A Mobility Model for Connected Vehicles Induced by the Fish School

Abstract

We proposed a mobility model of connected vehicles with wireless communications. The proposed mobility model is inspired by the fish school, which takes into account several significant attractive/repulsive potential fields induced by the expected mobility, the tendency to avoid collisions, and the road constraints. With this model, we present a comprehensive approach for designing and analyzing the potential fields acting on connected vehicles. Based on the Lyapunov stability theorem, a theoretical analysis is also provided to show that the cooperative collision avoidance and stable constrained flocking of connected vehicles can be performed by using this model. The numerical experiments prove the effectiveness of the wireless communication on the safety and efficiency of connected vehicles.

1. Introduction

In recent years, more and more researches on swarm intelligence have been advanced and utilized in the fields of engineering, artificial intelligence, robotics, and transportation [1]. The cooperative behaviors of vehicles have been researched for the sake of giving a deep insight into traffic models [2]. With the rapid growth of wireless communication technologies, a large number of vehicles form a vehicular ad hoc network and cooperate with each other; thus collision warning, road obstacle warning, intersection collision warning, and lane change assistance can be improved [3, 4]. Those connected vehicles move in a cooperative manner, which is similar to some cooperative behaviors of animal flocking in nature. Therefore, it is convenient for researchers to model the connected vehicles as a group by drawing an analogy to the animal flocking. The goals of this paper are twofold: (i) to mathematically model the cooperative movements of the connected vehicles with the wireless communications based on the fish school, and, (ii) in the meanwhile, to provide the comprehensive evaluation of the influence of connected vehicles on the safety and efficiency.

The wireless communication technologies expand the sensing range of vehicles and enhance the information interplays among vehicles moving in the same road section. For example, one popular application of these wireless communication technologies in vehicular system is vehicular ad hoc network (VANET) which is supported through vehicle-to-roadside (V2R) and vehicle-to-vehicle (V2R) communications. There are large numbers of studies on signal propagation in vehicle-to-roadside and vehicle-to-vehicle communications, such as works [5, 6], which focus on the mechanism of communication under various traffic situations. Besides, other studies are working on the impacts of the cooperated vehicles on the transportation system where the communication technologies are widely deployed. For instance, the work [7] investigates the impact of the number of cooperative vehicles on the network performance under Nakagami fading channel. Moreover, the wireless communication also can be used to assist in developing the driving-assistance system [8], whereas there are few studies paying attention to the behaviors of the cooperative behaviors of the connected vehicles moving in the same road section as a group. It is a large challenge to model the cooperative behaviors of connected vehicles, because the interplays among vehicles as well as some environmental factors that influence the vehicular movement should be carefully identified at first and then mathematically formulated. Nevertheless, this work aims at this issue by following the bioinspired modelling approach. Since the cooperation is one characteristic of the animal flocking such as fish school, we draw on the mechanism of flocking in biosphere (i.e., the fish school) to model the cooperative behaviors of the connected vehicles.

The cooperative behaviors of animal flocking are usually governed by the three rules: cohesion, separation, and alignment [9, 10]. Some other researchers turn to the evolutionary models to simulate the evolving animals. These studies, for instance, the selfish herd theory [11], the predator confusion effect [12], and the dilution effect [13, 14], attempt to answer the key question of how flocking behavior evolves. As illustrated in some early studies, the typical examples for biological behavior in the same movement pattern include bird flocks, fish schools [15], insect swarms [16], and quadruped herds [17]. Nowadays, as a collective behavior, flocking is not only exhibited by animals of similar size which aggregate together, but also presented in the crowd and the vehicles [18, 19]. In fact, cooperative behaviors are pervasive among all forms of self-propelled particles [20]. Hence, these fundamental works pave the way to applying the rules and approaches for modeling the biological cooperative behaviors to other engineering fields.

Moreover, bioinspired approaches have been widely adopted in a lot of existing literature in the field of modeling mobility of flocking individuals moving as a group [21–23]. In nature, a large number of fish swim as a disciplined phalanx and they can stream up and down at high speed, twist in different ways, vary school shapes, and avoid obstacles without collisions. In order to coordinate its behavior with the overall schooling, fish has a sensor-response system to steadily keep the relative position among their neighbors regardless of their topology changing all the time. The lateral-line system is very sensitive to changes in water currents and vibrations so that fish is able to dynamically and adaptively respond to their environmental changes in time [24]. The environmental signals can be propagated throughout the group, which results in a unified group decision making. To better understand the basic nature of the influences at work in a school of fish, many works have discussed and presented an algebraic approach to describe the interplay in the fish school [22].

Inspired by the aforementioned behaviors of fish school, we are allowed to draw an analogy between the connected vehicles and the fish school. With the assistance of the advanced sensors, such as velocity sensor, vehicular positioning system, and navigation system, a vehicle can be provided with the real-time information of velocity, acceleration, position, and other basic parameters and even can feed back the collected information to its neighboring vehicles through wireless communications. That is, the wireless communications system can make each vehicle able to sense their own and environmental information as well as share the collected information with their neighbors, which is similar to the environment-sensing behavior of fish school. Furthermore, with wireless communication, vehicles are able to coordinate their movement (velocity and direction) according to the environmental situation and the overall mobility of the vehicle group. Each vehicle can interact with its neighbors in real time via wireless communication system so that they can move in a cooperative manner. This is similar to the cooperative behavior of fish school which is induced by the interplays among different fish. In this sense, some rules characterizing the behaviors of fish school such as the cohesion, the separation, and the alignment can be adopted to present the cooperation of those connected vehicles to some extent. In addition, another important character of fish school behavior is danger avoidance (such as avoiding obstacles or predators), which can be analogous to the collision avoidance. Moving fish can form a coordinated school and shift back to an amorphous shoal within seconds when facing emergence or obstacles [24]. Obstacle avoidance has been studied in many flocking researches [25–27]. However, these studies have not considered the realistic applications of the models. Hence, it is significant to provide some mathematical models for describing the mobility of connected vehicles when taking obstacles avoidance into account. In the meanwhile, some important factors should be additionally considered, well defined, and formulated in modeling, which include the constraints of road and the constraints of traffic rules, when drawing an analogy between the vehicle group and the fish school.

In this paper, a novel model has been proposed to formulate the movement of connected vehicles with considering some realistic traffic situations where the constraints of the road and road obstacles exist. By analogy, connected vehicles also follow some similar regulations governing the fish school. The interplay among connected vehicles is mathematically modeled by introducing the potential field functions that are induced by fish school behaviors. In addition, a theoretical analysis framework is provided to verify the rationality of proposed model. Finally, some numerical experiments are also given to demonstrate the model as well as provide a better understanding of the improvement of the safety and efficiency of traffic by wireless communication.

The rest of the paper is as follows. Section 2 proposes a mobility model to describe the connected vehicles and presents the stability analysis. In Section 3, extensive simulations are performed to verify the proposed model. Section 4 concludes this paper.

2. Model of Cooperative Vehicles

2.1. Mobility Model of Connected Vehicles

Moving as a group in the same road section, each vehicle needs to maintain different velocity and accelerate under the effects of forces. In the model, we address the attraction of the goal, the repulsion of the obstacles, the constraints of the road, and the interplays among vehicles in the group including both attractive and repulsive forces. Moreover, we assume that vehicles can communicate with other neighbor vehicles via wireless communication technologies which reflect in sensing range which will be provided later. In this paper, the main notations used throughout the paper are given in Notations.

In the model, each vehicle i belongs to the set of vehicles denoted by $N = {1,2, \dots, n}$ with the total number of vehicles $n (n > 2)$ . In a vehicle group, we concentrate on the target vehicle i and its neighbor j, $\forall i \neq j \in N$ . The vector $p_{i} (t) = {[x_{i} (t), y_{i} (t)]}^{T}$ signifies the location of vehicle i at time t. Besides, we describe the force between mobility vehicles due to the distance changes among them so that the interplay ranges of each vehicle are significant. In this paper, each vehicle is treated as a virtual sphere with the radius $r_{i}$ in order to correct the drawbacks of overlap (see in Figure 1). The concept of the “local warning scope” is defined as the local circle zone with radius $r_{i}^{W}$ of the vehicle warning that the situation may be dangerous when other vehicles or obstacles enter this area. In this paper, the vehicle can get the information of the location, velocity, and other behaviors of neighbor vehicles via communicating with other vehicles in its sensing range with the radius as $d_{i}$ (see in Figure 1).

Figure 1

The interplay range of vehicle i.

In order to develop a mobility model which can reflect an accurate traffic situation, the model not only considers the interactive forces among vehicles in the group, but also takes the repulsive forces of obstacles and road boundaries into account. These forces are described by a vector at every location in the force field. Following these considerations, we defined the effects induced by accumulated forces exerted on the vehicle i as follows:

\begin{matrix} m_{i} {\dot{v}}_{i} (t) = F_{i}^{(1)} (t) + F_{i}^{(2)} (t) + F_{i}^{(3)} (t), \end{matrix}

(1)

in which

v_{i} (t)

is the actual velocity at time t of vehicle i with the mass

m_{i}

. In formula 1,

F_{i}^{(1)} (t)

is the attractive force by the goal of vehicle i,

F_{i}^{(2)} (t)

is the interactive force among connected vehicles, and

F_{i}^{(3)} (t)

is the environmental effects of vehicle i. Without special notation, the letters in bold represent the term which is a vector within a two-dimensional plane.

Under the effect of the goal, the vehicles are expected to move in the certain direction in the desired speed. Therefore, $F_{i}^{(1)} (t)$ stimulates the vehicle i with a certain desired velocity $v_{i}^{0}$ in the expected direction $e_{i} (t)$ . Hence, the expected velocity can be formulated as

\begin{matrix} v_{i}^{o} (t) = v_{i}^{o} e_{i} (t) . \end{matrix}

(2)

However, in reality, the velocity $v_{i} (t)$ at time t of vehicle may deviate from the desired velocity $v_{i}^{0} (t)$ due to the necessary deceleration, acceleration, or other unknowns. Thus, $F_{i}^{(1)} (t)$ as a stimulus to force the vehicle back to the $v_{i}^{0} (t)$ again with a relaxation time $τ_{i}$ can be given by

\begin{matrix} F_{i}^{(1)} (t) = m_{i} \frac{v_{i}^{o} (t) - v_{i} (t)}{τ_{i}} . \end{matrix}

(3)

One of the characteristics of connected vehicles is cooperative behavior. The vehicles benefit from the wireless communication so that they can cooperate in moving as a group. The vehicles aggregate together in the same movement pattern while keeping the certain safe distance from each other. For the purpose of formulating the group forces among vehicles, we set the interactive forces into two categories including the attractive force and the repulsive force which can make the vehicles run as a group while keeping a certain distance without collision. We lump the attractive and repulsive forces in $F_{i}^{(2)} (t)$ as follows:

\begin{matrix} F_{i}^{(2)} (t) = \sum_{\forall j \in N_{i} (t)} (F_{i j}^{A} (t) + F_{i j}^{R} (t)) . \end{matrix}

(4)

$F_{i}^{(2)} (t)$ is the accumulated forces among target vehicle i and all other neighbor vehicles j in its sensing range. $F_{i j}^{A} (t)$ is the attractive force on vehicle i from its neighbor vehicles. Under the influence of the attractive force, the vehicles will accelerate toward each other to keep them as a group. Meanwhile, the $F_{i j}^{R} (t)$ is the repulsive force on vehicle i from its neighbor vehicles to keep the safe distance and to avoid collisions among connected vehicles.

In the short distances like the warning scope, the repulsion is increasing with the decreasing of the distances among vehicle i and its neighbors. Likewise, the attraction will play a leading role in making vehicles stay as a group when the distances among vehicle i and its neighbors are larger than the warning scope. In the case that the neighbor vehicle is beyond the bounds of the sensing range $d_{i}$ , the attraction will not exist. Similarly, the repulsive force will not contribute to the acceleration of a vehicle if it is not in the sensing range of the target vehicle. In order to depict this situation, we use the concept of potential functions to formularize the attractive and repulsive forces. Besides, according to the research [19, 22], the potential functions are used to describe the interplays among individuals in fish school and vehicles, which also can control the individual to avoid the obstacle through the repulsive potential field.

The effects of two kinds of interplays among vehicles, that is, attraction and repulsion, are varying along with the changing relative distance between vehicles. And the comprehensive impact of the interplays can be appropriately modeled with the potential field functions. A bump function model in [28] is adopted to formulate the smooth potential functions with finite cutoffs. In this paper, for the sake of simplicity, the bump function used to indicate the proximity between the vehicles i and j is denoted by $φ_{i j} (t) = ψ_{l} (p_{i} (t), p_{j} (t))$ . By combining $φ_{i j} (t)$ and the attractive/repulsive potential functions $U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} (p_{i} (t), p_{j} (t))$ , the attractive/repulsive forces are defined as

\begin{matrix} F_{i j}^{A} (t) = - m_{i} \nabla U_{A} (p_{i} (t), p_{j} (t)) - ρ_{i} φ_{i j} (t) v_{i} (t), \\ F_{i j}^{R} (t) = - m_{i} \nabla U_{R} (p_{i} (t), p_{j} (t)) - μ_{i} φ_{i j} (t) v_{i} (t), \end{matrix}

(5)

where

ρ_{i}

and

μ_{i}

are two positive real coefficients used to scale the magnitude of the attractive/repulsive forces, and they are limited as

ρ_{i}, μ_{i} \in (0,1)

in this paper.

According to the form of the bump function, $φ_{i j} (t)$ can be expressed as

\begin{array}{l} φ_{i j} (t) \\ = \{\begin{array}{l} 1, when \frac{Δ d_{i j} (t) - (r_{i} + r_{j})}{d_{i} - (r_{i} + r_{j})} \in [0, l], \\ k_{v} (1  +  \cos ( π ( \frac{Δ d_{i j} (t)  +  (l - 1) (r_{i}  +  r_{j})  -  l \times d_{i}}{(1 - l) (d_{i} - (r_{i} + r_{j}))} ))), \\ when \frac{Δ d_{i j} (t) - (r_{i} + r_{j})}{d_{i} - (r_{i} + r_{j})} \in (l, 1], \\ 0, otherwise, \end{array} \end{array}

(6)

where

k_{v}

is similarly a positive parameter within (0,1) and the variable

Δ d_{i j} (t)

denotes the intervehicular distance between the vehicles i and j at the time t. In order to explore the influence of the different parameters

k_{v}

and l on the adopted bump function, we conduct the numerical experiment with the settings of

d_{i} = 300 m

r_{i} = r_{j} = 5 m

k_{v} = {0.2,0.5}

, and

l = {0.1,0.3,0.5,0.7,0.9}

. The results are shown in Figure 2. It can be found that this bump function is a scalar function that ranges between 0 and 1 and has different cutoffs with different values of

k_{v}

and l. It also can be seen in Figure 2 that when

k_{v} = 0.2

and

l = 0

the function has a sudden jump at a certain point. To guarantee the smoothness of the potential function, we fix

k_{o} = k_{L} = k_{v} = 0.5

and

l = 0.3

in the following experiments.

Figure 2

Bump function with different parameters.

The potential function of attractive force is designed as the form of a tangent function:

\begin{array}{l} U_{A} (p_{i} (t), p_{j} (t)) \\ = \{\begin{cases} \frac{1}{2} k_{A} \tan (\frac{π (‖p_{i} (t) - p_{j} (t)‖ - (r_{i} + r_{j}))}{2 (d_{i} - (r_{i} + r_{j}))}), \\ when ‖p_{i} (t) - p_{j} (t)‖ \in [(r_{i} + r_{j}), d_{i}), \\ 0, otherwise \end{cases} \end{array}

(7)

and the repulsive potential function is given as follows:

\begin{array}{l} U_{R} (p_{i} (t), p_{j} (t)) \\ = \{\begin{cases} \frac{1}{2} k_{R} (\frac{1}{{(‖p_{i} (t) - p_{j} (t)‖ - (r_{i} + r_{j}))}^{2}} \\ {- \frac{1}{{(d_{i} - (r_{i} + r_{j}))}^{2}})}^{2}, \\ when ‖p_{i} (t) - p_{j} (t)‖ \in ((r_{i} + r_{j}), d_{i}], \\ 0, otherwise, \end{cases} \end{array}

(8)

where the vectors

p_{i} = {[x_{i}, y_{i}]}^{T}

p_{j} = {[x_{j}, y_{j}]}^{T}

denote the position of vehicles

i, j

, respectively.

k_{A}

and

k_{R}

are two scaling positive real coefficients. By introducing the potential functions

U_{A} (p_{i} (t), p_{j} (t))

and

U_{R} (p_{i} (t), p_{j} (t))

, we can further assume that the magnitudes of the attractive and repulsive effects existing between two neighboring vehicles i and j are equal when distance

Δ d_{i j} (t) = ‖p_{i} (t) - p_{j} (t)‖

between them stays at a certain desired value. Let this desired distance be

Δ d_{desire} \in ((r_{i} + r_{j}), d_{i})

. Note that the attractive and repulsive effects are represented by the gradients of the potential functions

U_{A} (p_{i} (t), p_{j} (t))

and

U_{R} (p_{i} (t), p_{j} (t))

with respect to the relative distance between the vehicles i and j, respectively. Accordingly,

k_{A}

and

k_{R}

should be set to satisfy the partial differential equation

‖\partial U_{A} (p_{i} (t), p_{j} (t)) / \partial Δ d_{i j} (t)‖ = ‖\partial U_{R} (p_{i} (t), p_{j} (t)) / \partial Δ d_{i j} (t)‖

under the condition of

Δ d_{i j} (t) = Δ d_{desire}

. By substituting

Δ d_{i j} (t) = Δ d_{desire}

into 7 and 8 and solving the equation

‖\partial U_{A} (p_{i} (t), p_{j} (t)) / \partial Δ d_{i j} (t)‖ = ‖\partial U_{R} (p_{i} (t), p_{j} (t)) / \partial Δ d_{i j} (t)‖

, we can get the relationship of the magnitudes of

k_{A}

and

k_{R}

as follows:

\begin{array}{l} \frac{k_{R}}{k_{A}} \\ = (\frac{π}{4 (d_{i} - (r_{i} + r_{j}))} \\ \times (\cos^{- 2} (\frac{π}{2} (\frac{Δ d_{desire} - (r_{i} + r_{j})}{d_{i} - (r_{i} + r_{j})})))) \\ \times (2 {(Δ d_{desire} - (r_{i} + r_{j}))}^{- 3} \\ {\times [{(Δ d_{desire} - (r_{i} + r_{j}))}^{- 2} - {(d_{i} - (r_{i} + r_{j}))}^{- 2}])}^{- 1} . \end{array}

(9)

Based on 9, it can be seen that once the desired distance

Δ d_{desire}

among vehicles and the parameters

d_{i}

r_{i}

, and

r_{j}

are given, the settings on

k_{A}

and

k_{R}

are expected to satisfy the relationship

{k_{R} = ξ \times k}_{A}

where ξ is set to be the result of the right side of 9. In order to explore the impact of the parameters

r_{i}

and

r_{j}

on the potential functions

U_{A} (p_{i} (t), p_{j} (t))

and

U_{R} (p_{i} (t), p_{j} (t))

, we conduct the following numerical experiment where the sensing range

d_{i}

is fixed at 300 m and

r_{i}

has the same value with

r_{j}

for simplicity; that is,

r_{i} = r_{j} = 5 m

. In addition, the desired distance

Δ d_{desire}

is set to be 50 m, and the parameter

k_{A}

discretely varies from 0.01 to 0.5. According to 9, the ratio of

k_{R}

and

k_{A}

can be calculated as

ξ = 1.482 \times 10^{5}

. Hence, in this experiment, we can set the other parameter

k_{R}

{k_{R} = ξ \times k}_{A} = 1.482 \times 10^{5} k_{A}

. The relevant results are given in Figure 3.

Figure 3

The variation of the potential functions with different parameter settings.

From Figure 3, it can be found that the potential functions are both monotonously varying along with the value of the distance $‖p_{i} (t) - p_{j} (t)‖$ between vehicle i and vehicle j. The definitions of the functions $U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} (p_{i} (t), p_{j} (t))$ guarantee that the attractive and repulsive potential fields have the important properties, respectively: the smaller the distance $‖p_{i} (t) - p_{j} (t)‖$ is, the smaller the effect of the attractive potential field becomes, while the effect of the repulsive potential field is increasing along with decreasing the distance $‖p_{i} (t) - p_{j} (t)‖$ . As shown in Figure 3, when $‖p_{i} (t) - p_{j} (t)‖$ decreasingly approaches the value of $(r_{i} + r_{j})$ , $U_{A} (p_{i} (t), p_{j} (t))$ is reduced to zero while $U_{R} (p_{i} (t), p_{j} (t))$ dramatically increases. This indicates that when a vehicle tends to collide with another vehicle, the attractive potential effect should be weakened and the repulsive effect strengthened at the same time. Hence, the comprehensive effect resulting from both the attractive and repulsive potential fields will force the vehicle to avoid the collisions. Additionally, from Figure 3, it can be found that the larger the parameters $k_{R}$ and $k_{A}$ are, the steeper the slopes of $U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} (p_{i} (t), p_{j} (t))$ become. At this point, large parameter settings on $k_{R}$ and $k_{A}$ would make the potential fields more sensitive to the time-related change in the intervehicular distance. Hence, an appropriate value of $k_{R}$ and $k_{A}$ should be chosen carefully. According to the results presented by Figure 3, we fix $k_{A} = 0.1$ and ${k_{R} = ξ \times k}_{A}$ for our model in the other following experiments for the sake of controlling the sensitivity of the potential field functions $U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} (p_{i} (t), p_{j} (t))$ .

On the other side, some other environmental factors influencing the movement of the connected vehicles should be taken into account as well such as the road obstacles and the road constraints. That is, the vehicles on the road might face the obstacles like accidents, road maintenances, and other emergencies. Besides, the vehicles are also constrained by the road boundaries, which is different from the fish school. In this work, the effects arising from road obstacles and road boundaries are modeled as

\begin{matrix} F_{i}^{(3)} (t) = \sum_{\forall k \in M} F_{i k}^{o} (t) + F_{i}^{L} (t), \end{matrix}

(10)

where

F_{i k}^{o} (t)

is the repulsive forces of the obstacle k,

k = 1,2, \dots, M

(M is the amount of the road obstacles) and

F_{i}^{L} (t)

is the virtual repulsive force that forces a vehicle to be away from the boundary of the road and to stay within the road when the vehicle is too close to one boundary and would drive off the road. Since the decrease in the relative distance between a vehicle and an obstacle or between a vehicle and one road boundary would lead to the increase in the repulsive effect on the vehicle, the repulsive forces of road obstacles and road boundaries should be formulated as decreasing functions of such a relative distance. By referring to the bump function given in 6, we propose the potential field functions to model the repulsive effect arising from road obstacles and road boundaries as follows.

Define the vector $p_{k}^{O} = {[x_{k}^{O} (t), y_{k}^{O} (t)]}^{T}$ as the position of a certain obstacle denoted by $O_{k}$ . The obstacle in this paper is treated as a ball with the radius $r_{k}^{o}$ . When the vehicle i approaches to the obstacle $O_{k}$ in proximity and the distance $‖p_{i} (t) - p_{k}^{O}‖$ becomes smaller than the sensing range $d_{i}$ , the vehicle i will detect it and prepare for avoiding it. Furthermore, $F_{i k}^{o} (t)$ should be a monotonically decreasing function of the distance between the obstacle $O_{k}$ and the vehicle i in the sensing range, and the vehicle i should keep a certain distance from the obstacles so as to avoid the collision. Then the function of the repulsive force induced by a bump function defined in [28] is designed as

\begin{matrix} F_{i k}^{o} (t) = F_{i k}^{o} \times φ_{l} (p_{i} (t), p_{k}^{o} (t)) n_{i k} (t), \end{matrix}

(11)

where

F_{i k}^{o}

is the maximum magnitude of the obstacle repulsive force

F_{i k}^{o} (t)

, which is a scalar, and

n_{i k} (t)

is a unit vector used to indicate the repulsive acting direction at the time instant t:

\begin{matrix} n_{i k} (t) = \frac{(p_{i} (t) - p_{k}^{o} (t))}{‖p_{i} (t) - p_{k}^{o} (t)‖} \end{matrix}

(12)

and the bump function

φ_{l} (p_{i} (t), p_{k}^{o} (t))

is represented as

\begin{array}{l} \begin{array}{l} φ_{l} (p_{i} (t), p_{k}^{o} (t)) \\ =  \{\begin{array}{l} 1, when \frac{Δ d_{i k} (t) - (r_{i} + r_{k}^{o})}{d_{i} - (r_{i} + r_{k}^{o})} \in [0, l], \\ k_{o} (1  +  \cos (π (\frac{Δ d_{i k} (t)  +  (l  -  1) (r_{i}  +  r_{k}^{o})  -  l  \times  d_{i}}{(1 - l) (d_{i} - (r_{i} + r_{k}^{o}))}))), \\ when \frac{Δ d_{i k} (t) - (r_{i} + r_{k}^{o})}{d_{i} - (r_{i} + r_{k}^{o})} \in (l, 1], \\ 0, otherwise, \end{array} \end{array} \end{array}

(13)

where

Δ d_{i k} (t) = ‖p_{i} (t) - p_{k}^{o} (t)‖

and

k_{o}

and l are positive real coefficients whose values are limited within

(0,1)

Considering the constraints of road is essential for realistic modeling of connected vehicles moving as a group; the road constraints force which is similar to the force of the obstacle is defined as

\begin{matrix} F_{i}^{L} (t) = F_{i}^{L} ψ_{l} (p_{i} (t), p_{i}^{L} (t)) n_{i}^{L} (t), \end{matrix}

(14)

where

F_{i}^{L}

is the maximum magnitude of the force

F_{i}^{L} (t)

and

ψ_{l} (p_{i} (t), p_{i}^{L} (t))

is also designed as a bump function as follows:

\begin{array}{l} ψ_{l} (p_{i} (t), p_{i}^{L} (t)) \\ = \{\begin{cases} 1, when \frac{Δ d_{i}^{L} (t) - r_{i}}{R - r_{i}} \in [0, l], \\ k_{L} (1 + \cos (π (\frac{Δ d_{i}^{L} (t) + (l - 1) r_{i} - l \times R}{(1 - l) (R - r_{i})}))), \\ when \frac{Δ d_{i}^{L} (t) - r_{i}}{R - r_{i}} \in (l, 1], \\ 0, otherwise, \end{cases} \end{array}

(15)

where

k_{L} \in (0, 1)

and

p_{i}^{L} (t)

represents the position of the vehicle i's center projection point on the nearest road boundary line at the time instant t, and

Δ d_{i}^{L} (t) = ‖p_{i} (t) - p_{i}^{L} (t)‖

n_{i}^{L} (t)

is the unit direction vector from

p_{i}^{L} (t)

p_{i} (t)

that is perpendicular to the road boundary line, which can be calculated as

\begin{matrix} n_{i}^{L} (t) = \frac{p_{i} (t) - p_{i}^{L} (t)}{Δ d_{i}^{L} (t)} . \end{matrix}

(16)

The parameter R is the maximum range of the virtual road constraint force acting on a vehicle. From the equation above, it can be seen that when the relative distance between the vehicle i and its nearest road boundary (i.e.,

Δ d_{i}^{L} (t)

) is larger than R, this virtual road constraint force is set to 0; otherwise, the closer to the road boundary the vehicle is, the larger this repulsive force becomes.

In addition, in order to make the repulsive forces $F_{i k}^{o} (t)$ and $F_{i}^{L} (t)$ , respectively, induced by the obstacles and the road constraints dynamically coordinated with the attractive force arising from the desired velocity $v_{i}^{o} (t)$ , we combine them with $F_{i}^{(1)} (t)$ defined in 3 and then defined the maximum magnitudes of these two repulsive forces as follows:

\begin{array}{l} F_{i k}^{o} (t) = γ_{i k} \times |F_{i}^{(1)} (t) \cdot n_{i k} (t)| \\ = γ_{i k} \frac{m_{i}}{τ_{i}} |(v_{i}^{o} (t) - v_{i} (t)) \cdot n_{i k} (t)|, \\ F_{i}^{L} (t) = β_{i} \times |F_{i}^{(1)} (t) \cdot n_{i}^{L} (t)| \\ = β_{i} \frac{m_{i}}{τ_{i}} |(v_{i}^{o} (t) - v_{i} (t)) \cdot n_{i}^{L} (t)|, \end{array}

(17)

where

|*|

is the absolute value sign and

γ_{i k}

and

β_{i}

are positive sensitivity coefficients within

(0,1]

which are used to scale and coordinate the magnitudes of

F_{i k}^{o} (t)

and

F_{i}^{L} (t)

2.2. Stability Analysis of Connected Vehicles

Since connected vehicles are equipped with wireless communications, one is able to interact with the others in its sensing range. Thus, these mobile terminals constitute a local vehicular network. The communication topology of these connected vehicles at any time instant t can be presented by a bidirectional graph, which can also be called as communication graph. Let $G (t) = (V, E (t))$ denote this communication graph, and $V = {i | \forall i \in N}$ . As aforementioned in Section 2.1, we assume that the sensing range of all the vehicles is identical; that is, $d_{i} = d (\forall i \in V)$ . Subsequently, an edge in this communication graph $G (t)$ is used to indicate a bidirectional wireless communication interaction between a vehicle and one member in this vehicle's neighbor. That is, the set of those edges in $G (t)$ can be defined as $E (t) = \{(i, j) | ‖p_{i} (t) - p_{j} (t)‖ < d\}$ and $E (t) \in V \times V$ . For simplicity, we also denote the neighboring nodes of any node i (i.e., the vehicle i) in the given communication graph $G (t)$ at any time instant t as a set $N_{i} (t) = \{j | (i, j) \in E (t), j \in V\}$ . From the notation of $E (t)$ , it is obviously seen that the topology of this communication graph is varying over time due to the possibility of a vehicle entering into or out of the communication range of a host vehicle. Attempting to analyze the overall mobility of a given vehicle group moving in the same road, we assume that the initial velocity of all those vehicles is the same and their desired velocities at any time instant t are also identical; that is, $v_{i} (0) = v$ and $v_{i}^{o} (t) = v^{o} (t)$ for all vehicles $i \in V$ . It should be noted that this assumption is reasonable since vehicles would move with a constant velocity in the same direction when they are in an equilibrium traffic flow of the same road and encounter no emergencies or disturbances. Also, we assume that the sensing ranges of all the vehicles are equal; that is, $d_{i} = d_{j}$ for $\forall i \neq j \in V$ . We introduce the concept of the “group centroid,” which is represented as $p_{c} (t) = (1 / N) \times \sum_{i = 1}^{N} p_{i} (t)$ . Now, we state the analysis results of our proposed mobility model.

Corollary 1.

Consider that connected vehicles with wireless communications evolve under the mobility model defined by 1, and given that $\sum_{\forall k \in M}^{} γ_{i k} + β_{i} \leq 1$ , $v_{i} (0) = v$ , and $v_{i}^{o} (t) = v^{o} (t)$ for all vehicles $i \in V$ , the overall group can asymptotically converge to the desired velocity $v^{o} (t)$ which is defined in 2.

Proof.

We can lump all the forces of every vehicle as follows:

\begin{matrix} \sum_{\forall i \in V} m_{i} {\dot{v}}_{i} (t) = \sum_{\forall i \in V} (F_{i}^{(1)} (t) + F_{i}^{(2)} (t) + F_{i}^{(3)} (t)) . \end{matrix}

(18)

Since the condition is given that

d_{i} = d_{j}

for

\forall i \neq j \in V

and these forces

F_{i j}^{A} (t)

and

F_{i j}^{R} (t)

are bidirectional for all vehicles

i \in V

, we have

F_{i j}^{A} (t) = {- F}_{j i}^{A} (t)

and

F_{i j}^{R} (t) = - F_{j i}^{R} (t)

according to 5. At this point, we further have

\begin{array}{l} \sum_{\forall i \in V} F_{i}^{(2)} (t) = \sum_{\forall i \in V} (\sum_{\forall j \in N_{i} (t)} (F_{i j}^{A} (t) + F_{i j}^{R} (t))) \\ = \sum_{\begin{array}{l} \forall i \neq j \in V \\ \forall (i, j) \in E (t) \end{array}} (F_{i j}^{A} (t) + F_{i j}^{R} (t) + F_{j i}^{A} (t) + F_{j i}^{R} (t)) = 0 . \end{array}

(19)

Therefore, equation 1 can be rearranged as

\sum_{\forall i \in V}^{} m_{i} {\dot{v}}_{i} (t) = \sum_{\forall i \in V}^{} (F_{i}^{(1)} (t) + F_{i}^{(3)} (t))

. Now, recalling that the bump functions

φ_{l} (p_{i} (t), p_{k}^{o} (t))

and

φ_{l} (p_{i} (t), p_{i}^{L} (t))

are limited within

[0,1]

, we can get the inequality as follows:

\begin{array}{l} ‖F_{i}^{(3)} (t)‖ \\ = ‖(\sum_{\forall k \in M} F_{i k}^{o} (t) \times φ_{l} (p_{i} (t), p_{k}^{o} (t)) n_{i k} (t)) \\ + F_{i}^{L} (t) ψ_{l} (p_{i} (t), p_{k}^{L} (t)) n_{i}^{L} (t)‖ \\ \leq ‖(\sum_{\forall k \in M} F_{i k}^{o} (t) \times φ_{l} (p_{i} (t), p_{k}^{o} (t)) n_{i k} (t))‖ \\ + ‖F_{i}^{L} (t) ψ_{l} (p_{i} (t), p_{k}^{L} (t)) n_{i}^{L} (t)‖ \\ \leq \sum_{\forall k \in M} F_{i k}^{o} (t) + F_{i}^{L} (t) \\ = \sum_{\forall k \in M} γ_{i k} \frac{m_{i}}{τ_{i}} |(v_{i}^{o} (t) - v_{i} (t)) \cdot n_{i k} (t)| \\ + β_{i} \frac{m_{i}}{τ_{i}} |(v_{i}^{o} (t) - v_{i} (t)) \cdot n_{i}^{L} (t)| \\ \leq \sum_{\forall k \in M} γ_{i k} \frac{m_{i}}{τ_{i}} ‖v_{i}^{o} (t) - v_{i} (t)‖ + β_{i} \frac{m_{i}}{τ_{i}} ‖v_{i}^{o} (t) - v_{i} (t)‖ \\ = (\sum_{\forall k \in M} γ_{i k} + β_{i}) \frac{m_{i}}{τ_{i}} ‖v_{i}^{o} (t) - v_{i} (t)‖ \leq ‖F_{i}^{(1)} (t)‖ . \end{array}

(20)

Hence, the magnitude of the attractive force induced by the desired velocity

v_{i}^{o} (t)

is larger than that of the force induced by the obstacle and the road boundary. This implies that the mobility of the overall vehicle group is mainly dominated by the potential field that can force vehicles to keep moving with or asymptotically converge to the desired velocity even after being disturbed by the repulsive potentials resulting from obstacles and road constraints. At this point, we prove Corollary 1.

Now, we firstly provide some properties of the potential functions $U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} (p_{i} (t), p_{j} (t))$ related to the interactive force $F_{i}^{(2)} (t)$ as follows. For simplicity, we equivalently denote $U_{A} ({Δ d}_{i j} (t)) = U_{A} (p_{i} (t), p_{j} (t))$ and $U_{R} ({Δ d}_{i j} (t)) = U_{R} (p_{i} (t), p_{j} (t))$ , where ${Δ d}_{i j} (t) = ‖p_{i} (t) - p_{j} (t)‖$ . The following can be mathematically proven.

Corollary 2.

$U_{A} ({Δ d}_{i j} (t))$ satisfies the following properties: (i) it is continuously differentiable for ${Δ d}_{i j} (t) \in [(r_{i} + r_{j}), d_{i})$ and is a monotonously increasing function of ${Δ d}_{i j} (t)$ within $[(r_{i} + r_{j}), d_{i})$ that makes $U_{A} ({Δ d}_{i j} (t)) \to 0$ with ${Δ d}_{i j} (t) \to (r_{i} + r_{j})$ and $U_{A} ({Δ d}_{i j} (t)) \to \infty$ with ${Δ d}_{i j} (t) \to d_{i}$ ; (ii) $u_{i j}^{A} = u_{j i}^{A}$ when denoting $u_{i j}^{A} = \partial U_{A} ({Δ d}_{i j} (t)) / \partial {Δ d}_{i j} (t)$ for $\forall (i, j) \in E (t)$ ; and (iii) $\sum_{\forall (i, j) \in E (t)}^{} (\partial U_{A} ({Δ d}_{i j} (t)) / \partial p_{i} (t)) = \sum_{\forall i \neq j \in V}^{} (\partial U_{A} ({Δ d}_{i j} (t)) / \partial p_{i} (t))$ .

Proof.

It is easily obtained that the partial derivative of $U_{A} ({Δ d}_{i j} (t))$ with respect to ${Δ d}_{i j} (t)$ is expressed as

\begin{array}{l} u_{i j}^{A} = \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial {Δ d}_{i j} (t)} = \frac{π}{4 (d_{i} - (r_{i} + r_{j}))} \\ \times k_{A} {(\cos (\frac{π ({Δ d}_{i j} (t) - (r_{i} + r_{j}))}{2 (d_{i} - (r_{i} + r_{j}))}))}^{- 2} > 0 \end{array}

(21)

for

{Δ d}_{i j} (t) \in [(r_{i} + r_{j}), d_{i})

. At this point,

U_{A} ({Δ d}_{i j} (t))

is a monotonously increasing function that is defined on

[(r_{i} + r_{j}), d_{i})

. Consequently, we further get

\begin{matrix} \lim_{{Δ d}_{ij} (t) \to (r_{i} + r_{j})} U_{A} ({Δ d}_{i j} (t)) = U_{A} ((r_{i} + r_{j})) = 0, \\ \lim_{{Δ d}_{ij} (t) \to d_{i}} U_{A} ({Δ d}_{i j} (t)) = U_{A} (d_{i}) = \infty \end{matrix}

(22)

and

u_{i j}^{A} > 0

for

{Δ d}_{i j} (t) \in ((r_{i} + r_{j}), d_{i})

. Thus, property (i) is proven. Moreover, recall that we have assumed

d_{i} = d_{j}

for

\forall i \neq j \in V

. This means that the sensing ability (sensing range) of every vehicle is identical. Hence, according to

{Δ d}_{i j} (t) = {Δ d}_{j i} (t)

, it can be seen that

\partial U_{A} ({Δ d}_{i j} (t)) / \partial {Δ d}_{i j} (t) = \partial U_{A} ({Δ d}_{j i} (t)) / \partial {Δ d}_{j i} (t)

; that is,

u_{i j}^{A} = u_{j i}^{A}

. Thus, property (ii) is proven. Noting the definition of

U_{A} ({Δ d}_{i j} (t))

and the communication graph

G (t)

, when one vehicle j is not in the neighbor

N_{i} (t)

of another vehicle i, the relative distance between them

{Δ d}_{i j} (t)

is larger than the sensing range

d_{i}

. Then, the potential

U_{A} ({Δ d}_{i j} (t))

is set to 0 accordingly, as well as

\partial U_{A} ({Δ d}_{i j} (t)) / \partial {Δ d}_{i j} (t) = 0

. So, we have

\begin{array}{l} \sum_{\forall i \neq j \in V} \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial p_{i} (t)} \\ = \sum_{\forall j \in {V ∖ N}_{i} (t)} \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial p_{i} (t)} + \sum_{\forall j \in N_{i} (t)} \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial p_{i} (t)} \\ = 0 + \sum_{\forall j \in N_{i} (t)} \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial p_{i} (t)} = \sum_{\forall (i, j) \in E (t)} \frac{\partial U_{A} ({Δ d}_{i j} (t))}{\partial p_{i} (t)} . \end{array}

(23)

This finishes the proof of property (iii).

Similarly, we present Corollary 3 for the potential function $U_{R} ({Δ d}_{i j} (t))$ . Its proof can also be finished in the similar way of proving Corollary 2, so it is not needed to repeat them here in consideration of space.

Corollary 3.

$U_{R} ({Δ d}_{i j} (t))$ is (i) continuously differentiable for ${Δ d}_{i j} (t) \in ((r_{i} + r_{j}), d_{i}]$ while it is a monotonously decreasing function defined on $((r_{i} + r_{j}), d_{i}]$ that satisfies $U_{R} ({Δ d}_{i j} (t)) \to \infty$ with ${Δ d}_{i j} (t) \to (r_{i} + r_{j})$ and $U_{R} ({Δ d}_{i j} (t)) \to 0$ with ${Δ d}_{i j} (t) \to d_{i}$ ; (ii) the partial derivative $u_{i j}^{R} = \partial U_{R} ({Δ d}_{i j} (t)) / \partial {Δ d}_{i j} (t)$ ( $u_{i j}^{R} < 0$ ) also satisfies the symmetry $u_{i j}^{R} = u_{j i}^{R}$ for $\forall (i, j) \in E (t)$ ; and (iii) $\sum_{\forall (i, j) \in E (t)}^{} (\partial U_{R} ({Δ d}_{i j} (t)) / \partial p_{i} (t)) = \sum_{\forall i \neq j \in V}^{} (\partial U_{R} ({Δ d}_{i j} (t)) / \partial p_{i} (t))$ .

In addition, it is significant to analyze the intragroup mobility besides the basic characteristic of the overall group. From model 1, it can be noted that the connected vehicle group model can be decomposed into two terms: the attractive/repulsive forces induced by the external potentials including $F_{i}^{(1)} (t)$ and $F_{i}^{(3)} (t)$ , and the interactive force $F_{i}^{(2)} (t)$ that exists among vehicles. According to the principle of inertial frame of reference, when the group centroid $p_{c} (t)$ is set as an original point of the inertial frame of reference, each vehicle's motion relative to this reference frame is mainly dependent on the interactive force $F_{i}^{(2)} (t)$ induced by the attractive/repulsive potentials between vehicles instead of the external force potentials $F_{i}^{(1)} (t)$ and $F_{i}^{(3)} (t)$ . Thus, in order to analyze the intragroup mobility, we turn to focus on the influence of the interactive force $F_{i}^{(2)} (t)$ between vehicles. Then, the derivative of a vehicle's velocity relative to the reference frame (the relative velocity is denoted by $v_{i}^{c} (t)$ ) with respect to the time instant t can be expressed as

\begin{matrix} m_{i} {\dot{v}}_{i}^{c} (t) = F_{i}^{(2)} (t) . \end{matrix}

(24)

Also, let $p_{i}^{c} (t)$ be the relative position of the vehicle i in the reference frame. The vehicle dynamic system in the reference frame is then formulated as follows:

\begin{array}{l} {\dot{p}}_{i}^{c} (t) = v_{i}^{c} (t), \\ {\dot{v}}_{i}^{c} (t) = \frac{1}{m_{i}} F_{i}^{(2)} (t) \\ = - \sum_{\forall i \neq j \in V}^{} \{\nabla U_{A} ({Δ d}_{i j} (t)) + \nabla U_{R} ({Δ d}_{i j} (t))\} \\ - \sum_{\forall i \neq j \in V} \frac{(ρ_{i} + μ_{i})}{m_{i}} φ_{i j} (t) v_{i}^{c} (t) . \end{array}

(25)

Based on the vehicle dynamic system model above, one form of the Lyapunov function $L (t)$ for the whole group can be designed by combining the potential field energy and the kinematic energy as

\begin{matrix} L (t) = U (t) + V (t), \end{matrix}

(26)

where the potential field term is

U (t) = \sum_{\forall i \in V}^{} \sum_{\forall (i, j) \in E (t)}^{} (U_{A} ({Δ d}_{i j} (t)) + U_{R} ({Δ d}_{i j} (t)))

and the kinematic energy is

V (t) = (1 / 2) \sum_{\forall i \in V}^{} {‖v_{i}^{c} (t)‖}^{2}

. Thus, by differentiating

L (t)

with respect to the time variable t, one further gets

\begin{array}{l} \dot{L} (t) \\ = \sum_{\forall i \in V} \sum_{\forall (i, j) \in E (t)} {({\nabla U}_{A} ({Δ d}_{i j} (t)) + {\nabla U}_{R} ({Δ d}_{i j} (t)))}^{T} \\ \cdot {\dot{p}}_{i}^{c} (t) + \sum_{\forall i \in V}^{} {(v_{i}^{c} (t))}^{T} \cdot {\dot{v}}_{i}^{c} (t) \\ = \sum_{\forall i \in V}^{} \sum_{\forall (i, j) \in E (t)}^{} {({\nabla U}_{A} ({Δ d}_{i j} (t)) + {\nabla U}_{R} ({Δ d}_{i j} (t)))}^{T} \cdot v_{i}^{c} (t) \\ + \sum_{\forall i \in V}^{} \{{(v_{i}^{c} (t))}^{T} \\ \cdot (- \sum_{\forall i \neq j \in V}^{} \{\nabla U_{A} ({Δ d}_{i j} (t)) + \nabla U_{R} ({Δ d}_{i j} (t))\} \\ - \sum_{\forall i \neq j \in V}^{} \frac{(ρ_{i} + μ_{i})}{m_{i}} φ_{i j} (t) v_{i}^{c} (t))\} . \end{array}

(27)

According to Corollaries 2 and 3, it should be noted that

\begin{array}{l} \sum_{\forall (i, j) \in E (t)} {({\nabla U}_{A} ({Δ d}_{i j} (t)))}^{T} \cdot v_{i}^{c} (t) \\ - \sum_{\forall i \neq j \in V} [{(v_{i}^{c} (t))}^{T} \cdot {\nabla U}_{A} ({Δ d}_{i j} (t))] = 0, \\ \sum_{\forall (i, j) \in E (t)} {({\nabla U}_{R} ({Δ d}_{i j} (t)))}^{T} \cdot v_{i}^{c} (t) \\ - \sum_{\forall i \neq j \in V} [{(v_{i}^{c} (t))}^{T} \cdot {\nabla U}_{R} ({Δ d}_{i j} (t))] = 0 . \end{array}

(28)

Therefore, $\dot{L} (t) = - \sum_{\forall i \in V}^{} \sum_{\forall i \neq j \in V}^{} ((ρ_{i} + μ_{i}) / m_{i}) φ_{i j} (t) {{(v_{i}^{c} (t))}^{T} \cdot v}_{i}^{c} (t)$ . Since ${{(v_{i}^{c} (t))}^{T} \cdot v}_{i}^{c} (t) \geq 0$ and $((ρ_{i} + μ_{i}) / m_{i}) φ_{i j} (t) \geq 0$ , we have $\dot{L} (t) \leq 0$ , which means that those vehicles in the reference frame can asymptotically converge to a stable state. In the actual reference frame, this fact suggests that those vehicles having knowledge of some other vehicles' state in their local neighbor tend to move as an overall group. Now, based on this, we declare the following result to illustrate the potential of collision avoidance among vehicles equipped with wireless communications.

Theorem 4.

Consider that connected vehicles with wireless communications evolve under mobility model 1, and given that $\sum_{\forall k \in M}^{} γ_{i k} + β_{i} \leq 1$ , $v_{i} (0) = v$ and $v_{i}^{o} (t) = v^{o} (t)$ for all vehicles $i \in V$ . If $Δ d_{i j} (0) > (r_{i} + r_{j})$ for $\forall i \neq j \in V$ , those vehicles move as an overall group that tends to keep avoiding intrasystem collision.

Proof.

According to Corollary 1, those vehicles tend to keep the same desired velocity $v^{o} (t)$ . Furthermore, since we have $\dot{L} (t) \leq 0$ for $t \geq 0$ , $L (t) \leq L (0) < \infty$ . If there were at least two vehicles i and j extending to collide with each other, that is, $Δ d_{i j} (t) \to (r_{i} + r_{j})$ , $L (t) = (U (t) + V (t)) \to \infty$ due to $U (t) \to \infty$ under the consideration of the property (i) in Corollary 3. Hence, the contradiction occurs. At this point, any two vehicles i and j in the group do not extend to collide with each other. That is, this theorem is proven.

Theorem 5.

Consider that connected vehicles with wireless communications evolve under mobility model 1, given that $\sum_{\forall k \in M}^{} γ_{i k} + β_{i} \leq 1$ , $v_{i} (0) = v$ and $v_{i}^{o} (t) = v^{o} (t)$ for all vehicles $i \in V$ . Those vehicles that initially belong to the neighboring member of one certain vehicle i, that is, those ones $(i, j) \in E (0)$ ( $\forall i \in V$ ), tend to keep connecting to this vehicle i when all vehicles move as an overall group.

Proof.

Similar to the proof of Theorem 4, since $\dot{L} (t) \leq 0$ for $t \geq 0$ , $L (t) \leq L (0) < \infty$ . Recall property (i) in Corollary 2. When any one neighboring member j of a vehicle i tends to move out of the sensing range, that is, $Δ d_{i j} (t) \to d_{i} ((i, j) \in E (0))$ , $U_{A} ({Δ d}_{i j} (t)) \to \infty$ so that $U (t) \to \infty$ . Hence, $L (t) = (U (t) + V (t)) \to \infty$ when at least one pair $(i, j)$ is satisfying $Δ d_{i j} (t) \to d_{i}$ . This is contradictory to the fact that $\dot{L} (t) \leq 0$ for $t \geq 0$ . In this way, we prove Theorem 5.

3. Simulations

In this section, simulations achieved by MATLAB are provided to verify the model proposed in previous section. The analysis of the results is also given in this section in order to characterize the benefit of wireless communication among vehicles for safety and efficiency of traffic. We set up a traffic scene where an accident situation on the unidirectional road as the obstacle has to be avoided. Then we get the diagram of the trajectories of the connected vehicles to demonstrate that the model can achieve the obstacle avoidance effectively. In addition, we also have the comparative numerical experiment of vehicles with wireless communications and without wireless communications while travelling which characterize the impacts of the communications in the connected vehicles.

The simulation scene is assumed on a section of a unidirectional road. The unidirectional road we concentrate on is 2000 meters long and 50 meters wide. We suppose that an accident occurs on the road where the location of the center of it is marked as $O_{1}$ . The obstacle is a circle region with the center set to be 1000 meters away from the beginning boundary of the simulation range and with radius of $r_{i}^{O} = 5$ meters which needed to be detoured. Namely, a collision will occur when the vehicle enters the circle area.

The number and the velocity of vehicles are produced randomly at the start of simulations. During the simulations, those connected vehicles are controlled by our proposed model to move as a cooperative group, whose direction is consistent. In all simulations, we fix the time interval for updating the kinematic information of each vehicle at 0.1 s. Besides, the physical parameters of the vehicle are given as $r_{i} = 5 m$ , $d_{i} = 300 m$ , $m_{i} = 2000 kg$ , and $r_{i}^{W} = 7 m$ . Moreover, other parameters required in the simulations are set as $Δ d_{desire} = 50 m$ , $τ_{i} = 0.05 s$ , $ρ_{i} = 0.1$ , $μ_{i} = 0.1$ , $k_{v} = 0.5$ , $k_{A} = 0.1$ , $k_{R} = 1.482 \times 105 k_{A}$ , $k_{O} = 0.5$ , $l = 0.3$ , $k_{L} = 0.5$ , $R = 3 m$ , $γ_{i k} = 0.8$ ( $k = 1$ , and $\forall i \in V$ ), and $β_{i} = 0.6$ .

3.1. Obstacle Avoidance Trajectory

Figure 4 shows the various travelling trajectories of each vehicle when the vehicle group comes across the obstacle in different average velocities. In these simulations, the average velocities are set to 10 m/s, 15 m/s, and 20 m/s separately in order to explore the performances of avoiding obstacles in different velocities. What is more, the total number of vehicles is the same in these three simulations set as $n = 20$ . The axes in Figures 4(a), 4(b), and 4(c) are the referenced coordinates used to measure the real-time position of vehicles moving on the road, whose unit is meter. From Figure 4, it can be observed that when encountering the obstacle, the vehicles change into two subgroups like the behaviors of fish school when the established information is locally broadcasted in the whole group. Subsequently, they are unified into a group again as before after successfully avoiding this obstacle. In addition, comparing the travelling trajectory in different average velocities, the conclusion can be drawn that the lower the average velocity of vehicles is, the easier the vehicles unified together. Although various velocities have different travelling trajectory, all the vehicles have been successfully reached and the vehicles have stayed together as a group and avoided collisions or obstacles. This result indicates that the model of connected vehicles based on the behaviors of fish school is effective in describing the cooperative behaviors in avoiding obstacles.

Figure 4

The travelling trajectory of vehicles.

3.2. Impacts of Wireless Communications

In order to explore the influences of wireless communications on the overall mobility of connected vehicles, we also conduct other comparative numerical experiments. In the simulations, when any vehicle i is considered to be equipped with wireless communications, its sensing range is set to $d_{i} = 300 m$ ; otherwise, the sensing range of a driver is limited to $d_{i} = 100 m$ if the relevant vehicle i is not equipped with wireless communications and its driving decision is made dependent on visual perception. Thus, those experiment results shown in the following figures marked by “Equipped” are related to the specific case where every vehicle is considered to be able to communicate with its neighbors via wireless communications, while the results marked by “Unequipped” are obtained in the other case where all of the vehicles can only interact with others in the aforementioned shorter sensing range. For comparison, the following performance metrics are defined.

(i) Probability of a Vehicle Moving into the Local Warning Scope of Obstacle $P_{o}$ . Recalling that we assume there is an obstacle with the radius of 5 m on the road, it should be noted that even though a vehicle can successfully avoid colliding with the obstacle, its situation may be dangerous when it is too close to this obstacle. Thus, the probability of an obstacle entering into the predefined local warning scope of the vehicle i can be used to quantify the danger severity even when there are not any obstacle collisions occurring.

(ii) Probability of a Vehicle Moving into the Local Warning Scope of Other Vehicles $P_{v}$ . Similarly, the local warning scope of a vehicle is defined as the local range with a radius of 7 m that covers this vehicle. With this metric, we can also quantify the danger severity of vehicles equipped with or without wireless communications as moving even when no vehicle collisions occur among them.

Subsequently, let $Event (k)$ represent the total number of the events occurring at the kth simulation epoch in which the obstacle or other vehicles enter into the local warning scope of vehicle i, let $N = |N|$ be the total number of vehicles, and TotalEpoch the total number of simulation epochs. These metrics $P_{o}$ and $P_{v}$ are calculated as follows:

\begin{matrix} P_{o} or P_{v} = \frac{\sum_{k = 1}^{TotalEpoch} Event (k)}{TotalEpoch \times N} . \end{matrix}

(29)

In a real life traffic situation, the vehicle velocity will significantly affect the mobility of the overall vehicle group. In addition, it is necessary to evaluate the traffic efficiency of connected vehicles flocking as a group under different vehicle numbers (indicating the traffic density). Therefore, in this paper, the impacts of the wireless communications of connected vehicles by varying vehicle velocity and vehicle number are studied.

To analyze the impacts on road safety level, firstly, we vary the initial vehicle velocity from 10 m/s to 30 m/s and fix the total number of vehicles at $N = 30$ . At each vehicle velocity point, simulation has been performed 30 times repeatedly, and the metrics $P_{o}$ and $P_{v}$ are then averaged. Additionally, the standard deviations of these evaluation metrics are also calculated at each velocity. The relevant results are shown in Figure 5. As can be seen from these results, the danger severity indicated by $P_{o}$ , $P_{v}$ increases with increasing vehicle velocity. Because the higher the velocity a vehicle moves with on the road, the more the time this vehicle needs to decelerate to a certain lower velocity and the larger the possibility of the obstacle or other vehicles entering into the predefined local warning scope of vehicle i; the danger severity on the road indicated by $P_{o}$ and $P_{v}$ is higher with higher vehicular mobility. Nevertheless, assisted with wireless communications, connected vehicles are able to make a decision on braking in advance and larger distance is reserved for changing their motion trajectory. Therefore, the metrics $P_{o}$ and $P_{v}$ obtained in the case marked by “Equipped” are lower on the overall level than those in the “Unequipped” case. In Figure 5(b) especially, $P_{v}$ in the “Equipped” case is reduced by 57.89% on average compared with that in the “Unequipped” case when the vehicle velocity is larger than 10 m/s. Hence, this suggests that vehicles can benefit from wireless communications in terms of extending the perception range and enhancing traffic safety level.

Figure 5

The results of $P_{o}$ and $P_{v}$ under different initial velocity settings.

Next, we turn to focus on the effects of different vehicle numbers of the group on the safety level. For comparison, the following experiment is processed by varying the vehicle number from 10 to 70 and fixing the initial vehicle velocity at 20 m/s. Also, the simulations have been performed with 30 replications per vehicle number and, subsequently, the results of $P_{o}$ and $P_{v}$ obtained at each vehicle number are averaged. The error bars in Figure 6 also indicate the standard deviations of $P_{o}$ and $P_{v}$ . Some similar conclusions can be drawn from Figure 6. The safety level on the road can be more enhanced by wireless communications. When the vehicle number increases on the road, the traffic density is increased. Consequently, the distance between vehicles on average decreases; this would lead to a lower safety level. However, the averaged value of $P_{o}$ corresponding to different vehicle numbers in the “Equipped” case is about 11.5% lower than that obtained in the “Unequipped” case, while the overall averaged $P_{v}$ of the “Equipped” case is reduced by about 22.61% compared to that of the “Unequipped” case. The reason is that, as discussed before, wireless communications expand the sensing range of each equipped vehicle in a group so as to enable them to adjust their velocity to the overall velocity of the group ahead of time, in the meanwhile guaranteeing a relatively larger reserved distance among vehicles.

Figure 6

The results of $P_{o}$ and $P_{v}$ under different vehicle numbers.

In fact, there are no obstacle collisions or vehicle collisions occurring during simulations, which is because the proposed mobility model is adopted to simulate the cooperative behaviors of connected vehicles. What is more, as theoretically analyzed before, collisions can be avoided between vehicles evolving with the comprehensive effects of potential forces in the proposed mobility model. Even though no collisions are induced, the potential danger on the road in actual situations may exist. Thus, with considering the evaluation metrics $P_{o}$ and $P_{v}$ instead of collision-related statistics, we can quantify the impacts of wireless communications on the traffic safety level. This fact shown in Figures 5 and 6 implies that the road capacity could be increased by deploying wireless communications onboard when considering keeping the road danger severity under a certain level. This is also in accordance with the results in [29].

To provide a deep insight into the impacts of wireless communications on the road traffic efficiency, we would like to conduct some other simulations. Firstly, we fix the total vehicle number as $N = 30$ and then vary the initial vehicular velocity from 10 m/s to 30 m/s. At each velocity point, we repeat the simulation 30 times and calculate the flow rate on average. The flow rate is proposed here which is measured at the section of the obstacle. Its calculation is defined as follows: in one simulation we can record the time instant $t_{1}$ when the first vehicle of the vehicle group passes the section located at the obstacle and $t_{2}$ to indicate the time instant when the last vehicle passes this section. The simulation results obtained in the “Equipped” and “Unequipped” cases are given in Figure 7(a). From Figure 7(a), it can be seen that the flow rate increases along with increasing the initial vehicular velocity in both compared cases. Logically, the increase in the velocity of each vehicle will lead to the increase in the mean speed of the overall traffic flow. Consequently, whether vehicles are equipped with wireless communications or not, a high vehicular velocity will promote the road traffic efficiency. Nevertheless, when comparing the results of the “Equipped” and the “Unequipped” cases in Figure 7(a), the flow rate at each initial velocity setting obtained in the “Equipped” case is higher than that of the “Unequipped” case. The reason is that with the assistance of wireless communication those equipped vehicles are able to sense the traffic condition far away from their own positions so that they can smoothly react to potential collisions ahead of time and efficiently avoid collisions in a cooperative manner. Then, the overall mobility of the equipped vehicles can be better maintained than those unequipped vehicles during vehicles operating collision avoidance.

Figure 7

The results of flow rate.

In the next experiment, the initial velocity of all the vehicles is fixed at 20 m/s, and the total vehicle number is set to range from 10 to 70. Similarly, the simulations in this experiment have been performed with 30 replications per vehicle number point. The results are shown in Figure 7(b). The flow rate obtained in the “Equipped” case at each vehicle number point is obviously larger than that of the “Unequipped” case. This indicates that the vehicle group of a small or a large size can benefit more from the wireless communication, since those equipped vehicles can coordinate their velocity more smoothly according to the overall movement of the group via wireless communication systems. At this point, this result confirms that the wireless communications are promising to improve the overall efficiency of the road traffic.

4. Conclusions

In this paper, the connected vehicles are analogized to the fish schools in order to develop a more realistic model to describe the mobility of the connected vehicles with the wireless communications. The proposed model consists of the attraction of the goal, the repulsion of the obstacles, the constraint of the road, and the interplays of vehicles in the group including both attractive and repulsive affection. Many simulations are performed to verify that the wireless communications among vehicles can improve the safety and efficiency of traffic greatly, and the numerical experiments prove the reasonableness of the model in terms of describing the cooperative behaviors of connected vehicles with wireless communications.

Footnotes

Notations

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant nos. 61103098, 91118008 and 61302110 and the Foundation of Key Laboratory of Road and Traffic Engineering of the Ministry of Education in Tongji University.

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