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Abstract—This paper presents a method for an optimal wireless sensor network deployment with a quadratic cost function. It is assumed that the network is symmetric with an equal radio transmission range across the network. The optimization problem is formulated using a convex quadratic cost function that considers distances between mobile nodes and both static nodes and the target(s). The network constraints related to the network/graph connectivity conditions are presented. It is assumed that the base station can directly communicate with nodes that are to be deployed and with the mobile nodes. Simulation results are presented using a newly developed simulation platform for sensor network suboptimal deployment.

Key words—wireless sensor networks, optimal control, deployment, coverage.

I. INTRODUCTION Wireless sensor networks (WSNs) have integrated

computing, storing, networking, sensing, and actuating capabilities [7], [12], [13], [14], [17], [18], [22], [29], [33]. These networks consist of a number of sensor nodes (static and mobile) with multiple sensors per node that communicate with each other and the base station through wireless radio links (see Figure 1). The base station, or the gateway, is used for data processing, storage, and control of the sensor network. Sensor nodes are usually battery powered; hence the whole sensor network is limited by fundamental tradeoffs between sampling rates and battery lifetimes [22].

These limited resources in terms of costs, power, and bandwidth require the use of optimality principles in sensor network deployment. An effective and optimal coverage of the sensor field reduces costs, improves sensor network functionality, increases resilience to failures, and enables novel applications in the sensor network. We discuss here optimal wireless sensor network coverage control applied to both static and mobile sensor nodes. A novel method for optimal sensor network coverage control is presented that utilizes a linear quadratic cost function. The optimal sensor

node deployment problem has been formulated under network connectivity constraints.

Sensor networks and related coverage control problems have recently gained significant interest [3], [5], [6], [9], [10], [11], [16]. Several optimal control problems related to sensor networks were formulated in [5]. This survey paper also provides an excellent overview of existing results in this area. An optimal coverage under constraints of imprecise detection and terrain properties where the number of sensor nodes is minimized was presented in [10]. Reference [16] discusses the problems arising in maintaining coordination and communication between the group of robots and solutions to these problems. The paper presented three models of coordination (i.e., deployment) in order to maximize the coverage area within the close range of the robots, deployment to maximize the probability of detecting a source, and deployment to maximize the visibility of the network. A tutorial paper [29] provides an overview of control methods in multivehicle cooperative control using graph theory.

Mobile nodes

Static nodes

UAV nodes

Gateway

Mobile nodes

Static nodes

UAV nodes

Gateway

Fig. 1. Ad-hoc wireless sensor network with static and mobile nodes.

Rastko R. Selmic 1, Jinko Kanno 2, Jack Buchart 1, and Nicholas Richardson 3 1 Department of Electrical Engineering, 2 Department of Mathematics

3 Department of Computer Science Louisiana Tech University Ruston, LA 71272, USA

Tel: 318-257-4641 Fax: 318-257-4922 Email: rselmic@latech.edu, jkanno@coes.latech.edu,

jgb059522@gmail.com, nmr002@latech.edu

Quadratic Optimal Control of Wireless Sensor Network Deployment

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Optimal coverage control for mobile sensor networks was

presented in [9]. The paper uses a Voronoi partitioning and Lloyd descent algorithm but does not consider network connectivity constraints. Weighted directed graphs that relate averaging protocols in consensus problems and coverage control laws over Voronoi graphs were given in [15]. Two location functions that characterize coverage performance were provided in [10]. The paper studies their gradient properties via nonsmooth analysis. In all the above results the feasibility sets are assumed to be convex and the network (i.e., optimization) is not constrained.

General textbooks in the wireless sensor network area include recent publications [4], [19], [24], [34]; however none deal with the specific problem of network deployment optimization. A linear programming problem for sensor deployment was formulated in [32]. Moreover, the paper presents an excellent review of existing deployment algorithms. The considered optimization problem is a convex one and sensors are constrained into a grid-type of network topology. Results in [31] showed a fundamental relationship between coverage and connectivity by proving that if the communication radius rc of the nodes is at least twice their sensing radius rs, then complete sensing coverage (1-coverage) of a convex region implies that the related network graph is connected. Additionally, it is shown that if rc ≥ 2rs, then Ks-coverage of a convex region implies Ks-connectivity of the sensor nodes. Reference [20] minimizes the number of sensors to cover an area of interest. However, they assume that the sensing and communication ranges are identical, while our approach can easily be extended to a general case of different communication and sensing ranges. A virtual force algorithm for sensor network deployment that maximizes the sensor field coverage was proposed in [35]. The approach requires a grid-based sensor node placement and also requires that all nodes are mobile. This paper provides an excellent energy model for the network; however, is does not consider the network connectivity as a constraint in the optimization process. A deterministic and statistical sensor network coverage algorithm using Voronoi diagrams for sensor field partitioning was studied in [23]. A graph theory abstraction has been applied to study the coverage problem and to provide an algorithm complexity.

II. BACKGROUND WSNs are topologically similar to graphs. This similarity

can be used to map different deployment scenarios and conditions into graph theory problems. In such a mapping, every sensor node, including the base station, corresponds to a vertex in a graph that varies in time, i.e. ( ).)(),()( tEtNtG = , (1)

where N(t) is a set of vertices and E(t) is a set of edges. Nodes or vertices in a graph are connected with a graph edge if there is a direct communication link between corresponding nodes.

With direct communication link we mean that the nodes communicate directly, without multi-hopping. Note also that the graph G(t) is, in general, a function of time since nodes and links can be time dependant. This is especially true in a sensor network with mobile nodes as the set of edges will certainly change as the mobile nodes move throughout the network. Illustrative mapping is shown in Figure 2 where a WSN for chemical agent monitoring application, developed with Crossbow motes is represented as a graph.

Base Station

SN 2

Radio Link

SN 1

SN 5

SN 3SN 0

SN4⇔

2

1

0

3

4

5

Base Station

SN 2

Radio Link

SN 1

SN 5

SN 3SN 0

SN4⇔

2

1

0

3

4

5

2

1

0

3

4

5

Fig. 2. Wireless sensor network map into a graph.

A. k-Vertex Connectivity and k-Edge Connectivity A graph G is said to be connected if there is a path between

any two vertices of G. A graph G is called k-connected if the minimum number of vertices of G whose removal will disconnect G is at least k. A k-connected network graph will remain connected with the failure of (k-1) nodes. As a result of Menger’s Theorem, a graph G is k-connected if and only if there are k vertex-disjoint paths between any two vertices in G. The graph on the left (Figure 3) is 1-connected since the removal of vertex v would disconnect the subgraph from the base station. The graph on the right is 2-connected since any single vertex can be removed and the graph will remain connected. Note that there are two vertex disjoint paths between any vertex vi (i=1, 2, …, 5) and the base station, namely one path is B-v1-v2-…-vi and another is B-v5-v4-…-vi.

B

v

B

v1 v2

v3

v4v5

B

v

B

v1 v2

v3

v4v5

Fig. 3. The graph on the left is 1-connected, and the graph on the right is 2-connected.

III. SENSOR NETWORK MODEL AND PROBLEM FORMULATION Wireless sensor networks (WSNs) consist of sensors that

communicate between themselves and the base station using multi-hop routing protocols. Information hops between the

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nodes towards the base station using wireless on-board radios that have a limited transmission radius, often adjustable by microcontroller algorithm. In order to map such a network to an equivalent graph model, several assumptions are needed. Assumption 1. The WSN consists of sensor nodes all having an equal wireless radio transmission range r. Wireless radio and sensor nodes have omnidirectional antennas.

While this assumption is not always fulfilled in practical applications, one can always consider the minimum available transmission range as the least-common transmission range for sensor nodes [1]. Assumption 2. The WSN consists of symmetric links between the sensor nodes, meaning that if node i can transmit information to a node j, then node j can also send information to node i.

Assumptions 1 and 2 allow the use of undirected graphs in the sensor network modeling and simplify analysis by using uniform radio transmission range r.

A. Sensor Network Connectivity and Localization Conditions A wireless sensor network is connected if the corresponding

graph G(t) is at least 1-connected, which implies that data from any node in the network can multi-hop its data to the base station. A WSN is 1-connected (but not 2-connected) if there is a single node whose failure in the sensor network will disconnect the network. It is clear that such connectivity does not provide system robustness and fault redundancy in case of node failures. Bredin et al. [3] considered vertex k-connectivity algorithms to be used for robust and fault tolerant networks. In such cases up to k-1 nodes may fail and the network will still be connected. We will consider the sensor network deployment under the k-connectivity condition.

Localization of sensor nodes is an important problem in the wireless sensor network area. Sensor nodes can be equipped with GPS receivers in which case the location is known to the degree of accuracy of the receivers. If GPS is not available, a variety of methods exist to locate the sensor node in the wireless sensor network environment. A two-phase process for localization is a common choice, which includes gathering range information between nodes in the first phase and estimating positions in the second phase. Range information can be obtained by exploiting signal strength, as RADAR [1] and APS [25], time difference of arrival of two different signals, as Cricket [28], AHLoS [30], and angle of arrival, as given in [26]. However, this may be a challenging problem since a ranging error may propagate through the whole network.

Conditions needed to localize the unknown sensor node also have their equivalent counterparts in graph theory. Consider a sensor network with static and mobile sensor nodes where positions of static sensor nodes are known. Either a field technician is aware of the positions of the deployed static wireless sensor nodes or they have GPS receivers to report their geo-locations. In the case that the mobile nodes do not have GPS receivers, it is required to estimate the position of a mobile sensor node that is slowly moving in the sensor network field. In the case of 2-dimensional movement, a

mobile node should be in the communication neighborhood of at least 3 nodes that know their position in order to uniquely determine its position. Such an estimate is based on the received radio signal strength. Equivalently, each mobile vertex must have 3 incident edges (i.e. they must be of degree at least 3). Mobile Node Localization. Given a graph defined as in (1) that includes static and mobile nodes, we define M(t) to be a subset of N(t) where M(t) is the set of mobile nodes. The localization condition means that for each mobile node in M(t), the degree must be at least 3. Connectivity conditions in sensor networks do not usually follow crisp logic, i.e., when inter-node distances are less than r, the nodes are connected, otherwise they are disconnected. One natural question arises – what happens with connectivity when the distance between nodes is close to r? In order to mathematically deal with these kinds of situations we define strong and weak connectivity. A spanning subgraph of a graph G is a subgraph that contains all vertices of G. Definition (Strong Connectivity). The graph G(t) is strongly connected if there is a spanning subgraph H(t) such that H(t) is k-connected and all edges of H(t) are less than or equal to λr, where 0 < λ ≤ 1 and r is a sensor node transmission radius. Definition (Weak Connectivity). The graph G(t) is weakly connected if it is connected and it is not strongly connected. For example, let λ=0.75 for some graph G. If there is a spanning k-connected subgraph of G with every edges less than or equal to 0.75r, then G is strongly connected. If no such subgraph exists then G is weakly connected.

B. Optimal Sensor Network Deployment Problem Many practical applications using wireless sensor networks

have a common optimization problem that presents a tradeoff between opposite goals, e.g., uniformly distribute sensor nodes over the monitoring area while focusing on a specific area or phenomenon (Figure 4). A fundamental question that motivates this research can be expressed as follows: how to optimally position a large number of sensor nodes over the area of interest, while providing extensive coverage of the focus area of interest (e.g., a contamination cloud in case of chemical agent monitoring).

While it is desired to uniformly monitor an area of interest, it is also required to focus additional attention on an interesting phenomenon that has been detected. We introduce a novel mathematical problem formulation of a suboptimal sensor network coverage control problem, namely a linear quadratic (LQ) type of optimal coverage control [2], [21]. The LQ cost functions have been designed to balance the tradeoff between uniform sensor network coverage and focused coverage of specific areas.

Using graph theory terminology, the problem can be summarized as follows: find the optimal sensor network deployment that minimizes a given index of performance such that the corresponding sensor network graph G(t) is vertex k-connected and satisfies the mobile node localization condition.

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Note that if k is at least 3, the mobile node localization condition is met by default, and we only need to check for k-connectivity in this case. The problem formulation is given below.

Fig. 4. Focused coverage vs. uniform coverage of the monitoring area of interest.

Problem 1 (Optimal Sensor Node Placement Without Network Constraints). Let the sensors’ radio transmission range be given by r. Assume there is a focus point (XF(t), YF(t)) where several mobile sensor nodes should converge. Given the compact coverage area of interest S, a set of wireless sensor nodes N(t), and a subset of mobile sensor nodes M(t)⊂N(t), find an optimal vertex location of the mobile nodes M(t) such that the following cost function is minimized

[ ]

[ ]∑∑

∈∈

∈

+=

NjMijjii

MiFFii

yxyxQ

tYtXyxdistRJ

,

21

),(),,(

))(),((),,(min (2)

where R and Q are control design parameters. The parameter R rewards closeness to the focus point and the parameter Q rewards uniform distribution of vertices across the set S. For example, Q >> R means that the user is much more concerned with the uniform vertex distribution than to cover the specific area of interest, Figure 4 (right). On the other hand, R >> Q indicates that the user wants extensive coverage of the focus point rather than uniform vertex distribution, Figure 4 (left). Note that the cost function in (2) was chosen because the problem we have proposed is a multi-objective optimization problem. One common way to solve this type of problem is to combine the weighted sums of each individual objective function into an aggregate objective function (AOF). Specifically, the two objectives are to minimize the distance between the mobile nodes and the focus point and to maximize the distance between the each mobile node and all other nodes (including other mobile nodes). Rather than maximizing this distance, however, we have chosen to minimize the inverse of this distance so that we could easily combine the two terms into an AOF. Problem 2 (Optimal Sensor Node Placement With Network Constraints). Given the compact coverage area of interest S, a focus point (XF(t), YF(t)), the graph G(t) = (N(t),E(t)), and the mobile subgraph Gm(t) = (M(t),Em(t)), find an optimal vertex location of the mobile subgraph Gm(t) such that (2) is minizmized, the graph G(t) stays k-connected, and the mobile node localization condition is satisfied. Thus we

impose two constraints on Problem 1, namely the k-connectivity requirement and the mobile node localization condition requirement. Again, if k>2 the mobile node localization condition will, by default, be satisfied.

The next section provides assumptions needed to specify network connectivity constraints. It also provides a solution for the above presented optimization problems.

IV. OPTIMAL WIRELESS SENSOR NETWORK DEPLOYMENT The presented indices of performance are quadratic convex

functions. In optimization theory there are two subclasses of such problems, depending on the constraint conditions: a problem can be convex or non-convex, depending on the constraints and the corresponding constraint set or feasibility set. A feasibility set is a set of allowable solution, i.e., allowable sensor node locations in case of the problem described in Section 3. Assumption 3. The base station is one hop away from all mobile sensor nodes in the network. Note that this assumption means that the base station can directly see all mobile sensor nodes, therefore restricting the problem to a convex feasibility set, see Figure 5. A similar assumption was used in the literature [32] where it is assumed that all nodes communicate with the base station directly.

BB

Fig. 5. A suboptimal deployment of a mobile node (red) that preserves the network connectivity is a convex optimization in case the base station can directly communicate with mobile nodes (one hop away). Otherwise, it is a

non-convex optimization problem. A. Sensor Network Placement Algorithm

There are many algorithms that can be used for constrained (convex) optimization problems, see [27] for instance. We summarize two methods that can be easily implemented. Sequential Grid Search. This method provides a useful solution for problems with a relatively small number of moving UAV nodes, for instance less than 10, see [8]. For a preselected grid at the area of interest, the cost function is evaluated in the immediate neighborhood of the present state in m-dimensional space and decision is made about where the

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next state is, where m is the number of mobile UAV nodes, i.e. the order of the set M. The algorithm is as follows.

1. Place a grid of preselected size ∆x for each variable

(mobile sensor node) in the space of interest S. 2. Select the starting point for all mobile sensor nodes in

S. 3. Evaluate the cost function J in 9m surrounding points

including the existing point, where m is the number of mobile nodes.

4. As a next state select the point with the smallest cost function J.

5. Check if the next state is in S and if it satisfies the sensor network constraints. If yes, accept that point as a new state for mobile nodes and repeat.

One can see that the sequential grid search method will

work fine for a few mobile nodes, but if a larger number of mobile nodes needed to be optimally positioned, the method is inefficient in terms of computational requirements. More efficient and easier to implement is a gradient descent method that only requires a gradient computation at every step. Gradient Descent Algorithm for One Mobile Sensor Node Deployment. If there is only one mobile sensor node, the previous problem formulation can be expressed in equivalent form: find an optimal vertex location (x1, y1) such that the following cost function is minimized.

[ ]

[ ]∑∈

+⋅=

Njjj

FF

yxyxQYXyxdistRJ

),(),,(),(),,(min

11

112

1

(3)

The gradient descent algorithm is then given as follows.

1. Let designate the s-th iteration (where s=0, 1, 2, 3,...) and calculate

[ ]

[ ]⎪⎪⎩

⎪⎪⎨

⎧

−+−

−+−=

∂∂

−+−

−−−=

∂∂

∑

∑

∈

∈

Nj jj

jF

Nj jj

jF

yyxxyy

QYyRyJ

yyxxxx

QXxRxJ

221

21

11

1

2

221

21

11

1

2

)()()(2

)(2

)()()(2

)(2

2. Let the start point be given by (x10, y10) and let the tolerance be given by ε.

3. Choose step size λs to minimize J2, that is ),(

1

21

1

212

sss

sss y

Jy

xJ

xJ∂∂

−∂∂

− λλ

),(min1

21

1

212

ss

ss y

Jy

xJ

xJ∂∂

−∂∂

−= λλλ

4. The calculate the next point (x1(s+1), y1(s+1)) as

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

−=

+

+

ssss

ssss

yJyy

xJxx

1

21)1(1

1

21)1(1

λ

λ

5. Terminate the calculation if

ε≤− ++ ),(),( )1(1)1(12112 ssss yxJyxJ .

Then, we take (x1(s+1), y1(s+1)) to be the optimal position of the sensor node we are seeking; otherwise return to Step 2.

Simulation results presented in the next section show an

implementation of the gradient descent algorithm. Due to its simplicity, this algorithm is also suitable for fast, real-real time execution.

V. SIMULATION RESULTS Self-Adjusting Connected Sensor Network Simulation

(SACSeNS) is a simulation program that can test different cost functions and their effectiveness to distribute stationary and mobile sensor nodes in a given area, see Figure 6. The program also provides graph statistics such as connectivity and coverage percentage. Currently the simulation makes assumptions that all nodes have the same communication range, the mobile nodes move at the same rate, and the mobile nodes can move in any direction. Users can specify a communication range; however, it is the same for all nodes in the network.

Stationary and mobile nodes can be deployed simultaneously. The stationary nodes are moved to a specific location by the user and will remain there during the simulation. The mobile nodes automatically move into a position to minimize the cost function based on the position of the stationary nodes, other mobile nodes, and the focus point.

The movement of mobile nodes follows the simple gradient descent algorithm from the previous section such that the cost function is minimized. From these positions, an additional constraint of connectivity will be placed on the system that will require the node to maintain connectivity to the network. If a node’s movement will violate the network connectivity constraints, the program will skip the selected node position and check the other positions for the best movement of the node. The program uses a spanning tree algorithm to determine if the network graph is connected or disconnected.

The focus/tracking point is something that we may want additional sensor nodes to monitor and track. Based on parameters given in the cost function, mobile nodes can move to increase sensor coverage and track a target. There is a performance index graph over time that is updated periodically and shows the relative change based on the new sensor node positions. This chart shows how well the nodes distribute themselves based on the cost function.

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Fig. 6. Self-Adjusting Connected Sensor Network Simulation (SACSeNS) program for simulation of wireless sensor network.

Figure 7 shows two opposite cases discussed previously,

i.e., uniform vs. focused coverage. There is one target/focus point in the network. The sensor network is connected, however, weakly connected since communication links between a few sensor nodes are “weak”.

Presented results are an initial development of SACSeNS, while future steps will include:

• consider cases where mobile nodes do not have same speed, do not have same communication range, power, etc.

• implement various cost functions and optimization algorithms

• develop a web-based user interface such that the software can be used over the Internet.

Currently, the first version of SACSeNS is built in C++.

VI. CONCLUSIONS A novel linear quadratic-type of a cost function was

presented for a wireless sensor network deployment problem under network connectivity constraints. The convex optimization problem was solved using a gradient descent algorithm that was implemented as a part of new sensor placement simulation software for wireless sensor networks that is being developed at Louisiana Tech. The simulation tool was presented as well as simulation results for focused and uniform network coverage. Future research will consider the more general non-convex optimization problem, different cost functions, and novel features in the simulation software.

Fig. 7. SACSeNS snapshot of a uniform coverage and deployment of mobile

(blue) sensor nodes (left) and a focused coverage/tracking of mobile nodes (right). The red links emphasize weak connectivity of the network graph.

ACKNOWLEDGMENT This project is supported in part by the ARFL Sensor

Directorate through contract FA8650-05-D-1912 and by Louisiana Board of Regents through PKSFI grant LEQSF(2007-12)-ENH-PKSFI-PRS-03. The authors also acknowledge help by graduate student Grain Zhang.

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Network graph statisticsTarget to be tracked

Cost function in time

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Static sensor nodes (green)

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Cost function in time

Mobile sensor nodes (blue)

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