# Optimal Trajectory Planning for Connected and Automated Vehicles in Lane-
free Traffic with Vehicle Nudging

Venkata Karteek Yanumula, Panagiotis Typaldos, Dimitrios Troullinos, Milad
Malekzadeh, Ioannis Papamichail, Markos Papageorgiou Manuscript created May,
2022.This research received funding from the European Research Council under
the EU Horizon 2020 Programme / ERC Grant no. 833915, project TrafficFluid,
see: https://www.trafficfluid.tuc.grAll the authors are with Dynamic Systems
and Simulation Laboratory, Technical University of Crete, Chania, Crete,
73100, Greece, e-mail: {karteek, ptypaldos, dtroullinos, mmalek, ipapa,
<EMAIL_ADDRESS>Karteek Yanumula is with Electrical and
Instrumentation Engineering Department, Thapar Institute of Engineering and
Technology, Patiala, India.Markos Papageorgiou is also with Faculty of
Maritime and Transportation, Ningbo University, Ningbo, China.

###### Abstract

The paper presents a movement strategy for Connected and Automated Vehicles
(CAVs) in a lane-free traffic environment with vehicle nudging by use of an
optimal control approach. State-dependent constraints on control inputs are
considered to ensure that the vehicle moves within the road boundaries and to
prevent collisions. An objective function, comprising various weighted sub-
objectives, is designed, whose minimization leads to vehicle advancement at
the desired speed, when possible, while avoiding obstacles. A nonlinear
optimal control problem (OCP) is formulated for the minimization of the
objective function subject to constraints for each vehicle. A computationally
efficient Feasible Direction Algorithm (FDA) is called, on event-triggered
basis, to compute in real-time the numerical solution for finite time-horizons
within a Model Predictive Control (MPC) framework. The approach is applied to
each vehicle on the road, while running simulations on a lane-free ring-road,
for a wide range of vehicle densities and different types of vehicles. From
the simulations, which create myriads of driving episodes for each involved
vehicle, it is observed that the proposed approach is highly efficient in
delivering safe, comfortable and efficient vehicle trajectories, as well as
high traffic flow outcomes. The approach is under investigation for further
use in various lane-free road infrastructures for CAV traffic.

###### Index Terms:

Lane-free traffic, optimal control, trajectory planning, automated vehicles.

##  I Introduction

CAVs (Connected Automated Vehicles) are expected to revolutionize the
transportation sector, improving on traffic safety and efficiency by
overcoming the limitations of human driving. The current driving pattern
followed by human drivers is resulting in underuse of the road infrastructure,
excessive fuel consumption and reduced safety [1]. Human perception of the
surroundings is limited compared to the wide range of sensors [2] that are
available onboard CAVs. In these circumstances, accidents and congestion are
regular plaguing occurrences across the world, calling for comprehensive and
radical solutions. One potential solution would be introducing CAVs on the
roads with appropriate movement strategies. However, due to the strongly
dynamic nature of road traffic in both longitudinal and lateral directions,
complex algorithms, like optimal model predictive control, nonlinear feedback
control, reinforcement learning, and artificial intelligence, are necessary
[3, 4, 5, 6, 7]. With appropriate sharing of information via Vehicle-to-
Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communication, CAVs may be
enabled to drastically alleviate the problems of safety and congestion [8, 9].

Traffic lanes were introduced to aid the human driving, as they minimize the
need for lateral and rear monitoring; albeit with an exception during lane-
changing, which calls for simultaneous multi-direction monitoring, prediction
of multiple vehicle movements and fast decision making. Around 10% of all road
accidents occur while performing the lane-change by human drivers [10].
Besides, due to diverse vehicle dimensions, lanes are designed to fit the
widest vehicles in regular traffic, which are big trucks. Thus, a great deal
of lateral space is left unused when smaller vehicles are using the same lane
structure. At traffic level, lane-changes lead to capacity losses and are
known to trigger traffic breakdown at critical traffic conditions.

Automated driving systems that mimic human driving have been using several
approaches to generate the lane-changing maneuvers [11, 12], which are complex
procedure even for CAVs. For example, optimal control techniques within a
model predictive control (MPC) scheme have been proposed for lane-based
trajectory planning, including lane-changing [13, 14]. However, with the large
amount, quality and range of perception data available from various sensors,
with V2V and V2I communications, and with the development of efficient
trajectory planning algorithms, CAVs may drive without consideration of lane
structure on the roads. Moreover, several other traffic rules, designed for
human drivers, can be relaxed for CAVs, without compromising on safety, while
improving the efficiency of traffic.

For future traffic, dominated by fully automated vehicles, there is no
apparent reason to necessarily employ human-inspired driving restrictions,
such as lane-based driving or accounting of downstream vehicles only (as in
conventional car-following). Some works (e.g. [15]) proposed CAV lane-free
crossing of urban junctions, see [16] for a brief historical review. A general
traffic paradigm is proposed in [17], characterized by lane-free CAV driving
and vehicle “nudging”. In fact, CAVs have sufficient sensors to continuously
scan the $360^{\circ}$ surroundings reliably and fast. Thus, the CAVs, backed
with appropriate trajectory planning algorithms, may move on a lane-free road,
like particles constituting an artificial fluid, whereby vehicle movement may
be influenced not only by other vehicles in front of them (as with human
driving), but also by vehicles around and behind, something that is referred
to as nudging and seems to have beneficial implications for traffic flow
capacity and stability [18]. Lane-free structure improves the usability of
lateral road space by the vehicles and has additional potential benefits, such
as increased capacity with incremental road widening over limited road length,
as well as flexible (in space and time) incremental sharing of the total road
capacity among the two traffic directions in real-time [19]. Using appropriate
algorithms, the traffic on motorways, highways, arterials, and urban roads
could be freed from the obligation to drive on lanes and regain the lost
capacity without compromising on safety. The discontinuous lateral
displacements from one lane to another become obsolete, and smooth lateral
vehicle movement may be employed for overtaking and other driving tasks.

This article develops a nonlinear constrained Optimal Control Problem (OCP),
with both fixed and state-depended bounds on control inputs, for the movement
of CAVs in lane-free traffic involving vehicle nudging. Thanks to the lane-
free vehicle movement, real-valued displacements on the lateral direction are
considered in the OCP, in contrast to the introduction of discrete variables
engaged in most lane-based optimal trajectory planning approaches.

Due to the highly dynamic nature of traffic and the related uncertainty
increase with time, relatively short planning time-horizons (8 s) are
considered within an MPC framework with receding time-horizon. MPC is a time-
tested approach with applications in various fields [20]. In the present
study, an efficient Feasible Direction Algorithm (FDA) [21] is employed to
solve the OCP problem within an event-based MPC framework [22]. It should be
noted that the optimal trajectories of 8-s duration are computed in 12.4 ms of
CPU time on average, which allows for convenient application in real-time.

A lot of developments are ongoing in research and industry to enable reliable
V2I and V2V information sharing in real-time [23, 24, 25]. In the present
work, it is assumed that there is sufficient communication among adjacent
vehicles within a limited range, both upstream and downstream of each ego
vehicle (EV), which enables vehicles to share information related to their
position, speed, and the latest calculated optimal trajectories.

In the present work, the OCP is solved independently for hundreds of vehicles,
driving on a ring-road, based on event-triggered MPC for each vehicle, in
emulated real-time. This allows for investigation of both vehicle-level and
traffic-level implications of the proposed vehicle movement strategy. It is
found that a fundamental diagram (flow-density curve), with a typical
inverse-U shape, emerges at traffic level. It is also observed that the
vehicles move efficiently, smoothly and safely over the full range of possible
vehicle densities, while maintaining passenger convenience. Emerging flow
values are significantly higher than in conventional lane-based traffic.

A preliminary version of the approach with limited evaluation results was
reported in [4]. The concept of lane-free roads and vehicle trajectory
planning using optimal control within a nonlinear MPC framework is also
considered in [26], which features similarities, but also significant
differences, compared to [4] and the present work, with respect to the OCP
formulation, the numerical solution, the considered MPC procedures and the
application focus.

The major contributions of the article are as listed below;

  * •

Development and efficient numerical solution of an optimal control approach
for vehicle movement on a lane-free roads with nudging.

  * •

Comprehensive testing of the trajectory planning approach for hundreds of
vehicles, involved in myriads of driving circumstances.

  * •

Traffic-level simulation-based evaluation for a wide range of densities,
demonstrating sensible improvements.

The rest of the article is organized as follows: the kinematic model of the
vehicle dynamics for each EV is discussed in Section II-A, while constraints
on lateral and longitudinal accelerations are designed in Section II-B. The
OCP formulation, numerical solution and employed MPC features are detailed in
Section III. Simulation environment, vehicle-level and emerging traffic-level
results and related discussion are given in Section IV. Finally, the
concluding remarks are provided in Section V.

##  II Vehicle Model and Constraints

This section describes the mathematical vehicle model, along with the fixed
and state-dependent control constraints.

###  II-A Vehicle Model

A discrete-time vehicle model is considered, governed by discrete-time control
inputs. In the present work, vehicles are moving on an unfolded straight ring-
road, with small lateral speed compared to longitudinal speed. Therefore, the
heading angles of the vehicles are small, and, consequently, each vehicle is
represented by two decoupled double integrators in discrete-time, one for each
of the longitudinal and lateral directions. As a precaution, a specific term
is considered in the objective function to discourage infeasible heading
angles (see Section III-A4). Furthermore, the lateral movement uses real-
valued states thanks to the lane-free road structure, which allows for the
vehicles to be positioned within the road boundaries continuously in the
lateral direction. The mathematical model of each EV in the form of state
equations is:

$\displaystyle x_{1}(k+1)$
$\displaystyle=x_{1}(k)+Tx_{3}(k)+\frac{1}{2}T^{2}u_{1}(k)$ (1a)
$\displaystyle x_{2}(k+1)$
$\displaystyle=x_{2}(k)+Tx_{4}(k)+\frac{1}{2}T^{2}u_{2}(k)$ (1b)
$\displaystyle x_{3}(k+1)$ $\displaystyle=x_{3}(k)+Tu_{1}(k)$ (1c)
$\displaystyle x_{4}(k+1)$ $\displaystyle=x_{4}(k)+Tu_{2}(k)$ (1d)

where $x_{1}$ and $x_{2}$ are longitudinal and lateral positions,
respectively; $x_{3}$ and $x_{4}$ are speeds in longitudinal and lateral
directions, respectively; $k$ is the discrete-time integer index; and $T$ is
the discrete-time step. The index $k$ relates the step $T$ and continuous time
$t$ through $t=kT$. The control inputs $u_{1}$ and $u_{2}$ are the vehicle
accelerations in longitudinal and lateral directions, respectively. Both
control inputs are bounded, as discussed in section II-B. It should be noted
that the state equation above is the exact discrete-time version of the
corresponding continuous-time double-integrator movement equations, if the
control inputs are held constant for the duration $T$ of each time interval.

###  II-B Fixed and State-dependent Bounds on Control

In view of limited vehicle capabilities, passenger comfort considerations, and
the need to respect the road boundaries, the control inputs (longitudinal and
lateral accelerations) must be bounded appropriately.

####  II-B1 Longitudinal Bounds on Control

A state-dependent lower limit on the longitudinal acceleration is obtained
from the requirement for non-negative speed at the next time-step $k+1$; i.e.,
we have the requirement

$x_{3}(k+1)\geq 0.$ (2)

Using 1c, 2 yields $u_{1}(k)\geq-\frac{1}{T}x_{3}(k)$, hence a state-dependent
lower limit on longitudinal acceleration is
$U_{\text{min1}}(k)=-\frac{1}{T}x_{3}(k)$. In addition, the vehicle has
limited deceleration capabilities, hence a constant limit $U_{\text{min1}}$ is
also considered, and the overall lower limit of the longitudinal acceleration
is

$u_{\text{min1}}(k)=\text{max}\left\\{U_{\text{min1}}(k),U_{\text{min1}}\right\\}.$
(3)

Regarding the upper limit on the longitudinal acceleration, there are also two
kinds of bounds to account for, a constant bound and a state-dependent bound.
The constant upper bound $U_{\text{max1}}$ reflects the vehicle’s acceleration
capabilities; this bound is considered constant, but it could be readily
expressed as a function of longitudinal speed for more realism.

On the other hand, the state-dependent upper bound is used only in rare
emergency situations, as described later (Section III-D). For now, it suffices
to mention that, in such cases, an upper bound on the longitudinal
acceleration is obtained from the requirement of following the obstacle
vehicle downstream at a prespecified distance $\epsilon$. A corresponding
longitudinal position limit $\widehat{x}_{1}(k)$ behind the rear bumper of the
obstacle is defined, and the EV must not cross this limit. This longitudinal
position limit may be viewed as a moving boundary; Indeed, the longitudinal
position limit moves with the vehicle dynamics of the obstacle. Consider the
following dynamics for the moving boundary in the form of state equations,
where all $\widehat{\cdot}$ variables have similar meanings as in 1,

$\displaystyle\widehat{x}_{1}(k+1)$
$\displaystyle=\widehat{x}_{1}(k)+T\widehat{x}_{3}(k)+\frac{1}{2}T^{2}\widehat{u}_{1}(k)$
(4a) $\displaystyle\widehat{x}_{3}(k+1)$
$\displaystyle=\widehat{x}_{3}(k)+T\widehat{u}_{1}(k).$ (4b)

The EV, positioned behind the moving boundary at $x_{1}(k)$ and $x_{1}(k+1)$
should not cross the moving boundary at time step $k+2$ i.e. we must have

$\displaystyle x_{1}(k+2)\leq\widehat{x}_{1}(k+2).$ (5)

Note that, if the EV reaches the moving boundary, and constraint 5 is
activated, then the longitudinal speed of the EV should be equal to that of
the moving boundary, otherwise violation of 5 cannot be avoided eventually. By
substituting 1a, 1c and 4 in 5 and with some rearrangement, the following
upper limit on the longitudinal acceleration is obtained

$\displaystyle
u_{1}(k)\leq-\frac{1}{T^{2}}\left[x_{1}(k)-\widehat{x}_{1}(k)\right]-\frac{3}{2T}\left[x_{3}(k)-\widehat{x}_{3}(k)\right]+\widehat{u}_{1}(k).$
(6)

Defining the tracking errors

$\displaystyle e_{1}(k)$ $\displaystyle=x_{1}(k)-\widehat{x}_{1}(k)$ (7a)
$\displaystyle e_{2}(k)$ $\displaystyle=x_{3}(k)-\widehat{x}_{3}(k)$ (7b)

the right-hand side of 6 may be viewed as a tracking controller. In fact, if
the upper limit in 6 is activated and continuously applied, the EV’s position
and speed reach the moving boundary’s position and speed in exactly two time-
steps, i.e., the tracking controller exhibits a dead-beat behaviour. However,
the magnitude of the longitudinal acceleration required for such a dead-beat
behaviour could be very high and might violate the constant acceleration
bounds.

The required magnitude of the longitudinal acceleration can be moderated by
choosing appropriate feedback gain values $0<K_{\text{long1}}\leq 1/T^{2}$ and
$0<K_{\text{long2}}\leq 3/(2T)$. In place of 6, consider the maximum
longitudinal control input of EV as

$\displaystyle
u_{\text{max1}}(k)=-K_{\text{long1}}e_{1}(k)-K_{\text{long2}}e_{2}(k)+\widehat{u}_{1}(k).$
(8)

The choice of $K_{\text{long1}}$ and $K_{\text{long2}}$ should result in
aperiodic tracking of the set-points, when the limit is active, such that the
EV reaches the required position and speed asymptotically, reducing the
tracking error to zero without any overshooting. To choose appropriate gain
values for this behaviour, the following error dynamics are obtained using 1a,
1c, 4 and 8, when the limit is active, i.e., when
$u_{1}(k)=U_{\text{max1}}(k)$,

$\displaystyle\begin{bmatrix}e_{1}(k+1)\\\ e_{2}(k+1)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1-\frac{1}{2}K_{\text{long1}}T^{2}&T-\frac{1}{2}K_{\text{long2}}T^{2}\\\
-K_{\text{long1}}T&1-K_{\text{long2}}T\end{bmatrix}\begin{bmatrix}e_{1}(k)\\\
e_{2}(k)\end{bmatrix}.$ (9)

For stability and aperiodic behaviour, the gain values $K_{\text{long1}}$ and
$K_{\text{long2}}$ should ensure that the roots of the characteristic equation
of 9 are inside the unit circle (for stability) and non-negative real (for
aperiodic behaviour). The characteristic equation for 9 is given by

$\displaystyle z^{2}+$
$\displaystyle\left(\frac{K_{\text{long1}}T^{2}}{2}+TK_{\text{long2}}-2\right)+$
$\displaystyle\left(\frac{K_{\text{long1}}T^{2}}{2}-TK_{\text{long2}}+1\right)=0.$
(10)

To keep the design procedure simple, we choose
$K_{\text{long2}}=2\sqrt{K_{\text{long1}}}-K_{\text{long1}}T/2$, for which the
roots of the characteristic equation are non-negative real and identical. The
resulting feedback values (longitudinal accelerations) are kept moderate by
appropriately choosing the gain value $K_{\text{long1}}$ within the mentioned
range.

The emergency planning to avoid longitudinal collision will always result in
an upper limit on acceleration that is less than that of constant bound
$U_{\text{max1}}$, which is used in regular planning. Thus, the upper limit on
the longitudinal acceleration is either $u_{\text{max1}}(k)=U_{\text{max1}}$
or $u_{\text{max1}}(k)=U_{\text{max1}}(k)$ for regular or emergency planning,
respectively.

####  II-B2 Lateral Bounds on Control

The main requirement on the lateral direction is for the vehicle to stay
inside the road boundaries, or to drive exactly on a road boundary, without
violating them. In addition, some imaginary boundaries may be imposed in the
rare emergency cases. In both cases, the boundaries are stationary straight
lines in this study, while other shapes of road boundaries are currently under
investigation. The imaginary boundaries in the emergency cases are defined so
as to prevent collisions on lateral direction, and related details are
provided in Section III-D.

Using a similar approach as in Section II-B1, the EV, positioned within the
road boundaries at time steps $k$ and $k+1$, should also stay within the road
boundaries at time step $k+2$, i.e., we must have

$\displaystyle\widetilde{x}_{2}(k+2)\leq x_{2}(k+2)\leq\widehat{x}_{2}(k+2)$
(11)

where $\widehat{x}_{2}$ is the lateral position of the left road boundary
(equal to the road width minus half of the EV width); and $\widetilde{x}_{2}$
is the lateral position of the right road boundary (equal to half of the EV
width). Note that, if the vehicle actually reaches the left or right road
boundary, i.e., if the left or right constraint above is activated, then we
must have for the lateral vehicle speed $x_{4}\left(k+2\right)=0$, as
otherwise the vehicle would eventually exit the road. Thus, compared with
Section II-B1, we have, for fixed straight boundaries, the simpler case
that$\widehat{u}_{2}\left(k\right)=0$,
$\widehat{x}_{4}\left(k+1\right)=\widehat{x}_{4}\left(k\right)=0$,
$\widehat{x}_{2}\left(k+1\right)=\widehat{x}_{2}\left(k\right)=\widehat{x}_{2}$,
$\widetilde{u}_{2}\left(k\right)=0$,
$\widetilde{x}_{4}\left(k+1\right)=\widetilde{x}_{4}\left(k\right)=0$, and
$\widetilde{x}_{2}\left(k+1\right)=\widetilde{x}_{2}\left(k\right)=\widetilde{x}_{2}$.
By replacing 1b and 1d in 11 and after some rearrangements, we have that the
above constraints are equivalent to the following state-dependent inequalities
for the lateral acceleration

$\displaystyle-\frac{1}{T^{2}}[x_{2}(k)$
$\displaystyle-\widetilde{x}_{2}(k)]-\frac{3}{2T}x_{4}(k)\leq u_{2}(k)$
$\displaystyle\leq-\frac{1}{T^{2}}\left[x_{2}(k)-\widehat{x}_{2}(k)\right]-\frac{3}{2T}x_{4}(k)$
(12)

Similarly to 6, either side of section II-B2 can be considered, if taken as an
equality, as a respective dead-beat controller that drives the EV towards the
corresponding road boundary and its lateral speed to zero in two time steps.
Also here, the magnitudes of the resulting lateral accelerations produced by
such a controller may take unrealistically high values and may cause
discomfort to the passengers. Again, we can design more moderate controller
gains, such that the lateral acceleration is feasible and comfortable. Thus,
similarly to the generalization in 8, we have the state-dependent bounds on
lateral acceleration given by

$\displaystyle
U_{\text{min2}}(x_{2}(k),x_{4}(k))=-K_{\text{lat1}}[x_{2}(k)-\widetilde{x}_{2}(k)]-K_{\text{lat2}}x_{4}(k)$
(13a) $\displaystyle
U_{\text{max2}}(x_{2}(k),x_{4}(k))=-K_{\text{lat1}}[x_{2}(k)-\widehat{x}_{2}(k)]-K_{\text{lat2}}x_{4}(k)$
(13b)

where $0<K_{\text{lat1}}\leq 1/T^{2}$ and $0<K_{\text{lat2}}\leq 3/(2T)$ are
feedback controller gains, chosen by following a similar procedure as in
Section II-B1 and setting
$K_{\text{lat2}}=2\sqrt{K_{\text{lat1}}}-K_{\text{lat1}}T/2$ for two identical
non-negative real poles, so that we achieve asymptotic behaviour with moderate
lateral accelerations. Clearly, 13 apply to either the road boundaries or to
imaginary lateral boundaries (in emergency cases), with $\widehat{x}_{2}$ and
$\widetilde{x}_{2}\left(k\right)$ set accordingly.

It is interesting to note that, in a lane-free traffic environment,
guaranteeing that a vehicle will not violate the road boundaries is more
important than in a lane-based environment, for two reasons. First, lane-based
vehicles are typically guided around the middle of each lane, and therefore
the risk of departing from the road is less serious. Second, in lane-free
driving, there is a strong interest, e.g., at dense traffic, to let some
vehicles drive _exactly_ on a (left or right) road boundary (of course without
ever violating it), so as to maximize the usage of the available road width.
Note that exact road boundary tracking, under lateral pressure (nudging) by
laterally adjacent vehicles, e.g. due to high traffic density, is hardly
possible by use of related penalty or barrier terms in the optimisation
objective, as in lane-based driving [14].

These two endeavours of lane-free driving are fulfilled via the above two
boundary controllers, one for the left and another for the right road
boundary, which deliver appropriate upper and lower bounds, respectively, for
the lateral acceleration. As long as such a bound is not activated, the
vehicle drives according to other optimal control criteria (see Section
III-A); if one of the bounds is activated, it was designed above to navigate
the vehicle asymptotically to the corresponding road boundary and then have it
driving exactly on the road boundary, for as long as the lateral acceleration
is activating that bound.

For consistency of notation, we define the lower and upper limits as
$u_{\text{min2}}\left(k\right)=U_{\text{min2}}(k)$ and
$u_{\text{max2}}\left(k\right)=U_{\text{max2}}(k)$ respectively.

In summary, the bounds on accelerations given in 3, 8 and 13 are expressed by
the following state-dependent constraints, to be considered in the OCP

$\displaystyle h_{1}$
$\displaystyle=[u_{1}(k)-u_{\text{max1}}(k)][u_{1}(k)-u_{\text{min1}}(k)]\leq
0$ (14a) $\displaystyle h_{2}$
$\displaystyle=[u_{2}(k)-u_{\text{max2}}(k)][u_{2}(k)-u_{\text{min2}}(k)]\leq
0.$ (14b)

##  III Optimal Control Problem (OCP)

The OCP considered in this work minimizes the weighted sum of multiple sub-
objectives for the EV advancement in an efficient, safe and comfortable
manner. This section details the objective function, followed by the
formulation of the discrete-time OCP to be solved numerically. The section
also presents the FDA for numerical solution of the OCP and the MPC procedure
employed in real time.

###  III-A Objective Function

The objective function is composed by several sub-objectives that are designed
to mitigate discomfort for the passengers while applying accelerations on
longitudinal and lateral directions, to reach desired speeds, to avoid dynamic
obstacles and more. For application of the numerical algorithm, each of these
sub-objectives must be continuous and differentiable.

####  III-A1 Fuel Consumption and Passenger Comfort

Mitigation of fuel consumption results in accordingly reduced emissions and
environment-friendly CAV traffic. This is achieved by a quadratic cost term of
longitudinal acceleration $\left(u_{1}\left(k\right)\right)^{2}$ as
demonstrated in [27]. In addition, quadratic cost terms of longitudinal and
lateral accelerations $\left(u_{1}\left(k\right)\right)^{2}$ and
$\left(u_{2}\left(k\right)\right)^{2}$, respectively, result in moderate and
smooth accelerations and thereby improved passenger comfort.

####  III-A2 Desired Speed

Reaching desired speeds on both longitudinal and lateral directions is
essential for vehicle advancement and reaching of an exit. The road considered
in this study is an unfolded ring-road, requiring a positive longitudinal
desired speed and a zero lateral desired speed. The quadratic cost term
$\displaystyle\left[x_{3}(k)-v_{d_{1}}\right]^{2}$ with longitudinal desired
speed $v_{d_{1}}$ results in the EV trying to reach its desired speed.
Clearly, vehicles moving near their respective desired speeds reduce the
travel time of passengers. Unnecessary lateral movements can reduce traffic
efficiency, hence the quadratic cost term
$\displaystyle\left[x_{4}(k)-v_{d_{2}}\right]^{2}$, here with $v_{d_{2}}=0$ as
lateral desired speed, mitigates lateral movement. In ongoing work with
highways including on-ramps and off-ramps, non-zero lateral desired speeds may
be appropriately used for efficient vehicle merging on or departing from the
mainstream.

####  III-A3 Obstacle Avoidance

For obvious safety reasons, the EV should avoid collisions with other vehicles
that are treated as moving obstacles. This is an important requirement that
must be reflected in a carefully designed sub-objective with appropriate
consideration of the novel aspects stemming from the lane-free vehicle
movement, as well as from the possibility of vehicle nudging.

To avoid collisions, while generating the trajectory of the EV, it is
necessary that the current states (position and speed) of surrounding vehicles
are available via the EV sensors or via V2V communication. In addition, each
vehicle shares its generated trajectory to surrounding vehicles. Naturally,
short-term prediction of the movement of surrounding vehicles may have limited
accuracy, and this may trigger a new path generation, see Section III-D for
details.

While solving the OCP for finite-time horizons, an interaction zone (IZ) is
considered, and all the other vehicles inside the IZ are treated as obstacles.
The said IZ for each planning horizon has two equal-length parts, one
downstream and the other upstream of the EV, the latter serving the purpose of
enabling vehicle nudging, i.e., the possibility for upstream vehicles to
influence the movement of vehicles downstream. The length of each of these
parts is equal to the product of longitudinal desired speed times the planning
horizon duration. Since vehicles are assumed to broadcast their trajectory
decisions to other surrounding vehicles, each EV has such trajectory
information of all the obstacles in its IZ.

Maintaining longitudinal and lateral safety gaps from surrounding obstacles is
necessary in lane-free traffic. In lane-based driving, a longitudinal safety
gap to the front vehicle on the same lane, which is proportional to the EV
speed, is generally employed [28], and this is termed as constant time-gap
policy. In this work, an imaginary positive-valued, smooth, longitudinally and
laterally symmetric ellipsoid hemisphere around each obstacle is carefully
designed, which determines the cost incurred if the EV is positioned within
the ellipsoid range. This cost is aimed to discourage the EV from intruding
the ellipsoid range, i.e., from approaching too closely (or colliding with)
the obstacle. While designing the ellipsoid hemisphere, time-gap like safety
parameters $\left(\omega_{1},\omega_{2}\right)$ are considered in longitudinal
and lateral directions, respectively, along with the obstacle’s corresponding
physical dimensions. Assuming $n$ obstacles inside the IZ, we define for the
$i^{\mathrm{th}}$ obstacle the longitudinal and lateral positions
$\left(o_{i1},o_{i2}\right)$ and speeds $\left(o_{i3},o_{i4}\right)$.

The length and position (centre) of the ellipsoid in the longitudinal
direction are specified to satisfy the following requirements:

  * •

Physical dimensions of both ego and obstacle vehicles should be completely
covered by the ellipsoid, even for zero speed. We define
$L_{i}=\mu_{x}\left(l_{e}+l_{oi}\right)$, where $l_{e}$ and $l_{oi}$ are the
lengths of ego and obstacle vehicles, respectively, and $\mu_{x}$ is a
coefficient. When both ego and obstacle vehicles are at zero speed, the length
of the longitudinal axis of the ellipsoid equals $L_{i}$, hence $\mu_{x}$
ensures that vehicles are not bumper-to-bumper at stillstand.

  * •

If the EV is positioned behind the obstacle, a safety gap, proportional to its
current speed, should be considered (in addition to $L_{i}$), according to the
constant time-gap policy. Thus, a safety gap equal to $\omega_{1}x_{3}$ should
be maintained in front of the EV.

  * •

If the EV is positioned ahead of the obstacle, a safety gap proportional to
the current speed of the obstacle should be considered (in addition to
$L_{i}$). Thus, a safety gap equal to $\omega_{1}o_{i3}$ should be maintained
behind the EV. This is useful for preventing the EV from cutting-in
dangerously close in front of an obstacle. It is interesting to note that the
consideration of a safety gap not only in front, but also behind the EV is
creating a longitudinal “nudging” to the EV due to a moving obstacle behind
it. More generally, due to this gap, vehicle driving is not influenced only by
obstacles in front, as in manual driving or in most lane-based automated
vehicle concepts, but also by obstacles behind them, something that gives rise
to structurally new traffic flow characteristics at the macroscopic level [18,
29].

  * •

Due to the different length of the safety gaps in front and behind the EV,
which depend on the EV speed and the following obstacle speed, respectively,
the positioning of the centre of the symmetric ellipsoid in the longitudinal
direction depends on the speed difference of the ego and obstacle vehicles.
The ellipsoid centre coincides with the obstacle centre if both vehicles have
the same speed; but the ellipsoid centre is shifted longitudinally, to reflect
properly the above safety gap specifications, if the speeds of the ego and
obstacle vehicles are different.

All mentioned requirements are fulfilled, if the length of the longitudinal
ellipsoid axis is equal to $d_{1}=L_{i}+\omega_{1}x_{3}+\omega_{1}o_{i3}$,
while the ellipsoid’s longitudinal centre is positioned at

$\delta_{oi}=o_{i1}-\omega_{1}\left(x_{3}-o_{i3}\right)/2$ (15)

Consideration of the lateral direction in collision avoidance is of paramount
importance in lane-free driving, in contrast to lane-based driving, where
lateral vehicle collisions are not an issue (except in lane changing
manoeuvres). In the lateral direction, the safety gap between the vehicles is
necessary only when they are approaching each other. The safety gap considered
is proportional to the absolute value of speed difference and is equal to
$\omega_{2}\left|x_{4}-o_{i4}\right|$, when ego and obstacle vehicles are
approaching each other, else it is equal to 0. A smooth approximation of the
lateral safety gap, accounting for the mentioned specifications, is

$\displaystyle d_{2}=W_{i}+$
$\displaystyle\omega_{2}\Big{[}\tanh(o_{i2}-x_{2})\left(x_{4}-o_{i4}\right)$
$\displaystyle+\sqrt{\left[\tanh(o_{i2}-x_{2})\left(x_{4}-o_{i4}\right)\right]^{2}+\epsilon_{w}}\Big{]}$
(16)

where $\epsilon_{w}$ has a small value and
$W_{i}=\mu_{y}\left(w_{e}+w_{oi}\right)$ with $w_{e}$ and $w_{oi}$ the widths
of ego and obstacle vehicles, respectively. A coefficient $\mu_{y}$ slightly
greater than 1 is considered to ensure that the corners of the (rectangular)
vehicles are sufficiently covered by the ellipsoid, so as to suppress corner-
corner collisions. There is an interest that $\mu_{y}$ is not much bigger than
1, as this would waste lateral space and hence reduce the resulting traffic
flow capacity. The lateral midpoint of the ellipsoid coincides with the centre
of the obstacle in lateral direction. Note that the lateral dimension of the
ellipsoid, in particular the lateral safety gaps, create also a lateral
“nudging” to the EV due to other, longitudinally close, vehicles around it, if
their lateral distance from the EV is diminishing, i.e., if both vehicles are
approaching each other laterally.

The ellipsoid function, which covers sufficiently physical vehicle dimensions
and safety gaps, is essentially a potential function, whose value smoothly
increases as the EV approaches to the obstacle (or vice-versa), and acts as a
collision avoidance term in the objective function, separate for each obstacle
within the EV’s IZ. Besides the need for obstacle avoidance, the shape of the
ellipsoid should satisfy some additional requirements that stem from our
interest in achieving high traffic flow capacity, efficient vehicle
manoeuvring, as well as from the gradient-based nature of the numerical
solution algorithm. Specifically, we have the following two requirements for
the ellipsoid function:

  1. (i)

The shape of the ellipsoid’s iso-cost curves (contours) around the obstacle
should be sufficiently rectangular to cover the obstacle’s corners
(particularly at low speeds where safety gaps are short), mitigating the need
for larger coefficient $\mu_{y}$, which would waste lateral space and
capacity. On the other hand, the contour shape should be sufficiently “curvy”
at both longitudinal edges, so that the EV, guided by the ellipsoid gradient,
may “slide” around an encountered slower obstacle in front of it; rather than
being stuck behind the obstacle due to a quasi-straight contour edge.

  2. (ii)

The ellipsoid’s three-dimensional shape should be sufficiently steep, i.e.,
its gradient should be sufficiently far from zero, at all points of its range,
to facilitate faster convergence of the numerical solution algorithm to
collision-free trajectories.

While many functions may satisfy the above requirements, the function selected
here for the ellipsoid reads

$\displaystyle c_{i}(\boldsymbol{x},\ \boldsymbol{o}_{i})=$
$\displaystyle\left\\{1-\tanh\left[\left(\frac{x_{1}-\delta_{oi}}{0.5d_{1}}\right)^{p_{1}}+\left(\frac{x_{2}-o_{i2}}{0.5d_{2}}\right)^{p_{2}}\right]\right\\}$
$\displaystyle+\left\\{\left[\left(\frac{x_{1}-\delta_{oi}}{0.25d_{1}}\right)^{p_{3}}+\left(\frac{x_{2}-o_{i2}}{0.25d_{2}}\right)^{p_{4}}\right]^{p_{5}}+1\right\\}^{-1}$
(17)

where $\boldsymbol{x}$ and $\boldsymbol{o}_{i}$ are state vectors comprising
the four state variables of the ego and obstacle-$i$ vehicles, respectively.
The exponents $p_{1}$ to $p_{5}$ influence the ellipsoid shape. Specifically,
$p_{1}$ to $p_{4}$ take positive even integer values, while $p_{5}$ is a
positive integer. The function described in section III-A3 is a sum of two
ellipsoids and it is an extended version of the ellipsoid employed in [4].

Specifically, the first
$\left\\{1-\tanh\left[\left(\frac{x_{1}-\delta_{oi}}{0.5d_{1}}\right)^{p_{1}}+\left(\frac{x_{2}-o_{i2}}{0.5d_{2}}\right)^{p_{2}}\right]\right\\}$,
is designed such that its contours have a nearly flat lateral edge and a
curvy, sliding-prone longitudinal edge (see Fig. 1), in accordance with
requirement (i). This contour shape helps in accommodating more vehicles on a
given road width, while also allowing the vehicles to overtake, whenever
possible, thanks to the curvy shape and gradient on the longitudinal edge.
This is achieved by choosing a higher value of $p_{1}$ compared to $p_{2}$.
The drawback of this design (by itself) is that it does not fulfil requirement
(ii), as it creates a flat area over the top of the ellipsoid, with almost
zero gradient, which affects the convergence rate of the numerical solution
algorithm. This drawback may be repaired by adding a second ellipsoid,
$\left\\{\left[\left(\frac{x_{1}-\delta_{oi}}{0.25d_{1}}\right)^{p_{3}}+\left(\frac{x_{2}-o_{i2}}{0.25d_{2}}\right)^{p_{4}}\right]^{p_{5}}+1\right\\}^{-1}$,
which barely alters the opportune contours of the first ellipsoid around the
obstacle; while, with appropriate values of its exponents, it creates a strong
gradient of the potential function at all points up until the centre of the
ellipsoid, so that requirement (ii) is satisfied as well.

In summary, the sub-objective function in presence of obstacles is
$\sum_{i=1}^{n}\left[c_{i}\left(x,\ o_{i}\right)\right]$.

To illustrate the design, contour plots of the sub-objective section III-A3
for two obstacles, as seen by an EV, are shown in Fig. 1. All the three
vehicles have same dimensions of 4.25 m length and 1.8 m width. The EV is
positioned at (10, 5.5) m on a road of width 10.2 m, as shown in Fig. 1 with a
green box. The longitudinal and lateral speeds of the EV are 30 m/s and 0.75
m/s, respectively. Two obstacles 1 and 2 are positioned at (30, 2.5) m and
(40, 7.5) m with speeds (25, 0) m/s and (35, 0) m/s, respectively, and are
marked with cyan boxes. The combined dimensions of both ego and corresponding
obstacle vehicles are represented by black boxes. The time-gaps
$\left(\omega_{1},\ \omega_{2}\right)$ are (0.35, 0.5) s. Red boxes represent
the major and minor axes lengths $d_{1}$ and $d_{2}$ of section III-A3. The
safety coefficients $\mu_{x}$ and $\mu_{y}$ are 1.3 and 1.2, respectively.
Since the longitudinal speeds are different for each of the vehicles, the
position of the centre of the longitudinal axis of the ellipsoid is shifted
towards the EV for obstacle 1; and away from the EV for obstacle 2, according
to 15. It can also be observed that the width of the red box of obstacle 2 is
slightly larger that of obstacle 1, which is because the EV is approaching
obstacle 2, in accordance with section III-A3. The values of the exponents are
$p_{1}=6,p_{2}=p_{3}=p_{4}=p_{5}=2$. It can be observed that the contours are
flat at the lateral edge, facilitating better usage of lateral space; while
featuring strong curvature on the longitudinal edge, which facilitates smooth
overtaking by a faster EV. Furthermore, the displayed subsequent-contour
distances indicate a strong gradient up until the ellipsoid centre, where a
maximum value of function $c_{i}$ is reached. Finally, it may be seen that the
ellipsoid extends both upstream and downstream of each obstacle, hence it may
cause both repulsing and nudging influence on the EV.

Figure 1: Illustration of obstacle avoidance function

####  III-A4 Coupling of Longitudinal and Lateral Speeds

The longitudinal and lateral movements are assumed to be decoupled, according
to the vehicle dynamics in 1, which is largely a valid assumption in medium
and high longitudinal speeds without strong turning angles. In the case of
very low longitudinal speeds, however, there could be some infeasible lateral
movements, which would necessitate very high steering value for the real
vehicle. Such situations correspond to a high value of the rate of lateral
speed over longitudinal speed. To suppress such situations, a sub-objective
that penalizes the deviation of that rate, beyond a certain small parameter
$\beta$, as a quadratic function, is given as

$\displaystyle f_{c}=\begin{cases}\left[\beta
x_{3}(k)-|x_{4}(k)|\right]^{2}&\text{if }|x_{4}(k)|>\beta x_{3}(k)\\\
0&\text{otherwise}\end{cases}$ (18)

The sub-objective in 18 acts as an indirect coupling between longitudinal and
lateral speeds. It is noted that this helps in avoiding the usage of complex
nonlinear vehicle dynamics, which would be less efficient at obtaining a
numerical solution within a very small computation time.

####  III-A5 Deviation of Longitudinal Acceleration Between Planning Horizons

Each finite-time planning horizon, within the MPC scheme, can start with any
feasible longitudinal acceleration. This may result in a strong change of the
acceleration between the last applied value $u_{1\mathrm{prev}}$ and the first
value $u_{1}\left(0\right)$ of the new planning horizons. This can lead to
discomfort for the passengers and can be mitigated by penalizing the deviation
of the starting acceleration $u_{1}\left(0\right)$) of a planning horizon from
the last applied value of acceleration of the previous planning horizon
$u_{1\mathrm{prev}}$ , i.e.,

$\displaystyle f_{d}=\left[u_{1}(0)-u_{1\text{prev}}\right]^{2}$ (19)

###  III-B OCP Formulation and Necessary Conditions for Optimality

The OCP is defined as the minimization of the objective function, which is a
combination of all the sub-objectives discussed above, subject to the equality
constraints (state equations) in 1 and the inequality constraints (bounds on
control) in 14.

Summarizing Section III-A, we have the objective function

$\displaystyle J=$
$\displaystyle\sum_{k=0}^{K-1}\big{\\{}w_{1}\left(u_{1}(k)\right)^{2}+w_{2}\left(u_{2}(k)\right)^{2}+w_{3}\left(x_{3}(k)-v_{d_{1}}\right)^{2}$
$\displaystyle+w_{4}(x_{4}(k)-v_{d_{2}})^{2}+w_{5}\sum_{i=1}^{n}[c_{i}(\boldsymbol{x},\boldsymbol{o}_{i})]+w_{6}f_{c}\big{\\}}+w_{7}f_{d}$
(20)

where $K$ is the planning horizon; and $w_{1}$ to $w_{7}$ are weighting
factors of each of the sub-objectives to be chosen appropriately, so as to
reflect the relative importance and typical range of values of each sub-
objective. More specifically, the values of the weighting factors are decided
via a manual trial-and-error procedure. Using some initial weights, the
vehicle’s behaviour is observed and evaluated in preliminary simulations
involving multiple driving episodes; weights are increased or lowered if
corresponding sub-objectives are deemed to over-perform (to the detriment of
other sub-objectives) or to under-perform; and this is continued, until an
acceptable overall vehicle behaviour has been reached. Once appropriate
weights have been specified in this way, the sensitivity of optimal control
decisions with respect to different weight values of the same order is usually
low.

It should be noted that the use of an extremely high weight for a sub-
objective may lead to an accordingly ill-conditioned optimisation problem that
can hardly be solved via gradient-based numerical solution algorithms [30]. In
the present application, the most important sub-objective is the one referring
to collision avoidance. However, if the related weight $w_{5}$ is given a very
high value, so as to guarantee that a collision will literally never occur in
the produced optimal vehicle trajectory under any circumstances, then the
resulting OCP may be very difficult to solve numerically. Instead, one may
select a reasonably high value for $w_{5}$, which does not hinder the
numerical solution, but may, in very rare circumstances, result in a collision
trajectory. This is the approach pursued in this work. A possible produced
collision trajectory may be readily identified, in which case an emergency
situation is declared, and the OCP is reformulated with stricter constraints
to produce a collision-free vehicle trajectory according to Section III-D.

The general form of the objective function, which is used in the further
analysis, is given by

$J=\sum_{k=0}^{K-1}\Phi[\boldsymbol{x}(k),\boldsymbol{u}(k)]$ (21)

where $u$ is the control vector, comprising the longitudinal and lateral
accelerations. The general vector-based representation of 1 is

$\boldsymbol{x}(k+1)=\boldsymbol{f}[\boldsymbol{x}(k),\boldsymbol{u}(k)]$ (22)

The Hamiltonian function, including the equality 22 and inequality 14
constraints, is defined as

$\displaystyle
H[\boldsymbol{x}(k),\boldsymbol{u}(k),\boldsymbol{\lambda}(k+1),\boldsymbol{\mu}(k)]=\Phi[\boldsymbol{x}(k),\boldsymbol{u}(k)]+$
$\displaystyle\boldsymbol{\lambda}(k+1)^{T}\boldsymbol{f}[\boldsymbol{x}(k),\boldsymbol{u}(k)]+\boldsymbol{\mu}(k)^{T}\boldsymbol{h}[\boldsymbol{x}(k),\boldsymbol{u}(k)]$
(23)

where $\boldsymbol{\lambda}(k)$ are the co-states, associated with the state
equations, and $\boldsymbol{\mu}(k)$ are multipliers, associated with the
control constraints. Based on these, the necessary conditions of (local)
optimality, to be used in the numerical solution algorithm, are given next. We
have the state equation

$\displaystyle\boldsymbol{x}(k+1)=\frac{\partial
H}{\partial\boldsymbol{\lambda}(k+1)}=\boldsymbol{f}[\boldsymbol{x}(k),\boldsymbol{u}(k)].$
(24)

The control condition is given by

$\displaystyle\partial H/\partial\boldsymbol{u}(k)=\boldsymbol{0}.$ (25)

The co-state equation is given by

$\displaystyle\boldsymbol{\lambda}(k)=\partial H/\partial\boldsymbol{x}(k).$
(26)

The complementarity conditions are

$\displaystyle\mu_{i}(k)h_{i}[\boldsymbol{x}(k),\boldsymbol{u}(k)]=0,\;\mu_{i}(k)\geq
0$ (27)

Finally, the boundary conditions are given by

$\displaystyle\boldsymbol{x}(0)$ $\displaystyle=\boldsymbol{x_{0}}$ (28a)
$\displaystyle\boldsymbol{\lambda}(K)$ $\displaystyle=\boldsymbol{0}.$ (28b)

###  III-C Numerical Solution Algorithm

The solution of the described OCP must be obtained through an efficient
numerical solver that enables real-time feasibility. A very efficient
gradient-based feasible direction algorithm (FDA) [21, 31] is employed to
solve the present OCP. The underlying method leverages the explicit structure
of the state equations and uses the necessary conditions of optimality to map
the OCP into a Nonlinear Programming (NLP) problem in the reduced space of
control variables. Thus, the algorithm attempts to reach a control trajectory
$\boldsymbol{u}(k),k=0,\ldots,K-1$, which corresponds to a local minimum of
the cost function in the $mK$-dimensional space, where $m$ is the number of
control variables. This marks a substantial reduction of the problem
dimension, as the state variables are eliminated.

More specifically, FDA exploits the fact that
$\boldsymbol{g}(k)=\left[\partial\boldsymbol{f}/\partial\boldsymbol{u}(k)\right]^{\text{T}}\boldsymbol{\lambda}(k+1)+\partial\Phi/\partial\boldsymbol{u}(k)$
equals the reduced gradient in the $mK$-dimensional reduced space of the
control, if the states and co-states involved in this partial derivative
satisfy the state and co-state equations. Having this possibility to calculate
reduced gradients, FDA is an iterative procedure, starting with a feasible
initial-guess control trajectory specified by the user. The algorithm can be
readily extended to consider upper and lower (possibly state-dependent) bounds
on the control variables [21, 31].

Each FDA iteration uses the updated gradient $g\left(k\right)$ to compute an
appropriate descent direction (e.g., conjugate gradient or quasi-Newton
direction) and produce, based on line search, an improved feasible control
trajectory that reduces the objective function value, compared to the previous
iteration, while satisfying the state equations and control constraints. The
improved control trajectory is the starting point of the next iteration, and
so forth. The algorithm features global convergence, i.e., it reaches a local
minimum starting from any feasible initial-guess trajectory [30]. The minimum
is practically reached, and the algorithm’s iteration are stopped, when the
magnitude of the gradient approaches sufficiently a zero value. All necessary
conditions of optimality are satisfied at the algorithm;s convergence. It
should be noted that an initial-guess trajectory that is closer to the optimal
one may reduce the required number of iterations.

Among several conjugate-gradient and Quasi-Newton methods, the Polak-Ribiere
method was found most efficient for this OCP and is used for calculating
search directions inside FDA. The approach is fast enough to be considered for
real-time applications. With appropriate tuning of some algorithm parameters,
the algorithm’s runtime to generate vehicle trajectories, for a planning
horizon of 8 s with a time-step size of $T=250$ ms (i.e., for $K=32$), is 12.4
ms on average (less than 100 ms in 99.9% plans, less than 250 ms in 99.99%
plans) on a machine powered by Intel Core i5-8500 CPU operating at a maximum
of 3.00 GHz frequency. The maximum computation time observed is 1.71 s and the
minimum of 0.3 ms. It should also be noted that FDA iterations may be stopped
at any time, even before convergence, delivering a control trajectory that may
not be the optimal one, but is feasible, i.e., it satisfies all state
equations and control constraints.

In the OCP at hand, all equality and inequality constraints are linear, and
all sub-criteria are convex with one exception, namely the collision avoidance
term, which is non-convex. Thus, due to the collision avoidance term, the OCP
as a whole is non-convex, and this implies that FDA may converge to a local
minimum that satisfies all necessary conditions of optimality. Although it is
not possible to know, for any delivered minimum, whether it is a local or
global one, we never identified, in the extensive simulation investigations of
Section IV, any awkward EV manoeuvres that might correspond to a bad local
minimum.

(a)

(b)

(c)

Figure 2: A microscopic effect of vehicle nudging: The yellow vehicle nudges
the slower blue vehicles aside to pass.

One of the reasons why bad local optima are not encountered is the strongly
dynamic nature of traffic on a highway, as considered here. Another reason is
the utilised vehicle nudging. It is known from moving robotics applications
[32] that the potential-function like terms, as those used here for collision
avoidance, may create a local minimum on a trajectory, e.g., according to Fig.
2(a), where a fast car is trapped behind two slower vehicles that drive in
parallel symmetry. If no vehicle nudging would be present, the car would
follow the slow vehicles, taking a locally optimal trajectory. On the other
hand, if nudging is present, as in our OCP, the car may nudge both vehicles
aside (Fig. 2(b)) and pass (Fig. 2(c)). Many such microscopic episodes were
identified in the simulations reported in Section IV and they illustrate that
vehicle nudging is, in several situations, factually suppressing bad local-
minimum trajectories.

###  III-D Model Predictive Control (MPC)

MPC has been established as a quasi-feedback control scheme that enables the
deployment of optimisation and optimal control methods in real time [20]. The
present application is labelled as economic MPC [33], because it aims at
minimising a quasi-physical objective criterion, rather than tracking desired
values for system states.

In the present application, FDA is embedded in an MPC scheme to deliver in
real time numerical solutions of the formulated OCP that determine appropriate
vehicle trajectories. In order to investigate and assess the approach under
the novel requirements of lane-free traffic and vehicle nudging, as well as to
evaluate its impact at traffic level, the MPC scheme is applied, in the next
section, independently to all vehicles populating a ring-road infrastructure.
The OCP looks into the future, according to its time-horizon, thus avoiding
myopic actions by anticipating future action needs. This is deemed beneficial
for the pursued vehicle tasks, specifically for higher efficiency (vehicle
advancement), safety (collision avoidance) and passenger convenience (smooth
trajectories). On the other hand, due to the dynamic nature of traffic and the
inherent uncertainty about the movement of other vehicles around an EV, the
consideration of very long time horizons for the OCP is not sensible, as the
prediction of movement of other vehicles becomes increasingly inaccurate with
time. It was found empirically that time horizons in the order of 8 s are
suitable in terms of performance, as well as to account for the dynamic, but
also uncertain, nature of traffic.

MPC calls the FDA in an event-based manner for each vehicle, and this results
in asynchronous trajectory computation and application in emulated real-time
for each of the vehicles. Each EV receives the current states and the latest
computed trajectories of all other vehicles (obstacles) within its IZ. Since
the OCP is solved at different instances for each of the vehicles, each EV
usually holds, at the time it computes its next optimal trajectory, only
partial decided trajectories of the obstacles. These partial trajectories are
extrapolated, to cover the whole planning horizon of the EV, by considering
zero accelerations (both longitudinally and laterally) beyond the known
obstacle trajectory parts; and the completed obstacle trajectories are used in
the obstacle avoidance sub-objective of OCP (Section III-A3) and in the
emergency constraints (Section II-B) within the OCP.

Each FDA call is initiated by the MPC framework in an event-based mode, each
time with updated initial states of the EV and updated obstacle trajectory
predictions. The following events trigger the FDA calls, i.e., the computation
of a new trajectory for the EV:

  * •

The vehicle has already applied the part of the last generated trajectory,
which corresponds to a pre-specified application period, which is here half of
the planning horizon, and the reason why the planning horizon is larger than
the application period is in order to prevent the vehicle from taking myopic
actions.

  * •

There is a substantial deviation (0.2 m longitudinally or 0.1 m laterally) in
the trajectory of any of the dynamic obstacles. This is necessary to prevent
collisions due to inaccurate consideration of the actual obstacles’ movement.

  * •

A new obstacle enters the IZ of the EV.

In all the above events, the unused control trajectory part is used as a good
initial guess for FDA, so as to obtain faster convergence. The rest of the
guess trajectory, beyond the unused trajectory part, is assumed to have zero
accelerations. The trajectories generated at each of the iterations are always
feasible by construction, hence the FDA may be stopped at any time, especially
in the very rare case it is taking a lot of iterations.

As explained earlier, minimization of the objective function does not
guarantee generation of a collision-free trajectory for the EV in all possible
circumstances. Therefore, the trajectories generated by the FDA are checked
for possible collisions, based on the predicted trajectories of the obstacles.
The checks are performed for possible longitudinal or lateral collision, and,
if such a collision is indeed detected, a respective new OCP is specified,
including appropriate preventive measures. A longitudinal collision with
$i^{\text{th}}$ obstacle is detected if all the following checks return true:

  * •

$x_{1}\left(0\right)<o_{i1}\left(0\right)$, i.e., initial longitudinal
position of EV’s centre is behind the obstacle’s centre, otherwise no
longitudinal collision is conceivable.

  * •

$\displaystyle\left(o_{i1}\left(k\right)-\frac{e_{l}}{2}-\frac{o_{il}}{2}-\omega_{1em}x_{3}\left(0\right)\right)<x_{1}\left(k\right)<\left(o_{i1}\left(k\right)+\frac{e_{l}}{2}+\frac{o_{il}}{2}+\omega_{1em}x_{3}\left(0\right)\right)$,
for any $k$ within the time horizon $0\leq k<K$, to check longitudinal overlap
of length and emergency time-gap on rear and front sides of the EV, where
$\omega_{1em}=0.5\omega_{1}$. Fig. 3(a) depicts a case of longitudinal
overlap.

  * •

$\displaystyle\left(o_{i2}\left(k\right)-\frac{e_{w}}{2}-\frac{o_{iw}}{2}-\epsilon\right)<x_{2}\left(k\right)<\left(o_{i2}\left(k\right)+\frac{e_{w}}{2}+\frac{o_{iw}}{2}+\epsilon\right)$,
for any $k$ within the time horizon $0\leq k\ <\ K$, to check lateral overlap
of width on either side of the EV, with a small positive safety margin
$\epsilon$. Fig. 3(b) depicts a case of lateral overlap.

(a)

(b)

(c)

Figure 3: Illustration of collision detection: 3(a) Longitudinal overlap; 3(b)
Lateral overlap; 3(c) Longitudinal and lateral overlap (collision).

Fig. 3(c) depicts a case of both longitudinal and lateral overlaps, which is a
collision. If the initial position of the EV is behind the obstacle and
overlaps are detected both longitudinally and laterally in the trajectories,
this is categorized as a longitudinal collision. Note that, due to the
presence of $\omega_{1em}>0$ and $\epsilon>0$, the detected collision may not
be a physical one. Also, if the second and third conditions above are
satisfied for different times $k$, no physical collision may be present.
Nevertheless, such situations may reflect a risky vehicle interference that
should better be avoided.

Similarly to the longitudinal collision detection, a lateral collision is
detected if all the following checks return true:

  * •

$\displaystyle\left(o_{i1}(0)-\frac{e_{l}}{2}-\frac{o_{il}}{2}-\epsilon\right)<x_{1}(0)<\left(o_{i1}(0)+\frac{e_{l}}{2}+\frac{o_{il}}{2}+\epsilon\right)$,
to check the lateral alignment at the start of trajectory.

  * •

$\displaystyle\left(o_{i1}(k)-\frac{e_{l}}{2}-\frac{o_{il}}{2}-\epsilon\right)<x_{1}(k)<\left(o_{i1}(k)+\frac{e_{l}}{2}+\frac{o_{il}}{2}+\epsilon\right)$,
for any $k$ within the time horizon $0\leq k<K$, to check the overlap
longitudinally.

  * •

$\displaystyle\left(o_{i2}(k))-\frac{e_{w}}{2}-\frac{o_{iw}}{2}-\epsilon\right)<x_{2}(k)<\left(o_{i2}(k))+\frac{e_{w}}{2}+\frac{o_{iw}}{2}+\epsilon\right)$,
for any $k$ within the time horizon $0\leq k<K$, to check the overlap
laterally.

Thus, if the initial positioning of the vehicles is as shown in Fig. 3(a) and
evolves to a position as shown in Fig. 3(c) during the course of the
trajectory, this is categorized as a lateral collision.

It must be noted that the two checks are non-exclusive mutually, hence the
outcome may indicate both a longitudinal and a lateral collision. In this
case, the lateral collision characterisation prevails, since both vehicles are
certainly laterally aligned at the starting time ($k=0$) due to the first
bullet above.

When a longitudinal collision is detected with the $i^{\text{th}}$ obstacle,
the OCP is re-formulated to include a new state-dependent bound on the upper
limit of longitudinal acceleration, as described in Section II-B11, so that
the EV is instructed to follow the $i^{\text{th}}$ obstacle, with which it was
detected to interfere during the first run. When a lateral collision is
detected, the OCP is reformulated to include new state-dependent upper and
lower bounds on lateral acceleration, as described in Section II-B2. The new
bounds restrict the lateral movement of the EV within $\pm 0.15$ m of the
initial lateral position at the start of the respective planning horizon. This
is achieved by assuming two virtual boundaries at $x_{2}\left(0\right)\pm
0.15$ m, parallel to the actual road boundaries, but with a very narrow
corridor for lateral movement, which effectively prevents the lateral
collision observed in the first run.

In the investigations of Section IV, myriads of trajectory planning runs have
been executed. Collisions in the optimal trajectory were detected only for a
very tiny portion of them (see Table II). The triggered prevention measures in
those cases were always successful, i.e., there was no single case where the
second, more strongly restricted trajectory planning would deliver a collision
trajectory.

Each vehicle has a pre-defined constant longitudinal desired speed
$V_{{des}_{1}}$, and it is ideally desired that it moves close at that speed,
possibly by manoeuvring around encountered slower obstacles, whenever
possible. However, considering $V_{{des}_{1}}$ directly in the desired-speed
term of OCP’s objective function (Section III-A2) may result in unreasonably
high values for longitudinal acceleration if the initial longitudinal speed
$x_{3}(0)$ is low, because, in such a case, the sub-objective for desired
speed would assume accordingly high values that the optimisation would strive
to reduce by increasing the longitudinal acceleration, something that may
affect passenger convenience.

A similar issue arises in cases of high vehicle density, where there is very
little space for lateral movement and overtaking of downstream obstacles. In
such conditions, vehicles tend to move in clusters at similar longitudinal
speeds, that are clearly lower than everybody’s desired speed, and with only
small relative displacements. Similar conditions are encountered in
conventional lane-based traffic, when speed in all lanes is low, due to high
traffic density (congestion), and there is hardly any escape forward, even for
fast vehicles. Maintaining a high desired speed in the OCP in these
circumstances, may result in an aggressive forward-acceleration behaviour of
the EV, which is pointless, since overtaking is hardly possible. Such
aggressive behaviour may lead to collisions, but may also cause acceleration
bursts and hence discomfort to the passengers.

To address both these issues, an adaptive desired speed $v_{d_{1}}$ is defined
in 29, which is computed prior to every new trajectory calculation for use in
the desired-speed sub-objective of Section III-A2. Based on 29, $v_{d_{1}}$
may be limited to values lower than $V_{\text{des}_{1}}$, to address the above
two concerns. Specifically, the adaptive desired speed $v_{d_{1}}$ is always
limited to a positive increment $V_{{\text{incr}}_{1}}$ above the current
longitudinal ego speed $x_{3}(0)$, which addresses the first issue above,
since the desired speed targeted by the OCP is never higher than
$V_{{\text{incr}}_{1}}$ above the current vehicle speed $x_{3}\left(0\right)$,
hence the longitudinal acceleration will not take unreasonably high values.

To address the second issue above, a second limit is included in (29), which
is considered only at dense traffic, i.e., if $D_{d}>D_{\text{bar}}$ , where
$D_{d}$ (in veh/km) is the downstream vehicle density, defined as the number
of obstacles included in the downstream part of the IZ, divided by the zone’s
length; and $D_{\text{bar}}$ is a density threshold, selected around the
critical density value. Specifically, if $D_{d}>D_{\text{bar}}$ , the adaptive
desired speed $v_{d_{1}}$ is additionally limited to a positive increment
$V_{{\text{incr}}_{2}}$ above the current average downstream longitudinal
speed $D_{v}$ of the vehicles included in the downstream part of the IZ. This
second limit addresses the second issue above, as, at higher densities, the EV
has no strong incentive to accelerate beyond the average speed of surrounding
vehicles.

Of course, it may be seen in 29, that the adaptive desired speed $v_{d_{1}}$
may equal the actual desired speed $V_{{des}_{1}}$, e.g., at lower densities
and high initial speed $x_{3}(0)$.

$\displaystyle
v_{d_{1}}=\begin{cases}\text{min}\left\\{x_{3}(0)+V_{\text{incr}_{1}},V_{\text{des}}\right\\}&\text{if}\;D_{d}\leq
D_{\text{bar}}\\\
\text{min}\left\\{x_{3}(0)+V_{\text{incr}_{1}},D_{v}+V_{\text{incr}_{2}},V_{\text{des}}\right\\}&\text{if}\;D_{d}>D_{\text{bar}}\end{cases}$
(29)

##  IV Simulation Results

###  IV-A Simulation Environment and Scenarios

To run traffic-level simulations in a lane-free environment, a custom-made
extension, called TrafficFluid-Sim [34], was developed within the well-known
and broadly used Simulation of Urban Mobility (SUMO) simulator [35]. The open-
source codebase of SUMO is extended and used to communicate control inputs and
other information with SUMO. The MPC framework is incorporated into the API
and invokes the FDA algorithm when needed. Due to the lane-free structure of
the road, the default car-following and lane-changing models of SUMO are not
appropriate for the vehicle movement; therefore, the extension is utilized to
incorporate the proposed vehicle movement strategies. SUMO detectors,
measuring flow, are used, but, due to lane-free traffic, they are set to span
over the whole road width, at their specific locations, thus measuring the
total cross-section flow.

The described vehicle-movement strategy is based on the formulated OCP, its
numerical solution via FDA and its MPC deployment in real time. To test the
strategy’s vehicle-level properties, as well as to assess the emerging
traffic-level characteristics, the strategy is applied independently to all
vehicles driving on an unfolded lane-free ring-road of 1 km length and 10.2 m
width. Distinct simulations are carried out for corresponding vehicle
densities. Eight classes of vehicles are considered at random, the
corresponding dimensions of each class being given in Table I. At each
simulation, vehicles start with zero speed.

For homogeneous initial placement of the vehicles with appropriate two-
dimensional spacing between each other, the lateral space is divided into four
virtual lanes, and the longitudinal space is divided into $n/4$ sections,
where $n$ is the number of vehicles (density) being considered. This creates
exactly $n$ cells, homogeneously distributed on the two-dimensional ring-road
surface, and each vehicle is placed around the centre of a cell, with some
two-dimensional random deviations. The assignment of longitudinal desired
speeds to individual vehicles is based on their initial lateral positioning.
Specifically, the considered longitudinal desired speed range is [25, 35] m/s,
and this range is divided into four equal desired speed zones, one for each
virtual lane. Each vehicle has its own longitudinal desired speed, assigned
randomly with a uniform distribution within its speed zone, based on initial
lateral position. Note that, these desired speed zones are for initial
placement only, to aid the movement in high densities. Lateral desired speeds
are zero for all vehicles.

TABLE I: Vehicle classes and dimensions. Class | I | II | III | VI | V | VI | VII | VIII
---|---|---|---|---|---|---|---|---
Length (m) | 3.2 | 3.4 | 3.9 | 4.25 | 4.55 | 4.6 | 5.15 | 5.2
Width (m) | 1.6 | 1.7 | 1.7 | 1.8 | 1.82 | 1.77 | 1.84 | 1.88

After all vehicles are placed and are assigned their longitudinal desired
speeds, they start moving, from zero initial speed, according to the developed
strategy, and they strive to reach their desired speed, while avoiding other
vehicles in front and around them.

The following parameters are used in the simulations: In 20,
$\\{w_{1},w_{2},w_{3},w_{4},w_{5},w_{6},w_{7}\\}=\\{0.005,0.005,0.015,0.005,7.0,0.1,0.005\\}$;
longitudinal safety parameter $\omega_{1}=0.53$ s and lateral safety parameter
$\omega_{2}=0.5$ s; $\epsilon_{w}=0.1$ in 15 and III-A3; in section III-A3,
$p_{1}=6,\;p_{2}=p_{3}=p_{4}=p_{5}=2$; in 18, $\beta=0.03$; the length of each
part of the IZ is not allowed to fall below 100 m; density threshold
$D_{\text{bar}}=150$ veh/km in 29; discrete-time step $T=250$ ms; time-horizon
8 s, hence $K=32$; constant bounds on acceleration
$U_{\text{max1}}=0.5\;\text{m}/\text{s}^{2}$,
$U_{\text{min1}}=-2\;\text{m}/\text{s}^{2}$ for regular case and
$U_{\text{min1}}=-4\;\text{m}/\text{s}^{2}$ for emergency case.

###  IV-B Traffic-level Results

Simulations are run for densities ranging from 50 to 500 veh/km. For each
simulation with a specific density, a corresponding flow value is obtained
from loop detectors. Specifically, five loop detectors are placed at distances
of 200 m from each other, and the overall flow values are obtained by taking
the average of the flow values recorded at each of the loop detectors. For
each specific density value, five different random initializations (different
seed values) of desired speeds, within the specified range, and of starting
positions are considered in corresponding simulation replications. The average
flows (over the five replications) are displayed against the corresponding
density values to form a fundamental diagram (FD), with error bars reflecting
maximum and minimum deviations among the five replications, as provided in
Table II and Fig. 4.

Figure 4: Fundamental diagram TABLE II: Average flow outcomes (5 replications) for given densities. Density (veh/km) | 50 | 100 | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500
---|---|---|---|---|---|---|---|---|---|---
Flow (veh/h) | 5334 | 10350 | 14830 | 18332 | 15733 | 12879 | 10278 | 7137 | 4715 | 2799
Emergency % | 0.0 | 0.001 | 0.002 | 0.0 | 0.0 | 0.0 | 0.001 | 0.030 | 0.0 | 0.0143

A first, expected, qualitative observation is that the emerging FD has a
similar inverse-U (or, rather, inverse-V) shape, as in conventional lane-based
traffic. The left-hand (under-critical) side of the FD approaches a straight
line, particularly at lower densities, with a slight bending downwards when
the critical density (200 veh/km) is approached. The slope of this line is
around 28 m/s, which is slightly lower than the average desired speed of all
vehicles. This implies that, for under-critical densities, the vehicles,
guided by the developed movement strategy, find ways to drive close to their
desired speeds. Interestingly, the right-hand side of the FD also approximates
a straight line, with a slope of about $-14$ m/s.

For the road width of 10.2 m, conventional traffic has three lanes. In lane-
based traffic, a typical capacity flow is 2500 veh/km/lane, which amounts to
7500 veh/km for a three-lane road [36]. In this work, with an equivalent road
width of a three-lane road, more than the double flow has been achieved on the
account of lane-free and nudging effects. With lane-free structure, up to five
small-size vehicles are seen to occasionally fit in the lateral space, thereby
increasing the capacity. Furthermore, nudging also contributes to higher
flows, as also observed in [17]. Capacity of of 18332 veh/h (minimum 18213
veh/h and maximum 18446 veh/h) is reached here at a critical density of 200
veh/km. It may also be seen in Fig. 4 that change of seeds has a moderate to
low impact on the flow values at specific densities, with higher dispersion of
flows observed near the critical density and near the jam density. Finally, it
is interesting to note that no backward-moving jams are observed at any
density value.

###  IV-C Vehicle-level Results

Zooming on individual vehicle movements, it should first be mentioned that
there are no collisions in all the results reported, although there are very
rare occurrences of emergency situations, i.e., cases where the first computed
vehicle trajectory includes a collision, hence a new trajectory-planning
procedure is triggered with stringer control bounds according to Sections
II-B1 and II-B2. Such emergency cases appear only for a very minor percentage
of the total number of trajectory planning optimisations. Table II (second
row) displays the percentage of occurred emergency cases per traffic density
examined. All emergency situations delivered a collision-free vehicle
trajectory, after application of the stricter constraints.

Figure 5: Longitudinal movement in 100 veh/km density  Figure 6: Lateral
movement in 100 veh/km density

To assess the microscopic vehicle behaviour, representative movement
trajectories of three vehicles, driving in different densities, are depicted
in Figs. 5, 6, 7, 8, 9 and 10. Note that Figs. 5, 7 and 9 also display the
actual desired longitudinal speed (shown with a green dashed line) and the
adaptive desired speed (shown with a blue dotted line). The displayed vehicle
trajectories span the whole simulation period. In all the simulations, the
vehicles start from zero speed. It can be observed that, at a clearly under-
critical density (Fig. 5), the vehicle is able to reach its desired
longitudinal speed relatively fast, by incrementally approaching the adaptive
desired speed at moderate acceleration. Furthermore, the vehicle manages to
maintain the desired speed for most of the duration by applying appropriate
manoeuvres around encountered slower vehicles. On the other hand, in critical
or over-critical densities, as shown in Figs. 7 and 9, respectively, the
vehicle is not able to reach its desired longitudinal speed due to the lack of
manoeuvrable space. Instead, the vehicle reaches some quasi-stationary speed,
the value of which decreases with increasing densities, as expected, in
accordance with the FD of Fig. 4.

It can also be observed that, for all required manoeuvres at all densities,
the acceleration and jerk values in the longitudinal direction are well within
the comfortable range, which is important for passenger convenience and also
better for fuel economy. Histograms of jerk and acceleration values for all
vehicles at the same densities, given in Figs. 11, 12 and 13, confirm
convenient driving for all vehicles.

The demonstrated lateral vehicle movement is flexible and differs at different
vehicle densities. In under-critical densities, the vehicle is seen in Fig. 6
to leverage the complete available lateral space (road width) in its attempt
to attain or maintain the desired longitudinal speed. It is interesting to
note that lateral manoeuvring is more intensive at periods when the
longitudinal speed falls short of the desired speed, which happens due to
slower vehicles ahead. In such circumstances, the stronger employed lateral
displacements witness the striving of the vehicle to find appropriate
opportunities to accelerate longitudinally towards the desired speed. It may
also be seen that, at periods, the vehicle is moving exactly on the left or
right road boundaries, after having reached those asymptotically, and without
ever crossing the boundary, as designed and discussed in Section II-B2. Such
efficient vehicle manoeuvring is not only beneficial for vehicle advancement,
but also improves the overall traffic efficiency, contributing to the high
flows of the FD in Fig. 4. While applying these lateral manoeuvres, the
vehicle is seen in Fig. 6 to maintain moderate or small lateral speed and
acceleration.

Figure 7: Longitudinal movement in 200 veh/km density Figure 8: Lateral
movement in 200 veh/km density

In critical or over-critical densities, the lateral movement is limited, as
shown in Figs. 8 and 10, due to lack of lateral space, which is occupied by
other vehicles. Nevertheless, a few noticeable lateral displacements may be
observed, which are usually followed by a slight increase of longitudinal
speed. The lateral accelerations and speeds are very mild in all the cases and
hence convenient for the passengers, as it can be observed in the
corresponding trajectories depicted in Figs. 6, 8 and 10.

Videos showing the vehicle movement in different densities can be viewed at
https://bit.ly/TF-OPP. Note that, in the videos, the camera is tracking one
vehicle, pointed out by a mouse-pointer, hence all the surrounding vehicles
are seen to move with their relative motion with respect to the vehicle being
tracked.

Figure 9: Longitudinal movement in 300 veh/km density Figure 10: Lateral
movement in 300 veh/km density Figure 11: Acceleration and jerk histograms for
100 veh/km density Figure 12: Acceleration and jerk histograms for 200 veh/km
density Figure 13: Acceleration and jerk histograms for 300 veh/km density

##  V Conclusions

An optimal trajectory-planning methodology with fixed and state-dependent
bounds on control is developed and used to control the vehicle movement on
lane-free roads with vehicle nudging. An objective function with multiple sub-
objectives for safe, efficient and convenient vehicle movement is designed,
along with appropriate state equations and state-dependent control bounds in
the frame of an OCP formulation. The sub-objectives are designed to make the
EV move smoothly and flexibly in longitudinal and lateral directions without
compromising on safety and convenience of the passengers. State-dependent
bounds on control are designed, as constraints of the OCP, and are activated,
as and when they are needed, in both regular and emergency situations.

The OCP is appropriately transformed in an NLP problem and solved numerically
for a local minimum, using an efficient gradient-based FDA featuring global
convergence and polynomial convergence time. The OCP is solved independently
for multiple vehicles based on event-triggered MPC for finite time horizons of
8 s in emulated real-time. The vehicle trajectories are updated when there are
significant deviations in the predicted trajectories of obstacles or when
optimal control for half of the planned trajectory has been already applied.

Simulations are performed on a ring-road involving hundreds of vehicles with
different random initializations. It is observed that the vehicles move
efficiently and safely over the full range of possible vehicle densities,
while maintaining the passenger convenience. Flow values and critical density
values are significantly better than in lane-based traffic.

Ongoing and future work addresses:

  * •

Introducing on-ramps and off-ramps with appropriate merging and exiting
extensions of the present vehicle movement strategy.

  * •

Testing the performance of the approach at bottlenecks, that may result from
high inflow from an on-ramp, both at vehicle level and at traffic level

  * •

Implementing the internal boundary controller [19] for efficient road
utilization.

  * •

Using nonlinear Ackerman kinematic and the bicycle models [37] for roundabouts
and intersections.

#### Acknowledgement

This research received funding from the European Research Council under the EU
Horizon 2020 Programme / ERC Grant no. 833915, project TrafficFluid, see:
https://www.trafficfluid.tuc.gr

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|  Venkata Karteek Yanumula has been a Post Doc Associate with Dynamic Systems
& Simulation Laboratory, Technical University of Crete, Chania, Greece since
2020. He has been a faculty member at Electrical and Instrumentation
Engineering Department, Thapar Institute of Engineering and Technology,
Patiala, India since 2017. He received PhD in the domain of Systems Control
and Automation from Indian Institute of Technology, Guwahati in 2018. His
research interests include autonomous vehicles, optimal control, multi-agent
systems, and distributed control.
---|---
|  Panagiotis Typaldos completed his undergraduate studies in Applied
Mathematics at the University of Crete, Heraklion, Greece in 2014 and received
the M.Sc. degree in Operational Research from the School of Production
Engineering and Management, Technical University of Crete, Chania, Greece, in
2017. From October 2015, he is a research associate and since September 2017
he has been a PhD student with the Dynamic Systems & Simulation Laboratory,
Technical University of Crete.
---|---
|  Dimitrios Troullinos received the Diploma degree in Electrical and Computer
Engineering from the Technical University of Crete, Chania, Greece, in October
2019. His main research interests lie in the area of Multiagent Systems. From
November 2019, he is a research associate and since January 2020 he has been a
PhD student with the Dynamic Systems & Simulation Laboratory, Technical
University of Crete.
---|---
|  Milad Malekzadeh received the B.S. degree in Electrical engineering from
the Mazandaran University, Babolsar, Iran, in 2010, and the M.Sc. degree in
Control engineering from the Babol Noshirvani University of Technology, Babol,
Iran, in 2014. His research interests include observer design, optimal
control, and nonlinear dynamics. Since February 2020, he has been a Ph.D.
student with the Dynamic Systems & Simulation Laboratory, Technical University
of Crete.
---|---
|  Prof. Ioannis Papamichail is the Director of the Dynamic Systems and
Simulation Laboratory, at the Technical University of Crete, Chania, Greece.
He received the Dipl. Eng. degree in chemical engineering from the National
Technical University of Athens, in 1998 and the M.Sc. degree in process
systems engineering and the Ph.D. degree in chemical engineering from Imperial
College London, in 1999 and 2002, respectively. From 1999 to 2002, he was a
Research Assistant with the Center for Process Systems Engineering, Imperial
College London. He joined the Technical University of Crete in 2004 and has
served, since then, at all academic ranks. In 2010, he was a Visiting Scholar
with the University of California, Berkeley, CA, USA. He is the author of
several technical papers in scientific journals and conference proceedings.
His main research interests include automatic control and optimization theory
and applications to traffic and transportation systems. Dr. Papamichail is an
Associate Editor for IEEE Transactions on Intelligent Transportation Systems
and a Member of the Editorial Advisory Board for Transportation Research Part
C. He received the 1998 Eugenidi Foundation Scholarship for Postgraduate
Studies and the 2010 Transition to Practice Award from the IEEE Control
Systems Society.
---|---
|  Markos Papageorgiou (Life Fellow, IEEE) was a Professor of Automation with
the Technical University of Munich, Germany, from 1988 to 1994. Since 1994 he
has been a Professor (since 2021 Professor Emeritus) at the Technical
University of Crete, Chania, Greece. Since 2021 he has been a Professor at
Ningbo University, China. He was a Visiting Professor with the Politecnico di
Milano, the École Nationale des Ponts et Chaussées, MIT, the Sapienza
University of Rome, and Tsinghua University, and a Visiting Scholar with UC
Berkeley. His research interests include automatic control and optimisation
theory and applications to traffic and transportation systems, water systems,
and further areas. He is a fellow of IFAC. He received several distinctions
and awards, including the 2020 IEEE Transportation Technologies Award and two
ERC Advanced Investigator Grants.
---|---