Sensor Deployment and Target Localization Based
on Virtual Forces
Yi Zou and Krishnendu Chakrabarty
Abstract— The effectiveness of cluster-based distributed sensor
networks depends to a large extent on the coverage provided by
the sensor deployment. We propose a virtual force algorithm
(VFA) as a sensor deployment strategy to enhance the coverage
after an initial random placement of sensors. For a given
number of sensors, the VFA algorithm attempts to maximize
the sensor field coverage. A judicious combination of attractive
and repulsive forces is used to determine virtual motion paths
and the rate of movement for the randomly-placed sensors. Once
the effective sensor positions are identified, a one-time movement
with energy consideration incorporated is carried out, i.e., the
sensors are redeployed to these positions. We also propose a novel
probabilistic target localization algorithm that is executed by the
cluster head. The localization results are used by the cluster head
to query only a few sensors (out of those that report the presence
of a target) for more detailed information. Simulation results
are presented to demonstrate the effectiveness of the proposed
approach.
Index Terms— Sensor coverage, distributed sensor networks,
sensor placement, virtual force, localization.
I. INTRODUCTION
Distributed sensor networks (DSNs) are important for a
number of strategic applications such as coordinated target
detection, surveillance, and localization. The effectiveness of
DSNs is determined to a large extent by the coverage provided
by the sensor deployment. The positioning of sensors affects
coverage, communication cost, and resource management.
In this paper, we focus on sensor placement strategies that
maximize the coverage for a given number of sensors within
a cluster in cluster-based DSNs.
As an initial deployment step, a random placement of
sensors in the target area (sensor field) is often desirable,
especially if no aprioriknowledge of the terrain is available.
Random deployment is also practical in military applications,
where DSNs are initially established by dropping or throwing
sensors into the sensor field. However, random deployment
does not always lead to effective coverage, especially if the
sensors are overly clustered and there is a small concentration
of sensors in certain parts of the sensor field. The key idea
of this paper is that the coverage provided by a random
deployment can be improved using a force-directed algorithm.
Y. Zou and K. Chakrabarty are with the Department of Electrical and
Computer Engineering, Duke University, Durham, NC 27708, USA. E-mail:
{yz1, krish}@ee.duke.edu.
This research was supported in part by ONR under grant no. N66001-
00-1-8946. It was also sponsored in part by DARPA, and administered
by the Army Research Office under Emergent Surveillance Plexus MURI
Award No. DAAD19-01-1-0504. Any opinions, findings, and conclusions or
recommendations expressed in this publication are those of the authors and
do not necessarily reflect the views of the sponsoring agencies.
We present the virtual force algorithm (VFA) as a sensor
deployment strategy to enhance the coverage after an initial
random placement of sensors. The VFA algorithm is inspired
by disk packing theory [11] and the virtual force field concept
from robotics [5]. For a given number of sensors, VFA
attempts to maximize the sensor field coverage using a combi-
nation of attractive and repulsive forces. During the execution
of the force-directed VFA algorithm, sensors do not physically
move but a sequence of virtual motion paths is determined
for the randomly-placed sensors. Once the effective sensor
positions are identified, a one-time movement is carried out
to redeploy the sensors at these positions. Energy constraints
are also included in the sensor repositioning algorithm.
We also propose a novel target localization approach based
on a two-step communication protocol between the cluster
head and the sensors within the cluster. In the first step,
sensors detecting a target report the event to the cluster head.
The amount of information transmitted to the cluster head is
limited; in order to save power and bandwidth, the sensor
only reports the presence of a target, and it does not transmit
detailed information such as signal strength, confidence level
in the detection, imagery or time series data. Based on the
information received from the sensor and the knowledge of
the sensor deployment within the cluster, the cluster head
executes a probabilistic scoring-based localization algorithm
to determine likely position of the target. The cluster head
subsequently queries a subset of sensors that are in the vicinity
of these likely target positions.
The sensor field is represented by a two-dimensional grid.
The dimensions of the grid provide a measure of the sensor
field. The granularity of the grid, i.e. distance between grid
points can be adjusted to trade off computation time of the
VFA algorithm with the effectiveness of the coverage measure.
The detection by each sensor is modeled as a circle on the
two-dimensional grid. The center of the circle denotes the
sensor while the radius denotes the detection range of the
sensor. We first consider a binary detection model in which
a target is detected (not detected) with complete certainty by
the sensor if a target is inside (outside) its circle. The binary
model facilitates the understanding of the VFA model. We
then investigate a realistic probabilistic model in which the
probability that the sensor detects a target depends on the
relative position of the target within the circle. The details
of the probabilistic model are presented in Section III.
The organization of the paper is as follows. In Section II, we
review prior research on topics related to sensor deployment
in DSNs. In Section III, we present details of the VFA
algorithm. In Section IV, we present the target localization
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
algorithm that is executed by the cluster head. In Section
V, we present simulation results using the proposed sensor
deployment strategy for various situations. Section VI presents
conclusions and outlines directions for future work.
II. R
ELATED PRIOR WORK
Sensor deployment problems have been studied in a variety
of contexts [1], [2], [9]. In the area of adaptive beacon
placement and spatial localization, a number of techniques
have been proposed for both fine-grained and coarse-grained
localization [12].
Sensor deployment and sensor planning for military appli-
cations are described in [6], where a general sensor model
is used to detect elusive targets in the battlefield. However,
the proposed DSN framework in [6] requires a great deal of
aprioriknowledge about possible targets. Hence it is not
applicable in scenarios where there is no information about
potential targets in the environment.
The deployment of sensors for coverage of the sensing field
has been considered for multi-robot exploration [5]. Each robot
can be viewed as a sensor node in such systems. An incre-
mental deployment algorithm is used in which sensor nodes
are deployed one by one in an adaptive fashion. A drawback
of this approach is that it is computationally expensive. As the
number of sensors increases, each new deployment results in
a relatively large amount of computation.
The problem of evaluating the coverage provided by a given
placement of sensors is discussed in [7]. The major concern
here is the self-localization of sensor nodes; sensor nodes are
considered to be highly mobile and they move frequently. An
optimal polynomial-time algorithm that uses graph theory and
computational geometry constructs is used to determine the
best-case and the worst-case coverage.
Radar and sonar coverage also present several related chal-
lenges [13]. Radar and sonar netting optimization are of great
importance for detection and tracking in a surveillance area.
Based on the measured radar cross-sections and the coverage
diagrams for the different radars, the authors in [13] propose a
method for optimally locating the radars to achieve satisfactory
surveillance with limited radar resources.
Sensor placement on two- and three-dimensional grids has
been formulated as a combinatorial optimization problem, and
solved using integer linear programming in [3], [4]. This
approach suffers from two main drawbacks. First, compu-
tational complexity makes the approach infeasible for large
problem instances. Second, the grid coverage approach relies
on “perfect” sensor detection, i.e. a sensor is expected to yield
a binary yes/no detection outcome in every case. However,
because of the inherent uncertainty associated with sensor
readings, sensor detection must be modeled probabilistically
[10].
A probabilistic optimization framework for minimizing the
number of sensors for a two-dimensional grid has been pro-
posed recently [10]. This algorithm attempts to maximize the
average coverage of the grid points. Finally, there exists a
close resemblance between the sensor placement problem and
the art gallery problem (AGP) addressed by the art gallery
theorem [14]. Other related work includes the placement of a
given number of sensors to reduce communication cost [15],
optimal sensor placement for a given target distribution [16].
Our proposed algorithm differs from prior methods in
several ways. First, we consider both the binary sensor de-
tection model and probabilistic detection model to handle
sensors with both high and low detection accuracy. Second,
the amount of computation is limited since we perform a
one-time computation and sensor locations are determined at
the same time for all the sensor nodes. Third, our approach
improves upon an initial random placement, which offers a
practical sensor deployment solution. Finally, we investigate
the relationship between sensor placement within a cluster and
target localization by the cluster head.
III. V
IRTUAL FORCE ALGORITHM
In this section, we describe the underlying assumptions and
the virtual force algorithm (VFA).
A. Preliminaries
For a cluster-based sensor network architecture, we make
the following assumptions:
• After the initial random deployment, all sensor nodes are
able to communicate with the cluster head.
• The cluster head is responsible for executing the VFA al-
gorithm and managing the one-time movement of sensors
to the desired locations.
• In order to minimize the network traffic and conserve
energy, sensors only send a yes/no notification message
to the cluster head when a target is detected. The cluster
head intelligently queries a subset of sensors to gather
more detailed target information.
The VFA algorithm combines the ideas of potential field [5]
and disk packing [11]. In the sensor field, each sensor behaves
as a “source of force” for all other sensors. This force can
be either positive (attractive) or negative (repulsive). If two
sensors are placed too close to each other, the “closeness”
being measured by a pre-determined threshold, they exert
negative forces on each other. This ensures that the sensors are
not overly clustered, leading to poor coverage in other parts of
the sensor field. On the other hand, if a pair of sensors is too
far apart from each (once again a pre-determined threshold
is used here), they exert positive forces on each other. This
ensures that a globally uniform sensor placement is achieved.
Consider an n by m sensor field grid and assume that
there are k sensors deployed in the random deployment stage.
Each sensor has a detection range r. Assume sensor s
i
is
deployed at point (x
i
,y
i
). For any point P at (x, y),we
denote the Euclidean distance between s
i
and P as d(s
i
,P),
i.e. d(s
i
,P)=
(x
i
− x)
2
+(y
i
− y)
2
. Equation (1) shows
the binary sensor model [3], [4] that expresses the coverage
c
xy
(s
i
) of a grid point P by sensor s
i
.
c
xy
(s
i
)=
1, if d(s
i
,P) <r
0, otherwise.
(1)
The binary sensor model assumes that sensor readings have
no associated uncertainty. In reality, sensor detections are
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
imprecise, hence the coverage c
xy
(s
i
) needs to be expressed
in probabilistic terms. In this work, we assume the following,
motivated in part by [8]:
c
xy
(S
i
)=
0, if r + r
e
≤ d(s
i
,P)
e
−λa
β
, if r − r
e
<d(s
i
,P) <r+ r
e
1, if r − r
e
≥ d(s
i
,P)
(2)
where r
e
(r
e
<r) is a measure of the uncertainty in sensor
detection, a = d(s
i
,P)−(r−r
e
), and α and β are parameters
that measure detection probability when a target is at distance
greater than r
e
but within a distance from the sensor. This
model reflects the behavior of range sensing devices such as
infrared and ultrasound sensors. The probabilistic sensor detec-
tion model is shown in Fig. 1. Note that distances are measured
in units of grid points. Fig. 1 also illustrates the translation of
a distance response from a sensor to the confidence level as a
probability value about this sensor response. Different values
of the parameters α and β yield different translations reflected
by different detection probabilities, which can be viewed as
the characteristics of various types of physical sensors.
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance d(S
i
, P) between sensor and grid point
Detection probability
λ=0.5, β=1
λ=1, β=0.5
λ=0.5, β=0.5
λ=1, β=1
Fig. 1. Probabilistic sensor detection model.
B. Virtual Forces
We now describe the virtual forces and virtual force calcu-
lation in the VFA algorithm. In the following discussion, we
use the notation introduced in the previous subsection. Let the
total force action on sensor s
i
be denoted by
F
i
. Note that
F
i
is a vector whose orientation is determined by the vector sum
of all the forces acting on s
i
. Let the force exerted on s
i
by
another sensor s
j
be denoted by
F
ij
.
In addition to the positive and negative forces due to other
sensors, a sensor s
i
is also subjected to forces exerted by
obstacles and areas of preferential coverage in the grid. This
provides us with a convenient method to model obstacles and
the need for preferential coverage. Sensor deployment must
take into account the nature of the terrain, e.g., obstacles
such as building and trees in the line of sight for infrared
sensors, uneven surface and elevations for hilly terrain, etc.
In addition, based on relative measures of security needs and
tactical importance, certain areas of the grid need to be covered
with greater certainty.
In our virtual force model, we assume that obstacles exert
repulsive (negative) forces on a sensor. Likewise, areas of
preferential coverage exert attractive (positive) forces on a
sensor. Let
F
iA
be the total (attractive) force on s
i
due to
preferential coverage areas, and let
F
iR
be the total (repulsive)
force on s
i
due to obstacles. The total force
F
i
on s
i
can now
be expressed as
F
i
=
k
j=1,j=i
F
ij
+
F
iR
+
F
iA
(3)
We next express the force
F
ij
between s
i
and s
j
in polar
coordinate notation. Note that
f =(r, θ) implies a magnitude
of r and orientation θ for vector
f.
F
ij
=
(w
A
(d
ij
− d
th
),α
ij
) if d
ij
>d
th
0, if d
ij
= d
th
(w
R
1
d
ij
,α
ij
+ π), if otherwise
(4)
where d
ij
is the Euclidean distance between sensor s
i
and
s
j
, d
th
is the threshold on the distance between s
i
and s
j
,
α
ij
is the orientation (angle) of a line segment from s
i
to s
j
,
and w
A
(w
R
) is a measure of the attractive (repulsive) force.
The threshold distance d
th
controls how close sensors get to
each other. As an example, consider the four sensors s
1
, s
2
,
s
3
and s
4
in Fig. 2. The force
F
1
on S
1
is given by
F
1
=
F
12
+
F
13
+
F
14
. If we assume that d
12
>d
th
, d
13
<d
th
,
and d
14
= d
th
, s
2
exerts an attractive force on s
1
, s
3
exerts
a repulsive force on s
1
and s
4
exerts no force on s
1
.Thisis
shown Fig. 2.
Fig. 2. An example of virtual forces with four sensors.
If r
e
≈ 0 and we use the binary sensor detection model
given by Equation (1), we attempt to make d
ij
as close to
2r as possible. This ensures that the detection regions of two
sensors do not overlap, thereby minimizing “wasted overlap”
and allowing us to cover a large grid with a small number of
sensors. This is illustrated in Fig. 3(a). An obvious drawback
here is that a few grid points are not covered by any sensor.
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
Note that an alternative strategy is to allow overlap, as shown
in Fig. 3(b). While this approach ensures that all grid points are
covered, it needs more sensors for grid coverage. Therefore,
we adopt the first strategy. Note that in both cases, the coverage
is effective only if the total area kπr
2
that can be covered with
the k sensors exceeds the area of the grid.
Fig. 3. Non-overlapped and overlapped sensor coverage areas.
If r
e
> 0, r
e
is not negligible and the probabilistic sensor
model given by Equation (2) is used. Note that due to the
uncertainty in sensor detection responses, grid points are not
uniformly covered with the same probability. Some grid points
will have low coverage if they are covered only by only
one sensor and they are far from the sensor. In this case,
it is necessary to overlap sensor detection areas in order to
compensate for the low detection probability of grid points that
are far from a sensor. Consider a grid point with coordinate
(x, y) lying in the overlap region of sensors s
i
and s
j
.Let
c
xy
(s
i
,s
j
) be the probability that a target at this grid point is
reported as being detected by observing the outputs of these
two sensors. We assume that sensors within a cluster operate
independently in their sensing activities. Thus
c
x,y
(s
i
,s
j
)=1− (1 − c
x,y
(s
i
))(1 − c
x,y
(s
j
)) (5)
where c
xy
(s
i
) and c
xy
(s
j
) were defined in Section 3.1. Since
the term (1 − c
x,y
(s
i
))(1 − c
x,y
(s
j
)) expresses the probability
that neither s
i
nor s
j
covers grid point at (x, y), the probability
that the grid point (x, y) is covered is given by Equation (5).
Let c
th
be the desired coverage threshold for all grid points.
This implies that
min
x,y
{c
x,y
(s
i
,s
j
)}≥c
th
(6)
Note that Equation (5) can also be extended to a region which
is overlapped by a set of k
ov
sensors, denoted as S
ov
, k
ov
=
|S
ov
|, S
ov
⊆{s
1
,s
2
, ···,s
k
}. The coverage in this case is
given by:
c
x,y
(S
ov
)=1−
s
i
∈S
ov
(1 − c
x,y
(s
i
)) (7)
As shown in Equation (4), the threshold distance d
th
is used
to control how close sensors get to each other. When sensor
detection areas overlap, the closer the sensors are to each other,
the higher is the coverage probability for grid points in the
overlapped areas. Note however that there is no increase in the
point coverage once one of the sensors gets close enough to
provide detection with a probability of one. Therefore, we need
to determine d
th
that maximizes the number of grid points
in the overlapped area that satisfies c
xy
(s
i
) >c
th
. Let us
consider the three sensors s
1
, s
2
, and s
3
in Fig. 3(a), where
no overlap exists. Assume the three sensors are on a 31 by
31 grid, r =5and r
e
=3in units of grid points. Figures
4-6 show how the coverage is affected by d
th
and c
th
when
the threshold distance d
th
is changed from r + r
e
to r − r
e
.
The coverage for the entire grid is calculated as the fraction
of grid points that exceeds the threshold c
th
.We can use these
graphs to appropriately choose d
th
according to the required
c
th
.
2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coverage
c
xy
( S
1
, S
2
) for a sample point
Threshold disctance d
th
, r−r
e
≤ d
th
≤ r+r
e
λ=0.5, β=0.1
λ=0.5, β=0.5
λ=2, β=2
Fig. 4. Coverage vs. d
th
of a sample point inside the overlapped area of s
1
and s
2
.
2 3 4 5 6 7 8
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Threshold disctance d
th
, r−r
e
≤ d
th
≤ r+r
e
Coverage for the entire grid with
c
th
=0.7
λ=0.5, β=0.1
λ=0.5, β=0.5
λ=2, β=2
Fig. 5. Coverage vs. d
th
with c
th
=0.7 and different λ and β.
In order to prolong battery life, the distances between the
initial and final position of the sensors are limited in the
repositioning phase to conserve energy. We investigated two
approaches for incorporating energy constraints in the VFA
algorithm. The first approach disables any virtual forces on
a sensor whenever the current distance reaches the distance
limit. The second method records all virtual locations that
sensors are moved into during the VFA algorithm. When the
VFA algorithm terminates, a search procedure is used to find
the locations with maximum coverage, except those locations
that are already beyond the distance limit.
Note that the VFA algorithm is designed to be executed on
the cluster head, which is expected to have more computational
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
2 3 4 5 6 7 8
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Threshold disctance d
th
, r−r
e
≤ d
th
≤ r+r
e
Coverage percentage of a sample grid,
λ=0.5, β
=0.5
c
th
=0.5
c
th
=0.7
c
th
=0.9
Fig. 6. Coverage vs. d
th
with λ =0.5 and β =0.5 and c
th
=0.5,0.7
and 0.9.
VFA Data Structures: Grid, {s
1
, s
2
, ···, s
k
}
/* n
P
is the number of preferential area blocks (attractive
forces) and n
O
is the number of obstacle blocks (repulsive
forces). S
xy
, k
xy
and p table
xy
are used for localization. */
1 Grid structure:
2 Properties: width, height, k, c
th
, d
th
;
3 Preferential areas: PA
i
(x, y, wx, wy),
i =1, 2, ···,n
P
;
4 Obstacles areas: OA
i
(x, y, wx, wy),
i =1, 2, ···,n
O
;
5 Grid points, P
xy
:
c
xy
(s
1
,s
2
, ···,s
k
), S
xy
,k
xy
,p table
xy
;
6Sensors
i
structure: i, (x, y), r, r
e
, α, β;
Fig. 7. Data structures used in the VFA algorithm.
capabilities than sensor nodes. The cluster head uses the VFA
algorithm to find appropriate sensor node locations based on
the coverage requirements. The new locations are then sent to
the sensor nodes, which perform a one-time movement to the
designated positions. No movements are performed during the
execution of the VFA algorithm.
We next describe the VFA algorithm in pseudo-code form.
Fig. 7 shows the data structure of the VFA algorithm and Fig.
8 shows the implementation details. For a n by m grid with a
total of k sensors deployed, the computational complexity of
the VFA algorithm is O(nmk).
IV. T
ARGET LOCALIZATION
In our two-step communication protocol, when a sensor
detects a target, it sends an event notification to the cluster
head. In order to conserve power and bandwidth, the message
from the sensor to the cluster head is kept very small; in fact,
the presence or absence of a target can be encoded in just
one bit. Detailed information such as detection strength level,
imagery and time series data are stored in the local memory
and provided to the cluster head upon subsequent queries.
Based on the information received from the sensors within the
cluster, the cluster head executes a probabilistic localization
algorithm to determine candidate target locations, and it then
Procedure Virtual Fo rc e Algorithm (Grid, {s
1
, s
2
, ···, s
k
})
1Setloops =0;
2SetMaxLoops =MAX
LOOPS;
3 While (loops < M axLoops)
4 /* coverage evaluation */
5 For P (x, y) in Grid, x ∈ [1,width],y ∈ [1, height]
6 For s
i
∈{s
1
,s
2
, ···,s
k
}
7 Calculate c
xy
(s
i
,P) from the sensor model
using (d(s
i
,P),c
th
,d
th
,α,β);
8 End
9 If coverage requirements are met
10 Break from While loop;
11 End
12 End
13 /* virtual forces among sensors */
14 For s
i
∈{s
1
,s
2
, ···,s
k
}
15 Calculate
F
ij
using d(s
i
,s
j
), d
th
,w
A
,w
R
;
16 Calculate
F
iA
using d(s
i
,PA
1
, ···,PA
n
P
), d
th
;
17 Calculate
F
iR
using d(s
i
,OA
1
, ···,OA
n
O
), d
th
;
18
F
i
=
F
ij
+
F
iR
+
F
iA
,j ∈ [1,k],j = i;
19 End
20 /* move sensors virtually */
21 For s
i
∈{s
1
,s
2
, ···,s
k
}
22
F
i
(s
i
) virtually moves s
i
to its next position;
23 End
24 Set loops = loops +1;
25 End
Fig. 8. Pseudocode of the VFA algorithm.
queries the sensor(s) in the vicinity of the target. We assume
here that the sensor detection reports are time-labeled.
A. Detection Probability Table
After the VFA algorithm is used to determine the final
sensor locations, the cluster head generates a detection prob-
ability table for each grid point. The detection probability
table contains entries for all possible detection reports from
those sensors that can detect a target at this grid point. Let
us assume that a grid point P (x, y) is covered by a set of
k
xy
sensors, denoted as S
xy
, |S
xy
| = k
xy
, 0 ≤ k
xy
≤ k,
and S
xy
⊆{s
1
,s
2
, ···,s
k
}. The probability table is built on
the power set of S
xy
since there are 2
k
xy
possibilities for k
xy
sensors in reporting an event. These 2
k
xy
cases include the
event that none of the sensors detect anything (represented by
the binary string as “00...0”) as well as the event that all of
the sensors (represented by the binary string as “11...1”). Thus
the probability table for grid point (x, y) then contains 2
k
xy
entries, defined as:
p
table
xy
(i)=
s
j
∈S
xy
p
xy
(s
j
,i) (8)
where 0 ≤ i ≤ 2
k
xy
, and p
xy
(s
j
,i)=c
x,y
(s
j
) if s
j
detects
a target at grid point P (x, y); otherwise p
xy
(s
j
,i)=1−
c
x,y
(s
j
). Table I gives an example of the probability tables on
a 5 by 5 grid with 3 sensors deployed.
Consider the grid point (2, 4) in Fig. 9 which is covered
by all three sensors s
1
,s
2
and s
3
with probabilities as 0.57, 1,
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
