Barrier Coverage with Mobile Sensors

Changxiang Shen Weifang Cheng Xiangke Liao Shaoliang Peng

National University of Defense Technology, China

wfangch@nudt.edu.cn

Supported by Grant No. 2006AA01Z401 and 60673169.

Abstract

Barrier coverage, which guarantees that every

movement crossing a barrier of sensors will be de-

tected, is known to be an appropriate model of cover-

age for moving detection and boundary guard. The

related problems about barrier coverage with station-

ary sensors are extensively studied. When sensors are

randomly deployed, we require much more sensors to

achieve barrier coverage than deterministic deploy-

ment. In this paper we study barrier coverage with

mobile sensors, in which the sensors can be relocated

after deployment, and we are able to utilize much

fewer mobile sensors than stationary sensors to

achieve barrier coverage with random deployment. We

study the energy-efficient relocation problem for bar-

rier coverage, and propose a centralized barrier algo-

rithm, which computes the relocated positions based

on knowing the initial positions of all sensors. For

practicability and scalability, we further design a dis-

tributed barrier algorithm based on our proposed vir-

tual force model. We conduct extensive simulations to

study the effectiveness of the proposed algorithms.

1. Introduction

There has been tremendous work done for different

coverage problems in sensor networks [1-4], which is

called as full coverage by Kumar et al. [5]. In full cov-

erage, sensors deployed over the field monitor the en-

tire area. Any point within the area is ensured to be

covered by at least one or k sensors. A full coverage is

required usually when users need to fully monitor the

entire environment.

Wireless sensor networks are also widely applied to

many important applications involving boundary guard

or movement detection, such as when deploying sen-

sors along international borders to detect illegal intru-

sion, around forests to detect the spread of forest fire,

etc [6]. Barrier coverage, which guarantees that every

movement crossing a barrier of sensors will be de-

tected, is known to be an appropriate model of cover-

age for such applications [5-8].

As shown by Balister et al. [8], when sensors are

randomly deployed, we require much more sensors to

achieve barrier coverage than deterministic deploy-

ment. Unfortunately, deploying sensors deterministi-

cally is an expensive undertaking in terms of time,

effort, and money. When deploying sensors in inacces-

sible terrain (e.g., forests, mountains, enemy regions),

deterministic deployment may not even be an option.

Figure 1 shows an example of randomly deployed sen-

sors for barrier coverage. We can see that many sen-

sors are redundant, leading to significant waste of

sensors.

Based on this observation, we study barrier cover-

age with mobile sensors, in which sensors can be relo-

cated after random deployment. Thus we are able to

utilize much fewer mobile sensors to achieve barrier

coverage than using stationary sensors.

Figure 1. An example of randomly deployed sensors

for barrier coverage

In this work, we first formulate the problem of

finding the best positions of relocated sensors to mini-

mize the moving energy consumption, called as mini-

mum-energy barrier-coverage or MEBC. To address

the MEBC problem, we propose a centralized barrier

algorithm, CBarrier, which computes the relocated

positions for all sensors in a centralized way. In CBar-

rier, the initial positions of all sensors are assumed to

be known in advance. For practicability and scalability,

we have to design a distributed barrier algorithm based

The International Symposium on Parallel Architectures, Algorithms, and Networks

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The International Symposium on Parallel Architectures, Algorithms, and Networks

978-0-7695-3125-0/08 $25.00 © 2008 IEEEDOI 10.1109/I-SPAN.2008.8

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The International Symposium on Parallel Architectures, Algorithms, and Networks

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99

on locally obtained information.

As any individual sensor cannot locally determine

whether the given area is barrier covered or not, it is

challenging to design a distributed algorithm. To ad-

dress the problem, we first propose a virtual force

model, in which a pair of sensors exerts repulsive force

on each other the on the orientation of X axes, while

they exert attractive forces on each other on the orien-

tation of Y axes. Based on this model, we propose a

distributed barrier algorithm, DBarrier, which utilize

the two types of forces to relocate deployed sensors

into a barrier crossing the given area. Our simulations

show the effectiveness of the proposed algorithms.

The rest of this paper is organized as follows. Sec-

tion 2 defines the problem and presents the centralized

barrier algorithm. Section 3 proposes the virtual force

model. And based on this model, the distributed barrier

algorithm is described in detail. Performance evalua-

tion of our algorithms is presented in Section 4, and we

conclude the work in Section 5.

2. MEBC problem

2.1 Preliminaries

Assume that N mobile sensors

1 2, , NS s s s

are randomly deployed to provide barrier coverage for

a rectangle strip on 2-D plane. Each sensor si has initial

coordinates (xi, yi), and the new coordinates is (xi’, yi’)

after relocating. The length of the rectangle is L and

the width is W. The left boundary is set to be x = 0, the

right boundary is x = L, the top boundary is y = W and

bottom is y = 0. The sensors are assumed to have a

disk sensing model and the sensing range is R.

For such given area, if we deploy mobile sensors

into a straight line, the minimum number of required

sensors is 2L R , as shown in Figure 2. In such case,

all mobile sensors are aligned to a straight line which

is parallel with the bottom side of the given area. Fur-

ther, 2L R is also the minimum number of sensors we

have to deploy for achieving barrier coverage. This is

obvious since reducing any one sensor will lead to the

failure of providing barrier coverage, regardless of any

deployment strategy.

Figure 2. The deployment with minimum sensors

Figure 3. The deployment with maximum sensors

For such straight-line deployment, the maximum

number of mobile sensors is 2 2

2L W R , as shown in

Figure 3. In such case, the sensors are aligned into a

diagonal of the given rectangle. It is easy to see that

the number of sensors required by straight-line de-

ployment is much fewer than random deployment for

achieving barrier coverage[8].

2.2 Problem formulation

Given randomly deployed mobile sensors, how to

deploy them to provide required barrier coverage is not

very hard, but another problem exists. Since the reloca-

tion of sensors will waste energy, we should design an

energy-efficient scheduling for energy-constrained

sensors. In order to save energy consumption of relo-

cation, we expect the moving trips of sensors are as

short as possible. In other words, a mobile sensor is

expected to relocate at a position close to its initial one

if possible. At the same time, all sensors form a barrier

after relocating. Unfortunately, how to find the best

barrier position is non-trivial. We call this problem as

min-energy barrier-coverage problem or MEBC. To

exactly explain MEBC problem, we present its mathe-

matical formulation as follows.

To simplify the description, we set x0 = 0 and xN+1

= L. And the distance of two relocated sensors is 2 2

' ' ' ', i j i jd i j x x y y .

Then the goal of MEBC is

2 2

' '

1

minN

i i i ii

x x y y (1)

Subject to: ' ' , 2 1j ij x x d i j R i N (2)

' ' , 2 1i jj x x d i j R i N (3)

'0 jj x R (4)

'

jj L R x L (5)

The objective function in (1) minimizes the sum of

the moving distance of all sensors. The constraint (2)

guarantees that each sensor has a left neighbor and

their sensing range overlaps with each other. Similarly,

the constraint (3) guarantees that each sensor has a

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sensing-overlapped neighbor on the right side. The left

and right boundary coverage is restricted in the inequa-

tion (4) and (5) respectively. All the constraints make

the relocated sensors to form a barrier crossing the

given area.

2.3 CBarrier algorithm

In order to address the MEBC problem, we make

the following assumptions. We assume that the relo-

cated sensors are aligned into a straight line y = ax + b

and the distances between neighboring sensors are

identical. This implies that ' '

i iy ax b . And if we

order '

ix by sort ascending leading to

' ' '

1 2 Nx x x , we get x1’ = R, xN’ = L-R, and xi’ =

R + (i-1)(L-2R)/(N-1). Obviously the constraint (4) and

(5) are satisfied. Further, the distance between si and

si+1 is:

2 2' ' ' '

1 1

2

, 1

1 ( 2 ) ( 1)

i i i id i i x x y y

a L R N

.

As shown in Figure 3, if we relocate mobile sen-

sors into a straight line, the maximum number of re-

quired sensors is 2 2

2L W R . This implies that if

the relocated sensors are aligned into a straight line,

randomly deploying 2 2

2N L W R sensors are

enough for relocating to achieve barrier coverage. No-

tice that for a crossing line a W L , thus

, 1 2d i i R holds. Similarly, we can prove that

, 1 2d i i R holds. Therefore the constraints (2) and

(3) are satisfied.

Based on the above assumptions, we propose the

centralized barrier algorithm, CBarrier, as follows. As

we know, the key part of CBarrier is to compute the

relocated positions. Since the goal is to minimize the

sum of the moving distances of all sensors, we set 2 2

' '

1

,N

i i i ii

F a b x x y y

Then the goal in inequation (1) is equivalent to

minimize F(a,b). Based on extremum principle, we get:

0F F

a b,

which leads to

' '

1

'

1

2 0

2 0

N

i i ii

N

i ii

Fx y ax b

a

Fy ax b

b

.

We solve the above equations set, and get the fol-

lowing solution.

' ' '2 ' 2

1 1 1 1 1

'

1 1

( ) ( ( ) )N N N N N

i i i i i ii i i i i

N N

i ii i

a N x y x y N x x

b y N a x N

After getting the solution of a and b, we can com-

pute the new position for each sensor according to the

function ' '

i iy ax b . Further, sensor si is relocated

from (xi, yi) to (xi’, yi’), thus the given area is guaran-

teed to be 1-barrier covered.

3. Distributed barrier algorithm (DBarrier)

In this section, we propose a virtual force model to

move sensors in a distributed way. Based on the model,

we describe the design principle of DBarrier in detail

and address the above problems.

3.1 Virtual force model

To form a barrier with all sensors, we should let the

sensors to gather into certain positions nearby a line

crossing from the left boundary to the right boundary

of the given area. The X coordinates of sensors should

be distributed uniformly on [0, L], whereas the Y co-

ordinates values should be gathered closely. Thus, we

utilize repulsive and attractive forces individually for

X axes and Y axes to relocate sensors. On the orienta-

tion of X axes, a pair of sensors exerts repulsive force

on each other, while they exert attractive forces on

each other on the orientation of Y axes, as shown in

Figure 4.

In order to compute the virtual forces on a single

sensor itself in a distributed way, we only consider the

forces exerted by neighboring nodes. Given sensor si

with initial coordinates (xi, yi), the set of sensors which

are within the communication range of si are denoted

by Ni. Then the repulsive force between sensor si and sj

is computed as follows.

( ) ,,

0,

i j j i

R

j i

x x s NF i j

s N

Similarly, the attractive force between sensor si and

sj is computed as follows.

,,

0,

j i j i

A

j i

y y s NF i j

s N

The parameters and are positive constant value

utilized to normalize repulsive and attractive forces.

Further, repulsive forces are in inverse proportion to

the distances, while attractive forces are proportional

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to distances. It is easy to understand since repulsive

forces are utilized to impulse sensors to uniform distri-

bution. On the other hand, attractive forces are utilized

to gather sensors into the same horizon.

When the sensor is close to the boundary, the

boundary also exerts forces on this sensor. The left or

right boundaries exert repulsive forces on a sensor if

the distance between this sensor and the boundary is

less than the communication range Rc. The repulsive

force on sensor si exerted by boundaries is computed as

follows:

0,

, 1 ,

1 ( ) ,

c i c

R i i c

i i c

R x L R

F i B x x R

x L x L R

.

Then the total repulsive force of sensor si on the

orientation of X axes is:

, ,j i

R R Rs N

F i F i j F i B .

And the total attractive force of sensor si on the ori-

entation of Y axes is:

,j i

A As N

F i F i j .

1,2A

F

1, 2RF

2,1RF

2,1A

F

2,RF B

Figure 4. Virtual forces with two sensors

To explain the virtual force model exactly, an ex-

ample of virtual forces with two sensors is shown in

Figure 4. The distance between s1 and s2 is less than

the communication range. Then sensor s1 exerts repul-

sive forces 2,1RF and attractive forces 2,1AF on s2.

At the same time, s2 exerts repulsive forces 1, 2RF and

attractive forces 1,2AF on s1. Furthermore, the left

boundary exerts repulsive forces 2,RF B on sensor s2,

since s2 is close to the left boundary.

3.2 DBarrier algorithm

In DBarrier algorithm, the required barrier cover-

age is achieved step by step through the fine tuning of

positions.As mentioned above, the repulsive force is

used to adjust the X coordinates, and the attractive

force to adjust the Y coordinates. In order to control

the accuracy of fine tuning and the convergence time,

virtual forces move sensors a constant distance for

each step. Specifically, a sensor moves left or right

with a distance x along X axes, and moves up or

down with a distance y along Y axes. If the total re-

pulsive forces exerted on sensor si points to the left, si

moves to the left; otherwise it moves to the right. On

the other hand, if the total attractive forces exerted on

sensor si points up, si moves up; otherwise, it moves

down. In conclusion, the adjusting rule is as follows.

1

1

i i R R

i i A A

x t x t x F i F i

y t y t y F i F i

Denotations xi(t) and yi(t) represent the coordinates

of sensor si at step t. After the delicate adjustment

based on the adjusting rule, sensor si moves to new

coordinates (xi(t+1), yi(t+1)). Thus, we can relocate

sensors step by step using the proposed virtual force

model.

Obviously it is not energy-efficient to move sensors

step by step, since the sensor mostly moves on a maze,

as shown in

Figure 5. Sensor s1 relocates itself from location A

to B, resulting in a wandering trip which is represented

by the real line. It is easy to understand that the wan-

dering trip is much longer than the straight dash line.

Based on this observation, we do not really move sen-

sors for each step. In DBarrier, each sensor stores the

new position instead of moving to this position, which

is called as virtual moving. At the same time sensors

take the new position as current ones for iteration. Af-

ter the sensor gets the final relocated position, it per-

forms one-time moving directly from the initial

position to the relocated location.

Figure 5. An example of wandering trip

As we can see, virtual moving can save much en-

ergy, whereas it results in another problem. Since sen-

sors do not really move to corresponding positions on

runtime, how does a sensor get the exact information

of its neighbors based on its new position? To address

this problem, the sensor collects the information of

two-hop neighbors on runtime, from which the one-

hop neighbors are picked based on new positions after

their virtual moving. Since a sensor performs delicate

adjustment leading to very small motion, the new

neighbors should also be the two-hop neighbors before

virtual moving. Thus sensors can get the exact

neighbor information while performing virtual moving.

The pseudocode of DBarrier is shown in Figure 6.

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Figure 6. Pseudocode of DBarrier algorithm

4. Performance evaluation

We conduct simulation experiments using

MATLAB to test the performance of our algorithms.

We present the simulation results in this section.

For the following experiments, we set the sensing

range R = 4m and communication range Rc = 10m. We

evaluate the performance of DBarrier compared with

CBarrier, including the relocation effectiveness and the

average length of moving trip per sensor. Further, we

test the convergence time of DBarrier. In order to keep

the same iteration speed of X and Y coordinates, we

set x y L W .

To test the relocation effectiveness, 15 sensors are

randomly deployed on a15 60m m strip, i.e., W = 15m,

L = 60m, as shown in Figure 7. Figure 8 shows the

positions of relocated sensors after executing the

CBarrier algorithm. Obviously the relocated sensors

are aligned to a straight line to form a barrier crossing

the strip. In DBarrier, the moving unit 0.1x m .

Figure 9 shows the positions of relocated sensors after

executing the DBarrier algorithm. As we expect, relo-

cated sensors also forms a barrier for the given strip.

This implies that our algorithms are effective on relo-

cating sensors to provide barrier coverage.

The next group of simulations is used to compare

the average moving distance per sensor of CBarrier

and DBarrier. In DBarrier, the moving unit 0.1x m .

The width of the strip is fixed to be 15 meter, but the

length is multiplexed, so that the number of mobile

sensors is also multiplexed to provide the required bar-

rier coverage. This implies that we increase the net-

work scale. In Figure 10 we can see that CBarrier

always outperforms DBarrier. It is easy to understand

since CBarrier is a centralized algorithm and the global

information is known in advance. DBarrier, however,

is just based on locally obtained information. Further,

the average moving length per sensor does not change

much as the network scale increases in both CBarrier

and DBarrier. It implies that our algorithms can be

applicable to large scale networks.

Figure 11 and Figure 12 show the convergence

time of DBarrier. We set MAX_STEP = 100 and take

the average step in Figure 6 as the evaluation parame-

ter of convergence time. Varied network scale is tested

to evaluate its impact on the convergence time. The

width of the strip is fixed to be 15 meter, but the length

is multiplexed, so that the number of mobile sensors is

also multiplexed to provide the required barrier cover-

age. This implies that we increase the network scale.

The moving unit is fixed 0.1x m . Figure 11 shows

the convergence time does not change much with the

network scale. Thus DBarrier is practicable to large

scale networks.

In Figure 12, the varied moving unit is tested to

evaluate its impact on the convergence time. The

length of the strip is set to be 60m and 15 sensors are

randomly deployed. As we increase the moving unit,

the convergence time decreases quickly in the begin-

ning, but suddenly increases to 100 after the minimum

point. The main reason for above results is that when

the moving unit is below certain threshold, the DBar-

rier algorithm can converge and the convergence speed

is proportional to the moving unit; whereas it does not

converge when the moving unit is too large, since large

moving unit is inconsistent with the key idea fine-

tuning of DBarrier.

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0 10 20 30 40 50 600

10

20

30

40

50

60

X coordinate

Y c

oord

inate

Figure 7: Initially deployed sensors for

barrier coverage

0 10 20 30 40 50 600

10

20

30

40

50

60

X coordinate

Y c

oord

inate

Figure 8: Relocated sensors for barrier

coverage by CBarrier

0 10 20 30 40 50 600

10

20

30

40

50

60

X coordinate

Y c

oo

rdin

ate

Figure 9: Relocated sensors for

barrier coverage by DBarrier

60 120 180 240 300 360 420 4803

3.5

4

4.5

5

5.5

6

Length of the strip (L)

Avera

ge length

of

movin

g t

rips CBarrier

DBarrier

Figure 10. Average length of moving

trips of all sensors

60 120 180 240 300 360 420 48020

25

30

35

40

45

50

Length of the strip (L)

Avera

ge s

teps o

f converg

ence

Figure 11. The convergence time of

DBarrier vs. length of the strip

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10

20

30

40

50

60

70

80

90

100

Length of the moving unit

Avera

ge s

teps o

f converg

ence

Figure 12. The convergence time

of DBarrier vs. length of moving

unit

5. Conclusion

We study barrier coverage with mobile sensors, in

which the sensors can be relocated for achieving bar-

rier coverage. We discuss the problem of relocating

sensors with minimum energy consumption. Then we

propose a centralized barrier algorithm, which com-

putes the relocated positions based on knowing the

initial positions of all sensors. We further design a

distributed barrier algorithm based on our proposed

virtual force model. In DBarrier, the sensors adjust

their positions according to the total repulsive and at-

tractive forces. In the future work we will design the

relocation algorithms for the k-barrier coverage.

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