Elsevier

Neurocomputing

Volume 397, 15 July 2020, Pages 160-167
Neurocomputing

Circle formation control of second-order multi-agent systems with bounded measurement errors

https://doi.org/10.1016/j.neucom.2020.02.037Get rights and content

Abstract

In this paper, the circle formation control problem of second-order multi-agent systems which are constrained to move on a circle is considered. Each agent can only measure the positions and velocities of its neighbors with bounded measurement errors. Using these inaccurate measurements, a distributed formation control laws is developed for each mobile agent. It is shown that under the proposed control law the multi-agent system can be driven to a neighborhood of the desired circle formation. The upper bound on the agents’ formation errors in the final configuration is also provided. Note that collision between mobile agents can be prevented if the agents’ spatial order on the circle is always preserved. The conditions under which order preservation of the agents is guaranteed during the formation task are also given. Finally, a simulation example is provided to illustrate the effectiveness of the proposed formation control law.

Introduction

Multi-agent systems have drawn much attention recently due to their wide applications in robotics, unmanned vehicles, mobile sensor networks, just to name a few [1], [2], [3], [4], [5], [6]. One of the most important problems in cooperative control of multi-agent systems is the formation control problem [7], [8], [9], [10], [11]. In practice, multi-agent systems with certain geometric shapes can accomplish many prescribed tasks more efficiently and possess good robustness to environmental disturbances [12], [13].

Circle formation control is a typical formation problem, in which networked mobile agents need to converge to or move on a common circle [14], [15], [16], [17], [18], [19]. In [20], distributed formation control laws are developed for multi-agent systems with bearing-only measurements to enclose a static target. In [21], a limit-cycle-based decoupled-design approach is used to address the circle formation problem for mobile agents randomly deployed in a plane. In [22], the circular formation stabilization problem is considered for a group of dynamic unicycles via a hierarchical design approach.

In the past decade, the circle formation problem for multi-agent systems which are constrained to move on a circle has attracted increasing interest. The goal of the problem is to drive the angular distance between neighboring agents to the prescribed distance [23], [24], [25], [26], [27], [28], [29]. In [24], distributed control laws are developed for a group of anonymous mobile agents to form desired circle formation while preserving the agents’ spatial order on the circle. In [26], this work is extended to the case of mobile agents with limited interaction range and a common Lyapunov function is introduced to show the convergence of the proposed switching formation control law. To reduce the number of control updates and communication among the agents, distributed event-triggered algorithms are developed in [27], [28] and [29] for multi-agent systems with first-order and second-order dynamics, respectively.

Although much efforts have been devoted to circle formation control of multi-agent systems, most of them are based on the assumption that each agent can acquire the accurate position and/or velocity information of its neighbors. However, due to the existence of sensor noise and communication noise this assumption generally cannot be satisfied in practice [30], [31]. This motivates us to investigate the circle formation problem of multi-agent systems with second-order dynamics and bounded measurement errors. A distributed formation control law is firstly developed for each agent which only employs inaccurate position and velocity information of its neighbors. Using tools from matrix theory and linear algebra, it is shown that under the proposed formation control law the multi-agent system can be driven to a neighborhood of the desired circle formation and the upper bound on the agents’ formation errors in the final configuration is also provided. Note that collision between mobile agents can be prevented if the agents’ spatial order on the circle is always preserved. It is further shown that order preservation of the agents is guaranteed throughout the formation task if the agents’ initial configuration is close enough to the desired circle formation.

This main difference between this paper and our conference paper [32] lies in that the circle formation problem subject to measurement errors is addressed for mobile agents with second-order dynamics instead of first-order dynamics. Furthermore, both position and velocity measurement errors of the agents are taken into consideration in this paper, which bring difficulties in obtaining the upper bound on the formation errors in the final configuration and convergence analysis of the proposed formation control law.

The rest of the paper is organized as follows. In Section 2, the circle formation problem subject to bounded measurement errors is formulated. A distributed formation control law is developed in Section 3 and its convergence analysis is provided in Section 4. Finally, the simulation results and conclusion are given in Sections 5 and 6, respectively.

Section snippets

Problem formulation

Consider a group of N mobile agents which are constrained to move on a circle. Each agent’s position xi is described by the angle measured counterclockwise from the positive horizontal axis. The agents evolve according to the following second-order discrete-time dynamicsxi(k+1)=xi(k)+Tvi(k)+12T2ui(k),vi(k+1)=vi(k)+Tui(k),where T is the time-step size, ui(k) and vi(k) are the control input and velocity of agent i at step k, respectively [33]. In this paper, the agents are assumed to be static

Distributed formation control laws

Consider the following distributed formation control lawsu1(k)=dNd1+dN(x1,2(k)−x1(k)+v1,2(k)−v2(k))−d1d1+dN(x1(k)+2π−x1,N(k)+v1(k)−v1,N(k))−v1(k),ui(k)=di−di+di−(xi,i+(k)−xi(k)+vi,i+(k)−vi(k))−didi+di−(xi(k)−xi,i−(k)+vi(k)−vi,i−(k))−vi(k),i=2,…N−1,uN(k)=dN−dN+dN−(xN,1(k)+2π−xN(k)+vN,1(k)−vN(k))−dNdN+dN−(xN(k)−xN,N−(k)+vN(k)−vN,N−(k))−vN(k).Recalling the definitions of yi(k) and ξi(k) given by Eqs. (4) and (5), the above formation control law can be rewritten asui(k)=di−di+di−yi(k)−didi+di−yi−(k)

Convergence analysis

In this section, we first show that under the formation control law (6) the multi-agent system converges to a neighborhood of the desired circle formation. Then, the conditions under which order preservation of the mobile agents is guaranteed throughout the formation task are also provided.

Before we proceed, some useful lemmas are given as follows:

Lemma 1

[24]

Let λi be the eigenvalue of the matrix L(d). Then, 0 ≤ λi ≤ 2 holds and 0 is a single eigenvalue of L(d).

Lemma 2

[34]

All roots of the equation z2+az+b=0 with a, b

Simulations

In this section, a simulation example is given to illustrate the effectiveness of the proposed formation control law. Consider a group of six agents which are randomly deployed on a circle initially. The desired circle formation is given by d=[0.85,1.00,0.75,1.05,1.23,1.40]T.

Firstly, we consider the case where the agent’s spatial order on the circle is the same as their initial order when time goes to infinity. The step size is chosen as T=0.6 which satisfies 0<T<21+λ*. Let μ=0.9 and c=2.0 such

Conclusion

In this paper, the circle formation problem of multi-agent systems with second-order dynamics and bounded position and velocity measurement errors is addressed. It is shown that multi-agent systems can be driven to a neighborhood of the desired circle formation under the proposed formation control law. The conditions under which the spatial order of the agents on the circle is preserved during the formation task are also provided. In this paper, mobile agents are assumed to be located on the

CRediT authorship contribution statement

Yuanyuan Wang: Writing - original draft, Methodology. Tao Shen: Writing - review & editing, Formal analysis. Cheng Song: Supervision, Conceptualization, Writing - review & editing. Yijun Zhang: Resources, Software.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was partially supported by grants from the National Natural Science Foundation of China (Nos. 61773213, 61403203, 61973166), the Natural Science Foundation of Jiangsu Province (No. BK20181298), and the State Key Laboratory of Robotics and System (HIT) (Nos. SKLRS-2018-KF-08).

Yuanyuan Wang received the B.S. degree in Automation from Jiangsu University of Science and Technology, Zhenjiang, China, in 2017. She is currently pursuing the M.S. degree in Control Engineering from Nanjing University of Science and Technology, Nanjing, China. Her current research interest focuses on circle formation control of multi-agent systems.

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    • Cited by (0)

    Yuanyuan Wang received the B.S. degree in Automation from Jiangsu University of Science and Technology, Zhenjiang, China, in 2017. She is currently pursuing the M.S. degree in Control Engineering from Nanjing University of Science and Technology, Nanjing, China. Her current research interest focuses on circle formation control of multi-agent systems.

    Tao Shen received the B.Eng. degree in Automation from Nanjing University of Science and Technology, Nanjing, China. Now he is working toward the master degree in Control Theory and Control Engineering from Nanjing University of Science and Technology, Nanjing, China. His research interest focuses on multi-agent system.

    Cheng Song received the B.Eng. degree in Automation and the Ph.D. degree in Control Science and Engineering from the University of Science and Technology of China, Hefei, China, in 2007 and 2012, respectively, and the Ph.D. degree from the Department of Mechanical and Biomedical Engineering, at the City University of Hong Kong, Kowloon, Hong Kong, in 2012. He is currently an Associate Professor in the School of Automation, Nanjing University of Science and Technology. His main research interests include multi-agent systems, cooperative control, and optimal control.

    Yijun Zhang received the B.S. degree in Applied Mathematics from Nanjing Meteorological Institute, China, in 2002, the M.S. degree in Operational Research and Cybernetics from Nanjing Normal University, China, in 2005, and the Ph.D. in Control Theory and Control Engineering from Donghua University, China, in 2008. From 2008 to 2011, he was a post - doctoral fellow in Nanjing University of Science and Technology, China. He was a visiting scholar in the Center for Intelligent and Networked Systems, Central Queensland University, Australia, from December 2010 to June 2011, and a visiting scholar in the Engineering Department, University of Leicester, UK, from January 2012 to January 2013. Currently, he is an associate professor at Nanjing University of Science and Technology, China. His research interests include complex networks, networked control systems, and nonlinear system control.

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