# Multi-lane Unsignalized Intersection Cooperation with Flexible Lane
Direction
based on Multi-vehicle Formation Control

Mengchi Cai<EMAIL_ADDRESS>Qing Xu<EMAIL_ADDRESS>Chaoyi Chen<EMAIL_ADDRESS>Jiawei Wang  wang-
<EMAIL_ADDRESS>Keqiang Li<EMAIL_ADDRESS>Jianqiang Wang
<EMAIL_ADDRESS>Xiangbin Wu<EMAIL_ADDRESS>School of Vehicle
and Mobility, Tsinghua University, Beijing, China  Intel Lab China, Beijing,
China

###### Abstract

Unsignalized intersection cooperation of connected and automated vehicles
(CAVs) is able to eliminate green time loss of signalized intersections and
improve traffic efficiency. Most of the existing research on unsignalized
intersection cooperation considers fixed lane direction, where only specific
turning behavior of vehicles is allowed on each lane. Given that traffic
volume and the proportion of vehicles with different turning expectation may
change with time, fixed lane direction may lead to inefficiency at
intersections. This paper proposes a multi-lane unsignalized intersection
cooperation method that considers flexible lane direction. The two-dimensional
distribution of vehicles is calculated and vehicles that are not in conflict
are scheduled to pass the intersection simultaneously. The formation
reconfiguration method is utilized to achieve collision-free longitudinal and
lateral position adjustment of vehicles. Simulations are conducted at
different input traffic volumes and turning proportion of incoming vehicles,
and the results indicate that our method outperformances the fixed-lane-
direction unsignalized cooperation method and the signalized method.

###### keywords:

multi-lane unsignalized intersections, flexible lane direction, formation
control

##  1 Introduction

Intersection is one of the most common traffic scenarios where congestion
happens. Numerous intelligent intersection management methods are proposed to
reduce congestion and improve traffic efficiency [Bian et al., 2019, Chen et
al., 2021a, Ge et al., 2021]. Among the existing research about intersection
management, signalized intersections and unsignalized intersections are the
two main scenarios. At signalized intersections, researchers optimize traffic
signal timing, control vehicles to adapt to the signal timing, and combine the
two ideas to achieve cooperative optimization of both signal timing and
vehicle control [Mirchandani & Head, 2001, Xu et al., 2018a]. However, the
stop-and-go behavior of vehicles at signalized intersections can’t be avoided
due to the natural property of traffic signals, which may bring about green
time loss and result in inefficiency for traffic management. Hence,
researchers have started to focus on methods for unsignalized intersection
cooperation.

In the field of unsignalized intersection management, reservation methods,
planning methods, and optimization methods are three main categories.
Reservation methods centralizedly organize trajectories of vehicles where each
vehicle reserves a block of space-time in the intersection and the centralized
controller manages the reservations according to the “First Come, First
Served” policy [Dresner & Stone, 2008, Lee & Park, 2012, Huang et al., 2012].
Reservation is a simple way to achieve multi-vehicle collision-free passing
without traffic signal, but may not fully utilize the cooperation potential of
vehicles and limit the improvement of traffic efficiency. Planning methods
usually plan vehicles’ spatiotemporal trajectories to resolve conflict and
avoid collision between vehicles. Passing order is the main issue that most of
the planning methods consider, and some algorithms are designed to calculate
passing order of vehicles, e.g., depth-first searching methods [Xu et al.,
2018b], Monte Carlo Tree Searching methods [Xu et al., 2019], and other
graphical methods [Li & Wang, 2006, Chen et al., 2021b, c]. Optimization
methods centralizedly model the behavior of vehicles and solve the
optimization problem with collision-avoidance constraints. Some methods have
been proposed to solve the centralized optimization problem, e.g., dynamic
programming methods [Yan et al., 2009], model predictive control (MPC) methods
[Kamal et al., 2014], and other optimization methods [Wu et al., 2012b, Lee et
al., 2013].

Among the above intersection management methods, fixed lane direction is
commonly considered, where only specific turning is allowed on each lane. In
fact, fixed lane direction is a legacy from the signalized intersections to
guide vehicles with different turning expectation to distribute on different
lanes and avoid collision when passing the intersection. However, considering
the fluctuation of traffic input where proportion of vehicles with different
turning expectation may change with time, the capacity of some lanes can’t be
fully utilized in fixed-lane-direction intersections. Thus, removing the
constraint of fixed lane direction and making full use of intersection
capacity can further improve traffic efficiency with help of CAVs. Some
research has focused on dynamic lane use and lane-allocation-free intersection
management. Dynamic lane use methods usually adjust the lane-use configuration
of intersections according to the changing traffic inputs. Some research
combines dynamic lane use and signal control to improve traffic efficiency at
intersections [Alhajyaseen et al., 2017, Wu et al., 2012a]. Some other
research extends the idea of dynamic lane use and make some lanes reversable
(allowing reverse traffic flow) to balance traffic flow demand at different
time [Lu et al., 2018, Phan et al., 2019]. A cooperation method for lane-
allocation-free intersections is proposed in Yu et al. [2019] which optimizes
vehicles’ trajectories by mixed-integer linear programming. A lane-change-free
cooperation method is proposed in He et al. [2018] for future road
intersections, where vehicles don’t change lane and adjust their longitudinal
position to pass the intersection without collision. Other methods considering
dynamic lane use include Zhang & Wu [2012], Zhao et al. [2013].

Most of the existing intersection management methods ignore or simplify the
lane changing behavior of vehicles, e.g., lane changing is assumed to have
finished before vehicles enter the intersections in Chen et al. [2021b, c],
and is assumed to finish instantly in Yu et al. [2019]. Success rate of lane
changing has a great impact on the efficiency of intersections, and
unsuccessful and inefficient lane changing is one of the main reasons that
cause traffic congestion [Cai et al., 2021c]. Some cooperative lane changing
methods have been proposed in multi-lane scenarios, e.g., formation control
methods, vehicle sorting methods, and priority-based methods.

Multi-lane formation control methods consider vehicles in a certain range as a
group, and cooperatively control them to achieve formation forming, structure
reconfiguration, and vehicle joining/leaving management [Marjovi et al., 2015,
Cai et al., 2019, Zheng et al., 2021]. Both formation control and vehicle
sorting methods have focused on the problem of collision avoidance when a
group of vehicles perform cooperative lane changing [Wu et al., 2021, Cao et
al., 2021, Cai et al., 2021b]. Vehicles’ preference on specific lanes is
considered in Cai et al. [2021a], which is a key issue in intersection
management because vehicles need to change to the lanes where their desired
turning is allowed. A vehicle regrouping method based on multi-vehicle
formation control is proposed in Xu et al. [2021] to guide vehicles to change
lanes in a multi-lane intersection. Some priority-based methods are combined
with heuristic searching algorithms to plan trajectories for vehicles
considering lane changing at ramps and lane-drop bottlenecks [Xu et al.,
2020]. The above methods solve cooperative lane changing problems in multi-
lane scenarios, and have the potential to be applied to multi-lane
intersections.

This paper focuses on the problem of multi-lane unsignalized intersection
cooperation with flexible lane direction. The main contributions of this paper
include that:

  1. 1.

A two-dimensional distribution calculation method is proposed to calculate
both longitudinal (passing order) and lateral (lane occupation) position for
vehicles at intersections. Conflict relationship between different traffic
movement is defined and available position is calculated for each vehicle
considering achievability. The proposed method can be applied to multi-lane
unsignalized intersections, either with flexible lane direction or with fixed
lane direction.

  2. 2.

The formation reconfiguration method using relative motion planning and multi-
stage vehicle control is utilized for vehicles to achieve the calculated
distribution. Collision-free motion of vehicles is planned in the relative
coordinate system. Multi-stage trajectories are planned for vehicles and
trajectory following with spatiotemporal constraints is modeled and solved as
an optimal control problem. Compared with existing research that ignores or
simplify lane changing behavior of vehicles, this method enables efficient
lane changing at intersections.

  3. 3.

The performance of the proposed method is verified in simulations at different
input traffic volumes and different input turning proportion of vehicles.
Simulations show that removing lane direction constraints can significantly
imporve passing capacity of intersections because more vehicles are allowed to
pass the conflict zone of intersections simultaneously. The results indicate
that the efficiency of the proposed method outperforms both the unsignalized
cooperation method with fixed lane direction and the signalized method.

The rest of this paper is organized as follows. Section 2 introduces the
framework of multi-lane unsignalized intersection cooperation method. Section
3 and Section 4 propose the distribution calculation method and the formation
reconfiguration method respectively. Section 5 carries out simulations and
Section 6 gives the conclusion.

##  2 Framework of unsignalized intersection management

In this section, we introduce the scenarios studied in this paper and propose
the framework for multi-lane unsignalized intersection management with
flexible lane direction.

###  2.1 Scenarios

The scenario studied in this paper is the multi-lane unsignalized intersection
with flexible lane direction. An example of three-lane intersection where all
incoming lanes allow variable turning is shown in Fig. 1. A vehicle is allowed
to turning right, going straight, and turning left on every incoming lane. The
area where four arms of intersections overlap is conflict zone, shown as the
area inside the dashed lines in Fig. 1. The conflict zone is also the area
inside the stop line of intersections.

The incoming lanes are marked in gray. Lanes are indexed by integers from
right to left beginning with one, from the perspective of vehicles. In order
to simplify and regulate routes of vehicles, the assumption is made that
vehicles can only travel to the lane with the same index when passing the
conflict zone of an intersection, and the number of lanes of each arm is the
same.

Figure 1: A three-lane intersection where variable turning is allowed for all
the incoming lanes.

There are three types of traffic movements from one incoming arm, including
right-turning movement, straight-going movement, and left-turning movement,
and twelve movements in total from all the four arms. Three attributions are
defined for each vehicle to describe their movement: $r$, $t$ and $l$. $r$
represents the arm where the vehicle comes from, and is set to 1, 2, 3, and 4
if the vehicle is on the north, east, south, and west arm respectively. $t$
represents the turning behavior, and is set to 1, 2, and 3 if the vehicle is
going to turn right, go straight, and turn left respectively. $l$ represents
the lane that the vehicle is occupying, and is set equal to the index of the
lane. Then, in order to differentiate traffic movements of different vehicle,
the movement variable $M$ of a vehicle is calculated as:

$\displaystyle M=3n^{\text{L}}(r-1)+n^{\text{L}}(t-1)+l,$ (1)

where $n^{\text{L}}$ represents the number of lanes on each arm, and there are
$12n^{\text{L}}$ different movements in total. The definition of movement of
vehicles in (1) will be used to check conflict between vehicles later in
Section 3.1.

###  2.2 Framework

In order to control vehicles to safely pass an intersection without collision
with others, we need to determine two issues for every single vehicle: which
lane to occupy, and when to pass the conflict zone. Assuming that vehicles
pass the conflict zone with a constant speed [Xu et al., 2018b, Chen et al.,
2021b], the issue of determining when to pass the conflict zone is equal to
determining the relative longitudinal desired position of vehicles. Then the
key of controlling vehicles to safely pass an intersection becomes:
calculating longitudinal and lateral relative position of vehicles, guiding
them to drive to their desired position, and control them to pass the conflict
zone with a constant and safe speed.

For realizing the above steps, we propose a framework for multi-lane
unsignalized intersection management with flexible lane direction. The
framework consists of three parts: distribution calculation, formation
reconfiguration, and vehicle control.

  1. 1.

Distribution calculation. Both the longitudinal (passing order and following
distance) and lateral (lane occupation) distribution of vehicles are
calculated in this part, considering achievability (whether a vehicle is able
to catch up with a target position before reaching the stop line). The
calculated distribution guarantees that if all vehicles travel with the same
constant speed, they can pass the conflict zone without collision.

  2. 2.

Formation reconfiguration. In this part, all the vehicles driving on the same
incoming arm are considered as a formation, and paths for them to travel to
their desired position are calculated. A path consists of a sequence of points
for a vehicle to follow, and guides vehicles to arrive at their desired
position without collision.

  3. 3.

Vehicle control. In order to realize reconfiguration of formation, vehicles
plan their trajectories to pass through sequences of points, and calculate
their control inputs with spatiotemporal constraints.

The distribution calculation part is realized by the proposed iterative
grouping algorithm, and will be discussed in Section. 3. The formation
reconfiguration and vehicle control problems are solved by relative motion
planning and control of vehicles, and will be explained in detail in Section.
4.

###### Remark 1

The proposed intersection management framework is applicable to unsignalized
intersections either with or without flexible lane direction. The difference
of applying it to intersections with and without flexible lane direction lies
in the available position selection for vehicles in the distribution
calculation part, by defining selectable lanes for vehicles to occupy when
passing the conflict zone.

##  3 Conflict-free Distribution Calculation

In this section, we define conflict relationship between vehicles and propose
a conflict-free distribution calculation method to spatially arrange vehicles
to avoid collision when passing the conflict zone.

###  3.1 Conflict definition

A conflict represents that the motion of vehicles may lead to a collision. In
this section, we focus on the conflict relationship between vehicles in the
conflict zone of intersections. Previous research has defined conflicts in
intersections with fixed lane direction [Xu et al., 2018b]. In this paper,
since lanes are set free for all kinds of turning, the conflict relationship
is more complicated. As shown in Fig. 1, three types of conflict are defined:

  1. 1.

Crossing conflict. If the trajectories of two vehicles in the conflict zone
cross, the two vehicles are in crossing conflict. A crossing conflict may
exist between any two types of trajectories, including right-turning,
straight-going, and left-turning, shown as the yellow rectangles in Fig. 1.

  2. 2.

Diverging conflict. If the trajectories of two vehicles in the conflict zone
start from the same lane, the two vehicles are in diverging conflict, shown as
the yellow circles in Fig. 1.

  3. 3.

Merging conflict. If the trajectories of two vehicles in the conflict zone end
at the same lane, the two vehicles are in merging conflict, shown as the
yellow triangles in Fig. 1.

If two vehicles that are in conflict pass the conflict zone at the same time,
a collision may happen. Based on the definition of movement of vehicles in
(1), the conflict matrix is defined to check if vehicles with movement $i$ and
$j$ are in conflict. The conflict matrix $\mathcal{C}$, whose element on the
$i$-th row and the $j$-th column represents the conflict relationship between
movement $i$ and $j$, is defined as:

$\displaystyle\mathcal{C}=[c_{i,j}]\in\mathbb{R}^{12n^{\text{L}}\times
12n^{\text{L}}},\ c_{i,j}=\begin{cases}1,\ \text{if vehicle $i$ and $j$ are in
conflict},\\\ 0,\ \text{otherwise},\end{cases},\
i,j=1,2,3,...,12n^{\text{L}}.$

The number of rows and volumes of $\mathcal{C}$ is the same as the number of
movements. After defining conflict relationship, the key is to schedule
vehicles in conflict to pass the conflict zone at different time to prevent
collision.

###  3.2 Distribution calculation method

Based on the defined conflict relationship, we need to calculate longitudinal
and lateral distribution of vehicles to let them pass the conflict zone
without collision. In order to regulate motion of vehicles and simplify
planning, we make the following assumptions:

  1. 1.

Vehicles that are in conflict don’t exist in the conflict zone simultaneously.

  2. 2.

Vehicles that are not in conflict and are planned to pass the conflict zone
simultaneously arrive at the stop line at the same time.

According to the above assumptions, vehicles are scheduled into layers.
Vehicles that are in conflict are scheduled into different layers, and
vehicles in the same layer have no conflict and pass the conflict zone
simultaneously. The longitudinal gap between two layers is designed to let the
former layer completely pass the conflict zone before the latter layer reach
the stop line. Some methods have been proposed to calculate layers for
vehicles in intersections with fixed lane direction, e.g., depth-first
spanning tree method, and maximum matching method. In this paper, we proposed
the Iterative Grouping Algorithm (IGA) to calculate layers for vehicles in
intersections with flexible lane direction, shown as Algorithm 1.

Input: Vehicle sequence with ID from 1 to $N$.

Output: Lane distribution $\mathbb{L}$ and layer distribution
$\mathbb{V}_{k}(k=1,2,3,...)$ for vehicles.

1 Initialization Set $k\leftarrow 0$, and set $\mathbb{L}$ as the current lane
occupation of all the vehicles.

2 Calculate available layers for each vehicle.

3 while  _There is a vehicle not assigned to any layer._ do

4 Set $k\leftarrow k+1$ and $\mathbb{V}_{k}\leftarrow\emptyset$.

5 for  _$i\leftarrow 1$ to $N$_ do

6 if  _Vehicle $i$ is not assigned to any layer and layer $k$ is available for
vehicle $i$._ then

7 Search all the possible lane occupation for vehicle $i$ and $\mathbb{V}_{k}$
and check conflict.

8 if  _Conflict-free lane occupation is found for vehicle $i$ and
$\mathbb{V}_{k}$._ then

9 Set $\mathbb{V}_{k}\leftarrow\mathbb{V}_{k}\cup i$.

10 Update $\mathbb{L}$ as the one that has the minimum difference $e$ with the
current $\mathbb{L}$.

11 end if

12

13 end if

14

15 end for

16

17 end while

Algorithm 1 The Iterative Grouping Algorithm (IGA)

The input of IGA is a sequence of vehicles with ID indexed from $1$ to $N$,
and the output is conflict-free longitudinal (layer assignment) and lateral
(lane occupation) distribution of vehicles. IGA firstly calculates available
layers for all the vehicles, considering their maximum acceleration ability
and the distance towards the conflict zone. The detail of calculating
available layers for vehicles will be discussed later in Section 4.1. Then,
IGA iteratively assigns vehicles to layers and check their conflict
relationship. A new vehicle is assigned to a layer if it is not in conflict
with any other vehicles already in the layer, considering their lane
occupation. Finally, IGA outputs the conflict-free distribution of the $N$
vehicles. It is important to notice that IGA ignores vehicles’ motion to avoid
collision when calculating distribution, thus sometimes the calculated
distribution is not feasible to conduct because some vehicles have to take
extra steps to clear the way for other vehicles. The collision-free motion
planning for multiple vehicles will be discussed in Section 4. If a
distribution is found not feasible to conduct by multi-vehicle motion
planning, the available layers for vehicles is adjusted and IGA is run again
to calculate a new distribution.

In IGA, the set $\mathbb{L}$ represents the lane occupation of vehicles, where
the $i$-th element $\mathbb{L}_{i}$ represents the lane index that vehicle $i$
should occupy. From line 7 to line 11, IGA calculates lane occupation for
vehicles. In line 10, the difference between two lane occupation results
$\mathbb{L}$ and $\mathbb{L}^{\prime}$ is calculated as follows:

$\displaystyle e=\sum_{i=1}^{N}(\mathbb{L}_{i}-\mathbb{L}^{\prime}_{i})^{2}$
(2)

The lane occupation result that has the minimum difference with the current
occupation is updated as the new lane occupation, in order to reduce
unnecessary lane changes of vehicles.

###### Theorem 1

The IGA has the following properties:

  1. 1.

Every vehicle will be assigned to a layer.

  2. 2.

Vehicles that are assigned to the same layer are not in conflict with each
other.

  3. 3.

The algorithm will end in finite steps.

###### Proof 1

The three properties are proved as follows:

  1. 1.

IGA iteratively assigns vehicles to layers and there are three situations:

    1. (a)

The layer that a vehicle is trying to be assigned to is empty, so that no
conflict will exist and the vehicle is successfully assigned to the layer.

    2. (b)

The new vehicle is not in conflict with any other vehicles that are already in
the layer, and the vehicle is successfully assigned to the layer.

    3. (c)

Conflict exists between the new vehicle and other vehicles that are already in
the layer, and this vehicle will not be assigned to this layer. Now we
consider the worst case that the vehicle is in conflict with all the other
vehicles in the intersection, so that it can’t stay in the same layer with any
other vehicle. Since the number of vehicles is finite in an intersection, the
number of layers that the other vehicles are assigned to is also finite. After
checking those finite number of layers, the new vehicle will be successfully
assigned to the next available layer.

This completes the proof of the first property.

  2. 2.

This property is guaranteed by the assigning process. A vehicle will be
assigned to a layer if:

    1. (a)

The layer is empty and available for this vehicle. In this case, only one
vehicle exists in the layer, and no conflict exists.

    2. (b)

The vehicle is not in conflict with the vehicles that are already in the
layer. In this case, no conflict exists between the new vehicles and the
currently existing vehicles.

Since vehicles are assigned to layers one by one, no conflict will exist
between any two vehicles that are in the same layer. This completes the proof
of the second property.

  3. 3.

The algorithm will end if all the vehicles have been assigned to certain
layers. As is provided in Proof (1), every vehicle will be assigned to a layer
after finite steps, and the number of vehicles is finite. Thus, the total
number of steps that are needed to assign all the vehicles to certain layers
is also finite. This completes the proof of the third property.

###  3.3 Example

An example is provided to show the result of IGA. A sequence of ten vehicles
is given as input, and their traffic movement types are shown in Fig. 2. Note
that when calculating the distribution of vehicles for the example, we ignore
the reachability of vehicles and all layers are available.

Figure 2: A sequence of ten incoming vehicles. The numbers are ID of vehicles
and their traffic movements are shown in the black rectangles. Vehicles with
the smaller ID are closer to the stop line of the intersection.

In the intersection scenario with flexible lane direction, these vehicles are
assigned into two layers, as shown in Fig. 3. In the first layer, seven
vehicles including four right-turning vehicles, two straight-going vehicles,
and one left-turning vehicle have no conflict and pass the conflict zone
simultaneously. In the second layer, the other three vehicles, including two
straight-going vehicles and one left-turning vehicle, have no conflict and
pass the conflict zone simultaneously. In the intersection with fixed lane
direction, where only one type of turning is allowed for each lane, four
layers are needed to evacuate the ten vehicles, as shown in Fig. 4. This
indicates that cooperation at intersections with flexible lane direction can
achieve fewer layers, shorter passing time, and higher efficiency.

##  4 Multi-vehicle Formation Reconfiguration and Control

After being assigned to layers and lanes, vehicles need to plan trajectories
to drive to their desired position. Collision may happen between vehicles that
are driving on the same arm. Since the starting position and desired position
of vehicles are known, we can utilize multi-vehicle formation reconfiguration
method [Cai et al., 2021a] to solve the conflict-free motion planning problem,
by considering vehicles on the same arm as a group.

(a) Distribution of vehicles

(b) Layer 1

(c) Layer 2

Figure 3: Distribution results of vehicles at intersections with flexible lane
direction.

(a) Distribution of vehicles

(b) Layer 1

(c) Layer 2

(d) Layer 3

(e) Layer 4

Figure 4: Distribution results of vehicles at intersections with fixed lane
direction.

###  4.1 Bi-level formation control framework

The bi-level formation control framework is adopted to plan conflict-free
trajectories for vehicles to catch up with the planned distribution, where
relative motion planning is performed in the upper level to avoid collision
between vehicles, and trajectory planning with spatiotemporal constraints is
conducted in the lower level to calculate control inputs for vehicles. The
upper-level planning is performed in Relative Coordinate System (RCS), which
is a dynamic coordinate system moving with vehicles. In RCS, position of
vehicles is discretized by lanes laterally and by a constant gap
$d_{\text{F}}$ longitudinally. Every relative point in RCS corresponds to a
road point in Geodetic Coordinate System (GCS).

In order to prevent collision, time is also discretized by constant time
intervals $T_{\text{F}}$. Vehicles occupy relative points in RCS with integer
coordinates at fixed time points. During time period between two adjacent time
points, vehicles can move from one relative point to another, or maintain
their current relative position. If there are five available relative points
for a vehicle at each time step (four points around and its current point),
the motion map is four-connected. Given the relative position
$(x^{\mathrm{r,v}}_{i,j},y^{\mathrm{r,v}}_{i,j})$ of vehicle $i$ at step $j$,
the position $(x^{\mathrm{r,v}}_{i,j+1},y^{\mathrm{r,v}}_{i,j+1})$ at step
$j+1$ is limited by:

$\displaystyle|x^{\mathrm{r,v}}_{i,j+1}-x^{\mathrm{r,v}}_{i,j}|+|y^{\mathrm{r,v}}_{i,j+1}-y^{\mathrm{r,v}}_{i,j}|\leq
1.$ (3)

If there are nine available relative points for a vehicle at each time step
(eight points around and its current point), the motion map is eight-
connected, and the motion of vehicles is regulated by:

$\displaystyle|x^{\mathrm{r,v}}_{i,j+1}-x^{\mathrm{r,v}}_{i,j}|\leq 1,$ (4)
$\displaystyle|y^{\mathrm{r,v}}_{i,j+1}-y^{\mathrm{r,v}}_{i,j}|\leq 1.$ (5)

Denote the distance from the current position of a vehicle towards the stop
line of the intersection as $d_{i}$ and desired speed of the formation as
$v_{\text{F}}$, the maximum number of layers $N^{\text{y}}_{i}$ that vehicle
$i$ can accelerate to take forward before reaching the stop line is:

$\displaystyle
N^{\text{y}}_{i}=\lfloor\frac{d_{i}}{v_{\text{F}}T_{\text{F}}+d_{\text{F}}}\rfloor,$
(6)

where $\lfloor\cdot\rfloor$ is used to calculate the nearest integer that is
not bigger than a number. Layers that are behind the most forward achievable
layer for a vehicle are its available layers. The theoretical most forward
available layers for all the vehicles is the most forward layer for the
vehicle that is the closest to the stop line. Thus, the point whose relative
coordinate $x^{\mathrm{r}}$ is 0 is set on this layer. Suppose that ID of the
closest vehicle is 0, the relative coordinate $x^{\mathrm{r,v}}_{i}$ for
vehicle $i$ is calculated as:

$\displaystyle
x^{\mathrm{r,v}}_{i}=\left\langle\frac{d_{i}-d_{0}}{d_{\text{F}}}\right\rangle+N^{\text{y}}_{0},$
(7)

where $\left\langle\cdot\right\rangle$ is used to calculate the nearest
integer of a number. Note that different vehicles may have the same
$x^{\mathrm{r,v}}$ if they are close to each other longitudinally. In this
case, they are forced to drive to identical relative points before the
algorithm begins.

The relationship between relative states and real-world states are shown in
Fig. 5, where an example of movement during one time interval is provided. The
red arrow in the upper part of Fig. 5 shows an oblique move (if the relative
motion map is eight-connected), where the vehicle moves from relative
coordinate (2,3) to (1,2) to join the other four vehicles and forms an
interlaced five-vehicle formation.

Figure 5: The bi-level formation control framework.

Given the current and desired position of vehicles, we need to calculate
collision-free trajectories for those vehicles to travel to their target
position. This process is divided into two parts: the first part is to
calculate conflict-free relative paths for vehicles in RCS, and the second
part is to plan trajectories for vehicles to pass through those points one by
one with spatiotemporal constraints.

###  4.2 Relative motion planning

The Conflict-based Searching (CBS) method is utilized to calculate collision-
free relative paths for vehicles in RCS [Cai et al., 2021a]. CBS method
iteratively plan relative path for single vehicle, detect conflicts between
vehicles, and re-plan paths for vehicles with constraints. A conflict exists
between two vehicles and may lead to a collision. A constraint prevents a
certain movement of a vehicle. The searching process is shown in the left part
of Fig. 6, where each node represents a complete cycle of single-vehicle path
planning for all the vehicles in the formation. Conflicts are detected for
each node and are added as constraints for the child nodes. Each node is
evaluated by the total cost for vehicles to travel to their targets. The cost
can be defined as the total travelling time, total travelled distance, etc.
When finding a node without any conflict, a feasible conflict-free path
solution is got. If all the other unsearched nodes have higher cost than a
feasible node, this node is claimed to be the optimal solution with the
minimum cost.

For each node in the searching tree, individual path planning is performed for
every vehicle, considering the constraints. The path planning is conducted
without considering other vehicles, and may cause conflicts, which is then
added as the attributes of the node. The edge between nodes represents the
generating order of different nodes. Two nodes are called as parent node and
child node if there is a directed edge connecting them and starting from the
parent node to the child node. All the constraints of a parent node should
also be added to its child node, and one of the conflicts of the parent node
will be added as a new constraint to its child node. That is to say, the
number of child nodes of a node is equal to the number of the parent node’s
conflicts. Note that not every node is solid, because there may be conflicts
and the corresponding paths calculated for the node will cause collision.

The tree starts from the root node $N_{1}$, which doesn’t have constraints
since it has no parent node. By applying single-vehicle path planning method
to the initial and target position of vehicles, a path set, a conflict set,
and a cost is got. The single-vehicle path planning algorithm utilized in this
paper is the A* algorithm [Hart et al., 1968]. Child nodes are generated to
the root node and the tree iterates the process until an optimal solution is
found.

The CBS method is well known and widely used in the field of multi-robot path
planning. The properties of CBS method including completeness, optimality, and
time complexity have been proved and explained in numerous papers [Sharon et
al., 2013, 2015], and we omit them in this paper.

Figure 6: The process of conflict-based searching.

###  4.3 Multi-stage vehicle control

Vehicles arrive at relative points at each $t_{i}$ in
$\mathbb{T}=\\{t_{i}|t_{i}=iT,i=0,1,2,3...\\}$ in RCS. B$\acute{\text{e}}$zier
curves are generated for the vehicles to pass through the corresponding road
points in GCS. The key is to calculate longitudinal and lateral control inputs
for the vehicles to arrive at the desired position at desired time. The
trajectory in GCS passes through a sequence of road points and the inaccuracy
during the trajectory following process may accumulate by time. To prevent
severe following error which may lead to collision, a multi-stage motion
control framework is proposed and vehicles will replan their control inputs
after desired time intervals.

The vehicle model used in this paper is the 2-DOF vehicle model. The center of
the rear axle is chosen to represent the position of the vehicle. The yaw
angle and the steer angle are represented as $\theta$ and $\delta$
respectively. $L$ represents the wheelbase of the vehicle. The state variable
$\emph{{z}}^{\mathrm{v}}$ of the vehicle contains $x^{\mathrm{v}}$,
$y^{\mathrm{v}}$, $v$ and $\theta$, where $v$ represents the velocity of the
vehicle. The control input for the vehicle model contains the acceleration $a$
and the steer angle $\delta$, and the state is calculated as:

$\displaystyle\dot{\emph{{z}}^{\mathrm{v}}}=\begin{bmatrix}v\sin(\theta)\\\
v\cos(\theta)\\\ a\\\ \frac{v}{L}\tan(\delta)\end{bmatrix},$
$\displaystyle\emph{{z}}^{\mathrm{v}}=\begin{bmatrix}x^{\mathrm{v}}\\\
y^{\mathrm{v}}\\\ v\\\ \theta\end{bmatrix}.$ (8)

For the lateral control, a linear feedback preview controller is designed. The
vehicle looks ahead to the preview point and the closest point on the
trajectory. The controller calculates the angle between the vehicle velocity
and the desired velocity on the preview point, and the distance between the
vehicle and the closest point as feedbacks.

The discretized longitudinal state equation is given as:

$\displaystyle\begin{cases}&s(k+1)=s(k)+v(k)\text{d}t,\\\
&v(k+1)=v(k)+a(k)\text{d}t,\end{cases}$ (9)

where $s(k)$, $v(k)$ and $a(k)$ represent the longitudinal position, speed and
acceleration respectively. The time is discretized by sample interval
$\text{d}t$. The distance that the vehicle will travel from the starting point
to the final point is denoted as $S_{t}$. Since the vehicle moving in RCS
should pass a sequence of relative points, $S_{t}$ consists of series of
segments, whose length are denoted as $S_{1}$, $S_{2}$, …, $S_{N^{t}}$, where
$N^{t}$ is the number of segments. The time interval for the vehicle to cover
each segment is $T_{\text{F}}$. Taking the control energy of the whole control
process as the cost, the discretized longitudinal control can be described as:

$\displaystyle\min\quad$ $\displaystyle\sum_{k=0}^{N^{t}N^{k}-1}a^{2}(k),$
(10) s.t. $\displaystyle s(0)=0,$ $\displaystyle
s(k_{i})=\sum_{n=1}^{i}S_{n},\ k_{i}=iN^{k},\ i=1,2,3,...,N^{t},$
$\displaystyle v(0)=v(N^{t}N^{k})=v_{\mathrm{F}},$ $\displaystyle
v_{\text{min}}\leq v\leq v_{\text{max}},$ $\displaystyle a_{\text{min}}\leq
a\leq a_{\text{max}},$

where $N^{k}=\frac{T_{\text{F}}}{\text{d}t}$ is the steps in one segment,
$a(k)$ is the longitudinal control input (acceleration) of the vehicle at step
$k$, $v(k)$ is the longitudinal speed, $v_{\mathrm{F}}$ is the desired
longitudinal speed of the formation, and $v_{\text{min}}$, $v_{\text{max}}$,
$a_{\text{min}}$ and $a_{\text{max}}$ are the bounds of speed and
acceleration.

The optimal control problem in (10) is solved and the sequence of control
input is calculated to guide the vehicle to travel desired distance at desired
time. Since the linear feedback preview controller is designed for lateral
control, the inaccuracy of lateral motion may cause deviation for longitudinal
motion. In order to resolve the accumulated inaccuracy, the optimal problem is
reformed and solved every time the vehicle completes the following of one
segment at time $t=iT_{\text{F}}(i=1,2,3,...,N^{t})$.

(a) Process of formation reconfiguration

(b) State 1

(c) State 2

(d) State 3

Figure 7: Formation reconfiguration process of six vehicles.

###  4.4 Example

An example is provided in Fig. 7 to show the process of formation
reconfiguration. Six vehicles start from messy points and need to form a
parallel three-lane formation. The initial and expected position of vehicles
are shown in Fig. 7(b) and Fig. 7(c). The motion map for this example is
eight-connected, and the reconfiguration process takes two cycles. The
rectangles represent vehicles and the lines with the same color represent
trajectories of certain vehicles. Fig. 7(a) shows the state and trajectories
of vehicles in GCS, and the circles represent road points for vehicles to pass
through. Fig. 7(b), Fig. 7(c), and Fig. 7(d) show the states and movements of
vehicles in RCS, and the black circles represent discretized relative points.

##  5 Simulations

In this part, the simulation is conducted and the results are analyzed. The
settings of the simulation are presented, and the benchmark methods are
introduced. Then, numerical results and snapshots are presented to evaluate
the proposed method.

###  5.1 Simulation settings

The simulation is implemented with MATLAB 2017b and SUMO 0.32.0 [Lopez et al.,
2018] on a personal computer with CPU Intel CORE<EMAIL_ADDRESS>The average
travelling time of vehicles is analyzed to evaluate the performance. The
parameters chosen for this simulation are presented in Table 1. It is
important to notice that the following distance of two layers $d_{\text{c}}$
when passing the conflict zone is set twice as the safe following distance in
formation. Hence, there is a transitional relative point between two layers
for vehicles to temporally stay when moving in a formation. The optimization
problem in (10) is solved by the GPOPS solver [Patterson & Rao, 2014].

Table 1: Simulation parameters Safe one-lane following gap in formation | $d_{\text{F}}$ | $17.5\,\mathrm{m}$
---|---|---
Safe one-lane following gap in conflict zone | $d_{\text{c}}$ | $35\,\mathrm{m}$
Formation switching cycle | $T_{\text{F}}$ | $4\,\mathrm{s}$
Desired speed in formation | $v_{\text{F}}$ | $10\,\mathrm{m/s}$
Minimum speed of vehicle | $v_{\text{min}}$ | $0\,\mathrm{m/s}$
Maximum speed of vehicle | $v_{\text{max}}$ | $20\,\mathrm{m/s}$
Minimum acceleration of vehicle | $a_{\text{min}}$ | $-10\,\mathrm{m/s^{2}}$
Maximum acceleration of vehicle | $a_{\text{max}}$ | $5\,\mathrm{m/s^{2}}$

The scenario of simulation is a three-lane intersection with four incoming
arms, as shown in Fig. 1. The length of each arm is $400\,\mathrm{m}$. We
conduct simulation in different input traffic volumes, from
$1000\,\mathrm{vehicle/hour}$ to $5000\,\mathrm{vehicle/hour}$. Vehicles are
generated randomly on different lanes. At each input traffic volume, we
conduct simulation with four different input proportion of vehicles with
different turning expectation, where the proportion of right-turning vehicles,
straight-going vehicles, and left-turning vehicles is $(0.33,0.33,0.34)$,
$(0.25,0.25,0.5)$, $(0.25,0.5,0.25)$, and $(0.5,0.25,0.25)$ respectively. At
each input volume and turning proportion, we compare the results of three
scenarios: unsignalized intersection with flexible lane direction,
unsignalized intersection with fixed lane direction, and signalized
intersection with fixed lane direction. The signal timing cycle is given in
Fig. 8 and time for each phase is $20\,\mathrm{s}$. In order to clear vehicles
in conflict zone after each phase, a three-second clearing yellow-light phase
is conducted between each of the four phases. Similar signal timing cycle
design can be found in Xu et al. [2018a]. Random inputs are generated for each
cycle of simulation, and the results are the average of ten cycles. In total,
$200$ vehicles are generated for each case, where the first $100$ vehicles are
used for simulation warm up, and we only count the travelling time of the last
$100$ vehicles for result comparison.

(a) Phase 1

(b) Phase 2

(c) Phase 3

(d) Phase 4

Figure 8: Four phases of signal timing cycle.

###  5.2 Results

The results of average travelling time that vehicles take to travel on the
incoming arms are shown in Fig. 9. The four figures are the results under four
turning proportion of inputs. The standard deviation under different random
inputs is also marked in the figures. From the figures we can see that our
method at flexible-lane-direction scenarios achieves that best performance,
under all the input volumes and turning proportion. The reason is that our
method allows more vehicles to pass the conflict zone simultaneously than the
other two methods. At low traffic volumes (from 1000 to
$3000\,\mathrm{vehicle/hour}$) the unsignalized cooperation methods outperform
the signalized method, because vehicles don’t have to stop when arriving at
the conflict zone and can maintain a higher speed in the whole journey. As the
input volumes becomes higher, the performance of the unsignalized cooperation
with fixed lane direction become worse than the signalized method, because
vehicles have to slow down or even brake to achieve their desired layers to
pass the intersection. It is important to notice that the performance of the
methods differs at different input proportion, but the proposed method always
has the shortest travelling time results.

(a) Proportion $(0.33,0.33,0.34)$

(b) Proportion $(0.25,0.25,0.5)$

(c) Proportion $(0.25,0.5,0.25)$

(d) Proportion $(0.5,0.25,0.25)$

Figure 9: Results of average travelling time of vehicles. Figure 10: Snapshots
of the west incoming arm at four continuous cycles of the simulation. The
rectangles represent vehicles and their color shows their turning expectation,
where red for right-turning, green for straight-going, and blue for left-
turning. The full version video is available online at the address:
https://github.com/cmc623/flexible-lane-direction-intersection

Figure 11: Snapshots of conflict zone of the simulation. (a)-(d) are the
results of unsignalized intersection with flexible lane direction, (e)-(h) are
the results of unsignalized intersection with fixed lane direction, and
(i)-(l) are the results of signalized intersection with fixed lane direction.
The rectangles represent vehicles and their color shows their turning
expectation, where red for right-turning, green for straight-going, and blue
for left-turning. The full version video is available online at the address:
https://github.com/cmc623/flexible-lane-direction-intersection

Snapshots of the west incoming arm at four continuous cycles are presented in
Fig. 10. The rectangles represent vehicles and their color shows their turning
expectation, where red for right-turning, green for straight-going, and blue
for left-turning. The points represent trajectories of vehicles in the former
2.5 seconds. It shows the process of formation reconfiguration where vehicles
adjust their position to catch up with their desired layers and lanes.
Snapshots at input traffic volume $5000\,\mathrm{vehicle/hour}$ and turning
proportion $(0.33,0.33,0.34)$ of the three methods are provided in Fig. 11 to
show the process of simulation. We also show the trajectories of vehicles when
passing the conflict zone with lighter color. From the snapshots we can see
that our method allows more vehicles to pass the conflict zone simultaneously
than the other two methods, and vehicles have to stop and wait for signal
timing when using the signalized method.

##  6 Conclusions

This paper proposes a multi-lane unsignalized intersection cooperation method
that considers flexible lane direction. The two-dimensional distribution of
vehicles is calculated and vehicles that are not in conflict are scheduled to
pass the intersection simultaneously. The formation reconfiguration method is
utilized to achieve cooperative lane changing and longitudinal position
adjustment of vehicles. Simulations are conducted at different input traffic
volumes and turning proportion of vehicles, and the results indicate that: (1)
our method is able to cooperatively control vehicles to avoid collision and
pass unsignalized intersections with flexible lane direction, and (2) the
performance of our method is better than the unsignalized cooperation method
with fixed lane direction and the signalized method.

## Acknowledgment

This work is supported by the National Key Research and Development Program of
China under Grant 2018YFE0204302, the National Natural Science Foundation of
China under Grant 52072212, and Intel Collaborative Research Institute
Intelligent and Automated Connected Vehicles.

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