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Optimized Cooperative Dynamic Coverage in MixedSensor Networks

THEOFANIS P. LAMBROU,University of Cyprus

This article considers the problem of improving the dynamic coverage and event detection time of mixedWireless Sensor Networks (WSNs). We consider mixed WSNs that consist of sparse static sensor deploy-ments and mobile sensors that move continuously to monitor uncovered (vacant) areas in the sensor field.Mobile sensors move autonomously and cooperatively by executing a path planning algorithm. Using asimplified scenario, the article derives the optimal path strategy for a single mobile sensor to search twonon-connected uncovered regions with the minimum average detection delay or with the maximum dynamiccoverage. The resulting optimal strategy confirms that it is better to search areas that are less likely tohide a target but are located closer to the mobile node, rather than heading towards the most likely area.Based on the insights gained from the simplified scenario and the theory of coverage processes, the articleproposes a surrogate method to approximate the best searching neighborhood radius (a design parameterof the path planning algorithm) that optimizes the dynamic coverage and event detection time capabilitiesof Mixed WSN deployments. Extensive simulation results indicate that this approach can achieve very goodresults, both for a single and for multiple collaborating mobile sensors.

Categories and Subject Descriptors: C.2.2 [Computer-Communication Networks]: Network Protocols

General Terms: Algorithms, Performance, Theory

Additional Key Words and Phrases: Mixed Sensor Networks, Mobile Sensors, Dynamic Coverage, EventDetection, Path Planning, Distributed Decision Making, Search Strategies

ACM Reference Format:Theofanis P. Lambrou, 2014. Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks. ACMTrans. Sensor Netw. V, N, Article A (January YYYY), 35 pages.DOI:http://dx.doi.org/10.1145/0000000.0000000

1. INTRODUCTIONSensor networks have received considerable attention over the past decade for their po-tential as a cheap, easily deployed, distributed monitoring tool. Recently, researchershave begun to investigate the use of mobile sensor nodes. Mobile and Mixed SensorNetworks is a natural evolution of sensor networks where sensors can measure spa-tially and temporally distributed phenomena more efficiently. Emerging applicationdomains of such mixed WSN include ocean-marine monitoring, pollution monitoring,facility inspection, inspection of landfills and search and rescue operations.

This work is partly supported by the Cyprus Research Promotion Foundation under grantTΠE/OPIZO/0609 (BE) /06 and co-funded by the Republic of Cyprus and the European Regional De-velopment Fund.Author’s addresses: T.P Lambrou, KIOS Research Center for Intelligent Systems and Networks and theDepartment of Electrical and Computer Engineering, University of Cyprus, Nicosia, Cyprus.E-mail: [email protected] to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies show this notice on the first page or initial screen of a display along with the full citation. Copyrightsfor components of this work owned by others than ACM must be honored. Abstracting with credit is per-mitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any componentof this work in other works requires prior specific permission and/or a fee. Permissions may be requestedfrom Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212)869-0481, or [email protected].© YYYY ACM 1550-4859/YYYY/01-ARTA $15.00DOI:http://dx.doi.org/10.1145/0000000.0000000

ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

A:2 T.P. Lambrou

In all of these domains, the goal of mobile sensors is to efficiently navigate throughthe sensor field to maximize some metric of information (e.g., monitoring coverage,probability of event detection) while satisfying constraints on energy or detection time.This path planning problem is particularly challenging because it typically requiressearching over a large and complex space of possible trajectories. Such problems havebeen shown to be NP-hard [Singh et al. 2009b] depending on the form of the objectivefunction, the size of the environment and the space of possible trajectories. Most of themethods proposed to address such problems can be classified into two categories: Oneclass includes approaches that oversimplify the problem to derive a formal deriva-tion or proof of optimality but not scalable to large numbers of mobile sensors anddynamic or complex environments. The other class involves approaches that are de-centralized and scalable but heuristic. In this article, we address the problem withboth approaches.

This article considers the use of mobile sensors to improve the area coverage (mon-itoring capability) of a sparse static WSN. The main idea is that mobile sensors willcollaborate with the static sensors in order to sample the uncovered regions of thesensing field. The main objective of this work is to determine the best (near optimal)path that the mobile node (or a group of nodes) should autonomously follow in order toefficiently cover the monitored area and minimize the average event detection delay,assuming there is no a priori information about the covered and uncovered areas of thefield. In general, this is a difficult problem and it is not possible to guarantee optimalsolutions for any arbitrary instance of the problem.

To solve this path planning problem, a receding-horizon algorithm is proposed whereeach mobile sensor aims to visit and search the biggest coverage hole in a neighbor-hood around itself. An interesting question that needs to be addressed is the size ofthe neighborhood. Clearly, that neighborhood cannot be very small since this will leadto myopic strategies where the mobile sensor will search for very small holes ignoringmuch bigger holes that are a little further away. On the other hand, this article showsthat the neighborhood should not be very big either which is a rather counter-intuitiveresult. This result indicates that the mobile sensor should look for a “medium” (largeenough) size coverage hole located in the mobile sensor’s immediate neighborhood andignore the possibly larger holes that are located further away. This strategy is jus-tified because the mobile sensor will waste valuable time traveling towards a biggerhole when it can sample the smaller coverage holes that are located much closer to it.Formulating the problem to determine the optimal neighborhood size is not straight-forward, thus we resort to a surrogate metric that can lead us to the best neighborhoodsize. The main idea is to associate the neighborhood size with the radius that maxi-mizes the variance of vacancy (uncovered region) [Hall 1988; Chiu et al. 2013] in thenew region discovered by the mobile sensor. This approach is motivated by InformationTheory principles and here we associate information with the variance of vacancy. Thejustification behind this approach is that the mobile sensor needs to consider as muchnew information as possible when it will decide where it will go next. The proposed ap-proximation associates that best neighborhood radius with several other parametersused in the path planning method and thus each mobile node can set automatically oron-line the best neighborhood radius.

1.1. ContributionsThe contributions of this article are the following. In the context of mixed WSNs, itshows that it is not optimal to first search the largest coverage hole in the entire field(the most likely place to find a target); rather searching a big enough hole close to thecurrent mobile sensor location can yield faster coverage and faster event detection. Aproof is given to show that maximizing the dynamic coverage is equivalent to mini-


Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks A:3

mizing the event detection delay. Furthermore, the article proposes a surrogate metricthat can be used in order to approximate the best radius of the search neighborhoodwhere the mobile is searching for its target (centroid of the uncovered region) duringits next step. This best radius is approximated by finding the radius such that the vari-ance of vacancy (uncovered region) in the new region discovered by the mobile sensorduring the next step is maximized. Even though the proposed search approach cannotguarantee an optimal solution, the obtained solutions are satisfactory considering thatthe original problem is NP-hard.

1.2. Related WorkMobile sensors are widely studied in sensor networks for coverage improvement [Liuet al. 2013; Erdelj et al. 2013b; Hollinger and Sukhatme 2013; Deshpande et al. 2009;Li and Cassandras 2005; Bartolini et al. 2011; Lambrou and Panayiotou 2006; 2009b].

In the context of static coverage, several redeployment algorithms have been pro-posed that exploit mobility of nodes to achieve a better degree of static coverage [Wanget al. 2009]. These algorithms typically relocate nodes to optimal locations after aninitial deployment, and try to spread nodes in a uniform way so that coverage is max-imized. Algorithms based on the notion of potential fields, virtual forces, voronoi di-agrams and event distribution density functions have been proposed in [Deshpandeet al. 2009; Li and Cassandras 2005; Bartolini et al. 2011; Howard et al. 2002; Zou andChakrabarty 2003]. Another related problem is the space partitioning problem [Cortes2010; Cortes et al. 2004; Pavone et al. 2009] where the robots must autonomously di-vide the environment in order to balance search workload among themselves.

We point out that the underlying idea of our solution strategy is different from theaforementioned papers because in our approach, the mobile nodes are expected to con-tinuously move using the proposed path planning algorithm in order to enhance thedynamic coverage of the sensor network. In [Lambrou and Panayiotou 2009a], we pro-pose a path planning method that enables the collaboration of mobile and static sensornodes to enhance the dynamic coverage of the sensor network. This approach has alsobeen validated and evaluated experimentally using a mixed WSN test-bed with staticand mobile sensor nodes [Lambrou and Panayiotou 2012b]. Finally, in [Lambrou andPanayiotou 2012a; 2013] we have proposed a general framework that incorporates aprobabilistic sensing model and a dynamic speed policy that is applicable for mobilenodes with variable speed, events that may appear and disappear randomly, and sen-sor fields that include obstacles and we have investigated the tradeoffs between areacoverage and energy consumption or information-communication between nodes.

The problem presented in this article is related to sweep coverage problem, in whichmobile robots with finite sensor footprints travel through the environment to improvecoverage. Sweep coverage has recently been studied in [Liu et al. 2005; Brass 2007;Liu et al. 2013] and [Wimalajeewa and Jayaweera 2010], where authors study thedynamic coverage (sweep coverage) and event detection capabilities that result frommobile sensors moving according to the straight-line random mobility model. In thisarticle a coordinated movement is proposed for more efficient coverage. A preliminaryversion of this article has appeared in [Lambrou and Panayiotou 2011b]; this articlesignificantly extends the results of the aforementioned work. In particular, it has beenproved that maximizing the dynamic coverage is equivalent to minimizing the eventdetection delay and simulation results extended to validate the proof. Furthermore,additional simulation results have been included for the case of multiple mobile nodesas well as the case of transient events.

Persistent patrolling or surveillance of certain points of interest (POIs) with equalfrequency using mobile sensors has gained considerable attention recently. In [Erdeljet al. 2013b; Erdelj et al. 2013a] authors utilize a concentric mobile sensor movement


A:4 T.P. Lambrou

scheme for monitoring certain POIs and derive analytical expressions for mobility pa-rameters under this deployment. In [Nigam and Kroo 2008] the optimal control strat-egy for a two POIs problem is given and a heuristic method is proposed for problemswith more than two POIs based on space partitioning. An optimal control formulationfor persistent monitoring in one-dimensional spaces is given in [Zhong and Cassan-dras 2011]. The persistent monitoring problem is also related to robot patrol problems,where robots are required to visit points in the workspace with frequency constraints[Elmaliach et al. 2009]. Monitoring POIs periodically (but not regions as consideredin our paper) is also studied in [Li et al. 2011; Junzhao et al. 2010] where authorsreduce the problem to the Traveling Salesman Problem (TSP) and propose a central-ized segmentation method where each segment is assigned to a single mobile sensor.In [Ghaffarkhah and Mostofi 2012; 2014] the authors propose a MILP approach forthe persistent information collection problem (dynamic coverage of POIs) under fadingcommunication environments in order to minimize the total energy consumption ofmobile agents. We point out that the aforemention related works assumed that the lo-cations of the POIs are known and also do not apply to the problem considered here dueto the large and complex space of possible trajectories as well as due to the complexityof the environment (cluttered environment with large possibly connected uncoveredregions).

Theoretical work on searching for targets in unknown location was initiated by B.Koopman [Koopman 1956] during World War II to find enemy marine vessels for theU.S. Navy. Search theory as we know it today is based on work by Koopman [Koopman1956] and later work by L. Stone [Lawrence 1975] who especially study the movingtarget problem. However, in [Koopman 1956; Lawrence 1975], there is significant focuson how to allocate search effort across the environment instead of finding the bestsearch path to follow. A recent survey on search and pursuit-evasion in mobile roboticsis provided in [Chung et al. 2011].

Finally, we review various methods and control approaches dealing with multi-robotsystems for cooperative search. This task is also referred to as the cooperative surveil-lance problem using a collection of autonomous vehicles moving in a way that maxi-mizes the probability of finding the target(s). Hollinger et al. [Hollinger et al. 2011] pro-posed a distributed multi-vehicle coordination method under limited communicationand several data fusion techniques for merging vehicles’ estimates for the underwatertarget search scenario. In [Singh et al. 2009a] an informative path planning approachfor multi-robot systems is proposed to address the exploration-exploitation tradeoff forthe search and rescue scenario. Polycarpou et al. [Polycarpou et al. 2003] developed ageneral framework for directing a group of unmanned aerial vehicles (UAVs) to coop-eratively search a dynamic and uncertain environment. The search path generationproblem is separated into two parts: the on-line environment modeling process and areal-time path decision process. Along similar lines, a receding horizon approach withdynamic search is proposed in our previous work [Lambrou and Panayiotou 2009a].

The remaining of this paper is organized as follows. Section 2 presents the modelingassumptions as well as the required definitions. Section 3 presents the optimal searchpath problem and presents the optimal solution for the single mobile sensor - twocoverage holes problem. The solution shows that it is optimal to search smaller holeslocated closer to the mobile sensor rather than bigger holes far away. Based on thisidea, Section 4 presents the algorithm used by the mobile sensor in order to decide itspath when the general problem is considered. Section 5 introduces some basic resultsrelating to the coverage processes and presents the surrogate metric used. Section 6presents some simulation results with single and multiple collaborating mobile nodes.Finally the paper ends with the conclusions and future directions.



2. MIXED WSN MODEL AND DEFINITIONS

Fig. 1. Mixed sensor network model.

We consider a sparse1 wireless sensor network with a large number of static sensornodes and few mobile nodes, deployed in a large square area A as shown in Fig. 1. Forthe purposes of this paper we make the following assumptions:

— A set S with S = |S| static sensor nodes are randomly placed in A at positions xi =(xi, yi), i = 1, · · · , S.

— A set M of M = |M| mobile sensor nodes are available and their position after thek-th time step is xi(k) = (xi(k), yi(k)), i = 1, · · · ,M , k = 0, 1, · · · .

— All nodes have a common (known) sensing range rd and common communicationrange rc > rd and can communicate with the gateway (also referred to as sink) usingmulti-hop communication. Also all nodes know their position through a combinationof GPS and localization algorithms.

For notational convenience, we define the set of all sensor nodes N = S ∪ M andin this set the mobile nodes are re-indexed as m = S + 1, · · · , N , where N = S + M .Furthermore, we define the neighborhood of a sensor s to be the set of all sensor nodesthat are one hop away, i.e., the nodes that are located at a distance less than or equalto rc from s. This set is denoted by

Hrc(s) = {j : ∥xs − xj∥ ≤ rc, j ∈ N , j = s} (1)

where ∥ · ∥ denotes the Euclidean norm.The objective of the WSN is to detect a static point event that may occur at a random

position e = (xe, ye) in A. We also assume that the event emits a signal that can bedetected by near by sensors and is of the form

se(x, t) = I (∥x− e∥ ≤ rd) .(u(t− tON

e

)− u

(t− tOFF

e

))(2)

where x ∈ A, I(∥x − e∥ ≤ rd) is the indicator function that takes the value 1 if thecondition ∥x−e∥ ≤ rd is satisfied or 0 otherwise, u(t) is the step function and tON

e , tOFFe

are the times that the event is turned ON and OFF respectively. For the most part ofthis paper, we assume that tON

e = 0 and tOFFe = T where T denotes the simulation end

time, however, we point out that extensions to the case where tONe > 0 and tOFF

e < Tare possible as indicated in the simulation results of Section 6. If the event occursin the coverage area of at least one static sensor it is immediately detected by thenetwork. However, if the event occurs at a point that is not covered by any sensor, it will

1Complete coverage of the region of interest is not feasible either due to random static sensor deployment,failures of some sensors or budget constraints.


A:6 T.P. Lambrou

remain undetected. Thus the objective of all mobile nodes is to sample the uncoveredregions such that an event that has occurred in an uncovered region is detected as fastas possible.

Next, we define the dynamic area coverage C which will serve as an objective functionto be maximized by the mobile sensors. At any instant t, let I(x, t) be an indicatorfunction that takes the value 1 if point x ∈ A has been covered by at least one sensor(static or mobile) in the interval [0, t], and 0 otherwise. In other words, I(x, t) = 1 ifthere exist a sensor s ∈ S such that ∥xs − x∥ ≤ rd or if a sensor s ∈ M has passed froma point such that x was covered, i.e., ∥xs(k) − x∥ ≤ rd, where k · δt ≤ t and δt is thesampling interval. Thus, the coverage achieved by the network at t is given by

C(t) =1

A

∫AI(x, t)dx.

Assuming that the event can occur at any point of the field with equal probability, C(t)also defines the probability P (t) that a static event will be detected by at least onesensor node somewhere in the time interval [0, t], where t ≤ T and T defines the timethat is needed by mobile sensors to achieve full coverage of the uncovered regions2.

Uncovered Region is the set of points not covered by any sensor, i.e.,

U(t) = {x : I(x, t) = 0} (3)

This region may consist of one or more connected subsets. Each connected subset isreferred to as a coverage hole.

Fig. 2 illustrates the covered (see Fig. 2(a)) and uncovered regions (see Fig. 2(b)) is asensor field. The uncovered region is the complement of the covered region. If an eventoccurs within the sensing range of a sensor node, it can be detected immediately. If anevent occurs in an uncovered region then it is possible to be detected by mobile sensorswith some delay. The aim of this paper is to plan the path of the mobile nodes in orderto search the uncovered regions such that events are detected in minimum time.

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

X [m]

Y [

m]

(a) The covered regions in the sensor field

X [m]

Y [

m]

(b) The coverage holes in the sensor field de-tected using image processing techniques

Fig. 2. Covered and uncovered regions in a WSN of 400 randomly distributed stationary sensors with sens-ing radius rd = 5m.

2We consider applications that do not require or cannot afford simultaneous coverage of all locations butwant to cover the deployed region within a certain time interval T .



As mobile sensors move, they cover new areas, thus a reasonable objective functionthat needs to be maximized by the mobile sensors is denoted as dynamic area coverage3

and is defined by

C(T ) =∫ T

0

C(t)dt (4)

At this point, it is worth pointing out that maximizing (4) is equivalent to minimizingthe expected detection time E [τ ] of a static event that has occurred at one of the cov-erage holes, where τ is a random variable that denotes the time that the event is firstdetected by a mobile node. This result is formalized in the following lemma.

LEMMA 2.1. Given an event has occurred, and assuming that the event location isuniformly distributed over the entire area A, then

C(T ) ≈ T − E [τ ]

K

where K is a positive constant and T is the time required to achieve full area coverage.

PROOF. Using the result from [Washburn 2002; Washburn and Kress 2009] the ex-pected detection time is given by

E [τ ] =

∫ T

0

exp

(−∫ t

0

C(s)ds

)dt (5)

where C(t) is the rate of coverage and T is the time required to achieve full areacoverage. From the definition of coverage,∫ t

0

C(s)ds = C(t)− C(0)

substituting in (5), E [τ ] is given by

E [τ ] =

∫ T

0

e(−C(t)+C(0))dt

= eC(0)

∫ T

0

e−C(t)dt

Next we make the following approximation

e−C(t) = 1− C(t) + o(C2(t)) ≈ 1− C(t)

Note that 0 ≤ C(t) ≤ 1 thus higher order terms are becoming smaller and smaller.Therefore, substituting E [τ ] is given by

E [τ ] ≈ eC(0)

∫ T

0

(1− C(t)) dt

= eC(0)(T − C(T ))= K(T − C(T ))

3For a given T , C(T ) maximizes when the best trajectories are followed by mobile sensors, i.e. trajectoriesthat yield the best rate of coverage C(t) over time.


A:8 T.P. Lambrou

3. OPTIMAL SEARCH STRATEGYIn this section we consider the problem of finding the optimal path that maximizes thedynamic area coverage C(T ) or minimizes the expected detection time E [τ ] of an eventthat has occurred in either of h uncovered locations (coverage holes).

3.1. NP-completeness of the Optimal Search Path ProblemIn principle, finding the optimal path when h coverage holes are possible and whenM mobiles are available is a problem that is more complex compared to the travelingsalesman problem (TSP) or the multiple TSP (mTSP) [Bektas 2006] which are widelyknown to be NP-complete problems. In the TSP the objective is to find the minimumdistance tour through a set of h cities, visiting each city exactly once and returning tothe starting city. The TSP is an NP-hard problem in combinatorial optimization, theoptimal tour can be found using exhaustive search on (h − 1)! paths for h cities whichis computationally intractable (takes O(h!) running time) even for a small number ofcities. Dynamic programming yields a much faster solution, though not a polynomialone (takes O(h22h) running time).

In the following, it is shown that the Optimal Search Path Problem (OSPP) as de-fined above is NP-complete. This implies that we can not hope to find an optimal solu-tion of an arbitrary instance of the problem in reasonable time (intractable) but moreefficient search approaches are needed. Furthermore, it is desirable to determine solu-tion strategies that are distributed in nature, thus each mobile can determine its pathby itself utilizing only information that is locally available since such strategies willsignificantly reduce the communication overhead and can easily adapt to changes inthe environment (node failures).

LEMMA 3.1. The Optimal Search Path Problem (OSPP) is NP-complete

PROOF. We first show that OSPP ∈ NP. Given an instance of the problem, we useas a certificate the sequence of h coverage holes searched in the path. The verifica-tion algorithm checks that this sequence contains all holes (each hole exactly once),sums up the costs (euclidian distances for inter-hole traveling and hole searching),and checks whether the sum is the minimum. This process can certainly be done inpolynomial time.

To prove that OSPP is NP-hard, we describe a reduction [Cormen et al. 2001] froman arbitrary instance of a well-known NP-hard problem, namely the Euclidean pathTSP (EpTSP ) [Papadimitriou 1977], to a special instance of OSPP . Given an EpTSPinstance, (h, dij), where h is the number of cities and dij denotes the matrix with inter-city distances, we choose the following parameters of OSPP such that the achievedoptimal solution corresponds to that of the EpTSP :

(1) Number of mobile nodes M(2) Start and finish depots for all mobile nodes(3) Stationary node deployment(4) Areas Ah of the coverage holes to be searched(5) Sensor node detection range rd

We choose that M = 1 and this single mobile node is initially located in an arbitrarycoverage hole (note that there is not a requirement of returning to the starting city inEpTSP although such a requirement does not change the computational complexity ofthe ordinary TSP [Papadimitriou 1977]), the stationary node deployment is such thatenables a definite number of h coverage holes and the area of each Ai, i = 1, · · · , h isset to Ai = 0 and finally rd = 0. Thus the optimal path of the mobile node is the paththat visits all isolated uncovered points (city locations) with the minimum distance



cost. Due to these choices, the optimal solution of this specially designed instance ofthe OSPP will coincide with the optimal solution of the EpTSP . This completes theproof.

Finding the optimal solution to the problem by using technics as branch and boundor dynamic programming is still not computationally tractable as the algorithms areintended to run on tiny microcontrollers of sensor nodes. Therefore, knowing thatOSPP is NP-complete, we can not hope to solve all problem instances to optimality inreasonable time (as h is usually large) but we must adopt a heuristic solution methodthat provides a close to optimal solution.

3.2. Optimal Path for the 2 Hole ProblemIn this section we investigate the optimal path that a mobile sensor should take ifthere are only two well defined coverage holes in order to gain some insight on thecoverage problem in the context of mixed WSNs. Intuitively, assuming that an eventcan occur at any point in A with equal probability, one should start searching the areawhere it is more likely to find the target, i.e. one would expect that it is optimal tostart searching from the biggest coverage hole of the field. However, what our analysisin this section shows is that this is not the optimal strategy simply because valuabletime is lost until the mobile node reaches the biggest coverage hole of the field. Fur-thermore, this example shows that in an optimal path, the mobile sensor should visit“big enough” holes in an area around it before covering the biggest hole. This sectionaims to present a simple scenario which can be solved optimally and shows that byusing simple greedy strategies (going to the nearer or bigger hole first) are not goodheuristic solutions. Therefore this section presents the trade-off between small andlarge holes that motivates the heuristic path planning approach presented in the nextsection.

x

y

Cb

Ab

Cs

As

O

db

ds

dsb

dsb

x

y

θ

Fig. 3. Problem geometry.

Assume that the field has only two coverage holes with areas Ab and As (Ab ≥ As)and centroids, Cb and Cs respectively (see Fig. 3). For simplicity, it is also assumedthat there is no overlap between the two holes. A mobile node is initially placed atposition O at distance db = ∥Cb −O∥ from hole Ab and at distance ds = ∥Cs −O∥ from


A:10 T.P. Lambrou

hole As. The distance between the two coverage holes is indicated by dsb = ∥Cs −Cb∥.The objective of the mobile sensor is to maximize C(T ) given by eq. (4) and T is sometime instant such that in all of the paths considered, the mobile sensors achieve fullcoverage. Given that there are only two holes, the mobile sensor has only two options.

t

C(t)

C(0)

C(0) +As

As +Ab

C(0) +Ab

As +Ab

1

ts1 ts2 ts3 ts4tb1 tb2 tb3 tb4

W1

W2

W3

Fig. 4. Coverage over time.

First go to Ab, search Ab and then go to As or first go to As, search As and then goto Ab. Fig. 4 shows C(t) (where C(0) = 1 − (Ab + As)/A is the coverage of the staticWSN) under the two different paths thus C(T ) for each path is the area under thecorresponding curve from 0 until T ≥ tb4. As shown in this figure, when the mobilesensor travels over covered regions C(t) = dC(t)

dt = 0, meaning that coverage is notimproved (no new information is obtained) and thus the flat parts in Fig. 4, whilewhen it moves in coverage holes the coverage improvement has a constant rate of

C(t) =2rdv

(As +Ab)

This is because in an interval of length δt the mobile node will move a distance vδt thusits coverage disc will be shifted by vδt and thus the change in the area covered is givenby the “parallelogram” with sides vδt and the diameter of the node’s coverage disc 2rd.The denominator (As +Ab) is a normalization factor since the maximum coverage is 1.To obtain C we simply divide the change in coverage by δt and take the limit as δt goesto zero4.

When the mobile sensor follows the path from O → Cs → Cb, Csb(T ) is given by

Csb(T ) =1

2

As

2rdv

As

As +Ab+

dsbv

As

As +Ab+

Ab

2rdv

As

As +Ab+

1

2

Ab

2rdv

Ab

As +Ab+

db − dsv

(6)

These terms correspond to the area under the curve of Fig. 4 during different intervals.Specifically, the first two terms correspond to the area during the intervals [ts1, ts2] and[ts2, ts3] respectively. The next two terms correspond to the area during [ts3, ts4] while

4Fig. 4 implies that the mobile sensor does not start sampling until it reaches the centroid of the hole andthat the shape and size of the hole are such that constant C(t) is always applicable, however, we point outthat these simplifications do not significantly affect the final result.



the last term is the area of the last interval [ts4, tb4]. Similarly, if the mobile sensorfollows the path from O → Cb → Cs, Cbs(T ) is given by

Cbs(T ) =1

2

Ab

2rdv

Ab

As +Ab+

dsbv

Ab

As +Ab+

As

2rdv

Ab

As +Ab+

1

2

As

2rdv

As

As +Ab(7)

Comparing (6) and (7) or simply observing Fig. 4, the decision of the mobile sensoris to follow the path that maximizes C(T ) which is equivalent to comparing the threeareas W1, W2 and W3 in Fig. 4. Thus

Csb(T ) ≶ Cbs(T ) ⇔ W1 +W3 ≶ W2 (8)

which in turn, after some algebraic manipulations, is equivalent to

Csb(T ) ≶ Cbs(T ) ⇔ dsb (Ab −As) ≷ (db − ds) (Ab +As) (9)

Next, we consider the following cases:C1 {Ab = As}: The decision problem Csb(T ) ≶ Cbs(T ) reduces to ds ≷ db, i.e., the

mobile sensor should go to its nearest coverage hole first. The proof follows easily bysubstituting Ab = As in (9).

C2 {db = ds}: The decision problem Csb(T ) ≶ Cbs(T ) reduces to Ab ≷ As, i.e., themobile sensor should go to the biggest hole Ab first (since by assumption Ab ≥ As). Theproof again follows easily by substituting db = ds in (9).

C3 {Ab > As and db < ds}: The decision is to always go to the biggest hole which isalso located nearer to the mobile sensor. The proof follows by comparing the terms of(9).

C4 {Ab > As and db > ds}: The decision depends on the distance (db) and area ratio(ϱ = Ab/As) of the bigger hole with respect to the smaller one. Specifically, if the smallerhole is located inside an “egg shaped” area then the decision is to search the smallerhole first, otherwise, it is better to search the larger hole first. The proof follows bysolving (10) defined by the cosines rule of the triangle in Fig. 3. Using the cosines rulewe know that

dsb2 = ds

2 + db2 − 2dsdb cos(θ). (10)

Also, using some algebra, one can rewrite (9) such that the mobile sensor should visitthe smaller hole first. Thus, the mobile sensor should first visit the smallest hole if

dsb ≤ϱ+ 1

ϱ− 1(db − ds) (11)

where ϱ = Ab/As > 1. Substituting (11) in (10), we get a single equation (12) with oneunknown, ds which denotes the decision boundary that determines which hole will bevisited first. (

ϱ+ 1

ϱ− 1(db − ds)

)2

= ds2 + db

2 − 2dsdb cos(θ) (12)

Therefore, the mobile sensor should search the smaller hole first if its centroid is lo-cated within the egg-shaped region defined by the solution of (12). In polar coordinatesthe valid solution of (12) is eq. (13) where r = ds

r =db

((ϱ+1)2−(ϱ−1)2cos(θ)−

√((ϱ+1)2−(ϱ−1)2cos(θ))

2−(4ϱ)2)

4ϱ

θ = [0, 2π)(13)


A:12 T.P. Lambrou

This holds true when O = (0,0). Given the relative size between the two coverageholes, one can compute ϱ and thus use (13) to draw the region in polar coordinatesystem. The result is egg-shaped region illustrated in Fig. 5. If the centroid of the smallhole is located inside this region then coverage improvement rate is maximized whenfollowing the path O → Cs → Cb. Concluding, the analysis above demonstrates that

x

y

Cb

Ab

Cs

As

O

db

ds

dsb

dsb

x

y

θ

Fig. 5. The Egg-shaped region for ϱ = 3 and O = (0,0). If Cs is located inside the shaded region then amobile sensor should follow the path O → Cs → Cb to maximize coverage over time.

a mobile sensor should not go immediately to the largest hole in the field but it shouldfirst cover smaller holes that are closer to the mobile sensor (areas in the egg shapedregion). Also note that the precise size of the egg region, depends on the relative size ofthe two coverage holes (ϱ). If for example the smaller hole is significantly smaller thanthe larger one (As << Ab), then the egg will be significantly narrower, implying thatthe smaller one should be visited first only if it is exactly in the straight path from thecurrent mobile node position to the big hole. At this point it’s worth noting that thesame result (eq. (13)) is obtained when the analysis considers the problem of findingthe optimal path that minimizes the expected detection time of a static event that hasoccurred at one of the two coverage holes (see [Lambrou and Panayiotou 2011a]).

The problem of finding the optimal path for many non overlapped coverage holesis presented in appendix A. This problem is NP hard and hence the optimal solutionbecomes computationally intractable as the number of coverage holes increases. More-over, in many scenarios it may be difficult to clearly identify two holes (the uncoveredregion may be connected) and as already mentioned there may be more than two holesand more than one mobile sensors which makes it impractical to determine the optimalpaths. Thus, the implementation of such an algorithm is rather difficult, however, thefollowing insight can be obtained from the analysis: “Large enough holes close to themobile sensor should be searched first, before moving towards the biggest holes of thefield”. A simple heuristic to approximate this insight is by searching for the biggest



coverage hole in a neighborhood around the mobile sensor. By doing so, the mobilesensor avoids covering very small holes and in addition it covers the largest hole in itsarea which, in most cases, it will constitute a “large enough” hole. This efficient anddistributed heuristic search strategy which provides fast and satisfactory solutions forany arbitrary problem instance is presented in sequel.

4. HEURISTIC SEARCH STRATEGYIn this section we present an efficient path planning algorithm that can be used by mo-bile node(s) in order to improve the dynamic area coverage of sparse WSN. The mainidea of the algorithm is that the mobile sensor should aim to search large enough holesin its neighborhood and avoid points in its path that are sufficiently covered by othernodes. An important characteristic of the algorithm is that it is dynamic in the sensethat each mobile sensor determines its path at every step (using a receding horizonapproach) thus it is possible to take into account possible changes in the environment,e.g., recent failures of some static nodes. Before we proceed, let us present the infor-mation structure that is needed by mobile node(s) in order to run the path-planningalgorithm.

In order to keep track of the coverage state of the sensor field, we discretize theentire area into an X ×X grid. The current state of the sensor field is represented byan X2-matrix Gk, k = 0, 1, · · · , which corresponds to the network’s coverage of everycell of the grid. Initially G0(i, j) = 0 for all i, j. This matrix represents the accuratecoverage state of the sensor field and is updated as the mobile sensors move aroundin the field. At each step k, the updating rule for every element of matrix Gk can beexpressed as

Gk+1(i, j) =

{1, if (i, j) ∈ Drd(xs)f ·Gk(i, j), otherwise (14)

where xs are the coordinates of sensor s ∈ N in the Grid Gk and Drd(xs) is the set ofGrid cells covered by s with detection range rd. In other words, all cells that correspondto areas covered by any sensor (static or mobile) are assumed covered and thus take thevalue 1. For cells that correspond to areas not covered by static sensors, Gk(i, j) = 1 assoon as the area is covered by a mobile node but this value may be discounted as soonas the mobile sensor stops covering the cell area. This is captured by the “forgetting”factor 0 ≤ f ≤ 1 which discounts past observations by mobile nodes. If f = 1 thena cell previously covered is considered as covered in all future steps. This is sufficientwhen static point events are considered and thus the objective is to cover all uncoveredregions once. In the case when transient events are considered, e.g. when tON

e > 0 andtOFFe < T , then f < 1 should be set to take into account the fact that a covered point not

recently sampled is considered as “less” covered which will make the mobile sensorsrevisit points not recently covered and enabling them to detect transient events.

For the most part, the work of this paper assumes a static event and thus we setf = 1. Therefore, given the Grid Gk, the dynamic area coverage can be computed as

C(k) = δt

k∑κ=1

C(κ) =δt

X2

k∑κ=1

X∑i=1

X∑j=1

Gκ (i, j) (15)

where δt is the sampling interval. Note that Gk represents the accurate state of thefield which is used only for performance evaluation purposes but it is generally un-known.

Each mobile sensor has an estimate of Gk stored in matrix Pmk , m ∈ M where it keeps

the state of the field. Ideally Pmk should remain Pm

k = Gk at all times k, however, in adynamic environment where several sensors move, fail or more sensors are added, it is


A:14 T.P. Lambrou

impossible to guarantee that Pmk = Gk at all times. However, we emphasize, that the

proposed algorithm, that will run by a mobile sensor located at some position xm(k),computes its path based only on local information, i.e., information in the submatrixof Pm

k that corresponds to the cells Drc(xm(k)), and thus, it is sufficient to have accu-rate information only for the Drc(xm(k)) submatrix. This is easily attainable since therequired information can be obtained from the mobile sensor’s one-hop neighbors.

4.1. Path Planning AlgorithmThe path planning algorithm is based on Receding-Horizon approach where each mo-bile sensor computes its path on-line using only local information. The mobile sen-sor’s controller evaluates at each step the cost of moving to a finite set of candidatepositions and moves to the one that minimizes an overall cost (local to the mobilesensor) as shown in Fig. 6. Suppose that during the kth step, the mobile node is at

φρ

θ

yyyyi

yyyy1

yyyyν

xxxx(k)

Fig. 6. Evaluation of the mobile node’s next step.

position x(k) and is heading to a direction θ. The next candidate positions are theν ∈ {2n+ 1, ∀n ∈ Z+} points y1, · · · ,yν that are uniformly distributed on the arc thatis ρ meters away from x(k) and are within an angle θ − ϕ and θ + ϕ. The mobile nodeevaluates a cost function J(yi) for all candidate locations (y1, · · · ,yν) and moves to thelocation x(k + 1) = yi∗ = x(k) + ρ.ei(θ+φi∗ ) where i is the imaginary unit and i∗ is theindex that minimizes J(yi).

J(yi∗) = min1≤i≤ν

{J(yi)} (16)

In this model, θ is the direction that the mobile sensor is heading, ρ is the distancethat the mobile sensor can cover in one time step δt, ϕ is the maximum angle that themobile sensor can turn in a single step, and ν is the number of candidate positionsthat are being evaluated for the next step. Note that in this model, mobile nodes aremoving with constant velocity υ.

The objective function J(yi) that each mobile sensor is trying to minimize, is of theform

J(yi) =∑o∈O

woJo(yi) (17)

where O is a set of indexes such that the functions Jo, o ∈ O are normalized costfunctions with 0 ≤ Jo(·) ≤ 1 and are defined to achieve certain objectives. wo are



non-negative constant weights that are used to trade off these objectives. For the pur-poses of this paper, O = {t, s, a, b}, as explained below, but other functions can also beincluded (e.g. a cost function that depends on time or the residual energy of mobilenodes).

In order to maximize the area dynamic coverage over a time interval, the mobilesensors should move towards large uncovered regions in a neighborhood around themand at the same time, they should try to avoid areas that are covered by static sen-sors or have been covered by other mobile nodes. For the purposes of this article, thefollowing normalized functions have been used: Jt(·) which guides the mobile sensortowards the centroid of the largest coverage hole in its neighborhood, Js(·) which pe-nalize positions that are close to regions been covered by other sensors (stationary ormobile), Ja(·) which enable mobile sensors to avoid obstacles and Jb(·) which preventsmobile sensors moving outside the region under monitored. Next, we briefly presentthe formulas of these functions.

Target Cost Function Jt: This function guides the mobile sensor towards the cen-troid position xt (target point) of the largest coverage hole found in a neighborhood rzaround the mobile sensor and is given by

Jt(y) =∥y − xt∥

rz(18)

An efficient way of (approximately) quantifying and computing the centroid positionxt of the largest coverage hole in the neighborhood rz around the mobile sensor isachieved by the Zoom algorithm [Lambrou and Panayiotou 2007; 2009a] which is anefficient algorithm that can run at every step k in order to update the centroid posi-tion xt as the mobile sensor covers the hole. The main idea of the Zoom algorithm isto divide the submatrix of Pm

k that corresponds to the cells Drz (xm(k)) in four equalsegments, and choose the segment with the maximum number of empty cells (i.e. thesegment with the maximum number of cells with Pm

k (i, j) = 0) and repeats until eitherthe segment size is equal to a single cell or until all segments have the same number ofempty cells. In the first case, the hole center position will be the center of the cell. In thesecond case, the hole center position will be the center of the segment during the pre-vious iteration. The algorithm is based on the divide-and-conquer principle and thusis computationally efficient and can run repeatedly even on simple microcontrollers.

An important consideration for the above algorithm is the size of the neighborhoodrz where the mobile sensor needs to search for its target (largest coverage hole). Asindicated by the analysis of the previous section, the mobile sensor should not alwayshead towards the biggest coverage hole in the field since this may not result in thebest coverage performance with respect to the objective in (4). Rather, it should headtowards a big enough hole that is located “close” to itself. Therefore, the objective ofthe mobile sensor should be to look for a big enough (not necessarily the biggest) holethat is located relatively close to it (search for a “local” big enough hole rather thansearch for the biggest “global” one). In a sense, this is equivalent to searching for thebiggest hole in a small enough neighborhood around the mobile sensor. Also note that asmaller rz (rz ≤ rc−rd) is advantageous since it implies that less information is neededfor the coverage hole estimation (less communication and computation overhead).

Neighboring Sensor Cost Function Js: This function pushes the mobile sensoraway from areas covered by other sensors and is given by

Js(y) = maxj∈Hrc (m)

{exp

(− ∥y − xj∥2

r2d

)}(19)


A:16 T.P. Lambrou

where Hrc(m) is the set of all nodes in the communication range rc of the mobile sensorm. The detection range rd quantifies the size of the region around the mobile sensorm to be repelled by its neighbors. This objective forces the mobile sensor to pass fromareas not covered by other sensors. Note that other cost functions for avoiding coveredareas have been investigated, however its function achieves better performance.

Obstacle Avoidance Cost Function Ja: This function prevents the mobile nodefrom hitting obstacles that exist in the environment. The obstacle avoidance cost func-tion Ja is similar to Js and its form is given by

Ja(y) = exp

(−

(ro(y)− ∥y − x(k)∥

)10r10d

)(20)

where rd is the detection range and ro(y) indicates the distance of the obstacle’s bound-ary from the mobile sensor’s current position x(k) and its provided by the on boardrange-finding sensors of the mobile sensor. The information provided by range-findingsensors (that utilize an array or rotating detector) can be combined with each candi-date direction φi, i = 1, · · · , ν to provide the distance to obstacles associated with allcandidate locations yi.

Boundary Cost Function Jb: Finally, this function is used to prevent the mobilesensors from moving outside the field area A and is given by

Jb(y) ={1 if y /∈ A0 otherwise (21)

Note that this function is used along with projection which means that mobile sensorsreturn to the interior of the field whenever they reach to boundaries in a mannersimilar to that of a light wave reflecting on a mirror.

4.2. Asynchronous Distributed Collaboration between Mobile NodesWhen multiple mobile nodes are used, it is desirable to collaborate in order to enhancethe dynamic area coverage performance and avoid duplication of work. Due to the lo-calized nature of the proposed algorithm (it uses only the information within rz), if twomobile sensors are located sufficiently far apart, then they are guaranteed to searchdifferent coverage holes which is advantageous since duplication of work is avoided.However, when the two mobile sensors come sufficiently close to each other, it is verylikely that the information they will use to estimate the next target position will bethe same and as a result they will all estimate the same target location which resultsin coverage overlapping. To avoid this problem we utilize a collaboration protocol thatenables mobile nodes to exchange some information in order to avoid overlaps.

When two mobile sensors come into communication range rc for the first time (theyare out of communication range at step k − 1 but they are in communication rangeat k) they exchange their cognitive map Pm

k , thus they now know what areas eachone has searched so far. From this point onward, at every step, the mobile sensorsexchange their current locations to update their P i

k matrix as well as their computedtarget locations (i.e., the centroid of the biggest coverage hole in their respective rz ’s) inorder to avoid going towards the same point. Afterwards each mobile sensor i utilizestarget point information xj

t (k) received from its neighboring mobile sensors j = i inthe zoom algorithm [Lambrou and Panayiotou 2009a] to find a target point xi

t(k) thatis different from the target points of its neighboring mobile sensors.

With this simple scheme, the mobile sensors avoid exploring the same areas. Thisscheme has some important benefits. It is distributed (no need for a central controller),it finds spatially separated targets in an asynchronous manner (synchronization is not



needed) and utilizes only local information (the relevant information in the submatrixDrz (xi(k)), which corresponds to the neighborhood rz of the cognitive map P i

k).

4.3. Effectiveness of the Proposed Path Planning AlgorithmAt this point its worth pointing out that alternative path planning approaches likepotential function techniques usually fail to address the problem under considerationas they get stuck in local minima or oscillate between two closest points. Solutionsprovided to overcome the problem of local minima, when planning with potential func-tions, like wave-front planner [Barraquand et al. 1992], and navigation functions [Ri-mon and Koditschek 1992; Loizou and Kyriakopoulos 2008] still fail to address theproblem. Wave-front planner needs to search the entire space for a path each timethe path is updated which is computationally intractable. On the other hand, navi-gation functions are based on assumptions not satisfied in our problem (i.e. obstaclesare circular disks that do not intersect and the configuration space is bounded by asphere or a star space). Moreover other traditional path planning approaches do notsupport multiple robots and collaboration, navigation in dynamic and large environ-ments as well as complete coverage of free space. Instead, the proposed distributedheuristic path planning algorithm can provide coverage paths in an arbitrary randomsensor field under partial knowledge of the environment and can cope with compu-tation complexity (large free spaces, large number of uncovered regions and multiplemobile nodes) of the problem. However, the performance of the algorithm depends onthe size of neighborhood rz, which is a design parameter.

The solution to the single mobile, two-hole problem indicates that rather than firstsearching the areas that are most likely to hide an event, often it is optimal (withrespect to the detection delay) to search areas that are located closer to the currentposition of the mobile even if it is less likely to find an event in them. This can moti-vate a heuristic centralized approach to solve the single mobile - several hole problemas follows: Given that the mobile has global information regarding the coverage holesof the sensor field (i.e. knows the number, the centroid and area of each hole), it candecide whether to go and search the biggest uncover region in the field or the nearestuncovered region. The decision can based on egg-shape region. Once the mobile hassearched the decided coverage hole decides the next hole to search based on its cur-rent position and the remaining holes of the field. However, in the context of large andrandomly deployed WSNs, it is infeasible to have a central controller to solve the prob-lem and thus the proposed solution must be implementable in a distributed fashionand based on local-accurate information. In addition, it is needed to support multi-ple mobile nodes and to be dynamic because coverage holes might change their areasand centroids as some stationary or mobile sensors failed and/or multiple mobiles aresearching the WSN field.

This indicates that the proposed path planning algorithm, though does not explicitlysolve the OSPP , it provides a good heuristic solution and also tackles the more generalcase where several overlapping holes exist and multiple mobile nodes are searching theWSN field. In this approach, each mobile node searches for coverage holes in a smallcircular region of radius rz around the itself instead of the derived egg-shaped region.Therefore it is meaningful to study how the radius rz of the search neighborhood affectsthe event detection performance of the propose algorithm.

5. VACANCYThe performance of the proposed path planning algorithm depends on the size of theneighborhood rz which is used by the mobile sensor. An important question is how bigshould rz be. If rz neighborhood is too small, then the mobile sensor will waste timesearching insignificant holes missing much larger coverage holes. In other words, if rz


A:18 T.P. Lambrou

is too small, then there is a risk that the search strategy of the mobile sensor will be“myopic”, always searching in small insignificant holes and never reaching the largerholes. On the other hand, if the neighborhood is too big, then the mobile sensor willmove straight towards much larger holes avoiding significant holes that are locatedclose to it. In other words, if rz is too big, then the mobile sensor will give more priorityto larger holes that are located far away ignoring large enough holes that are locatedclose to it.

Therefore, there is an optimal neighborhood size. Given the difficulty in directlyfinding the optimal value for rz, in this section our objective is to derive a surrogatefunction that can be used to solve this problem (approximate the best neighborhoodsize rz).

5.1. Preliminaries on Coverage ProcessesIn this section we use the tools from coverage processes [Hall 1988; Stoyan et al. 1995;Bondesson and Fahlen 2003] in order to analyse the coverage holes that are generatedfrom the random deployment of sensors in A. Consider a two-dimensional point processwhere a collection of N random points is thrown in a square area A according to theprobability density f(x) = 1

A . Let the countable collection of randomly distributedpoints be P ≡ {x1,x2, · · · ,xN}. Assume that there exists a disc around each pointof radius r (in our case r = rd, the detection range) thus all points in the union ofall N discs are considered as covered while all non-covered points are considered asvacant. Vacancy is the collection of all vacant points within an arbitrary area R ⊆ Awhich constitutes a random variable with mean and variance that are defined in thesequel [Hall 1984]. Let I(x) be the indicator function of uncovered points such thatI(x) = 1 − I(x) = 1 if x ∈ A is not covered by any disk of radius rd or I(x) = 0otherwise. The vacancy within an arbitrary area R ⊆ A, VR = V (R) is given by

VR = V (R) ≡∫RI(x)dx (22)

and the mean of vacancy (expected uncovered area) is

E(VR) =

∫RE{I(x)}dx =

∫RP (x not covered) dx

=

∫R

(1− a

A

)N

dx = R(1− a

A

)N (23)

where p = aA is the probability that a point x ∈ A is covered by a disk of area a = πr2d

and (1− p)N is the probability that the point x is not covered by any of the N disks

(sensor positions are independent). Also R is the area of R.The variance of vacancy is

V ar(VR) = E(V 2R)− (E(VR))

2 (24)

where the mean square of vacancy is

E(V 2R) =

∫ ∫R2

E{I(x)I(y)}dxdy

=

∫ ∫R2

P (x,y both not covered)dxdy(25)

Thus, V ar(VR) can be computed by performing a numerical integration of the proba-bility P (x,y both not covered) (see [Hall 1988; Kendall and Moran 1963]).



Let the density λ ≡ NA of points per unit area of A converges to a constant value as

A increases. Hence, for N large and aA small(

1− a

A

)N

≈(1− λa

N

)N

≈ e−λa

Thus by (23) the mean of vacancy in a region R ⊆ A is

E(VR) ≈ Re−λa (26)

An approximation of the variance of vacancy in a subregion R ⊆ A with area R isderived in [Hall 1984] and is given by

V ar(VR) ≈ Rae−2λa

(8

∫ 1

0

x(eλ

aπB(x,1) − 1

)dx−Raλ2

)(27)

where B(x, r) is the intersection area of two disks with radius r and which are centered2x apart. This area is given by

B(x, r) =

{4r2

∫ 1

x/r

√1− y2dy if 0 ≤ x ≤ r

0 if x > r(28)

Hence B(x, 1) = 2 arccos (x)− 2x√1− x2. Even though (27) cannot be computed analyt-

ically, it can be computed numerically5.Let

Q(λ, rd) =

∫ 1

0

x(eλr

2dB(x,1) − 1

)dx (29)

independent of R, then the V ar(VR) can be written as

V ar(VR) ≈ Rπr2de−2πr2dλ

(8Q(λ, rd)−Rλ2πr2d

)(30)

which is a second order polynomial in R with a maximum at

R∗ =4Q(λ, rd)

πλ2r2d(31)

5.2. An Approximation of the Best Neighborhood r∗z

Next, we use the optimal area size R∗ in order to determine the best neighborhood sizer∗z that the mobile node should use in order to determine the biggest coverage hole tovisit next. Recall that the conjecture is that the neighborhood size rz should be largeenough such that the new information considered by the mobile sensor in making thisdecision is maximized. Assuming that at time k the mobile sensor is at position x(k),then it should search for the biggest hole in a circular area R1 with radius rz. Duringthe next step, the mobile sensor will move to a new location x(k+1) = x(k)+ρ, ρ ∈ R2,where the region that the mobile sensor will search for a coverage hole will be R2.Thus, the new information that the mobile sensor will consider from one step to thenext is ∆R = R2 \ R1 = πrz

2 − B(ρ/2, rz) (i.e Rc1 ∩ R2). This new region is illustrated

by region ∆R in Fig. 7.The objective then is to choose the size of the areas R1 and R2 (the radius rz) such

that the variance of vacancy in ∆R is maximized. As the variance of vacancy in ∆R

5Note that in order to avoid the edge effects, the above results assume that the square region A is a quadratictorus, i.e., when a disk protrudes out of one side of the region it re-enters from the opposite side. Also, notethat the approximation in (27) holds true under the assumption that aN converges to a constant value α(0 < α < ∞) as N → ∞ and a → 0. The proofs are provided in [Hall 1984] (see Case B)


A:20 T.P. Lambrou

Fig. 7. Illustration of the new region ∆R discovered by the mobile sensor at each step k.

is maximized between two consecutive steps, the mobile sensor can exploit, on average,the “maximum” difference in vacancy at each step k in order to take, on average, theoptimal best local decision when selecting its next position. In other words, when thevariance of vacancy in ∆R is maximized, the mobile sensor is able to have knowledgeof the largest vacant regions in its neighborhood (compared to the average vacancy)and hence, it takes a better decision in order to navigate towards that areas. The newregion ∆R depends on the current position of the mobile sensor and the next candidateposition. This means that by maximizing the variance of vacancy in the new region ∆R(between two consecutive steps) one also maximizes the amount of new information thatis used by the mobile sensor to decide its next target.

Given the result of (31), the best (optimal) radius r∗z is the solution to the equation

∆R(rz) = πrz2 − B(ρ/2, rz) =

4Q(λ, rd)

πλ2r2d(32)

where ∆R(rz) is the area of ∆R.

LEMMA 5.1. The solution to (32) is approximated by

r∗z ≈ 64Q2(λ, rd) + π2(ρλrd)4

32πρλ2r2dQ(λ, rd)(33)

where ρ = ∥ρ∥ is the distance travelled by the mobile sensor in one step.

PROOF. Assuming the mobile sensor has moved a distance ρ, the area of ∆R is givenby

∆R(rz) = πrz2 − B(ρ/2, rz)

= πr2z − 2r2z arccos(ρ

2rz) + ρ

2

√4r2z − ρ2

Thus, using (32), r∗z is the solution of the

πr2z − 2r2z arccos(ρ

2rz) +

ρ

2

√4r2z − ρ2 =

4Q(λ, rd)

πλ2r2d

This equation is difficult to solved due to the arccos term. Using a Taylor series expan-sion, one can approximate

arccos(ρ

2rz) ≈ π

2− ρ

2rz



Therefore, ∆R(rz) is approximated by

∆R(rz) ≈ρ

2

(2rz +

√4r2z − ρ2

)which is substituted in (32) and as a result, r∗z is the solution to

ρ

2

(2rz +

√4r2z − ρ2

)− 4Q(λ, rd)

πλ2r2d= 0

which, after some algebraic manipulations reduces to the lemma result.

Therefore, the surrogate metric proposed to approximate the best (near-optimal)searching neighborhood radius rz utilizes the information about the statistical prop-erties of the uncovered areas as well as several other parameters used in the pathplanning method.

6. SIMULATION RESULTSIn this section we present some numerical evaluations in conjunction with MonteCarlo simulation outcomes that support the main result of this paper, i.e., that thebest searching neighborhood radius rz is given by Lemma 5.1.

Unless otherwise stated, all experiments refer to a square sensor field of area A =40000m2. A set of N = 200 sensors are deployed where the coordinates are generatedaccording to a uniform distribution. The detection radius of all sensors is rd = 5mand the communication range rc = rz + rd. The weights are set to wt = ws = 0.5,wa = wb = 1 and the mobile sensor maneuverability parameters are set to ρ = 2.5mand ϕ = 35◦ while for every decision ν = 10 candidate next positions are considered.All simulations are performed in MATLAB and the outcomes are the averages of 100independent random deployments. We point out that in all experiments the simulationtime was long enough such that almost full coverage was achieved.

Before we proceed, we present some representative scenarios of the movement ofmobile sensors to illustrate the behavior of the proposed path planning algorithm.

In the first simulation, the path of a single mobile sensor is illustrated when navi-gating in an area where 300 randomly deployed stationary sensors and three obstacleswith different shapes exist. In this simulation, ρ = 2m and rz = 20m and all otherparameters remain the same as stated above. As shown in Fig. 8 the mobile sensornavigates through the sensor field, sampling points that are not adequately coveredby the stationary sensors and avoid collisions with the obstacles at the same time. Asseen from the path followed, there is collaboration between the mobile and stationarysensors in the sense that the mobile has found a path that is least covered by the sta-tionary sensors without colliding to obstacles. In the second simulation we considereda team of five mobile nodes navigating in a sensor field with 200 randomly deployedstationary sensors where ρ = 2m and rz = 25m. All other parameter are the same asdescribed previously. Fig. 9 shows how the five mobile nodes navigate collaborativelythrough the field, sampling points that are not adequately covered by the stationarysensors. As seen from the paths followed, there is collaboration between mobile andstationary sensors in the sense that the mobiles have found five different paths thatare least covered by the stationary sensors. Also notice how the five mobiles collaborateto select different targets at the beginning of their motion.

In the next simulations, we present some numerical results that strongly supportthe main contribution of this work, i.e.that the best searching neighborhood radius rzis given by Lemma 5.1. All simulations performed in MATLAB and the outcomes arethe averages of 100 independent random deployments.


A:22 T.P. Lambrou

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

X [m]

Y [

m]

Fig. 8. Dynamic path planning using M = 1 mobile node.

In the following simulation experiment we investigate the effect of the sensor de-tection range rd on the optimal neighborhood size rz. Using Lemma 5.1, the optimalneighborhood size for different rd is presented in Table I.

Table I. The optimal search neighborhoodr∗z for different rd values

rd N ρ rz∗ V ar(V∆R)

2 200 2.5 20.3 35.95 200 2.5 21.9 8478 200 2.5 25.7 2232.1

10 200 2.5 30.1 2417.6

As shown in Table I as the detection radius rd increases, the r∗z radius, whereV ar(V∆R) is maximized, also increases but remains small compared to the field size(e.g. 200m) which means that the best neighborhood should remain relatively small.This is reasonable because as the sensing radius of each sensor increases (and giventhat the number of sensors is fixed N = 200) it is possible to generate deploymentswith higher variation in the achieved coverage.

Fig. 10 presents the average dynamic coverage C(k) achieved by a single mobile nodeafter k = 2000 time steps when rd = 5m. The figure indicates that dynamic coverageis maximized when rz = 22m which is what was also predicted by Lemma 5.1 (see



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

X [m]

Y [

m]

Fig. 9. Dynamic path planning using a team of M = 5 mobile nodes.

Table I). Also note that the average event detection time6 is minimized when dynamiccoverage is maximized (rz = 22m) which was also predicted and proved in Lemma 2.1.

In the next simulation experiment we investigate how the best rz value is affectedby the density λ ≡ N

A of the static sensors. First, using Lemma 5.1 we compute theoptimal r∗z as shown in Table II.

Table II. The optimal search neighborhoodr∗z for different N values


5 100 2.5 41.9 1141.35 200 2.5 21.9 8475 300 2.5 15.4 630.45 400 2.5 12.1 470.7

Fig. 10 shows that for N = 200 the best r∗z = 22m which is in agreement with theresults of Table II. Furthermore, Fig. 11 presents the coverage achieved by the pathplanning algorithm when N = 300 sensors are deployed. The maximum coverage isachieved when rz = 15m which is again consistent with the Lemma 5.1 prediction as

6Assuming that each sensor field contains one initially undetected temporally static event, placed at randomposition in the sensor field.


A:24 T.P. Lambrou

0 10 20 30 40 501600

1620

1640

1660

1680

1700

1720

rz (m)

C(T

)(s)

0 10 20 30 40 50550

600

650

700

750

800

850

E(τ)(s)

Fig. 10. The average dynamic coverage and event detection time accomplished by a mobile node whenrd = 5m.

indicated in Table II. Notice again that the average event detection time is minimizedwhen dynamic coverage is maximized (rz = 15m).

The next simulation considers how the best rz value is affected by ρ, the distancethat the mobile sensor can move in one time step. Again, we evaluate the optimalradius r∗z using Lemma 5.1 as shown in Table III.

Table III. The optimal search neighborhoodr∗z for different ρ values


5 200 1 54.8 846.995 200 2.5 21.9 846.995 200 4 13.8 846.985 200 5 11.1 846.97

Fig. 10 shows that the best rz for ρ = 2.5m is about 22m while Fig. 12 indicatesthat for ρ = 4m the best rz is about 15m. Both of these results are consistent withthe Lemma 5.1 predictions shown in Table III. Therefore, when the mobile sensor issearching for targets, once it moves farther (bigger ρ) from its previous position the bestrz value decreases. Notice again that the average event detection time is minimizedwhen dynamic coverage is maximized (rz = 15m) as expected by Lemma 2.1.

Subsequently, under more exhaustive Monte Carlo simulation we obtain Table IVwhich presents a comparison between the optimal rz predicted by Lemma 5.1 to thebest rz obtained through brute force simulation for a large set of parameters. For eachset of parameters we selected a few values for rz and conducted multiple simulations(20) to compute the average dynamic coverage C(k) after k = 500 steps. The table



0 10 20 30 40 501720

1740

1760

1780

1800

rz (m)

C(T

)(s)

0 10 20 30 40 50350

400

450

500

550

E(τ)(s)

Fig. 11. The average dynamic coverage and event detection time accomplished by a mobile node whenN = 300.

shows the value of rz (among the ones used in the simulation7) with the maximumperformance. The last column of the table presents the percentage difference betweenthe results of the simulation and Lemma 5.1. As indicated by the results of the Table,the surrogate function can predict the best rz fairly accurately (at least for most of thescenarios investigated). Note that the difference between the two methods is generallybelow 10-15%. We point out that this difference may be the result of several factorslike the finite set of rz ’s and the discretization used in the simulation or the boundaryeffects. These effects are particularly pronounced for very small and very large valuesof rz. For very small rz the simulation results are affected by the discretization whilefor large rz the results are affected by the boundary effects.

In the previous simulations we have investigated the single mobile sensor case, how-ever we point out that the approximation method for obtaining r∗z also remains validfor the case of multiple mobile sensors given that the coverage process is mainly gov-erned by the initial distribution of stationary nodes (e.g. when the number of mobilesensors as well as their coverage rate are small enough). Figures 13 and 14 present theaverage area coverage C(k) (as a function of time) and the dynamic area coverage C(k)respectively achieved by the path planning algorithm after k = 1000 time steps whenfive mobile nodes are used and when rd = 5m, N = 400 and ρ = 1m. Figure 14 indi-cates that dynamic area coverage is maximized and event detection time is minimizedwhen rz = 25m among the values investigated. Using the Lemma 5.1 the r∗z ≈ 30m andhence the approximation remains valid.

7Note that these simulations are quite time consuming and it was not possible to extensively search for allvalues of rz .


A:26 T.P. Lambrou

0 10 20 30 40 501840

1860

1880

1900

1920

rz (m)

C(T

)(s)

0 10 20 30 40 50150

200

250

300

350

E(τ)(s)

Fig. 12. The average dynamic coverage and event detection time accomplished by a mobile node whenρ = 4m.

Finally, the last simulation experiment considers the case when the WSN monitorsthe area for transient events. Specifically, we consider scenarios where the events areactivated at random times and have finite duration, significantly shorter than the to-tal simulation time. In such scenarios an event may stay undetected even when itoccurs in areas searched by a mobile sensor because it may become active after themobile sensor has searched the area or it may become inactive before the mobile sen-sor searches the area. Our approach can address such scenarios using the forgettingfactor parameter f when updating their locally stored cognitive maps. At this point, it’sworth mentioning that the approximation method for obtaining r∗z is also valid whenthe forgetting factor is 0 ≤ f < 1 is used and hence the mobile sensor’s objective is toimprove the dynamic coverage rate over time in a small amount of time. This is dueto the fact that when f < 1 the coverage process is mainly governed by the initial dis-tribution of stationary nodes. We again consider 100 sensor fields with 200 stationarysensors and 10 mobile sensors randomly distributed in a 200m × 200m area and foreach scenario we assume 10 undetected dynamic events. We assume that the eventsare activated according to a Poisson process with rate µ1 = 1/200 (i.e. 200 time stepsis the expected interarrival time between consecutive activations of each event) andevents are uniformly distributed in the areas not covered by the static sensor nodes.Each event remains active for a time interval that it is exponentially distributed withrate µ2 = 1/100 (i.e. on average, the lifetime of each event is 100 time steps after itsactivation). In this simulation, the average probability of detection of transient eventsis used as a performance metric and the detection performance of the proposed pathplanning algorithm is evaluated as a function of the neighborhood rz for various for-



Table IV. Comparison between the optimal r∗z approximated byLemma 5.1 and best r∗z obtained by simulations.

Parameters rz∗ PErd N ρ Lemma Approx. Simulation Error

argmax{V ar(V∆R)} argmax{C(k)} %

2 200 1 50,7 45 11,32 200 2,5 20,3 19 6,52 200 4 12,8 14 9,72 300 1 34,1 37 8,62 300 2,5 13,7 14 2,52 300 4 8,6 10 15,82 400 1 25,7 28 8,82 400 2,5 10,3 11 6,42 400 4 6,6 8 21,45 200 1 54,9 49 10,75 200 2,5 22,0 21 4,45 200 4 13,8 15 8,85 300 1 38,4 38 1,15 300 2,5 15,4 15 2,55 300 4 9,7 11 13,35 400 1 30,3 27 10,95 400 2,5 12,1 12 1,25 400 4 7,7 9 16,8

10 200 1 75,3 61 1910 200 2,5 30,1 33 9,510 200 4 18,9 19 0,710 300 1 64,2 56 12,710 300 2,5 25,7 26 1,210 300 4 16,1 18 11,810 400 1 63,5 57 10,210 400 2,5 25,4 27 6,310 400 4 15,9 17 6,7

getting factor f values 8 and also compared with random search and standard searchapproaches using Monte Carlo simulation. In the random search, the next positionof a mobile node is ρ away from the previous one and at random heading directionθ−ϕ ≤ θ ≤ θ+ϕ whereas in the standard search the mobile sensors scan exhaustivelyand repeatedly the entire field using parallel S-shaped patterns. The standard searchis based on the so-called zamboni coverage pattern[Ablavsky and Snorrason 2000].The results are depicted in Fig. 15 where both the single and multiple (M = 10) mobilesensors cases are considered.

As shown in Fig. 15 the proposed algorithm outperforms random search for all fvalues considered and clearly when setting f < 1 the probability of detection is im-proved. In addition, when best rz and f values are selected the proposed algorithmachieves better performance compared to standard search. However, we indicate thatsetting appropriately the forgetting factor f value is a multi-parameter optimizationproblem as its value depends on many other parameters like the parameters µ1 andµ2 of the dynamic events, the rd and rc range as well as the number of mobile nodes.Nevertheless, when the best f value is selected (e.g. f = 0.5 in Fig. 15), the optimal r∗zapproximated by Lemma 5.1 (e.g. r∗z ≈ 22m) archives the best detection performanceof transient events. At this point we also point out that under extensive simulationsthat we could not present here due to space limitations, we reach the following con-clusions regarding the forgetting factor f : a) When f → 1 and rz is too small to catch

8Instead of constant values one can use a function of time k for the forgetting factor f , however this willrequire extensive memory and computation (each element of matrix Pk must have a time-stamp).


A:28 T.P. Lambrou

0 200 400 600 800 1000

80

85

90

95

100

k

C(k

)(%

)

rz= 10m

rz= 25m

rz= 50m

rz= 100m

rz= 200m

Fig. 13. Average coverage accomplished over 100 sensor fields by five mobile nodes for different rz valueswhen rd = 5m, N = 400 and ρ = 1m.

0 50 100 150 200900

920

940

960

980

rz (m)

C(T

)(s)

0 50 100 150 20050

100

150

200

250

E(τ)(s)

Fig. 14. The average dynamic coverage and event detection time accomplished by five mobile nodes fordifferent rz values when rd = 5m, N = 400 and ρ = 1m.



0 10 20 30 40 500

0.2

0.4

0.6

rz (m)

P(k)

f= 1.00f= 0.90f= 0.50standard searchrandom search

(a) When M = 1 mobile sensor is used.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

rz (m)

P(k)

f= 1.00f= 0.90f= 0.50standard searchrandom search

(b) When M = 10 mobile sensors are used.

Fig. 15. The average probability of detection of transient events accomplished by mobile sensors after 1000moving steps for different rz and forgetting factor f values when rd = 5m, N = 200 and ρ = 2.5m.


A:30 T.P. Lambrou

this degradation in the mobile sensor’s Pk mobile sensors will fail to identify regionsnot searched recently and thus once the field is searched they will trapped or moverandomly in some sense, e.g. see f = 1, rz = 10m in Fig. 15. b) For a given constant rz,the communication range rc is also a dominant factor and as rc increases, the averageprobability of detection also increases because it allows the mobile sensors to continu-ously communicate each other and thus maintain an accurate global knowledge of theregions been searched recently. c) The number of mobiles sensors and event character-istics indicate how ”smoothly” mobile nodes should forget their Pk map. For instance,when few mobile sensors exist it is desired to forget ”slowly” in order to explore theentire field, whereas, when many mobile sensors are available, its desired to forget”fast” in order to search more locally.

7. CONCLUSIONS AND FUTURE DIRECTIONSThis article addresses the problem of dynamic coverage improvement using mixedWSNs (consisting of static and autonomous mobile sensors). In this context, mobilenodes continuously navigate through the sensor field and sample areas not sufficientlycovered by the static sensors. We show that the optimal strategy for the mobile sen-sor is not always to cover the biggest coverage hole first, rather, it is better to sample“big enough” holes in the area around the mobile sensor before heading towards thebiggest hole. To determine the “big enough” hole, we developed a heuristic that looksfor the biggest coverage hole in a neighborhood around the mobile sensor and used asurrogate metric to determine the best size of the neighborhood that enhances the dy-namic coverage and event detection performance in the mixed WSN. Obtained resultsfrom numerical evaluations of the neighborhood size approximations have been veri-fied by extensive Monte Carlo simulation outcomes of the performance of the proposeddistributed algorithm.

The proposed mixed WSN framework considers applications that do not require orcannot afford simultaneous coverage of all locations but want to cover the deployedregion within a certain time interval. In the future, we plan to further investigate theperformance of the algorithm in scenarios that involve unreliable communication (de-layed or dropped packets) and imprecise sensor measurements. Moreover, we plan toderive bounds on dynamic coverage and event detection delay performance of mixedWSNs under various search strategies of mobile sensors. Finally, the developed collab-orative path-planning method can be extended to solve other types of problems suchas the mobile sink path-planning problem in WSNs or the coverage path-planningproblem for autonomous surface vehicles intended for pollution (e.g. oil) monitoring-cleanup applications.

APPENDICESA. OPTIMAL PATH FOR THE SINGLE MOBILE - H HOLE PROBLEMIn this appendix, we consider the scenario where there are h = 3 coverage holes (seeFig. 16). In such case, one can use the following technique to reduce the problem to theh = 2 case as follows: For each i, i = 1, · · · , 3 consider that the mobile is at Ai and hasalready search the Ai hole, then it decides where to go next using eq. (6) and eq. (7).Thus the expected event detection time for i = 1 can be given by

min (E [T123] ,E [T132]) =(

d1

v + 12

A1

2rdv

)A1

A1+A2+A3+(

min (E [T23] ,E [T32]) +d1

v + A1

2rdv

)A2+A3

A1+A2+A3

(34)

Finally the optimal path should be found by comparing the cases of all i, i = 1, · · · , 3 tofind the path that minimizes the expected detection time. Thought the proposed solu-



x

y

A1

A2

A3

O

d1

d2

d3

d12

d13

d23

Fig. 16. Problem geometry with three coverage holes

tion needs h! − 1 comparisons-evaluations to find the optimal path when h uncoveredholes exist, it easy to tackle the equations using the proposed technique. Note that theOSPP problem can not easy formulated in Dynamic Programming as the TSP prob-lem because the search cost between two holes is not a constant value but depends onthe previous holes that have been already searched.

Next, we present a centralized algorithm for finding the optimal tour for a singlemobile node when h non-overlapping holes exist using exhaustive search. In this al-gorithm we generate all possible permutations (paths) and compute the expected de-tection time associated with each path. The optimal solution is the path that resultsin the minimum average detection delay. The details of the algorithm are listed inAlgorithm 1.

B. LOWER & UPPER BOUNDS FOR THE AREA COVERAGE PERFORMANCEThis appendix presents probabilistic approximations for the lower and upper boundof the area coverage performance of the mixed WSN using the proposed path plan-ning algorithm. Using these analytical bounds, it is possible to determine the requirednumber of mobile or static sensors or the speed of mobile sensors to provide a prede-termined area coverage in a given period of time of the deployment area.

The coverage is defined as the ratio of covered area by the sensor network to thearea of interest A. Thus by using eq. (26) the coverage achieved by the mixed WSN attime step k is given by

C(k) = 1− E(VA(k))

A(35)

The coverage provided by static sensor nodes is given by

Cs = 1− exp

(−S

Aπr2d

)(36)

Based on [Liu et al. 2005], the coverage provided by mobile sensor nodes during a timeinterval [0, k] when they are moving around in the sensing field following a randommobility model is given by

Cm = 1− exp

(−M

A

(πr2d + 2rdvk

))(37)


A:32 T.P. Lambrou

ALGORITHM 1: Optimal Search Path Algorithm (function OSP (O, rd, v, h,C,A))Input: O: initial position of mobile sensor;

rd: detection range of mobile sensor;v: velocity of mobile sensor;h: number of uncovered regions;C: Centroids of uncovered regions;A: Areas of uncovered regions

Output: Optimal Path (ordered set of uncovered regions)/* Find all h! possible permutations (Johnson-Trotter algorithm)*/P(i, j) = Permutations(h); i = 1, ..., h!, j = 1, ..., h;ETP(i,:)(i) = 0; i = 1, ..., h!;/* Create distance matrix*/for i = 1 : h do

for j = 1 : h dod(i, j) = ∥C(i)− C(j)∥;if i == j then

d(i, j) = ∥C(i)− O∥;end

endend/* Find Optimal Path */for each path P(i, :), i = 1, ..., h! do

for each hole P(i, j), j = 1, ..., h dohf = P(i, 1); /* first hole in P(i, :) */if hf == P(i, j) then

t(j) = d(P(i, j),P(i, j)

)/v + (1/2)

(A(P(i, j))/(2rdv)

);

elset(j) = t(j − 1) + (1/2)

(A(P(i, j − 1))/(2rdv)

)+

d(P(i, j − 1),P(i, j)

)/v + (1/2)

(A(P(i, j))/(2rdv)

);

endEt(j) = t(j)

(A(P(i, j))/

∑(A)

);

endETP(i,:)(i) =

∑(Et);

endi∗ = argmini=1,...,h!

{ETP(i,:)(i)

};

Optimal Path= P(i∗, :);

B.1.1. A Lower Bound:. Assuming that there is no collaboration between the stationaryand the mobile sensor nodes in the WSN it turns out that a fraction of A can be in-dependently covered either by a mobile or a static sensor node during a time interval[0, k], thus the area coverage can be given by

Clb = 1− exp

(−Nπr2d

A

). exp

(−2Mrdvk

A

)(38)

B.1.2. An Upper Bound:. Assuming that there is perfect collaboration between the sta-tionary and mobile sensor nodes in the WSN and considering that the paths covered bymobiles may intersect, it turns out that a fraction of A can be only covered by mobilesor static sensor nodes (mutually exclusive events) during a time interval [0, k], thusthe area coverage can be given by

Cub = 2− exp

(−Nπr2d

A

)− exp

(−2Mrdvk

A

)(39)



Fig. 17 shows the coverage performance of the mixed WSN using the proposed pathplanning algorithm compared to the estimated theoretical upper and lower bounds.Monte Carlo simulation results indicate that the proposed approximations successfullybound the performance of the proposed path planning algorithm.

0 50 100 150 200 250 300 350 400 450

50

60

70

80

90

100

k

C(k)(%)

Path Planning AlgorithmAnalytical Upper BoundAnalytical Lower Bound

Fig. 17. The area coverage performance of the proposed path planning algorithm using Monte Carlosimulation among with the lower and upper bounds of the area coverage estimated probabilistically(A = 200× 200, N = S +M = 400,M = 10, rd = 4, v = 1 ).

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewers for their constructive comments that helped usto improve our work.

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Received November 4, 2014; revised ; accepted


Online Appendix to:Optimized Cooperative Dynamic Coverage in MixedSensor NetworksTHEOFANIS P. LAMBROU,University of Cyprus

© YYYY ACM 1550-4859/YYYY/01-ARTA $15.00DOI:http://dx.doi.org/10.1145/0000000.0000000


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