Journal of Robotics, Networking and Artificial Life
Vol. 7(1); June (2020), pp. 30–34
DOI: https://doi.org/10.2991/jrnal.k.200512.007; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Distributed Rotating Encirclement Control of Strict-Feedback
Multi-Agent Systems using Bearing Measurements
Tengfei Zhang, Yingmin Jia*
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

ARTICLE INFOABSTRACTArticle History

This paper focuses on the distributed multi-target rotating encirclement formation problem of strict-feedback multi-agent
systems using bearing measurements. To this end, an estimator is presented to localize agents’ neighbor targets. Then, basedon the trajectory planning method, a reference trajectory is constructed by three estimators, which are utilized to obtain thetargets’ geometric center, the reference rotating radius and angular. Finally, an adaptive neural dynamic surface control scheme
is proposed to drive all agents to move along their reference trajectories, which satisfy the multi-target rotating encirclement
formation conditions. A numerical simulation is provided to verify the correctness and effectiveness of our proposed control
scheme.

Received 21 October 2019
Accepted 22 February 2020
Keywords
Strict-feedback multi-agent systems
rotating encirclement control
target localization
trajectory planning
trajectory tracking© 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Recent years have witnessed considerable attention on the rotating
encirclement formation problem of multi-agent systems due to its
significant potential applications in both military and civilian areas
such as surveillance, search-and-rescue, reconnaissance, etc. [1].
Many interesting results have been achieved for the rotating formation or surrounding/encirclement control problem [1–7].
In practice, many physical systems, including manipulators, vessel,
unmanned aerial vehicles (UAVs) , can be written in the strict-­
feedback form [8]. Some important researches of the strict-feedback single/multi-agent system have been presented [8–11].
However, there is no research to date on the rotating encirclement
control of high-order multi-agent system.
Motivated by the above discussion, for the first time, we consider themulti-target rotating encirclement formation problem of strict-­feedback
multi-agent systems, and only bearing measurements of targets can beobtained. To this end, we divide the problem into three sub-problems:
target localization, trajectory planning and trajectory tracking. Four
estimators are designed to construct a reference trajectory for eachagent, and an adaptive neural dynamic surface control scheme is presented to drive all agents to move along their desired trajectories.

2. PRELIMINARIES AND PROBLEMSTATEMENT
2.1. Graph Theory
Let G ( V , E , A , B ) be a weighted undirected graph corresponding to n agents and m targets, where  = {v1 , v2 , , vn , s1 , , sm }Corresponding author. Email: ymjia@buaa.edu.cn; www.buaa.edu.cn*denotes the set of vertexes, E Ì V ´ V denotes the set of edges, = [aij ] ∈ R n × n denotes the weighted adjacency matrix of targets, = [bik ] ∈ R n × m denotes the weighted adjacency matrix from targets
to agents. Let d (vi, vj) denote the shortest distance from the vertex vi
to vj, for instance, d (vi, vj) = 1 if (vi , v j ) Î . The neighbor agents set ofthe agent vi is denoted by N i = {v j Î V | (vi , v j ) ÎE } and the neighbortargets set of the agent vi is denoted by N kT = {sk ∈ V | (vi , sk ) ∈E }.
The neighbor agents set of the target Sk is denoted by
N KI = {vi ∈ V | (vi , sk ) ∈E }.

2.2. Problem Statement
Consider a multi-target multi-agent system consisting of n agents(Index set  = {1, 2,, n} ) and m stationary targets (Index set
 = {1, 2,, m}) with bearing-only measurements, where the dynamicof agent vi is written in the following qi-order strict-feedback form.
ì x = f (x ) + x
ij
ij
ij +1
ï ij
í x iqi = fiqi (xiqi ) + ui
ïî y i = x i1(1)Twhere xij =  xiT1 ,, xijT  , and xiqi , yi , ui ∈R 2 represent the states,output and control input of agent v i, respectively. fij (xij ) is an
unknown continuous nonlinear function.
The objective of this note is to design the distributed control scheme
using bearing-only measurements of targets and the neighbor position information of agents, such that strict-feedback agents arecapable of achieving the multi-target rotating encirclement formation, which is properly formulated by Definition 1 using the polar
coordinate transformation yi = r + [li cos(q i ), lisin(q i )]T .

T. Zhang and Y. Jia / Journal of Robotics, Networking and Artificial Life 7(1) 30–34

Definition 14. The multi-agent system is said to achieve themulti-target rotating encirclement formation if

31i

2p ( j − i) qi = d i ∑ j ∈ aik q j − qi − +win(8)rk − r  = 0lim li − l maxk ∈(2)

2p (i − j) lim q i − q j −=0t →∞n(3)where bi, gi1, ..., giM, di are positive design parameters, andM = max i , j Î {d(i, j)} , which can be chosen as M = n − 1 if it is notprior information.

lim[q i − w ] = 0(4)Then, with the polar coordinate transformation, the reference trajectory of agent vi is provided as follows.

t →∞

•

t →∞

T

y i = pi + l i cos(qi ), l isin(qi )

where i, j Î , rk and r (t ) = 1 / må k Î rk denote the position of thek-th target and the geometric center of all targets respectively. Thedesign parameter l > 1 determines the radius of the desired rotation formation and w represents the desired angular velocity.

To facilitate the latter control design and analysis, we make somereasonable assumptions.

Define dynamic surface errors as follows.

Assumption 1. All agents are connected in some undirected communication topologies and each target connected to at least one agent via
the directed edge.

Assumption 2. The radius of the desired rotation formation
is bounded, i.e., there exists a positive constant d* satisfying
max k ∈ rk − r ≤ d *.
Assumption 3. The desired angular velocity w and angular acceleration w are continuous and bounded, i.e., there exists positive con**stants w* and w d* such that || w || ≤ w ,|| w || ≤ w d .

3. CONTROL DESIGN
3.1. Target Localization
With bearing-only measurements, the following estimator is proposed to obtain the neighbor target’s position of agent vi according
to Shao and Tian [7].
ir ik = a ik (I − j ikj ikT )(xi1 − r ik )

(5)

(9)

3.3. Trajectory Tracking
ìïzi1 = xi1 - yiíîïzij = xij - h ij

(10)

 (t ) is the first-order filter estimation of the virtual conwhere hijtroller hij(t) with the time constant τij > 0 and the filter error is -h .
denoted by h ij = hij
iji



 +h
 = h ,h
 (0) = h (0)t ij hij
ij
ij
ij
ij

(11)

Then, we present the following adaptive neural dynamic surface
control scheme.
h = −k z − W
 ijTS (z )
 ij
ij ij
ijij
T
ui = −k iqi ziqi − W iqi Siqi (z iqi ) ij = - G -1[s W ij - S (z )z T ]Wijijijij
ij

i(12)
(13)

T
 ij and S (z ) repwhere Γ ij = Γ ij > 0 is an adaptive gain matrix, Wij ij
resent the estimation of the optimal weight matrix Wij and the basis
function vector respectively. kij, sij are positive design parameters.

where k ÎN iT , aik is a positive design parameter, and jik is the unit
vector from xi to rk.

4. MAIN RESULTS

3.2. Trajectory PlanningWith the proposed control scheme in Section 3, we can easily
obtain the following reasonable results.

We design the following distributed estimators to obtain the estimations pi , l i and qi of the desired geometric center r , polarradius li and polar angle θi, respectively.

Lemma 1. Consider the estimator (5) under Assumptions 1–2. Thenfor any i Î I , k Î T , the estimation position r ik will asymptoticallyconverge to the actual position rk of the k-th target.

 p ik = b i ∑ aij[ p jk − pik ] + b ibik [r ik − pik ]j ∈ i1 m pi = ∑ k =1 pikm(((6))

 r = g max T r ik − pi − ri1i1k ∈N i
 i1
 r = g max
rj1 − ri 2i2j ∈N i ∪ {i} i 2 riM = g iM max j ∈N i ∪ {i} rjM −1 − riMl i = lriM

( )Proof. The proof is similar to Theorem 3.1 in Shao and Tian [7].

)Then, we define the estimation of the targets’ geometric center asfollows.
(7)1 n1 r = ∑ i =1∑ k ∈N Tr ikkmN KI(14)Apparently, r will asymptotically converge to the actual geometriccenter r .
Lemma 2. Consider the estimator (6) under Assumptions 1 and 2. For
any i Î, the estimation position p will asymptotically converge to r .
i

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32

Proof. The proof is similar to Lemma 4 in Zhang et al. [4].

 x11 = x11 + cos( x11 ) + x12
 x12 = x11x12 + u1

 x 41 = x 41 sin(x 41 ) + x 42
 x 42 = x 41 + x 42 + u4

 x 31 = 2 x31 + x32
 x 32 = x31 + 2 x32 + x33
 x = x cos( x ) + u
31
32
3
 33

 x 61 = 2 x61 + x62
 x 62 = x61x62 + x63
 x = x + sin(x ) + u
61
62
6
 63



a ik = 2, b i = 4, g ik = 4, d i = 2, k ij = 50, t ij = 0.01, s ij = 10

Then, combining Lemma 1 with Lemma 2, we can conclude thatthe estimation position pi of the i-th agent will asymptotically converge to the actual geometric center r .
Lemma 3. Consider the estimator (7) under Assumptions 1 and 2.
For any i Î , the following equation holds.
lim l i − lmax i ∈I , k ∈N Ti
t →∞ ( r − p ) = 0iki((15))

 = 0asymptotilim l ii −In other words, the estimation
ofltheradiusrk − r willmaxpolark ∈t →∞cally converge to the above value.

x 21 = sin( x21 ) + u2x 51 = 3 + cos( x51 ) + u5

Clearly, Assumptions 1–3 are satisfied. Figure 2 shows that allagents can precisely locate their neighbor targets. Figure 3 illustrates

Proof. The proof is similar to Lemma 5 in Zhang et al. [4].
Furthermore, with Lemma 1 and 2, it is easy to see thatlim l ii −satisfiesrk − r  = 0lmax k ∈thelim l i − lmax k ∈ rk − r  = 0 , which implies that
t →∞
t →∞
condition (2).

()()

Lemma 4. Consider the estimator (8) under Assumptions 1–3. Forany i, j Î , the following equations hold.
Figure 1 | Communication topologies.

2p (i - j) ùélim êqi - q j ú=0t ®¥në
û(16)

éi
ù
lim ëqi - w û = 0(17)t ®¥In other words, the estimation qi of the polar angle satisfies conditions (3) and (4).
Proof. The proof is similar to Lemma 6 in Zhang et al. [4].
Thus, from the polar coordinate transformation of (9), we cany i satisfies conditions
easily conclude that the reference trajectory of the multi-target rotating encirclement formation in Definition 1.
Then, with the same analysis process as given in our article atICAROB2020, we present the following theorem.
Theorem 1. Consider the multi-agent system (1) in the strict-feedback form with stationary multi-targets. Suppose that Assumptions
1–3 hold. For any bounded initial condition, if we choose design
parameters satisfy ci0 > 0, then all agents will achieve the multi-target
rotating encirclement formation with the proposed control scheme inControl Design.
Proof. The proof is similar to our article at ICAROB2020.

5. NUMERICAL SIMULATIONThe strict-feedback multi-agent system, design parameters are chosen
as follows and communication topologies are shown in Figure 1.

Figure 2 | Target localization.

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33

Figure 3 | Estimations of the targets’ geometric center, the desired polar radius and polar angle.

Figure 4 | Trajectory tracking and multi-target rotating encirclement formation in 3D.

that the reference trajectory satisfies conditions in Definition 1.
Finally, Figure 4 indicates that the proposed control scheme canforce all agents to track their reference trajectories and shows eachagent’s trajectory in 3D.

6. CONCLUSION
The collective multi-target rotating encirclement formation problem of strict-feedback multi-agent systems is investigated by dividing into three sub-problems. Our proposed control scheme cansolve this problem well.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

ACKNOWLEDGMENTS
This work was supported by the NSFC (61327807, 61521091,61520106010, 61134005) and the National Basic Research Programof China (973 Program: 2012C B821200, 2012CB821201).

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AUTHORS INTRODUCTION
Mr. Tengfei Zhang

Prof. Yingmin Jia

He received the B.S. degree in information and computational science from
Beihang University, Beijing, China, in2016. He is currently pursuing the PhD
degree with the Seventh Research Divisionand the Center for Information andControl, School of Automation Scienceand Electrical Engineering, Beihang
University. His current research interests include cooperative
control of multiagent systems.

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 and applications from Beihang University, China,
in 1990 and 1993, respectively. Then, hejoined the Seventh Research Division at
Beihang University where he is currently
Professor of automatic control. From February 1995 untilFebruary 1996 he was a Visiting Professor with the Institute ofRobotics and Mechatronics of the German Aerospace Center(DLR), Oberpfaffenhofen, Germany. He held an Alexandervon Humboldt (AvH) research fellowship with the Instituteof Control Engineering at the Technical University HamburgHarburg, Hamburg, Germany, from December 1996 until March
1998, and a JSPS research fellowship with the Department
of Electrical and Electronic Systems at the Osaka Prefecture
University, Osaka, Japan, from March 2000 until March 2002.
He was a Visiting Professor with the Department of Statistics
at the University of California Berkeley from December 2006
until March 2007. Dr. Jia was the recipient of the National
Science Fund for Distinguished Young Scholars in 1996,
and was appointed as Chang Jiang Scholar of the Ministry of
Education of China in 2004. He has been Chief Scientist of theNational Basic Research Program of China (973 Program) since2011, and in particular he won the Second Prize of National
Technology Invention Award in 2015. His current researchinterests include robust control, adaptive control and intelligent
control, and their applications in robots systems and distributed
parameter systems.