Journal of Robotics

Volume 2016 (2016), Article ID 8540761, 12 pages

http://dx.doi.org/10.1155/2016/8540761

## Design of Connectivity Preserving Flocking Using Control Lyapunov Function

^{1}School of Computing, Telkom University, Bandung, Indonesia^{2}School of Electrical Engineering & Informatics, Institut Teknologi Bandung, Bandung, Indonesia^{3}Faculty of Mathematical Sciences, Institut Teknologi Bandung, Bandung, Indonesia

Received 19 June 2016; Revised 28 August 2016; Accepted 14 September 2016

Academic Editor: Shahram Payandeh

Copyright © 2016 Bayu Erfianto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. These three rules describe how an individual agent moves according to position and velocity of the nearest agents in flocks. Flocks always move together with velocity converging to the same value and guaranteeing no collision among agents.

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. [2, 3] and Olfati-Saber [4]. Olfati-Saber [4] does not use flock centering rule to build flocking formation; instead of that he defines lattice formation to keep interagent distance.

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 [5]. Erfianto et al. [6] have extended Reynold flocking protocol with explicit communication using broadcast gossip protocol, where the explicit communication of flocking is simulated in Netlogo multiagent simulator. By equipping mobile robot agent with communication capability, it means that flocking protocol is then restricted by communication capability. Previous research works on flocking algorithm are mostly based on the assumption that the network topology is always connected without any mechanism to achieve this circumstance.

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 [7–9].

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 [10] further divide connectivity preservation into two main categories, that is, conservative connectivity preservation and flexible connectivity preservation. In conservative connectivity preservation, the approach is to preserve all existing links; thus the network topology is strictly fixed, as presented in [7, 11, 12]. In flexible connectivity preservation approaches, the underlying communication topology is allowed to switch into different topologies, which means that edges are allowed to be added or removed from a node as long as the global graph topology is always connected. It is done by maintaining the spanning tree of the underlying network. Regarding these two approaches, therefore, the challenge is to design cooperative control law where it depends on the assumption that the underlying communication topology is connected over the time during the whole flocking evolution.

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 [7, 9, 13, 14]. As a result, we conclude that, given the initially connected network, our flocking protocol can make all agents converge to a common velocity, collisions between agents, as well as between agent and obstacle, can be avoided, and the connectivity of underlying network can be preserved; thus the desired stable flocking behavior can be achieved.

This paper is organized as follows. Section 1 is the introduction of this paper. Section 2 of this paper is about problem definition to design connectivity preserving flocking. Section 3 discusses the design of artificial potential function that will be involved in control Lyapunov function. Flocking controller design is presented in Section 4, while the simulation to verify the controller is presented in Section 5.

#### 2. Problem Formulation and Assumption

Let us consider a group of points of mobile robots that consists of robot agents moving in 2D space. The dynamic of mobile robot agent is modeled aswhere denotes position vector, where is the Euclidean distance between two agents, which is clearly . Let be the velocity vector and be the control input of agent , respectively.

*Assumption 1. *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 1, interaction among agents can further be modeled by bidirectional communication network, that is, undirected graph topology.

*Definition 2 (flocking topology). *Topology of a flock of robots is a dynamic undirected topology, denoted as , that consists of(i)node or vertices ;(ii)set of edges or time varying communication links in which each undirected edge represents the communication between a pair of agents limited by communication range ;(iii)set of neighbors with symmetric property .

Without loss of generality, each agent is strictly not self-communicating agent . Regarding Definition 2, interagent communication can then be constructed by adjacency matrix , with elementwhere is the Euclidean distance between two agents, and each agent is supposed to have the same communication radius . Some fundamentals of graph theory used in the paper, including algebraic graph theory, can be found in the main reference [15].

From the adjacency matrix , the degree matrix is defined as . Laplacian matrix of can be further obtained by subtracting adjacency matrix from degree matrix, such that , where Laplacian matrix is positive semidefinite. Let the eigenvalues of be which is nondecreasing order with corresponding eigenvectors . We denote that the remarkable properties of Laplacian matrix are as follows.

Lemma 3 (see [15]). *Given the undirected graph *(i)* is always symmetric and positive semidefinite and satisfies sum of square property where ;*(ii)*; thus if graph is connected; then is referred to algebraic connectivity of ;*(iii)* has a single zero eigenvalue and the corresponding eigenvector is .*

*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.*

*Definition 4 (stable flocking protocol). *Group of robot agents is said to flock when all agents approach the same velocity, with no collisions among them, such thatwhere is the second smallest eigenvalue of Laplacian matrix .

*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 .*

*3. Potential Function Design*

*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 [16], the artificial potential function has been commonly used to solve the problems of motion planning in mobile robotic applications. Theoretically, the artificial potential function generates repulsive fields close to obstacles and attractive fields close target. In artificial potential fields, a noncollided motion is determined by how much the robot is attracted by other agents and repelled by either the obstacle or other agents. We add centroid function and attract-repel function to avoid local minima caused by goal attraction and obstacle function. The potential function method has several advantages, and the most important one is being easy to implement.*

*3.1. Environmental Potential Field*

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

*Definition 5 (workspace). *Suppose a point of robot moves in the workspace ; that is, the Euclidean distance between two points does not exceed the area of .

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

*Definition 6 (free space). *Free space is defined as , where a robot agent could move safely without colliding with any obstacles.

*Given the position of robot agents in stack vector , we assume all information of the environment in the is known by each robot agent. A target or goal position of agent is defined as , which is in the free area of configuration workspace. To attract the robot agent to reach the desired goal position, the attraction-to-target potential function is defined as follows.*

*Definition 7 (target attraction function). * is a metric that measures how close a robot agent is to the desired goal position. Target or goal attraction function is defined as

*Let be th obstacle in the workspace and be the th obstacle that has to be avoided by agent .*

*Definition 8 (obstacle). * is a th disc obstacle that exists in workspace with center and radius : such that the distance of any point to the center of obstacle does not exceed .

*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 . The obstacle avoidance function computed by ith agent is defined as follows.*

*Definition 9 (obstacle avoidance function). *Robot agent could avoid th obstacle with the following potential function:where is the center of obstacle and is radius of obstacle , such that robot agent could avoid obstacle if none of them intersect each other within .

*Using the definitions above, the environmental potential field is composed of goal attractiveness and obstacle repulsiveness, and the result is depicted in Figure 1. This figure illustrates two different sizes of obstacles indicated by two hills with high potential value, while the lowest potential, the darkest valley at the center of the basin, illustrates the desired position.*