DISTRIBUTED COVERAGE AND CONNECTIVITY PRESERVING ALGORITHM WITH SUPPORT OF DIFFERENT SENSING COVERAGE DEGREES IN3D MOBILE SENSOR ACTOR NETWORKS

International Journal of Computer Science, Engineering and Information Technology (IJCSEIT), Vol.2, No.4, August 2012
DOI : 10.5121/ijcseit.2012.2401 1
DISTRIBUTED COVERAGE AND CONNECTIVITY
PRESERVING ALGORITHM WITH SUPPORT OF
DIFFERENT SENSING COVERAGE DEGREES IN3D
MOBILE SENSOR ACTOR NETWORKS
Mohammad Javad Heydari1
and Saeid Pashazadeh2
1
Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
Mj.Heydari89@ms.tabrizu.ac.ir
2
Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
Pashazadeh@tabrizu.ac.ir
ABSTRACT
Given a 3D space where should be supervised and a group of mobile sensor actor nodes with limited
sensing and communicating capabilities, this paper aims at proposing a distributed self-deployment
algorithm for agents to cover the space as much as possible by considering non-uniform sensing coverage
degree constraint of environment while preserving connectivity. The problem is formulated as coverage
maximization subject to connectivity and sensing coverage degree constraint. Considering a desired
distance between neighbouring nodes, an error function which depends on pairwise distance between
nodes is described. The maximization is encoded to an error minimization problem that is solved using
gradient descent algorithm and will yield in moving sensors into appropriate positions. Simulation results
are presented in two different conditions that importance of sensing coverage degree support of
environment is very high and is low.
KEYWORDS
Mobile Sensor Networks, Coverage, Connectivity, Quality of Services, Sensing Coverage Degree
1. INTRODUCTION
Wireless sensor actor networks (WSANs), can be deployed in three dimensional (3D)spacein
many applications such as aerial defence systems [1], underwater WSNs for Ocean sampling and
tsunami warnings [2-6], intrusion detection systems, aerosphere pollution monitoring [7], etc.
Maximal Coverage of the target volume and having a connected network are important criteria in
terms of the Quality of Service (QOS) required for WSNs. In this paper, considering importance
of coverage and connectivity in 3D wireless sensor networks a distributed maximizing coverage
algorithm is proposed.
The network consists of mobile agents deployed to perform collaborative monitoring tasks over a
given volume.. The problem is modelled as maximization which is solved completely distributed.
In fact each agent tries to maximize its sensing volume while preserving connectivity and having
high quality communication links. First of all, Knowing that distance of two communicating
nodes have great impact on energy consumed for transmission, quality of their connection and

2
total coverage of network and regarding the remaining energy, required quality of coverage and
connectivity due to application and required link strength depend on environment situation, an
optimal distance for two neighbouring nodes was assumed. Then each agent knowing distance of
its neighbours defines an error function and tries to decrease the error by changing its positions
through gradient descent algorithm. In other words agents try to set their distance to their
neighbours something near desired value by changing their positions. After obtaining optimum
positions, all agents will go to optimal points which yields lower using of energy in deployment
and moving.
It is well–known that centralized coordination algorithms have potential to have bottleneck and
suffer from computational complexity and lack of adaptation, thus distributed algorithms are
preferred. In this type of solutions although there is no centralized coordinator in the network,
covered area by the network is increasing from one step to another.
Our contributions in this paper are as follows: 1) Proposedalgorithm is completely distributed and
there is no need to having extra information. 2) As computation consumed much lower energy in
comparison with moving, the approach is using lower energy and thus the lifetime of network
would increase. 3) The parameter of network is adjustable and can vary depending on application.
In fact, coverage can be increased by increasing the desired distance of two nodes and by
decreasing it the quality of connections and usually number of links will be increased which
means a better connectivity and lower power consumption.The reminder of this paper is
organized as follows: The next section is covering related work. The considered system model
and the problemare described in section 3. Section 4 and 5 discuss and analyse the proposed
approach. Simulation results and performance evaluation are discussed in section 6.section 7
concludes the paper.
2. RELATED WORK
Deferent approaches for providing maximum coverage are proposed in manyliteratures in context
of multi-robot networks, mobile sensor networks and wireless sensor and actor networks. In most
of papers, the network is considered in 2D space where mobile node deployed randomly and can
move around to adjust their positions after initial deployment in order to reduce their coverage
overlaps and maximize area coverage while preserving connectivity.
Many presented approaches reducing the 3D geometry to the 2D [8–12]. Different modelling and
strategies such as virtual forces, optimization and predefined pattern was proposed. In [13]
Megerian and Potkonjakmodelled the coverage problem in wireless sensor networks as Integer
Linear Programming (ILP) and solved the problem using a greedy algorithm. Similarly,
Nakamura et al [14] used ILP modelling and solve it with the commercial optimization package
CPLEX [15]. Yiannis and Tzes in [16] and[17] modelled the coverage problem as a constrained
maximization. Our proposed algorithm in [18] is mainly inspired from the approaches used to
model the problem in [16] and [17] and current paper develops the approach that is used in
[18].In previous paper, we assumed equal importance for deferent points in the environment
while this paper uses a non-uniform density function. This means that points of 3D space have
different importance level and some points need higher sensing coverage degree in comparison
with other areas.
In one of the other group of coverage improvement algorithms such as [19], pattern-based
strategies are presented in 2D and 3D space. In such algorithms target locations that can provide
both coverage and network connectivity requirements are computed based on a predefined pattern
and mobile nodes will deploy in the target positions. Alam and Haas [20] proposed a deployment

3
pattern that generates the Voronoi tessellation of truncated octahedron in 3D space. Achieving
full-coverage and k-connectivity in three-dimensional space, some lattice was proposed in [21]
and[22].Wu et al [23] used Delaunay Triangulation to solve the coverage problem and finding the
position of sensors in the environment. Vieira et al [24] use computational geometry and graph
theory to find a solution. In virtual force-based node movement strategies [25-27], mobile nodes
affect each-others. In detailed form, two mobile nodes expel each other if their distance is close
than a threshold or attract each other if their distance is too far away. Other types of forces such
as attractive force of boundaries and the repulsive forces of obstacles also can be considered. The
total force is vector addition of all forces which specifies the direction and distance of mobile
node movement. In [28] and[29], the coverage problem is modelled as an optimization problem
and is solved through evolutionary algorithms PSO and ant colony optimization respectively.
3. PROBLEM FORMULATION
As mentioned in advanced, in this paper previous proposed approach in [18] is developed. So
similar to [18], let assume that network includes set of mobile homogenous agents denotedbyA =
{aଵ, aଶ, … , ܽ݊}. Nodes are distributed with uniform distributionin 3D space called volume of
interest that we represent byआ ⊂ Rଷ
. Let assume that all nodes are location aware and each node
knows its position p୧using a GPS or other node localization approaches and is aware ofpositions
of its one-hop neighbours. All agents equipped with wireless transceiver and the transmission
range of all agents is equal and is denoted by ܴ௖and is much less than dimension of volume of
interest. The communication range of a node is defined as maximum Euclidean distance that its
radio signals effectively can reach to other agents. Considering communication range of each
agent, communication volume of agent ݅is denoted by ܿ௜and is a sphere centred in agent position
i.e.
ܿ௜ = {‫݌‬ ∈ ܴଷ| ∥ ‫݌‬ − ‫݌‬௜ ∥≤ ܴ௖} , ݅ ∈ ‫ܫ‬௡ (1)
where ∥. ∥ stands for standard Euclidean norm and I୬is defined according to following equation:
‫ܫ‬௔ = {݅ ∈ ℕ|݅ ≤ ܽ} , ܽ ∈ ℕ (2)
Similarly, the sensing volume of agent ݅is denotedby ‫ݏ‬௜ is a sphere of radius ܴ௦, having the sensor
node at its centre i.e.
‫ݏ‬௜ = {‫݌‬ ∈ ܴଷ| ∥ ‫݌‬ − ‫݌‬௜ ∥≤ ܴ௦} , ݅ ∈ ‫ܫ‬௡ (3)
Also it is assumed that sensing and transmission ranges of all agents are equal,but transmission
range is greater than sensing radius.Let the density function ߶: आ ⊂ ܴଷ
→ ℝା
demonstrate
probability of occurring an event in an arbitrary pointp ∈ आ. This function explains importance of
each point in 3D space of volume of interest. Some points require more sensing coverage degree
in comparison to other points. The assumed density function in this paper is 3D Gaussian function
which is defined by Equation (4). As it can be seen in Figure 1, the density function peaks at its
centre and the importance of points decrease as they get farther. The centre location and other
parameters determine the shape of function and peace of decrease in deferent dimensions.
Assuming an arbitrary point, ‫݌‬ሺ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ሻ ∈ ܴଷ
in the space, and ‫݌‬௖ሺ‫ݔ‬௖, ‫ݕ‬௖, ‫ݖ‬௖ሻ ∈ ܴଷ
as centre of
Gaussian function, the following equationrepresents our assumed density function:
߶ሺ‫݌‬ሻ = ݁
ሺି
ሺೣషೣ೎ሻమ
഑ೣ
మ ି
ሺ೤ష೤೎ሻమ
഑೤
మ ି
ሺ೥ష೥೎ሻమ
഑೥
మ ሻ
(4)
Let Assume that the volume of interest is much greater than sensing volume of an individual
agent and variation of density function occurs smoothly. So,each sensing volume can be

4
estimated by its center point. In fact importance of agent location is considered as mean of
covered volume of all points of its sensing sphere.Another important function is ݂௜: आ ⊂ ܴଷ
→
ℝା
which is determined by following equation:
݂௜ሺ‫݌‬ሻ = ൜
1 ݂݅ ‫݌‬ ∈ ‫ݏ‬௜
0 ݂݅ ‫݌‬ ∉ ‫ݏ‬௜
(5)
This function determines if agent ݅ covers the point ‫݌‬ or not.
(a)
(b)
Figure 1. (a) Cross section of density function in 3D view (b) Lateral view of cross
section of density function.

5
Considering aforementioned functions and notations the main objective of this paper can be
expressed as proposing a distributed algorithm to maximize following functionࣝ:
ࣝ = ራ න ݂௜ሺ‫݌‬ሻ ߶ሺ‫݌‬ሻ ݀‫, ݌‬ ݅ ∈ ‫ܫ‬௡
आ
௡
௜ୀଵ
(6)
If the position metric considered as ܲ = ሺ‫݌‬ଵ, ‫݌‬ଶ, … , ‫݌‬௡ሻ the maximization can be defined as
following constrained optimization:
݂݅݊݀ ܲ:
݉ܽ‫ࣝ ݁ݖ݅݉݅ݔ‬
‫:݋ݐ ݐ݆ܾܿ݁ݑݏ‬ ݊‫ݕݐ݅ݒ݅ݐܿ݁݊݊݋ܿ ݃݊݅ݏ݋݈ ݐ݋‬
(7)
The Voronoi diagram for a set of points in a given space ܴௗ
is the partitioning of that space into
regions such that all locations within any one region are closer to the generating point than to any
other. In three dimensions, a Voronoi cell is a convex polyhedron formed by convex faces, as
shown in the Figure 2.The ݅th node partition or ݅th node Voronoi cell is defined as follows:
‫ݒ‬௜ = ൛‫݌‬ ∈ आ , ∥ ‫݌‬ − ‫݌‬௜ ∥≤∥ ‫݌‬ − ‫݌‬௝ ∥, ∀ ݆ ∈ ‫ܫ‬ே, ݆ ് ݅ൟ (8)
Limited Voronoi cell of ݅th node is defined usingEquation (8) and is depicted in Figure2 as
follows:
‫ݒ‬௜
௟
= ‫ݒ‬௜ ∩ ‫ݏ‬௜ , ݅ ∈ ‫ܫ‬௡ (9)
Voronoi Cell
Limited Voronoi Cell
Sensing region
Figure 2. Demonstration of a Voronoi cell [18].

6
Denoting ܰ௜asone-hop neighbour of ݅th agent, the set of communication linksof ݅th node is
defined as:
‫ܮ‬௜ = {ሺ݅, ݆ሻ|݆߳ܰ௜} , ݅, ݆߳‫ܫ‬௡ (5)
Using similar approach to [18] the Equation 7 could be expressed as:
݂݅݊݀ ‫݌‬௜:
݉ܽ‫ݒ ݁ݖ݅݉݅ݔ‬௜
௟
‫:݋ݐ ݐ݆ܾܿ݁ݑݏ‬ ‫ܮ‬௜ ് ∅
(6)
In fact the maximization problem defined by Equation (14) can be divided into sub-problems
which are solvable through local information and in a distributed manner.
4. PROPOSED APPROACH
Considering Equation (11), each agent tries to increase volume of its Voronoi cell without losing
connectivity with the others while trying to keep itself closer to areas that requires more sensing
coverage degree. In this section a parallel constrained maximizing algorithm running in all agents
is proposed.
To perform the algorithm, first desired distance between two neighbours (α) is specified. The
value of α would determine importance of coverage and connectivity, such that by increasing or
decreasing α coverage or connectivity is made more important in other word having a greater α
means increasing coverage of each node and thus increasing total coverage of network and
decreasing α means having lower distance between nodes, so having a more compact and
connected network. As transmission is one of energy-consuming act of network, it should also be
noted that increasing or decreasing α will affect the energy consumption of network since
decreasing distance of two neighbour nodes means increasing signal power and decreasing node
distance means decreasing power used for transmission [18].
By determining α, a neighbourhooderror function easily could be obtained. The error functions
assign an error to each distance. In fact knowing the distance between nodes i and j represented
by dሺi , jሻ, the function assigns an error to dሺi , jሻ. The amount of error is zero when dሺi , jሻ equals
α and will increase as the distance become more or less. The error function is plotted in Figure 3
( α = 9.5 m ).
Figure 3. Shape of error function
Error(m)
Distance (m)

7
Denoting position of ݅th node or ݅th column of ܲ as‫݌‬௜, obtaining the following equations seems
straightforward.
‫݌‬௜ሺ‫ݔ‬௜, ‫ݕ‬௜, ‫ݖ‬௜ሻ ∈ ܴଷ
(7)
Let assume
∆‫݌‬௜௝ = ‫݌‬௜ − ‫݌‬௝ (8)
And
݀௜௝ = ට∆‫݌‬௜௝
்
∆‫݌‬௜௝ (9)
The error between nodes ݅ and ݆ is difference of their distance from predetermined threshold α.
݁௜௝ = ݀௜௝ − ߙ (10)
So the neighbourhood error of ݅th node is as follows:
‫ܧ‬ே௜ =
1
|ܰ௜|
෍ ݁௜௝
ଶ
௝∈ே೔
(11)
And the positionalerror of ݅th node is as follows:
Where݀௖represents the distance from centre of Gaussian. So the total error of ݅th node is:
‫ܧ‬௜ = ‫ܧܣ‬ே௜ + ‫ܧܤ‬௣௜ (13)
We call parameter ‫ܣ‬as spreading coefficient. It represents the importance of maximizing
coverage while preserving connectivity.Parameter ‫ܤ‬ represents the importance of supporting
required sensing coverage degree and we call it density coefficient. In other words, this parameter
represents tendency of agents to get closer to areas that requires more sensing coverage degree.
By appropriate adjustment of these parameters, non-uniform sensing coverage degree on the
environment could be supported. In asymptotic case, if the parameter ‫ܤ‬tends to zero and
parameter ‫ܣ‬ set to one, agents have no tendency to get closer to points that requires more sensing
degree and only spread in the environment such that supports sensing degree one for all points of
the environment. We name this as spreading case in future.
‫ܧ‬௣௜
= 1 − ݁
ష೏೎
మ
഑మ (12)

8
If the parameter ‫ܤ‬ set to one and parameter ‫ܣ‬ set to zero the agents act like consensus state. In
this conditions, Agents tends to focus in areas that requires more sensing coverage degrees. We
name this case as aggregating case in remaining of the paper.
Position of ݅th node is changed based onfollowing equation:
‫݌‬௜ሺ‫ݐ‬ + 1ሻ = ‫݌‬௜ ሺ‫ݐ‬ሻ − ߟ
߲‫ܧ‬௜
߲‫݌‬௜
(14)
To getting derivate from ‫ܧ‬ே௜ we have:
߲‫ܧ‬ே௜
߲‫݌‬௜
=
1
|ܰ௜|
෍ 2 ݁௜௝
௝∈ே೔
߲݁௜௝
߲‫݌‬௜
(15)
߲‫ܧ‬ே௜
߲‫݌‬௜
=
1
|ܰ௜|
෍ 2 ݁௜௝
௝∈ே೔
߲݁௜௝
߲݀௜௝
߲݀௜௝
߲‫݌‬௜
(16)
߲‫ܧ‬ே௜
߲‫݌‬௜
=
1
|ܰ௜|
෍ 2 ݁௜௝
௝∈ே೔
߲݁௜௝
߲݀௜௝
߲݀௜௝
߲∆‫݌‬௜௝
߲∆‫݌‬௜௝
߲‫݌‬௜
(17)
߲݀௜௝
߲∆‫݌‬௜௝
= 1 (18)
߲݀௜௝
߲∆‫݌‬௜௝
=
∆‫݌‬௜௝
݀௜௝
(19)
߲∆‫݌‬௜௝
߲‫݌‬௜
= 1 (20)
Using the above equations we get:
߲‫ܧ‬ே௜
߲‫݌‬௜
=
1
|ܰ௜|
෍ 2 ݁௜௝
௝∈ே೔
∆‫݌‬௜௝
݀௜௝
(21)
To getting derivate from ‫ܧ‬௣௜
we have:
߲‫ܧ‬௣௜
߲‫݌‬௜
=
2݀௖
ߪଶ
݁
ష೏೎
మ
഑మ
(22)
By substituting Equation (26) and(27) inEquation (19), the following equation is obtained:
‫݌‬௜ ሺ௧ାଵሻ = ‫݌‬௜ ሺ௧ሻ − 2ߟሺ
‫ܣ‬
|ܰ௜|
෍ ݁௜௝
௝∈ே೔
∆‫݌‬௜௝
݀௜௝
+ ‫ܤ‬
݀௖
ߪଶ
݁
ష೏೎
మ
഑మ
ሻ (23)
Agents by using Equation (28) can change their positions in each time step such that ࣝ is a non-
decreasing function of time. The modification will halt when:
‫݌‬௜ ሺ௧ሻ − ‫݌‬௜ ሺ௧ାଵሻ < ߳ (24)

9
5. ALGORITHM ANALYSIS
In this section the algorithm is investigated. The algorithm will be verified in maximizing
coverage especially around the centre and not losing connectivity. If the minimum distance
between two neighbours is set to ߙ , the volume of limited Voronoi cell of ݅th node can be
estimated by a sphere with radius equal toα
2ൗ :
‫ݒ‬௜
௟
≥
1
6
ߨߙଷ
(25)
Considering Equations (30) and (6) the following equations will tie α in total coverage of
network.
ࣝ ≥ ራ
1
6
ߨߙଷ
௡
௜ୀଵ
(26)
ࣝ ≥
݊
6
ߨߙଷ
(27)
So by increasing α, total coverage of network will be increased and vice versa.Number of agent in
a determined volume is called density of agent which is defined by following equation:
ߣ =
1
ଵ
଺
ߨߙଷ (28)
So by decreasing α, the agents density around the centre will be increased and vice versa. Due to
using Exponential model for communication links decreases the distance between two nodes
results better quality of connections or higher connectivity. The exponential model is plotted in
Figure4for parameterσଶ
= 0.5 . The vertical column represent remaining portion of signal power
versus distance from the node.
It is so clear that adjusting α lower than communication range prevent links from failing.By
considering the convexity of error functions and lack of any local minima, the gradient descent
algorithm will guarantee to find a position in which global minimum error occurred. We can
conclude that by setting α lower than ܴ஼the network will stay connected and by
considering α near sensing radius the sensing volumes are maximized and by decreasing α the
density of nodes are increase around the centre. So in determining above-mentioned parameters
should be considered.
Figure 4. Exponential model of connection
SignalStrengthPortion
Distance (m)

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6. SIMULATION RESULTS
Let assume a networkwhich includes 10agents deployed randomly with uniform distribution on
the space आ ⊂ Rଷ
. During simulation, the covered volume is increasing and agents get near to
centre. Two series of simulations is done as follows: considering a network with 1)high positional
error coefficient and 2) with low positional error coefficient. The simulation parameters are
summarized in table 1 and table 2.
Table1.Simulation parameters ofaggregating case.
Node Placement Random
No of agents 10
Dimension ofVOI 40m
Communication Range 10m
Sensing Range 5m
A 1
B 0.1
σx 5
σy 5
σz 5
α 9
η 0.2
ε 1
Considering a network with abovementioned parameters, the agent’s initial, intermediate and
final positions are shown in Figure5. Due to high positional error coefficient, the agents organize
themselves in a way that there is minimum overlapping between sensing volumes around a
centre.For better understanding an overall view of all 3D pictures are shown in the figures. As it
is seen the initial state of the network is so compact and sensing volume have overlaps but in the
final state the surveyed volume by the network is maximized around a centre without losing
connectivity of agents.
Table2.Simulation parameters forspreading case.
Node Placement Random
No of agents 10
Dimension ofVOI 100m
Communication Range 10m
Sensing Range 5m
A 1
B 0.001
σx 5
σy 5
σz 5
α 9.5
η 0.2
ε 1

11
Considering spreading case,total coverage is of network is some value near coverage upper bound
i.e.ࣝ =
୬
଺
πRୱ
ଷ
. In fact, by decreasing the positional error coefficient, the agents spread in the
environment. In simulation process making the simulation more realistic agents are selected
randomly to update their positions and the selected agent, updated its position with probability of
%95 and it can be mentioned that all simulations converged to optimum solutions in below thirty
iterations. In both cases sensing radii is considered half of communication radii which mean the
agents with tangent sensing volumes are connected. Comparing this work to previous one [18],
(d)(c)
(f)(e)
Figure 5. Deployment of agents in aggregating case (a, b) Lateral and top 3D view of
initial deployment (c, d) Lateral and top 3D view of intermediate deployment (e, f)
Lateral and top 3D view of final deployment.
(b)(a)

12
the algorithm try to herd the agents to centre while not getting disconnected and while
maximizing coverage volume.
7. CONCLUSIONS
In this paper a motion coordination algorithm for mobile agents deployed in a 3D wireless sensor
network is proposed. It is assumed that agents have spherical shaped sensing and communication
volume. The total coverage of network and quality of links can be controlled trough
parameterαand as seen in simulations by adjustingαnear sensing radii the approach would lead
the network to a configuration with maximum connected coverage around predefined centre. In
(d)(c)
(f)(e)
Figure 6. Deployment of agents in spreading case(a, b) Lateral and top 3D view of initial
deployment (c, d) Lateral and top 3D view of intermediate deployment(e, f) Lateral and top
3D view of final deployment.
(b)(a)

13
previous work [18],we proposed uniform importance for points in the environment but in this
paperwe considered non-uniform requirement for sensing degree for regions of the environment
and tried to drive agents to special points in the environment which are more important than the
others Simulation results show the efficiency and fast convergence of algorithm.
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Authors
Mohammad Javad Heydari received his B.Sc. in Computer Engineering from shahed
University of Tehran in 2009.He is currently pursuingthe M.Sc. degree in Artificial
Intelligence at University of Tabriz in Iran. His research interests mainly include
Wireless Sensor Networks, Multi-Robot Systems and Swarm Intelligence.
Saeid Pashazadeh is Assistant Professor of Software Engineering and chair of
Information Technology Department at Faculty of Electrical and Computer
Engineering in University of Tabriz in Iran. He received his B.Sc. in Computer
Engineering from Sharif Technical University of Iran in 1995. He obtained M.Sc. and
Ph.D. in Computer Engineering from Iran University of Science and Technology in
Iran in 1998, 2010 respectively.He was Lecturer in Faculty of Electrical Engineering in
Sahand University of Technology in Iran from 1999 until 2004. His main interest is in
the development, modeling and formal verification of distributed systems, and computer security and
wireless sensor and wireless actor networks. He is member of IEEE and senior member of IACSIT.

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