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Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks THEOFANIS P. LAMBROU, University of Cyprus

This article considers the problem of improving the dynamic coverage and event detection time of mixed Wireless Sensor Networks (WSNs). We consider mixed WSNs that consist of sparse static sensor deployments 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 a simplified scenario, the article derives the optimal path strategy for a single mobile sensor to search two non-connected uncovered regions with the minimum average detection delay or with the maximum dynamic coverage. The resulting optimal strategy confirms that it is better to search areas that are less likely to hide 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 article proposes a surrogate method to approximate the best searching neighborhood radius (a design parameter of the path planning algorithm) that optimizes the dynamic coverage and event detection time capabilities of Mixed WSN deployments. Extensive simulation results indicate that this approach can achieve very good results, 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, Event Detection, Path Planning, Distributed Decision Making, Search Strategies ACM Reference Format: Theofanis P. Lambrou, 2014. Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks. ACM Trans. Sensor Netw. V, N, Article A (January YYYY), 35 pages. DOI:http://dx.doi.org/10.1145/0000000.0000000

1. INTRODUCTION

Sensor networks have received considerable attention over the past decade for their potential as a cheap, easily deployed, distributed monitoring tool. Recently, researchers have begun to investigate the use of mobile sensor nodes. Mobile and Mixed Sensor Networks is a natural evolution of sensor networks where sensors can measure spatially and temporally distributed phenomena more efficiently. Emerging application domains 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 grant TΠE/OPIZO/0609 (BE) /06 and co-funded by the Republic of Cyprus and the European Regional Development Fund. Author’s addresses: T.P Lambrou, KIOS Research Center for Intelligent Systems and Networks and the Department of Electrical and Computer Engineering, University of Cyprus, Nicosia, Cyprus. E-mail: [email protected] Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from 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 $15.00 DOI:http://dx.doi.org/10.1145/0000000.0000000 ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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In all of these domains, the goal of mobile sensors is to efficiently navigate through the 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 requires searching over a large and complex space of possible trajectories. Such problems have been shown to be NP-hard [Singh et al. 2009b] depending on the form of the objective function, the size of the environment and the space of possible trajectories. Most of the methods proposed to address such problems can be classified into two categories: One class includes approaches that oversimplify the problem to derive a formal derivation or proof of optimality but not scalable to large numbers of mobile sensors and dynamic or complex environments. The other class involves approaches that are decentralized and scalable but heuristic. In this article, we address the problem with both approaches. This article considers the use of mobile sensors to improve the area coverage (monitoring capability) of a sparse static WSN. The main idea is that mobile sensors will collaborate with the static sensors in order to sample the uncovered regions of the sensing 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 to efficiently 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 the field. In general, this is a difficult problem and it is not possible to guarantee optimal solutions for any arbitrary instance of the problem. To solve this path planning problem, a receding-horizon algorithm is proposed where each mobile sensor aims to visit and search the biggest coverage hole in a neighborhood around itself. An interesting question that needs to be addressed is the size of the neighborhood. Clearly, that neighborhood cannot be very small since this will lead to myopic strategies where the mobile sensor will search for very small holes ignoring much bigger holes that are a little further away. On the other hand, this article shows that the neighborhood should not be very big either which is a rather counter-intuitive result. This result indicates that the mobile sensor should look for a “medium” (large enough) size coverage hole located in the mobile sensor’s immediate neighborhood and ignore the possibly larger holes that are located further away. This strategy is justified because the mobile sensor will waste valuable time traveling towards a bigger hole 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 straightforward, thus we resort to a surrogate metric that can lead us to the best neighborhood size. The main idea is to associate the neighborhood size with the radius that maximizes the variance of vacancy (uncovered region) [Hall 1988; Chiu et al. 2013] in the new region discovered by the mobile sensor. This approach is motivated by Information Theory principles and here we associate information with the variance of vacancy. The justification behind this approach is that the mobile sensor needs to consider as much new information as possible when it will decide where it will go next. The proposed approximation associates that best neighborhood radius with several other parameters used in the path planning method and thus each mobile node can set automatically or on-line the best neighborhood radius. 1.1. Contributions

The contributions of this article are the following. In the context of mixed WSNs, it shows 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 the current mobile sensor location can yield faster coverage and faster event detection. A proof is given to show that maximizing the dynamic coverage is equivalent to miniACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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mizing the event detection delay. Furthermore, the article proposes a surrogate metric that can be used in order to approximate the best radius of the search neighborhood where the mobile is searching for its target (centroid of the uncovered region) during its next step. This best radius is approximated by finding the radius such that the variance of vacancy (uncovered region) in the new region discovered by the mobile sensor during the next step is maximized. Even though the proposed search approach cannot guarantee an optimal solution, the obtained solutions are satisfactory considering that the original problem is NP-hard. 1.2. Related Work

Mobile sensors are widely studied in sensor networks for coverage improvement [Liu et 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 proposed that exploit mobility of nodes to achieve a better degree of static coverage [Wang et al. 2009]. These algorithms typically relocate nodes to optimal locations after an initial deployment, and try to spread nodes in a uniform way so that coverage is maximized. Algorithms based on the notion of potential fields, virtual forces, voronoi diagrams and event distribution density functions have been proposed in [Deshpande et al. 2009; Li and Cassandras 2005; Bartolini et al. 2011; Howard et al. 2002; Zou and Chakrabarty 2003]. Another related problem is the space partitioning problem [Cortes 2010; Cortes et al. 2004; Pavone et al. 2009] where the robots must autonomously divide the environment in order to balance search workload among themselves. We point out that the underlying idea of our solution strategy is different from the aforementioned papers because in our approach, the mobile nodes are expected to continuously move using the proposed path planning algorithm in order to enhance the dynamic coverage of the sensor network. In [Lambrou and Panayiotou 2009a], we propose a path planning method that enables the collaboration of mobile and static sensor nodes to enhance the dynamic coverage of the sensor network. This approach has also been validated and evaluated experimentally using a mixed WSN test-bed with static and mobile sensor nodes [Lambrou and Panayiotou 2012b]. Finally, in [Lambrou and Panayiotou 2012a; 2013] we have proposed a general framework that incorporates a probabilistic sensing model and a dynamic speed policy that is applicable for mobile nodes with variable speed, events that may appear and disappear randomly, and sensor fields that include obstacles and we have investigated the tradeoffs between area coverage and energy consumption or information-communication between nodes. The problem presented in this article is related to sweep coverage problem, in which mobile robots with finite sensor footprints travel through the environment to improve coverage. 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 the dynamic coverage (sweep coverage) and event detection capabilities that result from mobile sensors moving according to the straight-line random mobility model. In this article a coordinated movement is proposed for more efficient coverage. A preliminary version of this article has appeared in [Lambrou and Panayiotou 2011b]; this article significantly extends the results of the aforementioned work. In particular, it has been proved that maximizing the dynamic coverage is equivalent to minimizing the event detection delay and simulation results extended to validate the proof. Furthermore, additional simulation results have been included for the case of multiple mobile nodes as well as the case of transient events. Persistent patrolling or surveillance of certain points of interest (POIs) with equal frequency using mobile sensors has gained considerable attention recently. In [Erdelj et al. 2013b; Erdelj et al. 2013a] authors utilize a concentric mobile sensor movement ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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scheme for monitoring certain POIs and derive analytical expressions for mobility parameters under this deployment. In [Nigam and Kroo 2008] the optimal control strategy for a two POIs problem is given and a heuristic method is proposed for problems with more than two POIs based on space partitioning. An optimal control formulation for persistent monitoring in one-dimensional spaces is given in [Zhong and Cassandras 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 considered in our paper) is also studied in [Li et al. 2011; Junzhao et al. 2010] where authors reduce the problem to the Traveling Salesman Problem (TSP) and propose a centralized segmentation method where each segment is assigned to a single mobile sensor. In [Ghaffarkhah and Mostofi 2012; 2014] the authors propose a MILP approach for the persistent information collection problem (dynamic coverage of POIs) under fading communication environments in order to minimize the total energy consumption of mobile agents. We point out that the aforemention related works assumed that the locations of the POIs are known and also do not apply to the problem considered here due to the large and complex space of possible trajectories as well as due to the complexity of the environment (cluttered environment with large possibly connected uncovered regions). 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 the U.S. Navy. Search theory as we know it today is based on work by Koopman [Koopman 1956] and later work by L. Stone [Lawrence 1975] who especially study the moving target problem. However, in [Koopman 1956; Lawrence 1975], there is significant focus on how to allocate search effort across the environment instead of finding the best search path to follow. A recent survey on search and pursuit-evasion in mobile robotics is provided in [Chung et al. 2011]. Finally, we review various methods and control approaches dealing with multi-robot systems for cooperative search. This task is also referred to as the cooperative surveillance problem using a collection of autonomous vehicles moving in a way that maximizes the probability of finding the target(s). Hollinger et al. [Hollinger et al. 2011] proposed a distributed multi-vehicle coordination method under limited communication and several data fusion techniques for merging vehicles’ estimates for the underwater target search scenario. In [Singh et al. 2009a] an informative path planning approach for multi-robot systems is proposed to address the exploration-exploitation tradeoff for the search and rescue scenario. Polycarpou et al. [Polycarpou et al. 2003] developed a general framework for directing a group of unmanned aerial vehicles (UAVs) to cooperatively search a dynamic and uncertain environment. The search path generation problem is separated into two parts: the on-line environment modeling process and a real-time path decision process. Along similar lines, a receding horizon approach with dynamic search is proposed in our previous work [Lambrou and Panayiotou 2009a]. The remaining of this paper is organized as follows. Section 2 presents the modeling assumptions as well as the required definitions. Section 3 presents the optimal search path problem and presents the optimal solution for the single mobile sensor - two coverage holes problem. The solution shows that it is optimal to search smaller holes located closer to the mobile sensor rather than bigger holes far away. Based on this idea, Section 4 presents the algorithm used by the mobile sensor in order to decide its path when the general problem is considered. Section 5 introduces some basic results relating to the coverage processes and presents the surrogate metric used. Section 6 presents some simulation results with single and multiple collaborating mobile nodes. Finally the paper ends with the conclusions and future directions.

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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 sensor nodes and few mobile nodes, deployed in a large square area A as shown in Fig. 1. For the 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 the k-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 communication range rc > rd and can communicate with the gateway (also referred to as sink) using multi-hop communication. Also all nodes know their position through a combination of GPS and localization algorithms. For notational convenience, we define the set of all sensor nodes N = S ∪ M and in 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 nodes that are one hop away, i.e., the nodes that are located at a distance less than or equal to 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 , y e ) in A. We also assume that the event emits a signal that can be detected by near by sensors and is of the form ( ( ) ( )) F se (x, t) = I (∥x − e∥ ≤ rd ) . u t − tON − u t − tOF (2) e e where x ∈ A, I(∥x − e∥ ≤ rd ) is the indicator function that takes the value 1 if the OF F condition ∥x−e∥ ≤ rd is satisfied or 0 otherwise, u(t) is the step function and tON e , te are the times that the event is turned ON and OFF respectively. For the most part of F this paper, we assume that tON = 0 and tOF = T where T denotes the simulation end e e F time, however, we point out that extensions to the case where tON > 0 and tOF As and db < ds }: The decision is to always go to the biggest hole which is also 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 smaller hole is located inside an “egg shaped” area then the decision is to search the smaller hole first, otherwise, it is better to search the larger hole first. The proof follows by solving (10) defined by the cosines rule of the triangle in Fig. 3. Using the cosines rule we know that dsb 2 = ds 2 + db 2 − 2ds db cos(θ).

(10)

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

ϱ+1 (db − ds ) ϱ−1

(11)

where ϱ = Ab /As > 1. Substituting (11) in (10), we get a single equation (12) with one unknown, ds which denotes the decision boundary that determines which hole will be visited first. ( )2 ϱ+1 (db − ds ) = ds 2 + db 2 − 2ds db cos(θ) (12) ϱ−1 Therefore, the mobile sensor should search the smaller hole first if its centroid is located within the egg-shaped region defined by the solution of (12). In polar coordinates the valid solution of (12) is eq. (13) where r = ds ( ) √ 2 db (ϱ+1)2 −(ϱ−1)2 cos(θ)− ((ϱ+1)2 −(ϱ−1)2 cos(θ)) −(4ϱ)2

r= θ = [0, 2π)

4ϱ

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

y

As Cs

y

dsb

ds db

θ O

x

Ab Cb

x

Fig. 5. The Egg-shaped region for ϱ = 3 and O = (0, 0). If Cs is located inside the shaded region then a mobile 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 should first cover smaller holes that are closer to the mobile sensor (areas in the egg shaped region). Also note that the precise size of the egg region, depends on the relative size of the two coverage holes (ϱ). If for example the smaller hole is significantly smaller than the larger one (As 0 and e F tOF < T , then f < 1 should be set to take into account the fact that a covered point not e recently sampled is considered as “less” covered which will make the mobile sensors revisit 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 set f = 1. Therefore, given the Grid Gk , the dynamic area coverage can be computed as C(k) = δt

k ∑ κ=1

C(κ) =

k X X δt ∑ ∑ ∑ Gκ (i, j) X 2 κ=1 i=1 j=1

(15)

where δt is the sampling interval. Note that Gk represents the accurate state of the field which is used only for performance evaluation purposes but it is generally unknown. Each mobile sensor has an estimate of Gk stored in matrix Pkm , m ∈ M where it keeps the state of the field. Ideally Pkm should remain Pkm = Gk at all times k, however, in a dynamic environment where several sensors move, fail or more sensors are added, it is ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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impossible to guarantee that Pkm = 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 submatrix of Pkm that corresponds to the cells Drc (xm (k)), and thus, it is sufficient to have accurate information only for the Drc (xm (k)) submatrix. This is easily attainable since the required information can be obtained from the mobile sensor’s one-hop neighbors. 4.1. Path Planning Algorithm

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

y1

yi yν

φ ρ

θ

x(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 the ν ∈ {2n + 1, ∀n ∈ Z+ } points y1 , · · · , yν that are uniformly distributed on the arc that is ρ meters away from x(k) and are within an angle θ − ϕ and θ + ϕ. The mobile node evaluates a cost function J(yi ) for all candidate locations (y1 , · · · , yν ) and moves to the location x(k + 1) = yi∗ = x(k) + ρ.ei(θ+φi∗ ) where i is the imaginary unit and i∗ is the index that minimizes J(yi ). J(yi∗ ) = min {J(yi )} 1≤i≤ν

(16)

In this model, θ is the direction that the mobile sensor is heading, ρ is the distance that the mobile sensor can cover in one time step δt, ϕ is the maximum angle that the mobile sensor can turn in a single step, and ν is the number of candidate positions that are being evaluated for the next step. Note that in this model, mobile nodes are moving with constant velocity υ. The objective function J(yi ) that each mobile sensor is trying to minimize, is of the form ∑ J(yi ) = wo Jo (yi ) (17) o∈O

where O is a set of indexes such that the functions Jo , o ∈ O are normalized cost functions with 0 ≤ Jo (·) ≤ 1 and are defined to achieve certain objectives. wo are ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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non-negative constant weights that are used to trade off these objectives. For the purposes of this paper, O = {t, s, a, b}, as explained below, but other functions can also be included (e.g. a cost function that depends on time or the residual energy of mobile nodes). In order to maximize the area dynamic coverage over a time interval, the mobile sensors should move towards large uncovered regions in a neighborhood around them and at the same time, they should try to avoid areas that are covered by static sensors or have been covered by other mobile nodes. For the purposes of this article, the following normalized functions have been used: Jt (·) which guides the mobile sensor towards the centroid of the largest coverage hole in its neighborhood, Js (·) which penalize positions that are close to regions been covered by other sensors (stationary or mobile), Ja (·) which enable mobile sensors to avoid obstacles and Jb (·) which prevents mobile sensors moving outside the region under monitored. Next, we briefly present the formulas of these functions. Target Cost Function Jt : This function guides the mobile sensor towards the centroid position xt (target point) of the largest coverage hole found in a neighborhood rz around the mobile sensor and is given by Jt (y) =

∥y − xt ∥ rz

(18)

An efficient way of (approximately) quantifying and computing the centroid position xt of the largest coverage hole in the neighborhood rz around the mobile sensor is achieved by the Zoom algorithm [Lambrou and Panayiotou 2007; 2009a] which is an efficient algorithm that can run at every step k in order to update the centroid position xt as the mobile sensor covers the hole. The main idea of the Zoom algorithm is to divide the submatrix of Pkm that corresponds to the cells Drz (xm (k)) in four equal segments, and choose the segment with the maximum number of empty cells (i.e. the segment with the maximum number of cells with Pkm (i, j) = 0) and repeats until either the segment size is equal to a single cell or until all segments have the same number of empty cells. In the first case, the hole center position will be the center of the cell. In the second case, the hole center position will be the center of the segment during the previous iteration. The algorithm is based on the divide-and-conquer principle and thus is computationally efficient and can run repeatedly even on simple microcontrollers. An important consideration for the above algorithm is the size of the neighborhood rz where the mobile sensor needs to search for its target (largest coverage hole). As indicated by the analysis of the previous section, the mobile sensor should not always head towards the biggest coverage hole in the field since this may not result in the best coverage performance with respect to the objective in (4). Rather, it should head towards a big enough hole that is located “close” to itself. Therefore, the objective of the mobile sensor should be to look for a big enough (not necessarily the biggest) hole that is located relatively close to it (search for a “local” big enough hole rather than search for the biggest “global” one). In a sense, this is equivalent to searching for the biggest hole in a small enough neighborhood around the mobile sensor. Also note that a smaller rz (rz ≤ rc −rd ) is advantageous since it implies that less information is needed for the coverage hole estimation (less communication and computation overhead). Neighboring Sensor Cost Function Js : This function pushes the mobile sensor away from areas covered by other sensors and is given by { Js (y) =

max j∈Hrc (m)

( exp

−

∥y − xj ∥2 rd2

)}

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where Hrc (m) is the set of all nodes in the communication range rc of the mobile sensor m. The detection range rd quantifies the size of the region around the mobile sensor m to be repelled by its neighbors. This objective forces the mobile sensor to pass from areas not covered by other sensors. Note that other cost functions for avoiding covered areas have been investigated, however its function achieves better performance. Obstacle Avoidance Cost Function Ja : This function prevents the mobile node from hitting obstacles that exist in the environment. The obstacle avoidance cost function Ja is similar to Js and its form is given by )10 ) ( ( ro (y) − ∥y − x(k)∥ Ja (y) = exp − (20) rd10 where rd is the detection range and ro (y) indicates the distance of the obstacle’s boundary from the mobile sensor’s current position x(k) and its provided by the on board range-finding sensors of the mobile sensor. The information provided by range-finding sensors (that utilize an array or rotating detector) can be combined with each candidate direction φi , i = 1, · · · , ν to provide the distance to obstacles associated with all candidate locations yi . Boundary Cost Function Jb : Finally, this function is used to prevent the mobile sensors from moving outside the field area A and is given by { 1 if y ∈ /A (21) Jb (y) = 0 otherwise Note that this function is used along with projection which means that mobile sensors return to the interior of the field whenever they reach to boundaries in a manner similar to that of a light wave reflecting on a mirror. 4.2. Asynchronous Distributed Collaboration between Mobile Nodes

When multiple mobile nodes are used, it is desirable to collaborate in order to enhance the dynamic area coverage performance and avoid duplication of work. Due to the localized nature of the proposed algorithm (it uses only the information within rz ), if two mobile sensors are located sufficiently far apart, then they are guaranteed to search different 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 very likely that the information they will use to estimate the next target position will be the same and as a result they will all estimate the same target location which results in coverage overlapping. To avoid this problem we utilize a collaboration protocol that enables mobile nodes to exchange some information in order to avoid overlaps. When two mobile sensors come into communication range rc for the first time (they are out of communication range at step k − 1 but they are in communication range at k) they exchange their cognitive map Pkm , thus they now know what areas each one has searched so far. From this point onward, at every step, the mobile sensors exchange their current locations to update their Pki matrix as well as their computed target locations (i.e., the centroid of the biggest coverage hole in their respective rz ’s) in order to avoid going towards the same point. Afterwards each mobile sensor i utilizes target point information xjt (k) received from its neighboring mobile sensors j ̸= i in the zoom algorithm [Lambrou and Panayiotou 2009a] to find a target point xit (k) that is different from the target points of its neighboring mobile sensors. With this simple scheme, the mobile sensors avoid exploring the same areas. This scheme 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 ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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needed) and utilizes only local information (the relevant information in the submatrix Drz (xi (k)), which corresponds to the neighborhood rz of the cognitive map Pki ). 4.3. Effectiveness of the Proposed Path Planning Algorithm

At this point its worth pointing out that alternative path planning approaches like potential function techniques usually fail to address the problem under consideration as they get stuck in local minima or oscillate between two closest points. Solutions provided to overcome the problem of local minima, when planning with potential functions, like wave-front planner [Barraquand et al. 1992], and navigation functions [Rimon and Koditschek 1992; Loizou and Kyriakopoulos 2008] still fail to address the problem. Wave-front planner needs to search the entire space for a path each time the path is updated which is computationally intractable. On the other hand, navigation functions are based on assumptions not satisfied in our problem (i.e. obstacles are circular disks that do not intersect and the configuration space is bounded by a sphere or a star space). Moreover other traditional path planning approaches do not support multiple robots and collaboration, navigation in dynamic and large environments as well as complete coverage of free space. Instead, the proposed distributed heuristic path planning algorithm can provide coverage paths in an arbitrary random sensor field under partial knowledge of the environment and can cope with computation complexity (large free spaces, large number of uncovered regions and multiple mobile nodes) of the problem. However, the performance of the algorithm depends on the size of neighborhood rz , which is a design parameter. The solution to the single mobile, two-hole problem indicates that rather than first searching the areas that are most likely to hide an event, often it is optimal (with respect to the detection delay) to search areas that are located closer to the current position of the mobile even if it is less likely to find an event in them. This can motivate a heuristic centralized approach to solve the single mobile - several hole problem as follows: Given that the mobile has global information regarding the coverage holes of the sensor field (i.e. knows the number, the centroid and area of each hole), it can decide whether to go and search the biggest uncover region in the field or the nearest uncovered region. The decision can based on egg-shape region. Once the mobile has searched the decided coverage hole decides the next hole to search based on its current position and the remaining holes of the field. However, in the context of large and randomly deployed WSNs, it is infeasible to have a central controller to solve the problem and thus the proposed solution must be implementable in a distributed fashion and based on local-accurate information. In addition, it is needed to support multiple mobile nodes and to be dynamic because coverage holes might change their areas and centroids as some stationary or mobile sensors failed and/or multiple mobiles are searching the WSN field. This indicates that the proposed path planning algorithm, though does not explicitly solve the OSP P , it provides a good heuristic solution and also tackles the more general case where several overlapping holes exist and multiple mobile nodes are searching the WSN field. In this approach, each mobile node searches for coverage holes in a small circular 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 affects the event detection performance of the propose algorithm. 5. VACANCY

The performance of the proposed path planning algorithm depends on the size of the neighborhood rz which is used by the mobile sensor. An important question is how big should rz be. If rz neighborhood is too small, then the mobile sensor will waste time searching insignificant holes missing much larger coverage holes. In other words, if rz ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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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 larger holes. On the other hand, if the neighborhood is too big, then the mobile sensor will move straight towards much larger holes avoiding significant holes that are located close to it. In other words, if rz is too big, then the mobile sensor will give more priority to larger holes that are located far away ignoring large enough holes that are located close to it. Therefore, there is an optimal neighborhood size. Given the difficulty in directly finding the optimal value for rz , in this section our objective is to derive a surrogate function that can be used to solve this problem (approximate the best neighborhood size rz ). 5.1. Preliminaries on Coverage Processes

In 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 generated from the random deployment of sensors in A. Consider a two-dimensional point process where a collection of N random points is thrown in a square area A according to the probability density f (x) = A1 . Let the countable collection of randomly distributed points be P ≡ {x1 , x2 , · · · , xN }. Assume that there exists a disc around each point of radius r (in our case r = rd , the detection range) thus all points in the union of all N discs are considered as covered while all non-covered points are considered as vacant. Vacancy is the collection of all vacant points within an arbitrary area R ⊆ A which constitutes a random variable with mean and variance that are defined in the sequel [Hall 1984]. Let I(x) be the indicator function of uncovered points such that I(x) = 1 − I(x) = 1 if x ∈ A is not covered by any disk of radius rd or I(x) = 0 otherwise. The vacancy within an arbitrary area R ⊆ A, VR = V (R) is given by ∫ VR = V (R) ≡ I(x)dx (22) R

and the mean of vacancy (expected uncovered area) is ∫ ∫ P (x not covered) dx E(VR ) = E{I(x)}dx = R R ∫ ( ) ( N a a )N = 1− dx = R 1 − A A R

(23)

a where p = A is the probability that a point x ∈ A is covered by a disk of area a = πrd2 N and (1 − p) 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 ( ) 2 V ar(VR ) = E VR2 − (E(VR )) (24)

where the mean square of vacancy is ∫ ∫ 2 E(VR ) = E{I(x)I(y)}dxdy ∫ ∫ R2 = P (x, y both not covered)dxdy

(25)

R2

Thus, V ar(VR ) can be computed by performing a numerical integration of the probability P (x, y both not covered) (see [Hall 1988; Kendall and Moran 1963]). ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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Let the density λ ≡ N A of points per unit area of A converges to a constant value as a A increases. Hence, for N large and A small ( )N ( a )N λa 1− ≈ 1− ≈ e−λa A N 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 is derived in [Hall 1984] and is given by ( ∫ 1 ( ) ) a V ar(VR ) ≈ Rae−2λa 8 x eλ π B(x,1) − 1 dx − Raλ2 (27) 0

where B(x, r) is the intersection area of two disks with radius r and which are centered 2x apart. This area is given by { 2∫1 √ 4r x/r 1 − y 2 dy if 0 ≤ x ≤ r B(x, r) = (28) 0 if x > r √ Hence B(x, 1) = 2 arccos (x) − 2x 1 − x2 . Even though (27) cannot be computed analytically, it can be computed numerically5 . Let ∫ 1 ( ) 2 Q(λ, rd ) = x eλrd B(x,1) − 1 dx (29) 0

independent of R, then the V ar(VR ) can be written as ) 2 ( V ar(VR ) ≈ Rπrd2 e−2πrd λ 8Q(λ, rd ) − Rλ2 πrd2

(30)

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

4Q(λ, rd ) πλ2 rd2

(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 size rz∗ that the mobile node should use in order to determine the biggest coverage hole to visit next. Recall that the conjecture is that the neighborhood size rz should be large enough such that the new information considered by the mobile sensor in making this decision 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 . During the 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 the next 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 5 Note that in order to avoid the edge effects, the above results assume that the square region A is a quadratic

torus, i.e., when a disk protrudes out of one side of the region it re-enters from the opposite side. Also, note that 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)

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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, the optimal best local decision when selecting its next position. In other words, when the variance of vacancy in ∆R is maximized, the mobile sensor is able to have knowledge of 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 new region ∆R depends on the current position of the mobile sensor and the next candidate position. 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 that is used by the mobile sensor to decide its next target. Given the result of (31), the best (optimal) radius rz∗ is the solution to the equation ∆R(rz ) = πrz 2 − B(ρ/2, rz ) =

4Q(λ, rd ) πλ2 rd2

(32)

where ∆R(rz ) is the area of ∆R. L EMMA 5.1. The solution to (32) is approximated by rz∗ ≈

64Q2 (λ, rd ) + π 2 (ρλrd )4 32πρλ2 rd2 Q(λ, rd )

(33)

where ρ = ∥ρ∥ is the distance travelled by the mobile sensor in one step. P ROOF. Assuming the mobile sensor has moved a distance ρ, the area of ∆R is given by ∆R(rz ) = πrz 2 − B(ρ/2, rz ) √ = πrz2 − 2rz2 arccos( 2rρz ) + ρ2 4rz2 − ρ2 Thus, using (32), rz∗ is the solution of the πrz2 − 2rz2 arccos(

ρ ρ√ 2 4Q(λ, rd ) )+ 4rz − ρ2 = 2rz 2 πλ2 rd2

This equation is difficult to solved due to the arccos term. Using a Taylor series expansion, one can approximate ρ π ρ arccos( )≈ − 2rz 2 2rz ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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Therefore, ∆R(rz ) is approximated by ∆R(rz ) ≈

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) √ ρ( 2rz + 4rz2 − ρ2 2

which is substituted in (32) and as a result, rz∗ is the solution to ) 4Q(λ, r ) √ ρ( d 2rz + 4rz2 − ρ2 − =0 2 πλ2 rd2 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 properties of the uncovered areas as well as several other parameters used in the path planning method. 6. SIMULATION RESULTS

In this section we present some numerical evaluations in conjunction with Monte Carlo simulation outcomes that support the main result of this paper, i.e., that the best 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 generated according to a uniform distribution. The detection radius of all sensors is rd = 5m and 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 = 2.5m and 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 100 independent random deployments. We point out that in all experiments the simulation time was long enough such that almost full coverage was achieved. Before we proceed, we present some representative scenarios of the movement of mobile 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 navigating in an area where 300 randomly deployed stationary sensors and three obstacles with different shapes exist. In this simulation, ρ = 2m and rz = 20m and all other parameters remain the same as stated above. As shown in Fig. 8 the mobile sensor navigates through the sensor field, sampling points that are not adequately covered by the stationary sensors and avoid collisions with the obstacles at the same time. As seen from the path followed, there is collaboration between the mobile and stationary sensors in the sense that the mobile has found a path that is least covered by the stationary sensors without colliding to obstacles. In the second simulation we considered a team of five mobile nodes navigating in a sensor field with 200 randomly deployed stationary sensors where ρ = 2m and rz = 25m. All other parameter are the same as described previously. Fig. 9 shows how the five mobile nodes navigate collaboratively through the field, sampling points that are not adequately covered by the stationary sensors. As seen from the paths followed, there is collaboration between mobile and stationary sensors in the sense that the mobiles have found five different paths that are least covered by the stationary sensors. Also notice how the five mobiles collaborate to select different targets at the beginning of their motion. In the next simulations, we present some numerical results that strongly support the main contribution of this work, i.e.that the best searching neighborhood radius rz is given by Lemma 5.1. All simulations performed in MATLAB and the outcomes are the averages of 100 independent random deployments. ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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200 180 160

Y [m]

140 120 100 80 60 40 20 0 0

20 40 60 80 100 120 140 160 180 200 X [m] Fig. 8. Dynamic path planning using M = 1 mobile node.

In the following simulation experiment we investigate the effect of the sensor detection range rd on the optimal neighborhood size rz . Using Lemma 5.1, the optimal neighborhood size for different rd is presented in Table I. Table I. The optimal search neighborhood rz∗ for different rd values rd 2 5 8 10

N 200 200 200 200

ρ 2.5 2.5 2.5 2.5

rz ∗ 20.3 21.9 25.7 30.1

V ar(V∆R ) 35.9 847 2232.1 2417.6

As shown in Table I as the detection radius rd increases, the rz∗ radius, where V 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 given that the number of sensors is fixed N = 200) it is possible to generate deployments with higher variation in the achieved coverage. Fig. 10 presents the average dynamic coverage C(k) achieved by a single mobile node after k = 2000 time steps when rd = 5m. The figure indicates that dynamic coverage is maximized when rz = 22m which is what was also predicted by Lemma 5.1 (see ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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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 dynamic coverage 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 affected by the density λ ≡ N A of the static sensors. First, using Lemma 5.1 we compute the optimal rz∗ as shown in Table II. Table II. The optimal search neighborhood rz∗ for different N values rd 5 5 5 5

N 100 200 300 400

ρ 2.5 2.5 2.5 2.5

rz ∗ 41.9 21.9 15.4 12.1

V ar(V∆R ) 1141.3 847 630.4 470.7

Fig. 10 shows that for N = 200 the best rz∗ = 22m which is in agreement with the results of Table II. Furthermore, Fig. 11 presents the coverage achieved by the path planning algorithm when N = 300 sensors are deployed. The maximum coverage is achieved when rz = 15m which is again consistent with the Lemma 5.1 prediction as 6 Assuming

that each sensor field contains one initially undetected temporally static event, placed at random position in the sensor field.

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indicated in Table II. Notice again that the average event detection time is minimized when dynamic coverage is maximized (rz = 15m). The next simulation considers how the best rz value is affected by ρ, the distance that the mobile sensor can move in one time step. Again, we evaluate the optimal radius rz∗ using Lemma 5.1 as shown in Table III. Table III. The optimal search neighborhood rz∗ for different ρ values rd 5 5 5 5

N 200 200 200 200

ρ 1 2.5 4 5

rz ∗ 54.8 21.9 13.8 11.1

V ar(V∆R ) 846.99 846.99 846.98 846.97

Fig. 10 shows that the best rz for ρ = 2.5m is about 22m while Fig. 12 indicates that for ρ = 4m the best rz is about 15m. Both of these results are consistent with the Lemma 5.1 predictions shown in Table III. Therefore, when the mobile sensor is searching for targets, once it moves farther (bigger ρ) from its previous position the best rz value decreases. Notice again that the average event detection time is minimized when dynamic coverage is maximized (rz = 15m) as expected by Lemma 2.1. Subsequently, under more exhaustive Monte Carlo simulation we obtain Table IV which presents a comparison between the optimal rz predicted by Lemma 5.1 to the best rz obtained through brute force simulation for a large set of parameters. For each set 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 ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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shows the value of rz (among the ones used in the simulation7 ) with the maximum performance. The last column of the table presents the percentage difference between the 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 the scenarios investigated). Note that the difference between the two methods is generally below 10-15%. We point out that this difference may be the result of several factors like the finite set of rz ’s and the discretization used in the simulation or the boundary effects. These effects are particularly pronounced for very small and very large values of rz . For very small rz the simulation results are affected by the discretization while for large rz the results are affected by the boundary effects. In the previous simulations we have investigated the single mobile sensor case, however we point out that the approximation method for obtaining rz∗ also remains valid for the case of multiple mobile sensors given that the coverage process is mainly governed by the initial distribution of stationary nodes (e.g. when the number of mobile sensors as well as their coverage rate are small enough). Figures 13 and 14 present the average 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 when five mobile nodes are used and when rd = 5m, N = 400 and ρ = 1m. Figure 14 indicates that dynamic area coverage is maximized and event detection time is minimized when rz = 25m among the values investigated. Using the Lemma 5.1 the rz∗ ≈ 30m and hence the approximation remains valid.

7 Note

that these simulations are quite time consuming and it was not possible to extensively search for all values of rz .

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Finally, the last simulation experiment considers the case when the WSN monitors the area for transient events. Specifically, we consider scenarios where the events are activated at random times and have finite duration, significantly shorter than the total simulation time. In such scenarios an event may stay undetected even when it occurs in areas searched by a mobile sensor because it may become active after the mobile sensor has searched the area or it may become inactive before the mobile sensor searches the area. Our approach can address such scenarios using the forgetting factor parameter f when updating their locally stored cognitive maps. At this point, it’s worth mentioning that the approximation method for obtaining rz∗ is also valid when the forgetting factor is 0 ≤ f < 1 is used and hence the mobile sensor’s objective is to improve the dynamic coverage rate over time in a small amount of time. This is due to the fact that when f < 1 the coverage process is mainly governed by the initial distribution of stationary nodes. We again consider 100 sensor fields with 200 stationary sensors and 10 mobile sensors randomly distributed in a 200m × 200m area and for each scenario we assume 10 undetected dynamic events. We assume that the events are activated according to a Poisson process with rate µ1 = 1/200 (i.e. 200 time steps is the expected interarrival time between consecutive activations of each event) and events 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 with rate µ2 = 1/100 (i.e. on average, the lifetime of each event is 100 time steps after its activation). In this simulation, the average probability of detection of transient events is used as a performance metric and the detection performance of the proposed path planning algorithm is evaluated as a function of the neighborhood rz for various forACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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Table IV. Comparison between the optimal rz∗ approximated by Lemma 5.1 and best rz∗ obtained by simulations. rz ∗

Parameters rd 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10

N 200 200 200 300 300 300 400 400 400 200 200 200 300 300 300 400 400 400 200 200 200 300 300 300 400 400 400

ρ 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4 1 2,5 4

PE

Lemma Approx.

Simulation

arg max{V ar(V∆R )}

arg max{C(k)}

Error %

50,7 20,3 12,8 34,1 13,7 8,6 25,7 10,3 6,6 54,9 22,0 13,8 38,4 15,4 9,7 30,3 12,1 7,7 75,3 30,1 18,9 64,2 25,7 16,1 63,5 25,4 15,9

45 19 14 37 14 10 28 11 8 49 21 15 38 15 11 27 12 9 61 33 19 56 26 18 57 27 17

11,3 6,5 9,7 8,6 2,5 15,8 8,8 6,4 21,4 10,7 4,4 8,8 1,1 2,5 13,3 10,9 1,2 16,8 19 9,5 0,7 12,7 1,2 11,8 10,2 6,3 6,7

getting factor f values 8 and also compared with random search and standard search approaches using Monte Carlo simulation. In the random search, the next position of a mobile node is ρ away from the previous one and at random heading direction θ − ϕ ≤ θ ≤ θ + ϕ whereas in the standard search the mobile sensors scan exhaustively and repeatedly the entire field using parallel S-shaped patterns. The standard search is 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) mobile sensors cases are considered. As shown in Fig. 15 the proposed algorithm outperforms random search for all f values considered and clearly when setting f < 1 the probability of detection is improved. In addition, when best rz and f values are selected the proposed algorithm achieves better performance compared to standard search. However, we indicate that setting appropriately the forgetting factor f value is a multi-parameter optimization problem as its value depends on many other parameters like the parameters µ1 and µ2 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 rz∗ approximated by Lemma 5.1 (e.g. rz∗ ≈ 22m) archives the best detection performance of transient events. At this point we also point out that under extensive simulations that we could not present here due to space limitations, we reach the following conclusions regarding the forgetting factor f : a) When f → 1 and rz is too small to catch 8 Instead of constant values one can use a function of time k for the forgetting factor f , however this will require extensive memory and computation (each element of matrix Pk must have a time-stamp).

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this degradation in the mobile sensor’s Pk mobile sensors will fail to identify regions not searched recently and thus once the field is searched they will trapped or move randomly 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 average probability of detection also increases because it allows the mobile sensors to continuously communicate each other and thus maintain an accurate global knowledge of the regions been searched recently. c) The number of mobiles sensors and event characteristics 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 the entire field, whereas, when many mobile sensors are available, its desired to forget ”fast” in order to search more locally. 7. CONCLUSIONS AND FUTURE DIRECTIONS

This article addresses the problem of dynamic coverage improvement using mixed WSNs (consisting of static and autonomous mobile sensors). In this context, mobile nodes continuously navigate through the sensor field and sample areas not sufficiently covered by the static sensors. We show that the optimal strategy for the mobile sensor 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 the biggest hole. To determine the “big enough” hole, we developed a heuristic that looks for the biggest coverage hole in a neighborhood around the mobile sensor and used a surrogate metric to determine the best size of the neighborhood that enhances the dynamic coverage and event detection performance in the mixed WSN. Obtained results from numerical evaluations of the neighborhood size approximations have been verified by extensive Monte Carlo simulation outcomes of the performance of the proposed distributed algorithm. The proposed mixed WSN framework considers applications that do not require or cannot afford simultaneous coverage of all locations but want to cover the deployed region within a certain time interval. In the future, we plan to further investigate the performance of the algorithm in scenarios that involve unreliable communication (delayed or dropped packets) and imprecise sensor measurements. Moreover, we plan to derive bounds on dynamic coverage and event detection delay performance of mixed WSNs under various search strategies of mobile sensors. Finally, the developed collaborative path-planning method can be extended to solve other types of problems such as the mobile sink path-planning problem in WSNs or the coverage path-planning problem for autonomous surface vehicles intended for pollution (e.g. oil) monitoringcleanup applications. APPENDICES A. OPTIMAL PATH FOR THE SINGLE MOBILE - H HOLE PROBLEM

In this appendix, we consider the scenario where there are h = 3 coverage holes (see Fig. 16). In such case, one can use the following technique to reduce the problem to the h = 2 case as follows: For each i, i = 1, · · · , 3 consider that the mobile is at Ai and has already 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 ( ) A1 min (E [T123 ] , E [T132 ]) = dv1 + 21 2rAd1v A1 +A + 2 +A3 ( ) (34) d1 A1 A2 +A3 min (E [T23 ] , E [T32 ]) + v + 2rd v A1 +A2 +A3 Finally the optimal path should be found by comparing the cases of all i, i = 1, · · · , 3 to find the path that minimizes the expected detection time. Thought the proposed soluACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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y A2 d12 d2

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Fig. 16. Problem geometry with three coverage holes

tion needs h! − 1 comparisons-evaluations to find the optimal path when h uncovered holes exist, it easy to tackle the equations using the proposed technique. Note that the OSP P problem can not easy formulated in Dynamic Programming as the T SP problem because the search cost between two holes is not a constant value but depends on the previous holes that have been already searched. Next, we present a centralized algorithm for finding the optimal tour for a single mobile node when h non-overlapping holes exist using exhaustive search. In this algorithm we generate all possible permutations (paths) and compute the expected detection time associated with each path. The optimal solution is the path that results in the minimum average detection delay. The details of the algorithm are listed in Algorithm 1. B. LOWER & UPPER BOUNDS FOR THE AREA COVERAGE PERFORMANCE

This appendix presents probabilistic approximations for the lower and upper bound of the area coverage performance of the mixed WSN using the proposed path planning algorithm. Using these analytical bounds, it is possible to determine the required number of mobile or static sensors or the speed of mobile sensors to provide a predetermined 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 the area of interest A. Thus by using eq. (26) the coverage achieved by the mixed WSN at time step k is given by E(VA (k)) C(k) = 1 − (35) A The coverage provided by static sensor nodes is given by ) ( S 2 Cs = 1 − exp − πrd (36) A Based on [Liu et al. 2005], the coverage provided by mobile sensor nodes during a time interval [0, k] when they are moving around in the sensing field following a random mobility model is given by ( ) ) M( 2 Cm = 1 − exp − (37) πrd + 2rd vk A

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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 do d(i, j) = ∥C(i) − C(j)∥; if i == j then d(i, j) = ∥C(i) − O∥; end end end /* Find Optimal Path */ for each path P(i, :), i = 1, ..., h! do for each hole P(i, j), j = 1, ..., h do hf = 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))/(2rd v) ; else ( ) t(j) = t(j − 1) + (1/2) A(P(i, j − 1))/(2rd v) + ( ) ( ) d P(i, j − 1), P(i, j) /v + (1/2) A(P(i, j))/(2rd v) ; end ( ) ∑ Et(j) = t(j) A(P(i, j))/ (A) ; end ∑ ETP(i,:) (i) = (Et); end { } i∗ = arg mini=1,...,h! ETP(i,:) (i) ; Optimal Path= P(i∗ , :); B.1.1. A Lower Bound:. Assuming that there is no collaboration between the stationary and the mobile sensor nodes in the WSN it turns out that a fraction of A can be independently covered either by a mobile or a static sensor node during a time interval [0, k], thus the area coverage can be given by ( ) ( ) N πrd2 2M rd vk Clb = 1 − exp − . exp − (38) A A B.1.2. An Upper Bound:. Assuming that there is perfect collaboration between the stationary and mobile sensor nodes in the WSN and considering that the paths covered by mobiles may intersect, it turns out that a fraction of A can be only covered by mobiles or static sensor nodes (mutually exclusive events) during a time interval [0, k], thus the area coverage can be given by ) ( ) ( 2M rd vk N πrd2 − exp − (39) Cub = 2 − exp − A A ACM Transactions on Sensor Networks, Vol. V, No. N, Article A, Publication date: January YYYY.

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

100 90

Path Planning Algorithm Analytical Upper Bound Analytical Lower Bound

C(k)(%)

80 70 60 50

0

50

100 150 200 250 300 350 400 450 k

Fig. 17. The area coverage performance of the proposed path planning algorithm using Monte Carlo simulation 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 us to improve our work.

REFERENCES V. Ablavsky and M. Snorrason. 2000. Optimal Search for a Moving Target: a Geometric Approach. In Proceedings of AIAA Guidance, Navigation, and Control Conference. Denver, CO, USA. J. Barraquand, B. Langlois, and J.-C. Latombe. 1992. Numerical potential field techniques for robot path planning. IEEE Transactions on Systems, Man and Cybernetics 22, 2 (1992), 224 –241. N. Bartolini, T. Calamoneri, T.F. La Porta, and S. Silvestri. 2011. Autonomous Deployment of Heterogeneous Mobile Sensors. IEEE Transactions on Mobile Computing 10, 6 (2011), 753 –766. DOI:http://dx.doi.org/10.1109/TMC.2010.192 T. Bektas. 2006. The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 34, 3 (2006), 209 – 219. L. Bondesson and J. Fahlen. 2003. Mean and Variance of Vacancy for Hard-Core Disc Processes and Applications. Scandinavian Journal of Statistics 30, 4 (2003), 797–816. P. Brass. 2007. Bounds on Coverage and Target Detection Capabilities for Models of Networks of Mobile Sensors. ACM Trans. Sen. Netw. 3, 2 (2007). DOI:http://dx.doi.org/10.1145/1240226.1240229 S. Chiu, D. Stoyan, W. Kendall, and J. Mecke. 2013. Stochastic geometry and its applications. John Wiley & Sons.

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

A:34

T.P. Lambrou

T.H. Chung, G.A. Hollinger, and V. Isler. 2011. Search and pursuit-evasion in mobile robotics. Autonomous Robots 31, 4 (Nov. 2011), 299–316. T. Cormen, C. Leiserson, R. Rivest, and C. Stein. 2001. Introduction to Algorithms (2nd ed.). The MIT Press. J. Cortes. 2010. Coverage Optimization and Spatial Load Balancing by Robotic Sensor Networks. IEEE Trans. Automat. Control 55, 3 (march 2010), 749 –754. DOI:http://dx.doi.org/10.1109/TAC.2010.2040495 J. Cortes, S. Martinez, T. Karatas, and F. Bullo. 2004. Coverage Control for Mobile Sensing Networks. IEEE Transactions on Robotics and Automation 20 (2004), 243–255. A. Deshpande, S. Poduri, D. Rus, and G. Sukhatme. 2009. Distributed coverage control for mobile sensors with location-dependent sensing models. In Proceedings of the IEEE ICRA’09. 3493–3498. Y. Elmaliach, N. Agmon, and G. Kaminka. 2009. Multi-robot area patrol under frequency constraints. Annals of Mathematics and Artificial Intelligence 57, 3-4 (2009), 293–320. M. Erdelj, V. Loscri, E. Natalizio, and T. Razafindralambo. 2013a. Multiple point of interest discovery and coverage with mobile wireless sensors. Ad Hoc Networks (2013). M. Erdelj, T. Razafindralambo, and D. Simplot-Ryl. 2013b. Covering Points of Interest with Mobile Sensors. IEEE Transactions on Parallel and Distributed Systems 24, 1 (2013), 32–43. DOI:http://dx.doi.org/10.1109/TPDS.2012.46 A. Ghaffarkhah and Y. Mostofi. 2012. Optimal motion and communication for persistent information collection using a mobile robot. In IEEE Globecom Workshop, 2012. 1532–1537. A. Ghaffarkhah and Y. Mostofi. 2014. Dynamic Networked Coverage of Time-Varying Environments in the Presence of Fading Communication Channels. ACM Transactions on Sensor Networks (TOSN) 10, 3 (2014), 45. P. Hall. 1984. Mean and Variance of Vacancy for Distribution of k-Dimensional Spheres within kDimensional Space. Journal of Applied Probability 21 (1984), 738–752. P. Hall. 1988. Introduction to the Theory of Coverage Processes. John Wiley & Sons. G. Hollinger and G. Sukhatme. 2013. Sampling-based motion planning for robotic information gathering. In in Proc. Robotics: Science and Systems Conference (RSS). Berlin, Germany. G. Hollinger, Y. Yerramalli, S. Singh, U. Mitra, and G. Sukhatme. 2011. Distributed Coordination and Data Fusion for Underwater Search. In IEEE International Conference on Robotics and Automation. Shanghai, China, 349–355. A. Howard, M. Mataric, and G. Sukhatme. 2002. Mobile sensor network deployment using potential fields:A distributed, scalable solution to the area coverage problem. In DARS. D. Junzhao, L. Yawei, L. Hui, and S. Kewei. 2010. On Sweep Coverage with Minimum Mobile Sensors. In Proceedings of the 16th IEEE ICPADS. 283 –290. M. G. Kendall and P. A. P. Moran. 1963. Geometrical Probability. New York: Hafner. B. O. Koopman. 1956. The theory of search II - Target detection. Operations Research 4 (1956), 503–531. T.P. Lambrou and C.G. Panayiotou. 2006. Improving Area Coverage using Mobility in Sensor Networks. In International Conference on Intelligent Systems And Computing: Theory And Applications, ISYC 2006. Ayia Napa, Cyprus. T.P. Lambrou and C.G. Panayiotou. 2007. Collaborative Event Detection Using Mobile and Stationary Nodes in Sensor Networks. In the 3rd IEEE CollaborateCom 2007. New York, USA. T.P. Lambrou and C.G. Panayiotou. 2009a. Collaborative Area Monitoring Using Wireless Sensor Networks with Stationary and Mobile Nodes. EURASIP Journal on Advances in Signal Processing (2009), 1–16. ISSN: 1110-8657, doi:10.1155/2009/750657. T.P. Lambrou and C.G. Panayiotou. 2009b. Distributed collaborative path planning in sensor networks with multiple mobile sensor nodes. In 17th IEEE MED 2009. Thessaloniki, Greece. T.P. Lambrou and C.G. Panayiotou. 2011a. Area Coverage Vs Event Detection in Monitoring Applications using Mixed Sensor Networks. In 8th World Congress of the International Federation of Automatic Control, IFAC WC 2011. Milano, Italy. T.P. Lambrou and C.G. Panayiotou. 2011b. On the Optimal Search Neighborhood in Mixed Wireless Sensor Networks. In 50th IEEE Conference on Decision and Control and European Control Conference, CDCECC 2011. T.P. Lambrou and C.G. Panayiotou. 2012a. Online, Adaptive, and Distributed Multi-Robot Motion Planning for Collaborative Patrolling of Sparse Sensor Networks. In 25th IEEE IROS 2012 Workshop on Robot Motion Planning: Online, Reactive, and in Real-Time. T.P. Lambrou and C.G. Panayiotou. 2012b. A Testbed for Coverage Control using Mixed Wireless Sensor Networks. Journal of Network and Computer Applications 35, 2 (2012), 527–537. ISSN: 1084-8045, doi:10.1016/j.jnca.2011.05.010.

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

Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks

A:35

T.P. Lambrou and C.G. Panayiotou. 2013. Collaborative path planning for event search and exploration in mixed sensor networks. The International Journal of Robotics Research 32, 12 (2013), 1424–1437. S. D. Lawrence. 1975. Theory of optimal search. Academic Press, New York. 260 pages. M. Li, W. Cheng, K. Liu, Y. He, X. Li, and X. Liao. 2011. Sweep Coverage with Mobile Sensors. IEEE Transactions on Mobile Computing 10, 11 (nov. 2011), 1534 –1545. DOI:http://dx.doi.org/10.1109/TMC.2010.237 W. Li and C.G. Cassandras. 2005. Distributed Cooperative Coverage Control of Sensor Networks. In Proceedings of 44rd IEEE Conference on Decision and Control. B. Liu, P. Brass, O. Dousse, P. Nain, and D. Towsley. 2005. Mobility Improves Coverage of Sensor Networks. In Proceedings of the ACM MobiHoc 2005 Conference. B. Liu, O. Dousse, P. Nain, and D. Towsley. 2013. Dynamic Coverage of Mobile Sensor Networks. IEEE Transactions on Parallel and Distributed Systems 24, 2 (2013), 301–311. DOI:http://dx.doi.org/10.1109/TPDS.2012.141 S. Loizou and K. Kyriakopoulos. 2008. Navigation of Multiple Kinematically Constrained Robots. IEEE Transactions on Robotics 24, 1 (2008), 221–231. DOI:http://dx.doi.org/10.1109/TRO.2007.912092 N. Nigam and I. Kroo. 2008. Persistent surveillance using multiple unmanned air vehicles. In IEEE Aerospace Conference. 1–14. C. Papadimitriou. 1977. The Euclidean travelling salesman problem is NP-complete. Theoretical Computer Science 4, 3 (1977), 237 – 244. M. Pavone, A. Arsie, E. Frazzoli, and F. Bullo. 2009. Equitable partitioning policies for robotic networks. In Proceedings of the IEEE ICRA ’09. 2356 –2361. DOI:http://dx.doi.org/10.1109/ROBOT.2009.5152809 M. Polycarpou, Y. Yang, Y. Liu, and K. Passino. 2003. Cooperative Control: Models, Applications and Algorithms. Vol. 1. Kluwer Academic Publishers, Chapter Cooperative Control Design for Uninhabited Air Vehicles, 283–321. E. Rimon and D.E. Koditschek. 1992. Exact Robot Navigation Using Artificial Potential Fields. IEEE Transactions on Robotics and Automation 8, 5 (1992), 501–518. A. Singh, A. Krause, C. Guestrin, and W. Kaiser. 2009b. Efficient informative sensing using multiple robots. Journal of Artificial Intelligence Research 34, 2 (2009), 707. A. Singh, A. Krause, and W. Kaiser. 2009a. Nonmyopic Adaptive Informative Path Planning for Multiple Robots. In Proc. International Joint Conference on Artificial Intelligence. D. Stoyan, W. Kendall, and J. Mecke. 1995. Stochastic Geometry and Its Applications, 2nd Edition. John Wiley & Sons. B. Wang, H. Beng Lim, and D. Ma. 2009. A survey of movement strategies for improving network coverage in wireless sensor networks. Computer Communications 32, 13-14 (2009), 1427 – 1436. DOI:http://dx.doi.org/10.1016/j.comcom.2009.05.004 A. R. Washburn. 2002. Search and Detection (4th edition ed.). INFORMS. A. R. Washburn and M. Kress. 2009. Combat Models. Springer. T. Wimalajeewa and S.K. Jayaweera. 2010. Impact of mobile node density on detection performance measures in a hybrid sensor network. IEEE Transactions on Wireless Communications 9, 5 (may 2010), 1760 –1769. DOI:http://dx.doi.org/10.1109/TWC.2010.05.091012 M. Zhong and C.G. Cassandras. 2011. Distributed Coverage Control and Data Collection With Mobile Sensor Networks. IEEE Trans. Automat. Control 56, 10 (oct. 2011), 2445 –2455. DOI:http://dx.doi.org/10.1109/TAC.2011.2163860 Y. Zou and K. Chakrabarty. 2003. Sensor deployment and target localization based on virtual forces. In Proceedings of the IEEE Infocom. Received November 4, 2014; revised ; accepted

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Optimized Cooperative Dynamic Coverage in Mixed Sensor Networks THEOFANIS P. LAMBROU, University of Cyprus

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