SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 9(1); June (2022), pp. 77–80
ISSN (Online): 2352-6386; ISSN (Print): 2405-9021
https://alife-robotics.org/jrnal.html

Research Article

Formation control of rectangular agents using complex
LaplacianYaxin Li, Yingmin Jia, Jintao Hu

The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang
University (BUAA), 37 Xueyuan Road, Haidian District, Beijing, 100191, China

ARTIC LE INFO
Article History
Received 24 November 2021
Accepted 29 March 2021

Keywords
rectangular agentsformation control
communication maintenance eABSTRACT

A distributed control algorithm based on artificial potential function, coordinate transformationand complex Laplacian for formation control of second-order rectangular agents is presented inthis paper. The system can achieve the desired formation, avoid collision and maintainconnectivity between agents. The stability analysis of this algorithm is given. Finally, theeffectiveness of the algorithm is illustrated by the simulation results..

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

collision avoidanceconsider the problem of maintaining connectivity8. This
paper focuses on second-order multi-agent systems andpresents the control law to make the systems reach thedesired formation, avoid collision and maintaincommunication.
Notations: The length of the rectangle agent i is li, thewidth of the rectangle agent i is wi. Ls denotes the
Laplacian matrix of sensing graph. Lc denotes theLaplacian matrix of communication graph.

1. Introduction
Formation control of multi-agent systems has attracted alot of attention in recent years because of the progress ofscience and technology, especially the development ofartificial
intelligence,robots,sensors,andcommunication technology1-3.
Researches about formation control mainly focus onformation structure and the stability of the systems4-6.
Graph theory is usually used to define the positional
relationship between agents and the shape of formation.
The obstacle avoidance strategy between agents andenvironmental obstacles is the premise of the stability ofmulti-agent systems. Artificial potential function is often
used to realize collision avoidance and communication
maintenance between agents.
The shape of most of the agent models studied areparticle or circular. Considering some models intransportation system, rectangular agents are consideredin our paper, which can reduce redundancy when we
design collision avoidance areas. Formation control andobstacle avoidance of first-order rectangular multi-agent
systems have been studied7. On this basis, other articles

2. Preliminaries and Problem Statement
2.1. Coordinate transformation
Considering two rectangle agents (agent 1 and agent2)
and one obstacle (obs) as shown in Fig. 1, the coordinateof center of rectangle agent 1 is denoted by O1 (x1, y1)
and the coordinate of vertex P11 is denoted by P11 (x11,y11) in the frame OXY, the coordinate of center ofrectangle agent 1 is denoted by O11 (x11, y11) and thecoordinate of vertex P11 is denoted by P111 (x111, y111) in
the frame O1X1Y1. ϕ1 is the angle between the X axis andthe line between vertex and O1.

Corresponding author’s E-mail: lyx2018@buaa.edu.cn, ymjia@buaa.edu.cn www.buaa.edu.cn

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{Q1, Q2, …, Qn}, if there exists a path from nodes {Q1,Q2, …, Qn} to qi after removing any node in Q, we say qi
is 2-reachable from Q. If there exists a subset of twonodes, which is 2-reachable from any other node, we say
the directed graph is 2-rooted.

x  x1 Denote, P1k  1k  in the frame OXY, and=P1 = y1  y1k 1
x P11k =  11k  in the frame O1X1Y1 for k=1,2,3,4, y1k =
P1k R (ϕ1 )( P1k − P1 )1(1)2.3. Collision and connection region
Define two areas for obstacle avoidance ( Ψ ia ) and cos(ϕ1 ) sin(ϕ1 ) where R (ϕ1 ) = .
 − sin(ϕ1 ) cos(ϕ1 ) communication maintenance ( Ψ ic ) as shown in Fig. 2.

{} .If d < r , there may becollisions between agents. Ψ : { j ∈  : r ≤ d ≤ R } .
Ψ ia : j ∈  : ra ≤ dij ≤ RaijaiamijmIf dij ≥ Rm , agents will lose contact.
2.4. Desired formation
Given a desired relative position formation β :{0, β 2 , , β n }, β i ∈  , if the final position can be
expressed as lim p (t ) =ϖ 1 β + ϖ 2 1n + vn t1n , ϖ 1 ,ϖ 2 ∈  ,t →∞we say the target formation is formed.
Fig. 1. Coordinate transformation

The distance between two agents is defined as follows.
1l1l
 1
if x2 k > 1
x − ,
d x1 ( p2 k ) =  2 k 22
0,otherwisew1
 1
y − ,
d y1 ( p2 k ) =  2 k2
0,w12otherwise
1if y2 k >d1 ( p2 k ) = d x1 ( p2 k ) + d y1 ( p2 k )22Fig. 2. Communication and collision avoidance area

Define the distance between agent 1 and agent 2 is()d12 = min min ( d1 ( p2 k ) ) , min ( d 2 ( p1k ) ) .
kk

3. Control Design

2.2. Graph theory

Consider a system with n agents, and the dynamic modelis as follows: p i = vi(2)vi = ui

A directed graph G ( v, ε ) consists of a node setv = {1, 2, n} and a set of edges ε ⊂ v × v . If the edge
( j, i ) is in edge ε , j is the in-neighbor of agent i and iis the out-neighbor of agent j.
For a directed graph, if there exists a path from node
q2 to node q1, we say that node q1 is reachable from nodeq2. If there exists a node qi, which is reachable from anyother node qi, we say the directed graph is rooted.
Supposed existing a non-singleton subset of nodes Q:

where pi ∈  donates the position of agent i, vi ∈ donates the velocity of agent i, ai ∈  denotes theacceleration of agent i.
Consider the problem: how to design the control
protocol to make the system reach the desired formation
and avoid collision between agents.
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The control protocol is designed as follows:

The control protocol is designed as follows:
(3)

ui =
uif + uic , i =1, …, nui =uif + uic + uia , i =1, …, nwhere uif is used for formation maneuvering and uic is

Design a potential function Λ ij :used for collision avoidance.
uif is obtained from the following equation:2
22 
 tan µ2 pi − p j − rm   , if p j ∈ Ψ ic
Λ ij =

0, otherwisewhere µ2 is a constant.

(=vi ∑ aij ( v j − vi )
j∈M i


− ∑ sij ( p j − pi ) − α p i p i =j∈ i+u =− ∑ sij* p i + ∑ s*ji p j + vi − vi
 ifj∈ i+j∈ i−(4)

ζi = − ∑=uia

where p i is an auxiliary variable which is used to form
the desired shape, vi is an auxiliary variable which is

1, , n .The∑ s (β − β ) =0, i =…

j∈ i+ijji

1 + 4λmax ( Ls Ls ) − 1p j ∈Ψ ia∂pi

if

4, i = 1, , n( u + µ ) sgn (ζ )(7)i

Theorem 2. Using control protocols in Equations (4), (6),and (7), the system can achieve the desired formation,avoid collision and maintain communication.
Proof. The proof is similar to Theorem 1.

agent i in sensing graph, N i− is the out-neighbor of agent
i in sensing graph. aij can be any positive real number,satisfies∂Λ ij

where µ4 is a constant.

used to synchronize the velocity. M i is the neighbor ofagent i in communication graph, N i+ is the in-neighbor ofsij ∈ (6)4. Simulation*

variable α satisfies α >Design a potential function Θij :2.

The initial position of multi-agent system is arbitrary.
Using control protocols in Equations (4), (6) and (7), thesimulation results in Fig.3 shows changes of formation atdifferent times, which can prove the system can achieve
the desired formation, agents can avoid collision andmaintain communication.

1.4
2

pi − p j ,if p ∈ Ψ aji
Θij =
2−−tanµppr1ija

0, otherwise
where µ1 is a constant.

(ϑi = − ∑=uic∂Θijp j ∈Ψ ia

∂pi

if

3, i= 1, …, n( u + µ ) sgn (ϑ )(5)iwhere µ3 is a constant.
Theorem 1. Using control protocols in Equations (3), (4)and (5), the multi-agent systems can achieve the desired
formation and avoid collision.
Proof. The proof is similar to Theorem 3.1 in [9].
Consider that how to design the control law to makethe system reach the desired formation, avoid collisionand maintain communication. The solution is very
similar to the previous problem.

Fig. 3. Formation control for rectangular agents794. Conclusion

Conference on Artificial Life and Robotics, pp. 420-423,
2022.

For the formation control of rectangular multi-agent
systems, complex Laplacian is used to form the desired
formation and artificial potential function is used tomaintain communication, avoid collision. For
rectangular agents, global coordinate system is designed
which is used to get the position of the agent and local
coordinate system which is used to obtain the distance
between agents. The problem of high-order nonlinear
multi-agent systems can be considered in the future.

Authors Introduction
Mr. Yaxin Li

He received the B.S. degree inautomation from Shandong University,
China, in 2018. He is currently
pursuing the Ph.D. degree with theSeventh Research Division and theCenter for Information and Control,
School of Automation Science andElectricalEngineering,
Beihang
University, China. His researchinterests include distributed control and formation controlof multi-agent systems.

Acknowledgements
This work was supported by the NSFC (62133001,61520106010) and the National Basic Research Programof China (973 Program: 2012CB821201).

Prof. Yingmin Jia

He received the B.S. degree in control
theory from Shandong University,
China, in 1982, and the M.S. and PhD
degrees both in control theory andapplications from Beihang University,
China, in 1990 and 1993, respectively.
Then, he joined the Seventh Research
Division at Beihang University where
he is currently Professor of automaticcontrol. His research interests include robust control,intelligent control and their applications in robot systems.

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avoidance, Proceedings of the 2022 InternationalMr. Jintao Hu

He received the B.S. degree inautomation
fromChongqing
University, China, in 2019. He iscurrently pursuing the Ph.D. degreewith the Seventh Research Divisionand the Center for Information andControl, School of Automation
Science and Electrical Engineering,
Beihang University, China. His
research interests include consensus control of multi-agent
systems and event-triggered control.

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