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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
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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)
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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.
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(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.
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