## 1. Introduction

A driverless vehicle is a next big innovation in the transport industry. Many automotive companies are now working on five levels of autonomy [

1]. Most of these companies are doing research on developing the fifth level of autonomous car and set to launch their cars in upcoming years. The introduction of fully autonomous vehicles will perhaps constitute the largest change to everyday transportation in living memory and is predicted to deliver a wide range of environmental, social and economic benefits [

2]. Moreover, the United Kingdom has launched several projects to develop this technology further and created a Centre for Connected and Autonomous Vehicles (CCAV) [

3].

A vehicle formation control or vehicle platooning is an important part of traffic management because of several benefits such as improved safety, fuel efficiency, mileage, a time needed to travel and reduced road congestion. Vehicles on the road usually follow another vehicle and form a platoon-based formation. For the human-operated vehicle, a platoon based formation is simple and can be seen every day. However, for connected autonomous vehicles (CAV), they must stay in the lanes and follow nearby vehicles by maintaining safe distance and velocity. The aim of the platoon formation control is to confirm that all vehicles in a platoon move at the same speed while maintaining a desired formation shape or geometry, which is stated by a desired inter-vehicle spacing strategy [

4]. Therefore, for autonomous vehicles, forming a platoon formation requires specific algorithms, controllers and strategies consisting of longitudinal and lateral control.

Furthermore, for a formation control, vehicles need to communicate with each other; therefore, information sharing plays a vital role in the overall operation. This overall operation can be called collaboration within vehicles. A collaboration depends on sensing and information sharing within a vehicular network. The challenge here is to make these two operations, i.e. sensing and information sharing, autonomous to achieve desired control. Therefore, as autonomous vehicles are being developed, the next problem of collaboration between vehicles in dynamic environment needs to be addressed. The solution to these problems will secure the future of the CAVs.

Figure 1 shows a summary of current approachs and methods used for formation control.

In the leader-follower approach, a leader is assigned to the multi-robot formation and remaining robots are the followers. In this approach, a leader follows its desired trajectory while follower robots track the position of the leader. There are three kinds of leader in this approach, namely static leader, dynamic leader and virtual leader. The advantage of this approach is the reduced tracking errors and can be analysed using standard control techniques [

5]. Another benefit is that only the leader is responsible for planning trajectories and followers must follow the coordinates of the leader; therefore, it results in a simple controller. In terms of disadvantage, a leader’s fault can penalise the whole formation and feedback from followers to a leader is generally not applied in this approach.

In the behaviour-based approach, each individual robot shows several behaviours based on sensory inputs such as obstacle avoidance, goal seeking and formation keeping where final control is derived from the weighting of the relative importance of each behaviour. There are main four methods in this approach, namely motor scheme, potential field, swarm intelligence and flocking. This approach can be defined as a structured network of such interacting behaviours where the final action of each robot is derived by the behaviour coordinator. The behaviour coordinator multiplies the output of each behaviour by its relative weight, then summing and normalizing the results. One advantage of this approach is that it can operate in the unknown and dynamic environment because it is a parallel, real-time and distributed method, requiring less information sharing [

5]. Its another advantage is that each behaviour has its physical meaning and the formation feedback can be incorporated into group dynamics by coupling the outputs of each individual behaviour. This method has some disadvantage such as the need to deriving a mathematical model of group dynamics, studying the convergence of specific formation, and guaranteeing the stability of the whole formation [

6].

In the virtual structure approach, a virtual rigid structure is derived that represents a form of agents. Then, the desired motion of the virtual rigid structure is given, and agents’ motion is derived from the given rigid structure. Finally, to track the agents, a tracking controller for each individual agent is derived in which the formation is maintained by minimising the error between the virtual structure and the current agent position. In this approach, the desired trajectory is not assigned to the single agent, but it is shared by the whole formation team. In terms of advantage, this approach is easy to prescribe the coordinated behaviour for the whole group [

5]. In terms of disadvantage, this approach is centralized, therefore, a single point of failure can crash the whole system. Furthermore, heavy communication and computation burden is concentrated on the centralized location, which may degrade the overall system performance [

7].

The rest of this paper organised as follows. Some preliminaries are discussed in

Section 2 to help readers to understand several concepts in the multi-robot systems. In

Section 3, the leader-follower approach is presented, including the literature on vehicle platooning and formation controller design. The behaviour-based approach is described in

Section 4, including motor schema-based method, potential field method, flocking and swarm intelligence based method for vehicle formation control.

Section 5 analyses the virtual structure approach and resultant formation controllers. Finally, a number of technical and implementation challenges are discussed in

Section 6.

## 2. Preliminaries

In this section, the fundamental concepts which are frequently used in formation control are covered such as communication topologies, graph theory and consensus. A consensus control theory is used to study the interaction between a group of dynamical agents. A tool commonly used to analyse consensus control strategies is the graph theory, whereas the communication topologies indicate a potential interaction between neighbouring agents and are described by the graph.

#### 2.1. Topologies

To attain formation control, information sharing with other robots is a key task in the multi-robot system. Therefore, topologies are implemented in the vehicle platooning to address the information sharing problem. These topologies are responsible for the information exchange flow which describes how the vehicles in a platoon exchange information with each other. There are different types of communication topologies to choose from, and some of them are shown in

Figure 2.

In

Figure 2, several leader–follower topologies are presented. Here, topology A is the predecessor following (PF) topology, topology B is the predecessor–leader following (PLF) topology, topology C is the bidirectional (BD) topology, topology D is the bidirectional-leader (BDL) topology, topology E is the two-predecessor following (TPF) topology and topology F is the two-predecessor–leader following (TPFL) topology. Note that these topologies are for single platoon formation. During platoon operation, several scenarios can take place such as interaction between multiple platoons or loss of communication under existing topology. Therefore, dynamic or switching topologies should be considered for platoon stability and mobility [

9,

10]. Here, dynamic or switching topology means that the topology of platoon formation switch into a different topology over time.

#### 2.2. Algebraic Graph Theory

Graph theory is used in the multi-robot formation control for the information exchange between autonomous robots, to perform the stability analysis of the formation, and to achieve consensus. The topologies discussed previously are modelled as a graph where robots can be represented as nodes of a graph and intersections such as sensing and communication can be represented as edges of the graph. Here, two basic graph theories are presented, directed graph or diagraph and undirected graph. More information on graph theory can be found in these books [

11,

12].

A diagraph is called strongly connected if there is a directed path from every node to every other node. A diagraph g is defined as pair $(V,E)$, where V denotes the set of nodes and $E\subseteq VxV$ denotes the set of ordered pairs of the nodes, called edges. It is assumed that there is no self-edge, i.e., $(i,i)\notin E$ for any $i\in V$. The set of neighbours of $i\in V$ is defined as a set ${N}_{i}:=j\in V:(i,j)\in E$. A directed path of g is an edge sequence of the form $({v}_{{i}_{1}},{v}_{{i}_{2}}),({v}_{{i}_{2}},{v}_{{i}_{3}}),\dots ,({v}_{{i}_{k-1}},{v}_{{i}_{k}})$. If $(i,j)\in E$, j is called a parent of i and i is called a child of j. A tree is a directed graph where a node, called the root, has no parent and the other nodes have exactly one parent. A spanning tree of a directed graph is a directed tree containing every node of the graph.

Given a directed graph

$g=(V,E)$, where

${w}_{ij}$ is a real number associated with

$(i,j)$ for

$i,j\in V$ and assume that

${w}_{ij}>0$ if

$(i,j)\in E$ and

${w}_{ij}=0$, otherwise. The Laplacian matrix

$L=\left[{l}_{ij}\right]\in {\mathbb{R}}^{\left|v\right|x\left|v\right|}$ of

g is defined as:

An undirected graph is called connected if there is a path between any distinct pair of nodes. Let g be a directed graph such that $(i,j)\in E$ if and only if $(j,i)\in E$ and ${w}_{ij}={w}_{ji}$ for all $(i,j)\in E$. Then, g is said to be undirected. The Laplacian matrix L of g is symmetric and positive semi-definite. If g is connected, the second smallest eigenvalue of L is positive. Furthermore, the Laplacian matrix L can be defined as, $L=D-A$. Here, D is the degree matrix and A is the adjacency matrix.

#### 2.3. Consensus in Multi-Robot Systems

In multi-robot systems, reaching a consensus can be considered as one of the important requirements where autonomous vehicles can reach an agreement to form a formation by sharing information locally with their neighbour vehicles. Generally, convergence to a common value is called consensus, and this depends on communication between autonomous vehicles. Analysing emergent of consensus behaviour as a result of local interactions among mobile agents who share information only with their neighbours according to some designed distributed protocols is a critical issue. A consensus is helpful in several ways such as:

Alignment (Pointing in the same direction),

Synchronization (Agreeing on the same time),

Distributed Estimation (Agreeing on the estimation/measurement of the distributed quantity),

Rendezvous (Meeting at a common point).

Earlier, a consensus problem was targeted by a first-order dynamical system [

13,

14]. Afterwards, the second order consensus where all agents are governed by second-order dynamics, position and velocity, and the third-order consensus where all agents are governed by third-order dynamics, position, velocity and acceleration, have attracted much of the interest and several publications target second-order and third-order consensus to derive sufficient and necessary conditions to reach consensus.

An analysis on second-order consensus algorithms was carried out using algebraic graph theory and it concluded that the real and imaginary parts of the eigenvalues of the Laplacian matrix have a key role in reaching consensus [

15]. Moreover, the effect of communication delay was investigated in this study and it was shown that the second-order consensus can be achieved in the multi-agent systems with a directed spanning tree only when the time delay is less than a critical value. Another study proposed a second-order consensus protocol for multi-agent systems with synchronous intermittent information feedback and directed topology [

16]. In this study, algebraic theory and Lyapunov control approach were used for the analysis and proved that second-order consensus can be reached if the general algebraic connectivity of the communication topology is larger than a threshold value and each agent communicates with its neighbours frequently enough as the network evolves.

A third-order consensus of a dynamic system containing position, velocity and acceleration was studied for the case of an undirected graph [

17] and directed networks [

18] using algebraic graph theory in a theoretical study. In these papers, necessary and sufficient conditions for consensus was established in terms of scaling strength and the eigenvalues of the Laplacian matrix. Moreover, a consensus problem for the second-order multi-agent and third-order multi-agent systems were considered in which this heterogeneous system was converted into an equivalent error system for the stability analysis [

19]. In this study, both fixed and switching topologies were considered and a consensus was achieved by obtaining sufficient and necessary conditions based on Markovian jump system theory.

## 7. Conclusions

In this paper, many academic publications are reviewed to understand the current developments on CAV technology, including controller design and stability analysis. The leader–follower approach is found to be used by many researchers for the vehicle platooning problem. A second-order consensus control is widely targeted to reach the consensus, whereas third-order consensus control is a growing field. Furthermore, the string stability theorem is used to evaluate the platoon formation performance. It can be observed that simulation work is completed to prove the derived controllers and theories and few publications target actual implementation on mobile robots.

Moreover, the challenges discussed in this paper require extensive testing before implementing the CAV technology for general uses. Whatever approach one chooses to implement out of these discussed methods, these challenges remains the same and need to be solved. On the other hand, the application challenges provide a good ground to analyse the developed algorithms. After reviewing the vehicle formation control field using several approaches, it can be concluded that:

Few controllers and algorithms in a leader–follower approach targets third-order consensus control and real robot experiments. Furthermore, a dynamic role changing leader should be targeted in a leader–follower approach.

Considering the swarm intelligence based algorithms, it can be observed that they can handle a population of robots, but they do not offer formation control ability; therefore, new parameters or another algorithm should be implemented along with these algorithms.

A machine learning approach is not applied to the formation control problem compared to other techniques and deep learning remains unexplored for multi-robot formation control problems.