Content uploaded by Ali Marzoughi

Author content

All content in this area was uploaded by Ali Marzoughi on Dec 21, 2021

Content may be subject to copyright.

Distributed Control of a Robotic Network for

Protection of a Region from Intruders⇤

Andrey V. Savkin,1Ali Marzoughi,2

Abstract—This paper addresses a problem of intruder inter-

ception on the boundary of a protected region through the use of

a network of mobile robots. A necessary and sufﬁcient condition

for the existence of a solution of this problem is obtained. We

propose a decentralized motion control algorithm for the mobile

robots to intercept an intruder entering the protected region.

The algorithm is developed based on some simple rules that are

computationally efﬁcient and easily implementable in real time.

I. Introduction

The important recent technological developments in robotics

greatly increase the number of real world applications that

are suitable for multi-robot teams. Therefore, in recent years,

the use of teams of autonomous unmanned vehicles in pa-

trolling, monitoring and surveillance tasks has been increased

signiﬁcantly. A fundamental problem of robotics research is

navigation of mobile robots for patrolling a boundary of a

region in various border-security missions; see e.g. [1]–[10].

The most common approach to protection a region from

intruders is coverage control in which the barrier coverage

problem and the sweep coverage problem are studied. The

barrier coverage problem is to deploy a group of mobile robots

with sensing capabilities to form a static sensor barrier that

detects any object trying to enter a protected region [11],

[12]. On the other hand, the sweep coverage problem is to

steer a group of mobile robots along the boundary of the

protected region so that every point in some neighbourhood

of the boundary is detected by some robot [13], [14].

In this paper, we consider a team of mobile robots moving

along the boundary of a protected planar region. The robots

move in a decentralized fashion, i.e. each robot navigates

independently and has information about current coordinates

of just several closest other robots. Unlike coverage control

problems of [11]–[14], we assume that the intruder becomes

visible to the robots at some time , i.e. all the robots know

the planar coordinates of the intruder after a certain time

moment. The objective of the multi-robot team is to intercept

the intruder which means that when the interceptor crosses

the boundary of the protected region, there should be several

robots close to the interception point. The proposed problem

statement is relevant to various problems of asset guarding

in which a team of autonomous unmanned surface vehicles

*This work was supported by the Australian Research Council.

1.A.V.Savkin is with the School of Electrical Engineering,and Telecom-

munications, The University of New South Wales, Sydney 2052, Australia,

e-mail: a.savkin@unsw.edu.au.

2.A.Marzoughi is with the School of Electrical Engineering,and Telecom-

munications, The University of New South Wales, Sydney 2052, Australia,e-

mail: a.marzoughi@student.unsw.edu.au

(USVs) patrols and guards an asset in an environment with

hostile boats. Such problems require the team of USVs to co-

operatively patrol the area around the asset, identify intruders,

and actively block them [15]–[17]. An important example is

safeguarding civilian harbours from terrorist attacks coming

from the blue border (i.e. the sea-side) [18].

In this paper, we give a necessary and sufﬁcient condition

for existence of decentralized navigation strategy for the multi-

robot team that always results in intercepting the intruder

and proposed such a strategy. The proposed decentralized

navigation law is computationally non-demanding and easily

implementable in real time. Mathematically rigorous proofs of

the main theoretical results of the paper will be given in its

extended journal version.

The remainder of the paper is organized as follows. Section

II. states the problem under investigation. The main results are

presented in Section III.. Examples illustrating the proposed

navigation strategy are given in Section IV.. Finally, Section

V. presents a brief conclusion.

II. Problem Statement

Let Rbe a bounded closed connected and linearly con-

nected planar region with a piecewise smooth boundary. Fur-

thermore, let Sbe a segment of the boundary of the region

Rbetween some points P1and P2; see Fig.(1). We consider

a moving point-wise intruder Ithat aims to enter the region

Rthrough the segment Sfrom outside of R; see Fig.(1).

Let xI(t)be planar coordinates of the intruder. The intruder

is moving in the plane with an arbitrary time-varying vector

velocity vI(t)= ˙xI(t)satisfying the constraint

kvI(t)kVmax

I8t0(1)

where Vmax

I>0is a given constant, k·kdenotes the

standard Euclidean vector norm. Also, xI(t)denotes the planar

coordinates of the intruder.

Moreover, let nbe a given positive integer. We consider

nmobile point-wise robots labelled 1,2,...,n that defend

the region Rfrom the intruder. Unlike the intruder that can

move in any direction in the plane, the robots can move

only along the segment Sin the both directions. Furthermore,

x1(t),x

2(t),...,x

n(t)denote the planar coordinates of the

robots 1,2,...,n.

We introduce the curvilinear coordinate c(P)for any point

P2Ssuch that c(P)is the length of the portion of the

segment Sbetween the points P1and P; see Fig.(1). This

implies that c(P1)=0and c(P2)=Lwhere Lis the length

978-1-5386-3742-5/17/$31.00 © 2017 IEEE

Authorized licensed use limited to: Western Sydney University. Downloaded on May 20,2020 at 06:09:55 UTC from IEEE Xplore. Restrictions apply.

Fig. 1. The region R, the boundary segment Sand the curvilinear coordinate

c(P).

of the segment S. We assume that c1(t):=c(x1(t)),c

2(t):=

c(x2(t)),...,c

n(t):=c(xn(t)) are the curvilinear coordinates

of the robots 1,2,...,n at time t0.

Furthermore, we suppose that the robots labelled according

to their curvilinear coordinates so that

0c1(t)c2(t)...cn(t)L8t0.(2)

The requirement (2) means that the robots never change their

order on the segment S. We assume that the motion of the

robots along Sis described by the equation

˙ci(t)=ui(t)8k=1,2,...,n (3)

where ui(t)is the control input of the robot i. We assume that

the control inputs ui(t)satisfy the constraint

|ui(t)|Vmax

R8t0(4)

where Vmax

R>0is a given constant.

Available measurements: Let 1k<n

2be a given

positive integer. At any time t, each robot iknows the

curvilinear coordinates cj(t)of the robots jfor all ik

ji+k. Moreover, the robot knows its own coordinate

ci(t). Furthermore, the intruder becomes visible to the robots

at some time t00, i.e. all the robots know the planar

coordinates xI(t)of the intruder for all tt0.

Deﬁnition 1: Let ✏>0be a given constant. Suppose that

the intruder crosses the segment Sat time t?, i.e. xI(t?)2S.

We say that the multi-robot team kintercepts the intruder

at time t?if there exists some index i=1,2,...,n such

that |c(xI(t?)) cj(t?)|✏for all iji+k1.

Furthermore, a multi-robot team navigation strategy that is

based on the available information is called kintercepting

if for any movement of the intruder, the multi-robot team

kintercepts it when the intruder crosses the segment S.

In other words, kintercept means that when the interceptor

crosses the segment S, there should be at least krobots close

to the interception point.

Problem statement: The intruder’s objective is to enter the

region Rthrough the segment Swhile avoiding kintercept

by the multi-robot team. The objective of the multi-robot team

is to kintercept the intruder. The problem under considera-

tion in this paper is to derive a necessary and sufﬁcient con-

dition under which kintercept is possible for any motion of

the intruder, and design a decentralized navigation strategy for

the multi-robot team that is based on the available information

and will always result in kintercept of the intruder.

III. The Main Results

Let P2Sand xbe a point outside the region R. Then

P(x, P )denotes the set of all continuous piecewise smooth

paths L(x, P )connecting xand Psuch that the intersection

of L(x, P )and Rcontains only the point P. Furthermore,

let ↵(x, P )denote the inﬁmum of the lengths of all paths

L(x, P )from P(x, P ). Moreover, let i(1),i(2),...,i(n)be a

permutation of indices 1,2,...,n such that

|ci(1)(t)c(P)||ci(2) (t)c(P)|···|ci(n)(t)c(P)|.

Then, introduce the variable (t, P ):=|ci(k)(t)c(P)|.

In other words, (t, P )is the kth among all the robots

according to the length of the sub-segments of the segment

Sbetween the robots’ current locations and the point P.

Introduce the function F(s)from the interval [0,L]to the

segment Ssuch that for any number s2[0,L],F(s)is the

point P2Ssuch that c(P)=s. Furthermore, let [A1,A

2]

denote the closed sub-segment of the segment Sbetween the

points A1and A2. For i=k, k +1,...,nk+1, introduce

sub-segments Si(t),S

i(t)+of the segment Sas follows:

Si(t):= [F(ci(t)+cik(t)

2),F(ci(t)+cik+1(t)

2)]

if i =k+1,...,nk+ 1;

Si(t):= [P1,F(ci(t)+cik+1(t)

2)]

if i =k;

Si(t)+:= [F(ci(t)+ci+k1(t)

2),F(ci(t)+ci+k(t)

2)]

if i =k, . . . , n k;

Si(t)+:= [F(ci(t)+ci+k1(t)

2),P

2]

if i =nk+1.(5)

Moreover, for i=k,k +1,...,nk+1, introduce the numbers

M

i(t)and M+

i(t)as

M

i(t):= sup

P2Si(t)✓(t, P )↵(xI(t),P)Vmax

R

Vmax

I◆;

M+

i(t):= sup

P2Si(t)+✓(t, P )↵(xI(t),P)Vmax

R

Vmax

I◆.(6)

Now we can introduce the following decentralized navigation

law:

ui(t):=Vmax

Rif i < k

ui(t):=Vmax

Rif i > n k+1 (7)

Authorized licensed use limited to: Western Sydney University. Downloaded on May 20,2020 at 06:09:55 UTC from IEEE Xplore. Restrictions apply.

For i=k, k +1,...,nk+1,

ui(t):=Vmax

Rif M

i(t)<M

+

i(t)

ui(t):=Vmax

Rif M

i(t)>M

+

i(t)

ui(t):=0 if M

i(t)=M+

i(t).(8)

Remark 1: The intuition behind the decentralized navi-

gation law can be explained as follows. The sub-segments

Si(t),S

i(t)+are sets of point of the curve Sfor which the

robot iis the kth furthest robot at time t. The robot moves

with the maximum allowed speed towards the one of these

segments that is more ”dangerous” at the current time, i.e. it

has the biggest possible distance between the intruder and the

kth robot at the moment of crossing Sby the intruder. This

biggest possible distance is described by (6).

We will also need the following assumption.

Assumption 1: For any trajectory [xI(t),c

1(t),...,c

n(t)] of

the intruder-multi-robot team system, there exists no more than

ﬁnite number of time instants ⌧such that M

i(⌧)=M+

i(⌧).

Theorem 1: Consider the multi-robot team (3) and the

intruder satisfying (1), (4) and Assumption 1. Let 1k<n

2

be a given positive integer. Then there exists a kintercepting

multi-robot team navigation strategy if and only if

sup

P2S✓(t0,P)↵(xI(t0),P)Vmax

R

Vmax

I◆✏(9)

where t00is the time at which the intruder becomes visible

to the robots. Moreover, if the inequality (9) holds, then the

navigation law (7), (8) is a kintercepting navigation strategy.

The proof of Theorem 1 will be given in the full version of

the paper.

The inequality (9) and the navigation law (7), (8) can be

made computationally simpler under the following assump-

tion.

Assumption 2: The following inequality holds:

Vmax

IVmax

R.

For i=k, k +1,...,n k+1, introduce points

Di(t),D

i(t)+of the segment Sas follows:

Di(t):= F(ci(t)+cik(t)

2)(10)

if i =k+1,...,nk+ 1;

Di(t):= P1if i =k;

Di(t)+:= F(ci(t)+ci+k(t)

2)(11)

if i =k,...,nk;

Di(t)+:= P2if i =nk+1.(12)

Moreover, for i=k,k +1,...,nk+1, introduce the numbers

H

i(t)and H+

i(t)as follows:

H

i(t):=ci(t)cik(t)

2↵(xI(t),D

i(t))Vmax

R

Vmax

I

if i =k+1,...,nk+ 1;

H

i(t):=ci(t)↵(xI(t),P

1)Vmax

R

Vmax

I

if i =k;

H+

i(t):=ci+k(t)ci(t)

2↵(xI(t),D

i(t)+)Vmax

R

Vmax

I

if i =k,...,nk;

H+

i(t):=Lci(t)↵(xI(t),P

1)Vmax

R

Vmax

I

if i =nk+1.(13)

For i<kand i>nk, the navigation law (7) stays the

same. For i=k, k+1,...,nk+1, the simpliﬁed navigation

law (7) becomes:

ui(t):=Vmax

Rif H

i(t)<H

+

i(t)

ui(t):=Vmax

Rif H

i(t)>H

+

i(t)

ui(t):=0 if H

i(t)=H+

i(t).(14)

Also, we modify Assumption 1 as follows.

Assumption 3: For any trajectory [xI(t),c

1(t),...,c

n(t)] of

the intruder-multi-robot team system, there exists no more than

ﬁnite number of time instants ⌧such that H

i(⌧)=H+

i(⌧).

Theorem 2: Consider the multi-robot team (3) and the in-

truder satisfying (1), (4) and Assumptions 2, 3. Let 1k<n

2

be a given positive integer. Furthermore, let Hbe the set of

numbers H

i(t0),H+

i(t0)where i=k, k +1,...,nk+1

and t00is the time at which the intruder becomes visible

to the robots. Then there exists a kintercepting multi-robot

team navigation strategy if and only if

max H✏.(15)

Moreover, if the inequality (15) holds, then the navigation law

(7), (14) is a kintercepting navigation strategy.

The proof of Theorem 2 will be given in the full version of

the paper.

IV. Illustrative Examples

In this section, we consider a team of ﬁve mobile robots

deployed on the boundary of a planar region to intercept

the intruder that aims to enter this region. Our illustrative

examples include the problems of kintercepting the intruder

by the multi-robot team with k=1and k=2. In our

examples, Vmax

I=4.2and Vmax

R=3.0, hence, Assumption

2 holds. Therefore, we apply Theorem 2 and the navigation

law (7), (14). Fig. (2) illustrates the reaction of the robots to

the intruder’s motion when it starts moving from the initial

position ithrough the trajectory to the points a,b, and c

where k=1. Similarly, Fig. (3) represents the trajectory of

the intruder and the robots’ position when k=2.

Authorized licensed use limited to: Western Sydney University. Downloaded on May 20,2020 at 06:09:55 UTC from IEEE Xplore. Restrictions apply.

(a) (b)

(c) (d)

Fig. 2. The intruder trajectory and the positions of the robots when the

intruder at points i,a,b and c, k=1.

(a) (b)

(c) (d)

Fig. 3. The intruder trajectory and the positions of the robots when the

intruder at points i,a,b and c, k=2.

Fig.(4a) shows the evolution of the ycoordinates of the

intruder and the robots during the trajectory shown in Fig.(2)

with k=1. Analogously, Fig.(4b) shows the evolution of

the ycoordinates of the intruder and the robots during the

trajectory shown in Fig.(3) with k=2.

V. Conclusion

We proposed a decentralized motion control algorithm for

a network of mobile robots to intercept an intruder on the

boundary of a protected planar region. A necessary and

sufﬁcient condition for the existence of such an algorithm was

derived. The proposed algorithm is based on some simple rules

that only require information about the closest neighbours of

(a)

(b)

Fig. 4. The evolution of the ycoordinates of the intruder and the robots for

k=1and k=2.

each robot and the intruder. Computer simulations conﬁrmed

the efﬁciency of the developed navigation algorithm.

References

[1] A. S. Matveev, H. Teimoori, and A. V. Savkin, “A method for guidance

and control of an autonomous vehicle in problems of border patrolling

and obstacle avoidance,” Automatica, vol. 47, no. 3, pp. 515–524, mar

2011.

[2] D. Portugal and R. P. Rocha, “Distributed Multi-Robot Patrol: A Scalable

and Fault-Tolerant Framework,” Robotics and Autonomous Systems,

vol. 61, no. 12, pp. 1572–1587, 2013.

[3] A. S. Matveev, A. V. Savkin, M. Hoy, and C. Wang, in Safe Robot

Navigation Among Moving and Steady Obstacles. Elsevier, 2015.

[4] M. Hoy, A. S. Matveev, and A. V. Savkin, “Algorithms for collision-

free navigation of mobile robots in complex cluttered environments: a

survey,” Robotica, vol. 33, no. 03, pp. 463–497, mar 2015.

[5] A. Marino, L. E. Parker, G. Antonelli, and F. Caccavale, “A decentralized

architecture for multi-robot systems based on the null-space-behavioral

control with application to multi-robot border patrolling,” Journal of

Intelligent and Robotic Systems: Theory and Applications, vol. 71, no.

3-4, pp. 423–444, 2013.

[6] A.Marzoughi, “Maximizing the probability of intrusion detection by a

ﬂeet of mobile robots using an intelligent game theoretic approach,” in

36th Chinese Control Conference (CCC),2017. Dalian, China: IEEE,

Sep 2017.

[7] O. Zaki and M. Dunnigan, “A navigation strategy for an autonomous

patrol vehicle based on multi-fusion planning algorithms and multi-

paradigm representation schemes,” Robotics and Autonomous Systems,

vol. 96, pp. 133–142, oct 2017.

[8] A. Khan, B. Rinner, and A. Cavallaro, “Cooperative Robots to Observe

Moving Targets: Review,” IEEE Transactions on Cybernetics, vol. 99,

pp. 1–12, 2016.

[9] D. Panagou, D. M. Stipanovic, and P. G. Voulgaris, “Distributed Coordi-

nation Control for Multi-Robot Networks Using Lyapunov-Like Barrier

Functions,” IEEE Transactions on Automatic Control, vol. 61, no. 3, pp.

617–632, mar 2016.

[10] A. A. Semakova, K. S. Ovchinnikov, and A. S. Matveev, “Self-

deployment of mobile robotic networks: an algorithm for decentralized

sweep boundary coverage,” Robotica, vol. 35, no. 09, pp. 1816–1844,

sep 2017.

[11] T. M. Cheng and A. V. Savkin, “A distributed self-deployment algorithm

for the coverage of mobile wireless sensor networks,” IEEE Communi-

cations Letters, vol. 13, no. 11, pp. 877–879, 2009.

[12] A. V. Savkin, T. M. Cheng, Z. Li, F. Javed, Z. Xi, A. S. Matveev,

and H. Nguyen, Decentralized Coverage Control Problems For Mobile

Robotic Sensor and Actuator Networks. Hoboken, NJ, USA: John Wiley

& Sons, Inc., jun 2015.

[13] T. M. Cheng, A. V. Savkin, and F. Javed, “Decentralized control of a

group of mobile robots for deployment in sweep coverage,” Robotics

and Autonomous Systems, vol. 59, no. 7-8, pp. 497–507, jul 2011.

[14] T. M. Cheng and A. V. Savkin, “Decentralized control for mobile robotic

sensor network self-deployment: barrier and sweep coverage problems,”

Robotica, vol. 29, no. 02, pp. 283–294, mar 2011.

[15] P. Mahacek, C. A. Kitts, and I. Mas, “Dynamic guarding of marine assets

through cluster control of automated surface vessel ﬂeets,” IEEE/ASME

Transactions on Mechatronics, vol. 17, no. 1, pp. 65–75, 2012.

[16] E. Raboin, P. Svec, D. Nau, and S. K. Gupta, “Model-predictive target

defense by team of unmanned surface vehicles operating in uncertain

environments,” in 2013 IEEE International Conference on Robotics and

Automation. IEEE, may 2013, pp. 3517–3522.

[17] E. Raboin, P. ˇ

Svec, D. S. Nau, and S. K. Gupta, “Model-predictive asset

guarding by team of autonomous surface vehicles in environment with

civilian boats,” Autonomous Robots, vol. 38, no. 3, pp. 261–282, mar

2015.

[18] E. Simetti, A. Turetta, G. Casalino, E. Storti, and M. Cresta, “Protecting

assets within a civilian harbour through the use of a team of usvs:

Interception of possible menaces,” in IARP workshop on robots for risky

interventions and environmental surveillance-maintenance (RISE’10),

2010.