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This paper investigates cooperative flocking control design with connectivity preserving mechanism. During flocking, interagent distance is measured to determine communication topology of the flocks. Then, cooperative flocking motion is built based on cooperative artificial potential field with connectivity preserving mechanism to achieve the common flocking objective. The flocking control input is then obtained by deriving cooperative artificial potential field using control Lyapunov function. As a result, we prove that our flocking protocol establishes group stabilization and the communication topology of multiagent flocking is always connected.

Flocking in nature is a coordinated movement of group of living entities. Recently, flocking behavior has been adopted in a multiagent system to resemble collective movement of living system, for example, collective movement of group of multirobot system and autonomous vehicles. Those multiple agents together form a cooperative system to solve complex problems in distributed manner that is impossible to be carried out by a single agent.

With respect to the literature of multiagent system, many theories denote that flocking model is built using three basic heuristic rules, that is, cohesion, separation, and velocity alignment, which is initiated for the first time by [

In multiagent robot, common flocking rule can further be described technically as centroid attraction, collision avoidance, and velocity alignment, respectively. Centroid attraction rule tries to pull the agent to stay close to the nearby agents in the flocks if the agents are moving away. Meanwhile, collision avoidance rule forces the agent to avoid collision with nearby agents in the flock, while velocity synchronization rule will drive the agents to synchronize their velocities while moving in the flock. The most notable flocking controllers are based on the works of Tanner et al. [

The synchronization of movement in flocking can be carried out through local or global communication, with explicit communication and without communication (stigmergic) as summarized in [

Network communication is mainly used to exchange information state between an agent and the nearby agents. Regarding this matter, the problem is that the initially connected network can not guarantee connectivity over the time during flock movement. Once disconnected, flock movement can be separated and that causes collective flocking goal not being achieved. Thus, it is important to prevent separation of the flock into multiple separate groups when the network is being disconnected. Regarding this issue, many kinds of research have been conducted to design flocking control input (flocking protocol) to ensure that the flocks achieve the collective goal while preserving connectedness during the movement. Connectivity preserving flocking of multiagent mobile robot then becomes a new trending area, and various flocking protocols have been proposed including both centralized and decentralized approaches, for example, presented in [

Since connectivity of agents in flocking formation depends on its communication range, then the communication topology among agents can be connected or disconnected affected by the movement of an agent in flocks. The changes of signal power may also affect to the actual interagent distance since the distance is correspondingly perceived from measured signal power. Regarding this issue, a connectivity preserving mechanism is required to maintain connectivity during the flock movement. Fang et al. in [

Formation of flocking is commonly used in the various practical implementation of flocking protocol, such as in coverage control using multiagent mobile robot, a formation of multiple UAV, and cooperative mobile sensor networks, as well as cooperative mobile robot, for a surveillance system. Therefore, the design of formation flocking needs to consider navigation capability along with separation, cohesion, velocity alignment, and connectivity of a multiagent network.

In this paper, our work is to preserve network connectivity by using an artificial potential function, where flocking protocol can also perform collision avoidance and obstacle avoidance function. The artificial potential function is then considered as control Lyapunov function candidate to be derived as continuous flocking control input. We also define the algebraic connectivity gradient controller and blend this gradient controller into flocking control input. This is the major difference of our work from [

This paper is organized as follows. Section

Let us consider a group of points of mobile robots that consists of

We assume that each agent obtains its position from sensors. Each agent is also equipped with a wireless communication network; therefore information state can be exchanged within the neighboring agents via a communication network.

Regarding Assumption

Topology of a flock of robots is a dynamic undirected topology, denoted as

node or vertices

set of edges or time varying communication links

set of neighbors

Without loss of generality, each agent is strictly not self-communicating agent

From the adjacency matrix

Given the undirected graph

This paper will focus on the design of flocking protocol that can preserve connectivity in terms of algebraic graph representation.

We define the stable flocking protocol as the necessary condition to preserve connectivity of the network all the time.

Group of robot agents is said to flock when all agents approach the same velocity, with no collisions among them, such that

The connectivity maintenance of flocking topology is approximately determined by the distance between agents in the flock. The movement of agents that might cause the connectivity can be unavailable during flocking motion. Therefore, our problem is dealing with designing a cooperative flocking protocol using only local information to drive all robot agents to achieve the same velocity, avoiding collision with other agents in flock, avoiding static obstacle, driving the agents to the goal position, and guaranteeing that the network is always connected during the flocking, indicated by

This section introduces the design of artificial potential functions for the purpose of target attraction, obstacle avoidance, and interagent attraction and repulsion. Since first used in [

Consider the multiagent robots move in configuration workspace, which is defined as follows.

Suppose a point of robot moves in the workspace

We define the free space, where there is no obstacle within the area.

Free space

Given the position of robot agents in stack vector

Let

Using the definitions above, all robot agents have to avoid obstacles that block the robot route toward the desired position. Therefore, each robot has to execute an obstacle avoidance function that is constructed by measuring the distance between the robot agent and the obstacle center

Robot agent

Using the definitions above, the environmental potential field is composed of goal attractiveness and obstacle repulsiveness, and the result is depicted in Figure

Example of environment model used in this paper. Goal position is represented by the lowest potential value; two hills represent different size of obstacles.

We assume that the sensing capability of an agent equals its communication radius. Suppose there exists pair interaction between robot agents

Let

Open disk radius is an area covered by

Close disk radius is an area covered by

The idea of flock centroid is initiated in [

Illustration of an agent disk radius and centroid attraction among the flockmates.

The long-range attraction occurs between agents in the flocks. To make sure that robot agents in the flocks are attracted to each other, as well as attracted by the center of the flock, an attractive function is constructed by measuring the distance between individual agents and the center of the flocks. The flock centering attraction function is simply given by

This paper uses the definition of closed disk coverage, where two interagent distances may occur within

Collision avoidance function is used to keep

Interagent collision avoidance is implemented by attract-repel function. The long-range attraction occurs between two agents in the flocks. It is to make sure that the agents in the flock are attracted by the following function:

Attractive potential function can be viewed as potential energy between agents in the flock.

Let

Interagent collision avoidance is implemented by short range repulsive function, which is formulated from the distance between agents. The objective of this function is to repel neighboring agents if they are too close; therefore collision can be avoided. The short range repulsive potential function is defined as

Repulsive potential function can be viewed as potential energy between agents in the flock within the range

Let

We combine attraction and repulsion potential function to generate attract-repel function as

The attractive-repulsive potential function may guarantee that agent

Interagent attract-repel potential function with various repulsive constant

This section describes the design of flocking controller by utilizing control Lyapunov function. We use the direct method of Lyapunov to obtain the flocking control input which is derived from overall artificial potential function that has been defined in the previous section. The reader may refer to [

Regarding [

We extend the control law of formation control problem in [

A control Lyapunov function is continuously positive-definite function; that is,

Regarding Definition

To extract the control input from candidate of Lyapunov function, we differentiate all potential function components in

The

To prove the above theorem, we consider (

It is clear that since (

In this subsection, we present the analysis of velocity convergence which is the necessary condition of stable flocking controller as in Definition

Using system (

Let

Let us consider the system

We begin with following theorem.

Robot agent that uses control input (

To prove this theorem, we rewrite the sum of potential function

The derivative of Lyapunov function along

By replacing

In this section, we add the term of gradient of algebraic connectivity along

Let us define the following theorem.

Suppose

To proof this theorem, we follow [

Lemma

Second smallest eigenvalue can be written in the form of

We further define the gradient of algebraic connectivity as control input as follows:

To analyze the stability of flocking controller after

This section demonstrates the simulation results of connectivity preserving flocking in multiagent system using the stable control input obtained from the previous section. The simulation verifies flocking protocol, which is derived using control Lyapunov function. The common algorithm to simulate multiagent robot with connectivity preserving flocking is depicted in Algorithm

capture position

compute

build

compute

compute

compute

update

update

All scenarios in simulation are conducted in Matlab. The initial position of each robot agent is generated randomly with initial connected topology at

multiagent flocking without obstacle,

multiagent flocking with single obstacle,

multiagent flocking with multiple obstacles,

multiagent flocking with algebraic connectivity controller.

Multiagent flocking trajectories without obstacle.

From simulation, it can be seen that the multiagent graph is always connected during flocking evolution. The convergence of agent’s velocity is determined by consensus term as described in control input in (

Multiagent velocity trajectory in no obstacle scenario.

Multiagent robot trajectories with single obstacle.

Multiagent velocity trajectory in the scenario with obstacle.

The robot trajectories during simulation of multiagent with multiple size obstacles can be seen in Figure

Multiagent robot trajectories with multiple size multiple obstacles.

Multiagent velocity convergence in multiple size multiple obstacles.

Multiagent robot trajectories of 10 agents with multiple obstacles.

Multiagent robot trajectories with single and multiple obstacles with algebraic connectivity controller.

We also compare the velocity convergence where the gradient of

Multiagent velocity with multiple obstacles scenario. (a) Without

Algebraic connectivity value during simulation of multiagent velocity convergence in multiple size multiple obstacles.

In this paper, the control Lyapunov function is successfully used to construct flocking control input with connectivity preservation in a group of networks of mobile robot agents. We model the agents as a dynamic point mass, and the proposed control strategy involves potential force and velocity consensus. Several potential forces are designed to steer the neighbor agents to predetermined distance while avoiding collision among them such that connectedness of network topologies can be guaranteed. Under the assumption that the initial flocking topology is connected, the overall flocking topology is preserved to be connected while obstacle and collision avoidance are also guaranteed.

Simulation results verify the new proposed control input with gradient of algebraic connectivity. From the multiagent flocking point of view, it implies that the overall controller could maintain connectivity of agents, forming cohesive multiple agents, and matching velocities. In order to guarantee goal achievement of multiagent flocking, connectivity is still preserved by keeping the second smallest eigenvalue of flocking topology

The authors declare that there are no competing interests regarding the publication of this article.