s: Vehicle
trajectories on the priority road (a) and secondary road (b); dotted curves
are vehicle locations of simulated microscopic traffic situations, solid
curves are trajectories of the predicted vehicle trajectories. Parameters of
the model of human driving vehicles are given in Table I of Appendix A of
KKl2022_Pred ; in (5), $\tau_{1}=$ 2.0 s and $\tau_{2}=$ 0.5 s; in model of
the automated vehicle (see formulas (7)–(11) in Sec. II of KKl2022_Pred ), we
have used $\tau_{\rm d}^{\rm(AV)}=$ 1.5 s, $K_{1}=$ 0.3 $\rm s^{-2}$ and
$K_{2}=$ 0.6 $\rm s^{-1}$, $a^{\rm(AV)}_{\rm max}=$ 2.5 and $b^{\rm(AV)}_{\rm
max}=$ 3 ${\rm m/s^{2}}$, all other parameters of the model of the automated
vehicle are the same as those in KKl2022_Pred . Results of calculations:
$t_{\rm min}=$ 154 s and $t_{\rm max}=$ 159 s. A single-lane priority road is
2.5 km long; the intersection of this road with a single-lane secondary road
that is 0.5 km long is at location $x_{\rm ints}$= 0.5 km; in mixed traffic
flow, there are $1\%$ of automated vehicles randomly distributed between human
driving vehicles. Horizontal blue lines show road regions satisfying
conditions (11) of KKl2022_Pred  within which microscopic traffic situations
have been used for the prediction. In simulations, the flow rate on the
priority and secondary roads are, respectively, 1161 and 100 vehicles/h.
Poisson distribution for entering vehicles has been used.

Time instant $t_{p}=t_{1}$ of the beginning of the prediction in the simulated
scenario (Fig. 3) is $t_{1}=$ 148 s (vertical red line $t_{1}$ in Fig. 3).
This is because at this time instant the preceding vehicle for the automated
vehicle on the secondary road labeled by $``$i” (Fig. 3) has merged onto the
priority road222In simulations, in addition to this condition, to determine
the beginning of the microscopic traffic prediction $t_{\rm p}=t_{\rm 1}$, we
have used the condition that the distance from the current automated vehicle
location $x^{\rm(AV)}(t_{\rm p})$ to the location $x_{\rm ints}$ of the road
intersection is less than some given value $L_{1}$ (where $L_{1}=$ 0.15 km),
i.e., $x_{\rm ints}-x^{\rm(AV)}(t_{\rm p})<L_{1}$. This condition is satisfied
in all simulation results presented.. Thus, the following automated vehicle
(black curves labeled by $``$automated vehicle” in Fig. 3) can choose its
deceleration freely while approaching the intersection.

Figure 4: Simulations of the prediction of time instants $t_{\rm min}$ and
$t_{\rm max}$. Calculations are made for prediction time instant
$t_{p}=t_{1}=$ 148 s: Virtual trajectories of the automated vehicle on the
secondary road related to the prediction of $t_{\rm min}$ (dashed-dotted curve
1) and to the prediction of $t_{\rm max}$ (dashed-dotted curve 2),
respectively. Other model parameters are the same as those in Fig. 3.

The result of these calculations of microscopic traffic prediction made for
$t_{p}=t_{1}=$ 148 s (Fig. 3) shows that in the scenario there are no pair of
vehicles moving on the priority road for which both safety conditions (5) and
conditions (4) for the merging of the automated vehicle without a stop at the
intersection are satisfied. Therefore, the automated vehicle must stop at the
intersection and wait as long as a pair of vehicles appears, which satisfying
safety conditions (5), between which the automated vehicle can merge on the
priority road.

Thus, in the scenario under consideration without cooperative driving the
automated vehicle cannot merge on the priority road without a stop at the
intersection. For this reason the application of cooperative driving can have
a sense. Simulations of the application of cooperative driving will be
considered in Secs. IV.1 and V. Before the presentation of numerical
simulation results, it is useful to give a qualitative explanation of the
application of the general methodology for cooperative driving (Sec. II) for
this case (Figs. 2(c, d)). First, in accordance of item (ii) of the general
methodology, a cooperating vehicle is chosen (blue colored vehicle in Fig.
2(c)). The choice of this vehicle as the cooperating vehicle is as follows:
From Fig. 2(b) we see that if this vehicle through its addition deceleration
increases time headway $\tau^{-}$ to the automated vehicle, then condition
$\tau^{-}\geq\tau_{1}$ can be satisfied. In this case, the automated vehicle
can merge onto the priority road without stopping at the intersection. Second,
in accordance of item (iii) of the general methodology, both motion
requirements for the cooperating vehicle (like a required deceleration) and
related characteristics of automated vehicle control are calculated with the
use of the microscopic traffic prediction (Fig. 2(d)). Then, stages (ii) and
(iii) of the general methodology explained in Figs. 2(c, d) for $t_{p}=t_{1}$
are repeated at the next time instant $t_{p+1},\ p=1,2,3,\ldots$, when a next
microscopic traffic situation is known, and so on. The prediction of the
vehicle locations and vehicle speeds are used for the calculation of the
future trajectory of the automated vehicle as well as all other vehicles
involved in cooperative driving.

Before in Sec. V we consider simulation results of the scenario qualitatively
explained in Fig. 2, we discuss how the cooperating vehicle is chosen in
numerical simulations (Sec. IV.1) as well as models of motion of the automated
vehicle (Sec. IV.2) and cooperative vehicle (Sec. IV.3) based on microscopic
traffic prediction.

##  IV Characteristics of Cooperative Driving based on Microscopic Traffic
Prediction

###  IV.1 Choice of Cooperating Vehicle based on Microscopic Traffic
Prediction

At time instant $t_{\rm 1}$, based on the calculated microscopic traffic
prediction (Fig. 3), a cooperating vehicle is found. We assume that all
vehicles can communicate and are willing to cooperate with the automated
vehicle. The choice of the cooperating vehicle is illustrated with Fig. 5.

Figure 5: Continuation of Fig. 3. Simulations of the prediction of time
instants at which vehicles 1–6 moving on the priority road reach the
intersection location $x_{\rm ints}=$ 0.5 km. Numbers 1–6 are related to
vehicles 1–6 in Fig. 3, respectively. Calculated predicted time instants at
which locations of vehicles 3, 4, and 5 are at the intersection location
$x_{\rm ints}=$ 0.5 km are, respectively, $t^{(3)}=$ 153.5 s, $t^{(4)}=$ 157.2
s, and $t^{(5)}=$ 158.8 s.

We can see (Fig. 5) that time instant, at which location of vehicle 3 is at
the intersection location $x_{\rm ints}=$ 0.5 km, is slightly smaller than
$t=t_{\rm min}$. Therefore, vehicle 3 can be chosen as the preceding vehicle
for the automated vehicle merging onto the priority road without a stop at the
intersection. Respectively, vehicle 4 that follows vehicle 3 can be chosen as
the cooperating vehicle. The cooperating vehicle 4 should decelerate to ensure
a safety merging of the automated vehicle onto the priority road without a
stop at the intersection. This means that at time instant $t=t_{\rm E}$ the
cooperating vehicle should be behind the automated vehicle at time gap
$\tau^{-}=\tau^{(\rm c)}$ satisfying condition

$\tau^{(\rm c)}\geq\tau_{1}.$ (8)

###  IV.2 Motion of Automated Vehicle in Cooperative Driving based on
Microscopic Traffic Prediction

A required deceleration of the cooperating vehicle that satisfies condition
(8) will be considered in Sec. IV.3. Firstly, we should calculate deceleration
$b_{\rm p}$ of the automated vehicle and related predicted time instant
$t=t_{\rm E}$ at which the vehicle merges onto the priority road. Predicted
time instant $t=t_{\rm E}$ is found from condition $\tau^{+}\geq\tau_{2}$ in
(5) related to a safety merging of the automated vehicle behind of vehicle 3.
The model of the calculation of the predicted time instant $t_{\rm E}$ and
deceleration $b_{\rm p}$ of the automated vehicle is the same as that in Sec.
III and Appendixes B and C of KKl2022_Pred .

In accordance with KKl2022_Pred , there is the repetition of the prediction of
the automated vehicle trajectory made at each next time instants $t_{p},\
p=2,3\ldots$. Therefore, for each time instant $t_{p}$, the predicted time
instant $t_{\rm E}$ and the associated predicted deceleration $b_{p}$ are
calculated. The automated vehicle motion with deceleration $b_{p}$ is only
valid until the next traffic prediction for $b_{p+1}$ is calculated at
$t=t_{p+1}$. In other words, deceleration $b_{\rm p}$ of the automated vehicle
at time instant $t_{\rm p}$ is applied only during time interval (3). At
$t=t_{p+1}$, the same procedure of microscopic traffic prediction as made for
$t=t_{p}$ is repeated leading to a new deceleration $b_{p+1}$ and to a new
predicted time instant $t_{\rm E}$ at which the vehicle should turn right onto
the priority road. Then, during time interval $t_{p+1}\leq t<t_{p+2}$ the
vehicle moves in accordance with formulas (20) and (21) of KKl2022_Pred  in
which instead of the value $b_{p}$ the new predicted deceleration $b_{p+1}$ is
used, and so on. Because microscopic traffic situations depend on time
$t_{p}$, all predicted values, like the predicted time instant $t_{\rm
E}(t_{p})$, are functions of time instant $t_{p}$ at which the prediction is
made.

In accordance with formula (2), time instants $t_{p}$ at which the microscopic
traffic prediction is made can be written as

$t_{p}=t_{1}+(p-1)\tau,\ p=1,2,\ldots,p_{\rm E}.$ (9)

The repetition of the predictions of the automated vehicle trajectory is made
up to some maximum time instant $t_{p}$ denoted in (9) by $t_{p}=t_{p_{\rm
E}}$. Conditions for the maximum time instant $t_{p}=t_{p_{\rm E}}$ and for
the related time instant denoted by $t^{\rm(real)}_{\rm E}$ at which the
automated vehicle really merges onto the priority road are presented in
Appendix C of KKl2022_Pred . At $t\geq t_{p_{\rm E}}$ the automated vehicle
decelerates with the last predicted deceleration (Appendix C of KKl2022_Pred )
moving up to the road intersection at which it turns right onto the priority
road; later, no cooperating driving and no prediction is made.

###  IV.3 Motion of Cooperating Vehicle based on Microscopic Traffic
Prediction

####  IV.3.1 Deceleration of Cooperating Vehicle

For each of the time instants $t_{p}$, the deceleration $b_{p}^{\left(\rm
c\right)}$ of the cooperating vehicle that satisfies safety conditions (8) is
found from condition:

$x_{\rm ints}-x^{(\rm c)}_{n}=v^{(\rm c)}_{\rm E}\tau^{(\rm c)}+d+X^{(\rm
c)}_{\rm b},$ (10)

where index $n$ corresponds to discrete time $n\tau,\ n=0,1,\ldots$ of the
model, $x^{(\rm c)}_{n}$ is the position of the cooperating vehicle at time
$t^{(\rm c)}_{n}$, where $t^{(\rm c)}_{n}=t_{p}$; $v^{(\rm c)}_{\rm E}\geq 0$
is the calculated speed of the cooperating vehicle at time instant $t_{\rm
E}(t_{p})$, $d$ is the vehicle length, $X^{(\rm c)}_{\rm b}$ is a deceleration
distance of the cooperating vehicle. The distance $X^{(\rm c)}_{\rm b}$ and
the speed $v^{(\rm c)}_{\rm E}$ are found from the condition that the
cooperating vehicle decelerates with constant deceleration $b_{\rm
p}^{\left(\rm c\right)}<0$ from the speed $v^{(\rm c)}_{n}$ to the speed
$v^{(\rm c)}_{\rm E}$ during time interval $T_{\rm c}$:

$T_{\rm c}(t_{p})=t_{\rm E}(t_{p})-t^{(\rm c)}_{n},$ (11) $X^{(\rm c)}_{\rm
b}=\ v^{(\rm c)}_{n}\left(T^{\rm(c)}+\delta
T^{\rm(c)}\right)-\frac{1}{2}b_{\rm p}^{\left(\rm
c\right)}T^{\rm(c)}(T^{\rm(c)}+\tau+2\delta T^{\rm(c)}),$ (12) $v^{(\rm
c)}_{\rm E}=\max\left(0,v^{(\rm c)}_{n}-b_{p}^{\left(\rm
c\right)}T^{\rm(c)}\right),$ (13)

where $T^{\rm(c)}$ and $\delta T^{\rm(c)}$ are, respectively, the integer and
the fractional parts of time $T_{\rm c}$: $T^{\rm(c)}=\tau\left\lfloor{T_{\rm
c}}/{\tau}\right\rfloor$, $\delta T^{\rm(c)}=T_{\rm c}-T^{\rm(c)}$.
Deceleration $b_{\rm p}^{\left(\rm c\right)}$ is found from a solution of
(10)–(13):

$\displaystyle b_{\rm p}^{\left(\rm c\right)}=$ (14)
$\displaystyle{=}-\frac{2\left(x_{\rm
ints}{-x}^{\rm(c)}_{n}-v^{\rm(c)}_{n}\left(T^{\rm(c)}+\delta
T^{\rm(c)}+\tau^{(\rm c)}\right)-d\right)\
}{T^{\rm(c)}\left(T^{\rm(c)}+\tau+2\delta T^{\rm(c)}+2\tau^{(\rm c)}\right)}$
,

where

$\displaystyle p=1,2,\ldots,p_{\rm E}-1.$ (15)

The deceleration (14) is calculated for each time instant
$t^{\rm(c)}_{n}=t_{\rm p}$. Then, under condition (15) the next prediction for
the time $t_{\rm E}(t_{p+1})$ (see formula (31) of KKl2022_Pred ) is made at
time instant $t_{p+1}$, and the subsequent calculation of $b_{p+1}^{\left(\rm
c\right)}$ is made.

####  IV.3.2 Minimum Distance for Cooperative Driving

At time instant $t^{\rm(c)}_{n}$ the cooperating vehicle is at some distance
from the intersection location $D=x_{\rm ints}-x^{\rm(c)}_{n}$. It should be
emphasized that there should be some minimum distance $D=D_{\rm min}$ for
cooperative driving; in other words, cooperative driving is possible when
condition

$D\geq D_{\rm min}$ (16)

is satisfied. Indeed, first the deceleration of the cooperating vehicle is
limited by a given maximum deceleration $b_{p}^{\left(c\right)}=b_{\rm
max}^{\left(c\right)}$. In this case $D_{\rm min}=D^{\rm(b)}_{\rm min}$, where

$\displaystyle D^{\rm(b)}_{\rm min}=d+v^{\rm(c)}_{n}\left(T^{\rm(c)}+\delta
T^{\rm(c)}+\tau^{\rm(c)}\right)-$ (17) $\displaystyle-\frac{1}{2}b_{\rm
max}^{\left(c\right)}T^{\rm(c)}\left(T^{\rm(c)}+\tau+2\delta
T^{\rm(c)}+2\tau^{\rm(c)}\right).$

However, $D_{\rm min}$ cannot exceed some value $D_{\rm min}^{(0)}$, because
at the end of deceleration the final speed $v^{\rm(c)}_{\rm E}$ in (13) cannot
be negative. Therefore, time $T^{\rm(c)}$ in (17) is limited by some value
$T^{\rm(c)}_{\rm b}$. As a result, $D_{\rm min}$ is equal to

$D_{\rm min}=\left\\{\begin{array}[]{ll}D^{\rm(b)}_{\rm min}\ \textrm{at}\
T^{\rm(c)}\leq\ T^{\rm(c)}_{\rm b}\\\ D_{\rm min}^{(0)}\ \textrm{at}\
T^{\rm(c)}>T^{\rm(c)}_{\rm b}.\\\ \end{array}\right.$ (18)

In order to find values of $D_{\rm min}^{(0)}$ and $T^{\rm(c)}_{\rm b}$ in
(18), a case is considered when during some time steps $N^{\rm(c)}_{\rm b}$
the cooperating vehicle decelerates with the maximum deceleration $b_{\rm
max}^{\left(c\right)}$ from the speed $v^{\rm(c)}_{n}$ to a very low speed
$v_{\rm c}$ that satisfies conditions $0\leq v_{\rm c}<b_{\rm
max}^{\left(c\right)}\tau$. Then,

$N^{\rm(c)}_{\rm b}=\left\lfloor\frac{v^{\rm(c)}_{n}}{b_{\rm
max}^{\left(c\right)}\tau}\right\rfloor,$ (19) $T^{\rm(c)}_{\rm b}=\tau
N^{\rm(c)}_{\rm b},$ (20) $v_{\rm c}=v^{\rm(c)}_{n}-b_{\rm
max}^{\left(c\right)}T^{\rm(c)}_{\rm b}.$ (21)

From (21), value $T^{\rm(c)}_{\rm b}$ can also be written as $T^{\rm(c)}_{\rm
b}={\left(v^{\rm(c)}_{n}-v_{\rm c}\right)}/{b_{\rm max}^{\left(c\right)}}$. At
$T^{\rm(c)}=T^{\rm(c)}_{\rm b}$, from Eq. (17) we get:

$\displaystyle D_{\rm
min}^{(0)}=d+\frac{1}{2}v^{\rm(c)}_{n}\left(T^{\rm(c)}_{\rm b}-\tau\right)$
$\displaystyle+\frac{1}{2}v_{\rm c}\left(T^{\rm(c)}_{\rm b}+\tau+2\delta
T^{\rm(c)}+2\tau^{\rm(c)}\right).$ (22)

Using $v^{\rm(c)}_{n}=v_{\rm free}$ (where $v_{\rm free}$ is a maximum vehicle
speed) and condition $v_{\rm c}\ll\ v_{\rm free}$, formulas (20) and (22) read

$T^{\rm(c)}_{\rm b}=\tau\left\lfloor\frac{v_{\rm free}}{b_{\rm
max}^{\left(c\right)}\tau}\right\rfloor,$ (23) $D_{\rm
min}^{(0)}=d+\frac{1}{2}v_{\rm free}\left(T^{\rm(c)}_{\rm b}-\tau\right).$
(24)

Eqs. (23) and (24) determine the parameters $T^{\rm(c)}_{\rm b}$ and $D_{\rm
min}^{(0)}$ in formula (18) for $D_{\rm min}$.

Deceleration $b_{\rm p}^{\left(\rm c\right)},\ p=1,2,\ldots,p_{\rm E}$ of the
cooperating vehicle is found for each microscopic traffic prediction at time
instants $t_{\rm p}$. The calculated deceleration $b_{\rm p}^{\left(\rm
c\right)}$ can be different for different time instants $t_{\rm p},\
p=1,2,\ldots,p_{\rm E}$.

##  V Simulation Results of Cooperative Driving

At the beginning of microscopic traffic prediction $t_{1}=$ 148 s, the
following predictions have been calculated:

  * (i)

Time instant of automated vehicle merging $t_{\rm E}(t_{1})$ onto the priority
road behind vehicle 3.

  * (ii)

The deceleration of automated vehicle $b_{\rm 1}$ (see Appendix C of
KKl2022_Pred ).

  * (iii)

A proof whether condition $D\geq D_{\rm min}$ (16) is satisfied (Sec. IV.3.2).
If $``$Yes”, then next calculations of predicted characteristics (items (iv),
(v)) are made.

  * (iv)

The deceleration of cooperating vehicle $b^{\rm(c)}_{\rm 1}$ (vehicle 4) (Sec.
IV.3).

  * (v)

The speeds and trajectories of all vehicles that locations correspond to
conditions (11) of KKl2022_Pred  (Figs. 6 and 7).

Figure 6: Simulations of cooperative driving based on microscopic traffic
prediction made for time instant $t_{1}=$ 148 s: Trajectories of vehicles on
the priority road (a) and secondary road (b). Dotted parts of vehicle
trajectories are related to the microscopic traffic situation at $t\leq
t_{1}=$ 148 s, solid parts of vehicle trajectories are predicted vehicle
trajectories. Model parameters: $\tau^{\rm(c)}=$ 2.4 s, $b^{\rm(c)}_{\rm
max}=$ 1 ${\rm m/s^{2}}$. Calculated values $t_{\rm min}=$ 154 s, $t_{\rm
max}=$ 159 s, $t_{\rm E}=$ 154.6 s, $\tau^{-}=$ 2.36 s, $\tau^{+}=$ 0.97 s,
$D_{\rm min}=$ 78.5 m, $T_{\rm c}=$ 6.6 s, $b_{\rm 1}=-0.21\ {\rm m/s^{2}}$,
$b^{\rm(c)}_{\rm 1}=-0.56\ {\rm m/s^{2}}$. Other parameter designations,
vehicle numbers, and model parameters are the same as those in Fig. 3.  Figure
7: Continuation of Fig. 6: (a) Time-functions of microscopic speeds of some of
the vehicles; vehicles $i$ and 3 are the preceding vehicles for the automated
vehicle, respectively, on the secondary and priority roads. (b, c) Time-
functions of deceleration (acceleration) of automated vehicle (b) and the
cooperating vehicle (c). In (a, c), vehicle 4 is the cooperating vehicle.
Vehicles numbers are the same as those in Fig. 6. Dashed parts of curves are
related to real vehicle motion at $t\leq t_{1}=$ 148 s, solid parts of curves
are predicted characteristics.

Calculations of the microscopic traffic prediction at the beginning of the
prediction at time instant $t_{p}=t_{1}$ are shown in Fig. 6. In this Fig. 6,
dotted parts of vehicle trajectories are related to the microscopic traffic
situation, whereas solid parts of vehicle trajectories are predicted vehicle
trajectories. Time-functions of microscopic speeds of some of the vehicles of
Fig. 6 are presented in Fig. 7. It should be emphasized that the prediction of
values shown in Fig. 6 and listed in items (i)–(v) above, in particular,
$t_{\rm E}(t_{1})$, $b_{\rm 1}$, and $b^{\rm(c)}_{\rm 1}$ as well as of
trajectories of all vehicles have been calculated during a negligible short
time interval $\theta<$ 0.005 s in comparison with time step $\tau=$ 1 s of
the model. Corresponding to condition (3), the predicted values are used by
the automated vehicle and cooperating vehicle during the time interval

$t_{1}\leq t<t_{2}.$ (25)

At time instant $t=t_{2}$, the next microscopic traffic situation is
available. Therefore, the next prediction of values $t_{\rm E}(t_{2})$,
$b_{\rm 2}$, $b^{\rm(c)}_{\rm 2}$ as well as the new prediction of
trajectories of all vehicles should be calculated and used for the motion of
the automated vehicle and cooperating vehicle during the next time interval

$t_{2}\leq t<t_{3},$ (26)

and so on.

Figure 8: Simulations of repetitions of prediction procedure for cooperating
driving. Time-functions of deceleration (acceleration) of automated vehicle
(left column) and the cooperating vehicle (right column) (vehicle 4 in Figs.
3, 6, and 7): (a) $t_{2}=$ 149 s. (b) $t_{3}=$ 150 s. (c) $t_{4}=$ 151 s. (d)
$t_{5}=$ 152 s. (e) $t_{6}=$ 153 s. (f) $t_{7}=$ 154 s. Calculated predicted
characteristics: $t_{\rm E}(t_{2})=$ 154.7 s, $b_{2}=-0.26\ {\rm m/s^{2}}$,
$b^{\rm(c)}_{2}=-0.59\ {\rm m/s^{2}}$ (a), $t_{\rm E}(t_{3})=$ 154.4 s,
$b_{3}=-0.07\ {\rm m/s^{2}}$, $b^{\rm(c)}_{3}=-0.5\ {\rm m/s^{2}}$ (b),
$t_{\rm E}(t_{4})=$ 155.1 s, $b_{4}=-0.63\ {\rm m/s^{2}}$,
$b^{\rm(c)}_{4}=-0.7\ {\rm m/s^{2}}$ (c), $t_{\rm E}(t_{5})=$ 154.7 s,
$b_{5}=-0.17\ {\rm m/s^{2}}$, $b^{\rm(c)}_{5}=-0.59\ {\rm m/s^{2}}$ (d),
$t_{\rm E}(t_{6})=$ 154.7 s, $b_{6}=-0.17\ {\rm m/s^{2}}$,
$b^{\rm(c)}_{6}=-0.59\ {\rm m/s^{2}}$ (e), $t_{\rm E}(t_{7})=$ 154.7 s,
$b_{7}=-0.17\ {\rm m/s^{2}}$, $b^{\rm(c)}_{7}=-0.58\ {\rm m/s^{2}}$ (f).
Dashed parts of curves are related to real vehicle motion at $t\leq t_{p}$,
where $p=2$ for (a), $p=3$ for (b), $p=4$ for (c), $p=5$ for (d), $p=6$ for
(e), and $p=7$ for (f); solid parts of curves are predicted characteristics.

A dependence of predicted characteristics of the motion of the automated
vehicle and cooperating vehicle on time instants $t_{p},\ p=2,3,4,5,6,7$ is
illustrated in Fig. 8. With the used of the microscopic traffic prediction, it
has been found that at $t_{2}=$ 149 s, to satisfy safety conditions (5),
cooperating vehicle 4 should decelerate with $b^{\rm(c)}_{2}=-0.59\ {\rm
m/s^{2}}$; then, while decelerating with $b_{2}=-0.26\ {\rm m/s^{2}}$, the
automated vehicle can safe merge onto the priority road without stopping at
the intersection at time instant $t_{\rm E}(t_{2})=$ 154.7 s (Fig. 8(a)).
However, both the decelerations of the cooperating vehicle 4 and of the
automated vehicle has been applied during the time interval $t_{2}\leq
t<t_{3}$ only, where $t_{3}=$ 150 s.

At $t_{3}=$ 150 s, when a new microscopic traffic situation has been measured,
through the use of new microscopic traffic prediction calculations of the
decelerations of the cooperating vehicle 4 and of the automated vehicle as
well as the predicted time instant of the automated vehicle merging $t_{\rm
E}(t_{3})=$ 154.4 s have been made (Fig. 8(b)). However, both the
decelerations of the cooperating vehicle 4 and of the automated vehicle have
been applied during the time interval $t_{3}\leq t<t_{4}$ only, where $t_{4}=$
151 s. Calculations of the decelerations of the cooperating vehicle 4 and of
the automated vehicle as well as the predicted time instant of the automated
vehicle merging $t_{\rm E}$ have been repeated for further time instants
$t_{3}=$ 151 s, $t_{4}=$ 152 s, $t_{5}=$ 153 s, and $t_{6}=$ 154 s, at which
new microscopic traffic situations have been measured, respectively (Figs.
8(c–f)).

Thus, predicted decelerations $b_{p}$ and $b^{\rm(c)}_{p}$ as well as other
predicted characteristics of vehicle motion can be, respectively, different
for different time instants $t_{p},\ p=2,3,4,5,6,7$. This explains why both
the automated vehicle deceleration and the deceleration of the cooperating
vehicle can be complex time functions within time interval $t_{1}\leq
t<t^{\rm(real)}_{\rm E}$ (Figs. 7–10).

To illustrate this conclusion, we compare the deceleration of cooperating
vehicle 4 found from microscopic traffic predictions at time instants $t_{p},\
p=2,3$: Whereas at $t_{2}=$ 149 s cooperating vehicle 4 decelerates with
$b^{\rm(c)}_{2}=-0.59\ {\rm m/s^{2}}$, at $t_{3}=$ 150 s cooperating vehicle 4
should slightly reduce its deceleration to $b^{\rm(c)}_{3}=-0.5\ {\rm
m/s^{2}}$. The same effect is valid for the deceleration of the automated
vehicle; for example, whereas at $t_{2}=$ 149 s the automated vehicle should
decelerate with $b_{2}=-0.26\ {\rm m/s^{2}}$, however, at $t_{3}=$ 150 s the
automated vehicle should also reduce its deceleration to $b_{3}=-0.07\ {\rm
m/s^{2}}$.

Figure 9: Simulations of automated vehicle control through cooperating driving
based on traffic prediction (final results): (a, b) Vehicle trajectories on
the priority (a) and secondary roads (b); dotted curves are related to
microscopic traffic situations, solid curves are motion of automated vehicle;
in (a), vehicles 3 and 4 are, respectively, the preceding and cooperating
vehicles. (c, d) Time-functions of speed (c) and deceleration (acceleration)
of automated vehicle. Calculated value $t^{\rm(real)}_{\rm E}=$ 154.7 s. Other
parameter designations and model parameters are the same as those in Fig. 3.
Figure 10: Simulations of characteristics of motion of cooperating driving 4
based on traffic prediction (final results): Time-functions of speed (a),
deceleration (acceleration) (b), and time headway of cooperating vehicle (c).
In (a), dashed curve is time-dependence of automated vehicle speed taken from
Fig. 9(c). Other parameter designations and model parameters are the same as
those in Fig. 3.  Figure 11: Cooperative driving versus no cooperative
driving: (a, b) Time-functions of speed (a) and deceleration (acceleration)
(b) of the automated vehicle. (c) Trajectories of the automated vehicle. Solid
curves – automated vehicle motion through cooperative driving; dashed curves –
no cooperative driving is applied. Simulated data are taken from Fig. 3 for
dashed curves and from Figs. 9 and 10 for solid curves.  Figure 12:
Cooperative driving (a) versus no cooperative driving (b): Time-functions of
some of the vehicles. Simulated data are taken from Figs. 9 and 10 for (a) and
from Fig. 3 for (b). Vehicle numbers 4–8 are, respectively, the same as those
in Fig. 3.

In Figs. 9 and 10, simulations of cooperating driving based on final traffic
prediction are presented. As explained in Sec. IV.2, the final traffic
prediction is made at time instant $t_{p}=t_{p_{\rm E}}$ (9). At $t\geq
t_{p_{\rm E}}$, the automated vehicle decelerates with the last predicted
deceleration moving up to the road intersection at which it turns right onto
the priority road. Later, no cooperating driving and no prediction is made,
i.e., the automated vehicle moves on the priority road in accordance with
rules of automated vehicle motion of KKl2022_Pred .

Simulations of a speed harmonization in city traffic through the use of
cooperative driving based on the microscopic traffic prediction are shown in
Figs. 11 and 12. As can be seen from Fig. 11, the use of cooperative driving
based on the microscopic traffic prediction leads to a speed harmonization in
city traffic. However, the deceleration of the cooperating vehicle (vehicle 4)
forces the following vehicles 5–8 also to decelerate (Fig. 12). Nevertheless,
without cooperative driving the minimum speed of vehicles 4–8 is 33.5 km/h
(Fig. 12(b)), whereas due to cooperative driving this minimum speed decreases
only to 31.3 km/h (Fig. 12(a)). Thus, the decrease in the speed in vehicles
following the automated vehicle caused by cooperative driving can be very low
(in the case under consideration it is only 2.2 km/h) in comparison with a
considerably larger effect of cooperative driving on the motion of the
automated vehicle (Fig. 11).

##  VI Discussion

###  VI.1 About Limitations of Cooperative Driving

As shown above, the use of cooperative driving based on the microscopic
traffic prediction leads to a speed harmonization in city traffic. However,
there is a considerable limitation of the applicability of cooperative
driving.

Figure 13: Explanation of limitations of cooperative driving. Time-functions
of speed of vehicle 4 and the automated vehicle along trajectories of these
vehicles presented in Fig. 3. Calculated values: $t^{\rm(4)}_{\rm cross}=$
157.2 s, $v^{\rm(AV)}_{\rm cross}=$ 3.89 km/h.

To understand this limitation, we assume that vehicle 4 in Fig. 3 cannot
cooperate or it does not want to cooperate. We assume that the next upstream
vehicle 5 can cooperate and it wants to cooperate. We would like to understand
in which degree the cooperation of vehicle 5 can improve traffic flow
characteristics.

With this objective, we consider Figs. 5 and 13. Because we have assumed that
vehicle 4 does not cooperate, the automated vehicle should merge onto the
priority road behind vehicle 4. Therefore, the automated vehicle must
decelerate before vehicle 4 is located already downstream of the intersection
to satisfy safety condition $\tau^{+}\geq\tau_{2}$. However, from Fig. 5 we
can see that vehicle 4 reaches the intersection location $x_{\rm ints}=$ 0.5
km at time instant $t^{\rm(4)}_{\rm cross}=$ 157.2 s (Fig. 13). At this time
instant the automated vehicle speed is already equal to a very low value
$v^{\rm(AV)}_{\rm cross}=$ 3.89 km/h (Fig. 13). Thus, the automated vehicle
can merge onto the priority road at some time instant $t_{\rm
E}>t^{\rm(4)}_{\rm cross}$ while decelerating further to some lower speed
$v^{\rm(AV)}_{\rm E}<$ 3.89 km/h that is close to zero. Although even in this
case of cooperative driving the automated vehicle can merge onto the main road
without a stop at the intersection, it is realized on the cost of the
deceleration of vehicle 5 and subsequent deceleration of some other following
vehicles on the priority road. It seems that such cooperative driving is not
effective enough to be used in the reality.

We see that even if in traffic flow only a few of the vehicles cannot
cooperate or they do not want to cooperate, this can lead to considerable
decrease in the effective use of cooperative driving. For this reason, a
statistical analysis of cooperative driving based on microscopic traffic
prediction for the case when some of the vehicles do not cooperate could be a
very interesting task for further traffic studies. However, such a statistical
analysis is a separate study, which is out of the scope of this paper. As
above-mentioned (Sec. I), the objective of the paper is limited to the
presented methodology of cooperative driving based on microscopic traffic
prediction in mixed traffic flow with 100$\%$ of vehicles that participate in
cooperative driving.

###  VI.2 Short-Time Prediction: Statistical Algorithms of Artificial
Intelligence (AI) versus Microscopic Physical Traffic Modeling

Leaning algorithms of AI are the important basis of many future technologies.
This is also the case for future transportation technologies. In particular,
as mentioned, in traffic science the short-time prediction of vehicle
variables (vehicle locations and speeds) in the neighborhood of the automated
vehicle, which is required for the planning of the trajectory of the automated
vehicle, is made based on a diverse variety of statistical approaches of AI.
Some of these statistical approaches have been mentioned above Brechtel2014 ;
Lin2019 ; Schorner2019 ; Klimenko2014 ; Hubmann2019 ; A-Lombard-A_2023A ;
J-Luo-T_2023A ; Markov_Shi_2024A ; Isele2018 ; Qiao2018 ; Sama2020 ;
D-Chen_2023A ; X-Jiang-J_2023A ; Learn_Pred_AV_2024A ; Zhou-Z-Cao_2023A ;
Zhang-S-Li_2023A ; Shupei-Wang_2024A ; Renteng-Yuan_2023A ; Du-Y-Zou_2023A ;
K-Guo-M-Wu_2023A ; Chunyu-Liu_2023A ; J-Guo_2023A ; Bharti-Redhu_2023A ;
Wensong-Zhang_2023A ; Shun-Wang_2023A ; Siyuan-Feng_2023A ; Li-Z-Huang_2023A ;
Jiaxin-Liu_2023A ; Zelin-Wang_2024A ; C-J-Hoel_2023A ; S-Fang-C_2023A ;
Orzechowski2018 ; Althoff2016 ; Naumann2019 ; Tas2018 ; Akagi2015 ;
Morales2017 ; Yoshihara2017 ; Takeuchi2015 ; Yu2019 ; Hoermann2017 ; Guojing-
Hu_2023A ; Changxi-Ma_2023A ; S-Akhtar_2023A ; Xiaoxue-Yang_2023A ; Gao-X-
Li_2023A ; Z-Gu_2023A ; Y-Zhang_2023A ; Xiaoyong-Sun_2023A ; Weibin-
Zhang_2023A ; H-Chen-Y-Liu_2023A ; W-Shao_2023A ; T-Qie-W_2023A .

In the paper, rather than one of the statistical approaches of AI, we follow a
qualitative different approach to traffic prediction KKl2022_Pred  in which no
statistical analysis of a historical traffic database is used for traffic
prediction. To explain our motivation, we should explain some basic physical
traffic features. First, traffic occur in time and space. Second, vehicle
locations and speeds are continuous time-variables. Third, traffic prediction
needed for the confident planning of the trajectory of the automated vehicle
can include a large enough number of vehicles whose locations and speeds
should be predicted. Therefore, we can assume the following basic problems for
applications of the statistical approaches of AI for the planning of the
trajectory of the automated vehicle in real traffic:

  1. (i)

For each part of a traffic network and even at the same initial traffic
conditions at the network part boundaries, there can be infinity number of
microscopic traffic situations. Therefore, a statistical database cannot
include all possible microscopic traffic situations. This is true independent
of the time duration within which traffic data included in this database have
been measured.

  2. (ii)

Because the statistical database cannot include all possible microscopic
traffic situations (item (i)), in general case, the probability of the
prediction of a particular microscopic traffic situation with a given high
enough accuracy needed for the safety planning of automated vehicle trajectory
is probably very difficult to determine.

  3. (iii)

Due to differences in road infrastructure of different network parts within
which the short-time prediction of vehicle variables in the neighborhood of
the automated vehicle is made, it can occur that different historical traffic
databases are needed for each of the network parts 333Examples of the
differences in road infrastructure include different numbers of on- and off-
ramps and distances between them, road gradients, road curves, changes in the
number of road lanes, etc..

All these problems of the statistical approaches of AI for short-time
microscopic traffic prediction in real traffic are absent in the approach to
traffic prediction of KKl2022_Pred . This is because in this approach rather
than statistical analysis of a historical traffic database, the microscopic
modeling of the motion of each of the vehicles is used for microscopic traffic
prediction. We see the following advantages of cooperative driving based on
microscopic traffic prediction:

  1. 1.

No statistical database for microscopic traffic prediction is needed.

  2. 2.

A microscopic traffic flow model based on measured microscopic traffic
situations (locations and speeds of vehicles) can predict future microscopic
traffic situation. In accordance with this prediction, a decision for the
motion of the cooperating vehicles and the safety planning of automated
vehicle trajectory can be made.

  3. 3.

Parameters of the microscopic traffic flow model can include a variety of
vehicle behaviors adjustable for different road infrastructure, like on- and
off-ramps, road curves and road gradients etc. Therefore, in general there is
no need to use different traffic models for each of the network parts.

It must be emphasize that statistical approaches of AI can also be very useful
for the further development of the microscopic traffic prediction approach of
KKl2022_Pred . Indeed, parameters of a microscopic traffic model used for the
prediction can be automatically adapted over time with the use of learning
approaches of AI. However, this task, which can be very interesting for
further traffic studies, is out of the scope of this paper.

###  VI.3 Other Applications of Methodology for Cooperative Driving

We have illustrated and studied numerically the methodology of cooperative
driving for a particular simple traffic scenario, in which a subject vehicle
will to turn right at a unsignalized city intersection (Sec. III). From the
stages of the methodology (Sec. II), it is clear that its applicability for
other traffic scenarios is limited by the applicability of the microscopic
traffic prediction of KKl2022_Pred . In a general case, the prediction of the
motion of the local neighbors for the subject vehicle can depend on the
behavior of the subject vehicle, whereas the behavior of the subject vehicle
depends on the motion of the neighbor vehicles. In other words, without
cooperative driving the microscopic traffic prediction can be applied at least
when the predicted motion of the local neighboring vehicles does not depend on
the subject vehicle motion. As explained in Sec. VII A of KKl2022_Pred , there
can be a number of other other possible applications of the microscopic
traffic prediction in which these conditions are satisfied. Based on the
prediction, cooperative vehicles and their behavioral parameters are chosen as
explained in Sec. II.

Figure 14: Qualitative explanation of the application of the methodology for
cooperative driving based on microscopic traffic prediction for a traffic
scenario, in which a subject vehicle will to turn left at the unsignalized
city intersection.

As follows from Sec. II, the physical basis of the methodology for cooperative
driving presented in the paper is the microscopic traffic prediction approach
of KKl2022_Pred . Indeed, the prediction approach is used for all states of
the application of cooperative driving that are as follows:

(i) To make a decision whether cooperative driving is required for vehicle
trajectory planning.

(ii) To make a choice of required cooperative vehicles.

(iii) To calculate the motion behaviors of the cooperative vehicles that
satisfy driving safety conditions.

The microscopic traffic prediction approach of KKl2022_Pred  used in the
methodology of cooperative driving exhibits a general character. For this
reason, the methodology of cooperative driving presented in this paper can be
applied for different traffic scenarios in which the prediction approach is
applicable. To explain these conclusions, below we consider qualitatively one
of the traffic scenarios (Fig. 14).

Contrary to the simple scenario studied above, in which the automated vehicle
tries to turn right at the unsignilized intersection (Fig. 2), in the new
considerably more complex scenario the automated vehicle tries to turn left at
this intersection (Fig. 14).

In accordance with item (i) of the general methodology (Sec. II), we assume
that in a traffic situation of this scenario (Fig. 14(a)) the application of
microscopic traffic prediction without cooperative driving does not lead to
the turning left of the automated vehicle onto the priority road without
stopping at the intersection. This case is qualitatively shown in Fig. 14(b),
in which the first of safety conditions (5) is not satisfied: Condition
$\tau^{-}<\tau_{1}$ is valid for both right and left traffic directions.

In accordance of item (ii) of the general methodology, cooperating vehicles
are chosen (blue colored vehicles in Fig. 14(c)). The choice of these
cooperating vehicles is as follows: From Fig. 14(b) we see that if these
vehicles through their addition deceleration increase time headway $\tau^{-}$
to the automated vehicle, respectively, in the right and left directions, then
condition $\tau^{-}\geq\tau_{1}$ can be satisfied for both directions.
Therefore, the automated vehicle can merge onto the priority road without
stopping at the intersection.

In accordance of item (iii) of the general methodology, motion requirements
for the cooperating vehicles (like required decelerations) and related
characteristics of automated vehicle control are calculated with the use of
the microscopic traffic prediction (Fig. 14(d)). Then, stages (ii) and (iii)
of the general methodology explained in Figs. 14(c, d) for $t_{p}=t_{1}$ are
repeated at the next time instant $t_{p+1},\ p=1,2,3,\ldots$, when a next
microscopic traffic situation is known, and so on. The prediction of the
vehicle locations and vehicle speeds are used for the calculation of the
future trajectory of the automated vehicle as well as all other vehicles
involved in cooperative driving.

These qualitative conclusions should not be necessarily to be proven through
numerical simulations of the traffic scenario shown in Fig. 14. This is due to
the general character of the microscopic traffic prediction approach of
KKl2022_Pred  used for the methodology of cooperative driving of this paper.
Indeed, the procedure of the prediction of values $t=t_{\rm min}$, $t=t_{\rm
max}$, and $t=t_{\rm E}$ in (4) for both potential pairs of vehicles moving in
the right and left traffic directions on the priority road as well as the
procedure of the calculation of requirements for the cooperating vehicles and
automated vehicle control presented in KKl2022_Pred  are general ones. This
means that the microscopic prediction is independent of whether right or left
traffic direction is considered. The difference of the scenario $``$left
turning” (Fig. 14(b)) in comparison with the scenario $``$right turning” (Fig.
2(b)) is as follows: In Fig. 14, safety conditions (5) should be proven for
both the right and left traffic directions444It can occurs that rather than
the first of safety conditions (5) is not satisfied for both right and left
traffic directions (Fig. 14(b)), this safety condition is not satisfied only
for one of the directions. Then, only one cooperating vehicle should be chosen
for the traffic direction for which the first of safety conditions (5) is not
satisfied..

Thus, due to a general character of the microscopic traffic prediction
approach of KKl2022_Pred , the methodology presented in the paper can be used
for different traffic scenarios in which the prediction approach is
applicable. For this reason, the physics of the methodology for cooperative
driving could be understood from simulations of a simple scenario of right
turning (Fig. 2) as made in the paper. Further simulations of the general
methodology of cooperative driving for more complex scenarios can be very
interesting for traffic engineering that are out of the scope of the paper.

###  VI.4 Effect of Data Uncertainty on Prediction Reliability

Because measurements of microscopic traffic situations for unsignalized
intersections are not available, microscopic traffic situations have been
simulated with the same microscopic traffic flow model shortly discussed in
Sec. I.3. Moreover, we have assumed that vehicle speeds and locations in
microscopic traffic situations that are used as initial conditions for the
microscopic traffic prediction at time instants $t_{p},\ p=1,2,\ldots$ (see
formula (22) of KKl2022_Pred ) have no errors in comparison with real vehicle
speeds and locations of microscopic traffic situations.

As shown in KernerBook ; KernerBook2 , the Kerner-Klenov microscopic
stochastic traffic flow model used in all simulations in this paper is able to
simulate both microscopic and macroscopic measured (empirical) traffic data
with a sufficient accuracy555See also references to origin papers cited in the
books KernerBook ; KernerBook2  about the empirical proof of the Kerner-Klenov
microscopic stochastic traffic flow model.. Nevertheless, the following
question can arise: Is there a potential discrepancy between the predicted
vehicle trajectories and the actual trajectories. Indeed, in reality there can
be data latency errors and/or random errors in measurements of the vehicle
speeds and locations in microscopic traffic situations used in the microscopic
traffic prediction. We can call the errors the uncertainty of traffic data
used in the prediction model.

The related question how the robustness of the prediction model is maintained
in the face of significant prediction errors caused by the uncertainty of
traffic data has already been addressed in details in Secs. IV–VIII of
KKl2022_Pred . It has been found that the microscopic traffic prediction can
guarantee collision avoidance and safety traffic even when there is a
considerable data uncertainty caused by data latency, random errors of the
vehicle locations, and/or random errors of the vehicle speeds in the data of
microscopic traffic situations. However, there is a critical uncertainty,
i.e., the maximum amplitude of errors in the data of microscopic traffic
situations at which the microscopic traffic prediction can still reliably be
used for the automated vehicle control. There is probability $P_{\rm app}$ of
the reliability of the application of the microscopic traffic prediction: when
the data uncertainty does not exceed the critical uncertainty, then
probability $P_{\rm app}=$ 1 KKl2022_Pred . In this case, the microscopic
traffic prediction is applicable for automated vehicle control.

###  VI.5 Conclusions

1\. We have introduced a methodology for cooperative driving in vehicular
traffic based on time-limited microscopic traffic prediction that is performed
based on the microscopic traffic modeling. Contrary to most approaches based
on statistical algorithms of AI, no historical database is required for our
approach.

2\. With the use of microscopic traffic simulations of a simple city traffic
scenario, we have proven the availability of the introduced methodology for
the cooperative driving and studied its physical features.

3\. Cooperative driving based on microscopic traffic prediction enables the
control of the automated driving vehicle in mixed traffic in a complex city
traffic scenario leading to a speed harmonization and, therefore, to increase
in traffic comfort and safety in city traffic.

4\. Due to a general character of the microscopic traffic prediction approach
of KKl2022_Pred , the methodology presented in the paper can be used for
different traffic scenarios in which the prediction approach is applicable.

5\. If the rate of communicating and cooperative vehicles in traffic flow
becomes significantly less than 100%, the possibility of cooperative driving
is likely to be very limited.

###### Acknowledgements.

We thank our partners for their support in the project $``$LUKAS – Lokales
Umfeldmodell für das Kooperative, Automatisierte Fahren in komplexen
Verkehrssituationen” funded by the German Federal Ministry for Economic
Affairs and Climate Action.

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