Journal of Control Science and Engineering

Volume 2018, Article ID 9162358, 11 pages

https://doi.org/10.1155/2018/9162358

## Circular Formation Control of Multiagent Systems with Any Preset Phase Arrangement

^{1}School of Computer and Communication Engineering, Liaoning Shihua University, Fushun, China^{2}College of Marine Electrical Engineering, Dalian Maritime University, Dalian, China

Correspondence should be addressed to Shuanghe Yu; nc.ude.umld@ehgnauhs

Received 17 September 2017; Revised 4 December 2017; Accepted 6 December 2017; Published 1 February 2018

Academic Editor: Yongji Wang

Copyright © 2018 Lina Jin 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 deals with the circular formation control problem of multiagent systems for achieving any preset phase distribution. The control problem is decomposed into two parts: the first is to drive all the agents to a circle which either needs a target or not and the other is to arrange them in positions distributed on the circle according to the preset relative phases. The first part is solved by designing a circular motion control law to push the agents to approach a rotating transformed trajectory, and the other is settled using a phase-distributed protocol to decide the agents’ positioning on the circle, where the ring topology is adopted such that each agent can only sense the relative positions of its neighboring two agents that are immediately in front of or behind it. The stability of the closed-loop system is analyzed, and the performance of the proposed controller is verified through simulations.

#### 1. Introduction

Imitating the collective behaviors that occur in nature, the distributed control of multiagent systems (MAS), such as multiple autonomous underwater vehicles (AUVs) and unmanned aerial vehicles (UAVs) [1–3], has attracted a great deal of attention in control and robotic communities [4, 5] and has been extensively explored with different settings, including consensus [6], formation control [7, 8], flocking [9], distributed sensor networks [10], rendezvous [11], and source seeking [12, 13], through coordinating multiple autonomous mobile agents.

As one of these fundamental problems, the pattern-forming problem has attracted a considerable amount of research interest, where the agents are required to cooperatively generate and maintain the desired geometric patterns to perform various teamwork tasks. Formation patterns are typically limited to a point (rendezvous), line (flocking), or circle. The circular formation is a design method for steering the agents to orbit around a target along a common circle, which provides a simple geometric shape to collect data with a desired spatial and temporal distribution. In the community of systems and control, research efforts have been devoted to the circle formation problem for multiagent systems modeled as single or double integrators [14–18] and unicycles [19–26] under different communication topologies. Circular motion has been studied in the scenario of cyclic pursuit with ring topology [14–23]. A collective circular motion is addressed with a jointly connected communication condition [24, 25]. Under all-to-all communication condition, the phase potentials are used for uniform phase arrangement of particles along a circle [26]. In the aforementioned works, all the agents can enclose a fixed or moving target with position, distance, or bearing measurements in an equally circular distribution manner. However, for some special robotic application occasions, uniform distribution is unable to meet the practical demands; for example, AUV formation detects the concentration of oil pollution and UAV formation performs special escort missions in a nonuniform distribution [16]. Only a few works have presented the distributed control laws for a group of agents to formulate any given phase arrangement on a circle, and it should be noted that the agents are restricted to move in the one-dimensional space of a circle [27, 28].

The problem of circular formation of multiple agents with any preset phase arrangement in the two-dimensional space is addressed in this paper. The contributions can be summarized as follows. (1) Through introducing a rotated affine transformation, a tracking control strategy is proposed to achieve circular motion of agents by tracking a rotating matrix, where two cases of circling a target or not [29] are, respectively, considered. (2) Through combining the above control strategy with a multiagent phase cooperation mechanism, the circular formation task with any preset phase arrangement is implemented. Then, the phase arrangement algorithm without circle forming part in [28] is expanded to a two-dimensional model; that is, the positions of all the agents can be initialized out of the circle instead of being initialized on the assumed given circle. Furthermore, the advantage of order preservation is inherited because the circular motion control does not change the phase distribution during the entire motion. (3) An extension of the phase control law in [28] is presented to solve the positioning problem of the agents on the circle, such that some agents are located in the particular directions of the surrounded target according to the practical situation.

The paper is organized as follows. Some necessary preliminaries are presented and the control problem is formulated in Section 2. The tracking control law of circular motion is designed and analyzed in Section 3. Section 4 combines the above circular tracking control algorithm with a phase arrangement control law to achieve any phase arrangement along the circle. Simulation results given in Section 5 validate the strategy.

#### 2. Preliminaries and Problem Formulation

##### 2.1. Model of the Agents

Consider single-integrator-modeled agents moving in the planewhere and denote the position and the control input of agent , respectively. denotes the stacked column vector. As shown in Figure 1, for a target with position , denotes the relative displacement between agent in (1) and the target, denotes the radius of the desired circle, and is a unit vector on the line passing through agent and the target; that is,