Relocation of Hopping Sensors

Transcription

1 28 IEEE Internationa Conference on Robotics and Automation Pasadena, CA, USA, May 9-23, 28 Reocation of Hopping Sensors Zhiwei Cen Googe Inc. 6 Amphitheatre Pky Mountain View, CA 9443, USA Matt W. Mutka Dept. of Computer Science and Engineering Michigan State University East Lansing, MI 48824, USA Abstract Hopping sensors are a cass of mobie sensors whose mobiity design are inspired by creatures such as grasshoppers. Such sensors are abe to maintain mobiity in harsh terrains but may ack movement accuracy of those sensors that are powered by whees. We examine the opportunities and chaenges for utiizing the mobiity of ow cost hopping sensors to ensure coverage and maintain energy efficiency within a sensing fied. We focus on the probem of transporting a number of hopping sensors from mutipe sources to a destination. Probabiistic methods are used to contain the movement inaccuracies aong the hopping course. We aso consider the impact of wind under an aerodynamic setting. Two transport schemes are designed to minimize the number of hops needed whie considering other constraints, such as sustaining the capabiity of reocating sensors within the whoe network. In one scheme we use upper and ower hopping imits to appy the network mobiity constraints. The other scheme uses a baancing coefficient to construct a new optimization target to meet the requirement of path optimaity and network mobiity dynamicay. Simuation resuts show that both schemes work we regardess of the wind factors, whie the dynamic scheme is aso shown to be resiient to topoogica changes of the network. Index Terms Hopping sensors, Sensor reocation, Sensor networks, Energy efficiency I. INTRODUCTION Large scae sensor networks ca for automatic depoyment and maintenance. Mobie sensors are important for faciitating sensor depoyment and maintaining coverage and communication during runtime. Hopping sensors are a cass of mobie sensors with a bionic mobiity design that is inspired by creatures, such as grasshoppers. In order to move to a different ocation, a hopping sensor throws itsef high and toward the destination direction. Hopping sensors are capabe of maintaining mobiity in terrains where wheeed mobiity is not possibe. Mobiity powered by the concepts of hopping are widey discussed in the area of panetary exporation [], [2]. Bergbreiter et a., proposed to use rubber bands to power sma jumping sensors [3]. The work of Feddema, et a. [4] described a prototype minefied hopping robot. In this paper, we assume that the hopping sensors are capabe of adjusting the hopping direction. A fixed propeing force for hopping is aso assumed. In reaity, the hopping range may vary due to the physica oad of the sensor, This paper is supported in part by the Nationa Science Foundation under grants No. CNS-7244 and CNS different terrain conditions, and oca aerodynamic settings. In order to faciitate routing, the sensors are aso assumed to have a ocaization capabiity. Hopping enabed mobiity can be used to faciitate the sensor network depoyment and maintain coverage and connectivity during runtime. In the ifetime of a sensor network, it often happens that the sensors in a certain area are depeted faster than other areas. Those areas that have depeted sensors are caed sensing hoes or sensing wes. A we panned depoyment may aocate redundant sensors in the fied, thus when sensing wes are detected, sensors can be migrated from those regions that have redundant sensors (referred as suppiers or sources) to sensing wes. We consider the probem of transporting a certain number of hopping sensors from mutipe sources to a detected sensing we. To faciitate sensing we detection and the matching of sources to the we, we organize the sensor network fied as a set of custers. Quorum or broadcast based approaches can be used to match the suppier and consumer custers. We mode the hopping inaccuracy using a mutivariate norma distribution. In the transporting stage we empoy cascaded movement to speed the migration and argue the distance between reay custers is crucia in determining the routing path ength. We propose two schemes to minimize the tota number of hops needed to fi a certain sensing we, whie at the same time maintaining the reocation capabiity of the whoe network. One scheme uses upper and ower reay edge hop imits, whie the other uses a baancing coefficient to construct a new optimization target dynamicay. Simuation resuts indicate that both agorithms are effective in baancing the requirement of path optimaity and maintaining the reocation capabiity of the network. The dynamic agorithm is aso shown to be resiient to topoogica changes of the network. The paper is organized as foows. Section II briefy covers reated work. In section III, we study the muti-hop anding of the sensors under a mutivariate norma distribution mode. Under such premises we present two route panning schemes in section IV. We evauate the performance of the route panning schemes using Matab simuation and the resuts are presented in section V. Section VI provides the concusions. II. RELATED WORK Much work has utiized the mobiity of genera mobie sensors to faciitate depoyment, coverage maintenance and /8/$ IEEE. 569

2 improving energy efficiency [5], [6], [7], [8], [9], []. The use of hopping or fipping based sensors in sensor network depoyment is aso studied in [9]. However, the work in [9] does not expicity consider the characteristics of hopping sensors we investigate in this paper, and more importanty does not buid the reocation scheme based on the understanding of the movement of arge number of mutihop sensors. The work of Wang, et a. [8] aso comes cose to our paper in terms of the objectives and system settings. They foow a simiar approach for migrating mutipe wheeed sensors between a singe source and destination within the sensor network. In contrast, we focus on the probem of reocating hopping based sensors, which have a different mobie and dynamica mode compared with wheeed sensors. In addition, we consider the case where a singe source cannot provide sufficient sensors and mutipe sources are needed. III. SYSTEM MODEL Hierarchica modes are widey used in sensor network design. The work of Zou, et a. [] assumed that a custer head was avaiabe to coordinate the sensor depoyment based on virtua forces. Wang, et a. [8] aso used a grid based quorum method to match the source and destination grids. In our hopping sensor network mode, we assume that the sensors are organized into custers. A custer head assumes the responsibiity of eveny distributing the sensors, detecting sensor deficiency, and detecting redundant sensors within the custer. There are we estabished methods to detect redundant sensors in a sensor network based on computing Voronoi diagrams [2]. The custers that are detected to have redundant sensors mark themseves as sources and identify themseves through the custer head over the sensor network. The probem of detecting sensing hoes or sensing wes are aso studied [5], [3], [4]. Matching the sources and hoes coud be compicated in the presence of mutipe sources and hoes. The work of Cheapan, et a. reduces the matching probem in the sensor depoyment stage as a muti-commodity maximum fow probem [9]. We take the approach of fiing the sensing hoes one by one and consider the probem of migrating hopping sensors from mutipe sources to one destination. A. Norma Distribution Mode of Inaccurate Hopping Compared with wheeed mobie sensors, hopping sensors ack the accuracy of movement. We use a mutivariate norma distribution mode to determine the anding accuracy of mutipe sensors that hop together. Based on the mode, we derive the upper bound of the number of hops needed to migrate a number of sensors to a target ocation. The resut is aso used to derive the number of sensors that can reach the consumer from the sources for a given demand when astray sensors are considered. A sensor is regarded as astray when its anding point is outside a range of the projected point. In a two dimensiona case, the anding accuracy is characterized by the dispacement between the targeted ocation, represented by vector T, and the actua anding ocation represented by vector L. The dispacement vector D can be represented as D = T L. D is modeed by a two dimensiona norma distribution with mean (, ), standard deviation (σ, σ), and correation ρ. Note that we used a nondiscriminating standard deviation vector for the two dimensions. The probabiity density function of D in terms of (x, y) can be given as f(x, y) = 2πσ 2 ρ exp + y 2 2ρxy 2 ( x2 2σ 2 ( ρ 2 ). () ) In order to mode the uncertainty of the anding ocation, we define an acceptabe anding area as a disk S around the targeted ocation. As shown in Figure, the radius of the disk nσ is determined by mutipying a factor n to the standard deviation σ. For the migration distance that requires one hop ony, the acceptabe anding area ies within the target ocation of the sensor. For migrations that need more than one hop, if the sensor ands within the acceptabe anding area, the sensor is recharged for another hop towards the target; otherwise the sensor is deemed as astray and wi seek to join a oca custer. Fig.. Upper bound point Source Hop Hop 2 nσ Modeing the Hopping Accuracy Using Norma Distribution The probabiity that the hopping sensor ands in the acceptabe anding area S can be represented as P (S) = f(x, y)dx dy. (2) S It is known that Equation (2) does not have a cosed form soution. Numerica methods can be used to cacuate the probabiity [5]. Based on the probabiity that the hopping sensor ands in the acceptabe anding area, we derive the upper bound of the number of hops needed for a migration distance. Assuming the distance covered by one sensor hop is r, the ower bound of the number of hops N needed to cover distance is N = r. The upper bound of N, however, is determined by the situation when, for every hop, the sensor ands at the farthest point from the target and on the direct ine segment between the source and the target ocation, as indicated in Figure. This is equivaent to reducing the hopping range of the sensor from r to r nσ. Thus the upper bound of the number of hops given distance and hopping range r can be given by N u = r nσ. (3) Assume we know the number of sensors demanded by the target custer and the number of sensors that are tasked to the current source is E t. As wi be discussed in section IV- A, the consumer estimates the number of astray sensors and demands some additiona sensors to cover those sensors. For S 57

3 each hop, the number of hopping sensors wi decrease due to straying. For the first hop, the number of hopping sensors is E t. For the second hop, the number is E t P (S), etc. Thus the tota number of hops needed for a sensors is H = N u i= N u E t P i (S) =E t i= P i (S) = P Nu (S) P (S) E t. B. Aerodynamica Mode of Hopping under Air Disturbance Unike wheeed movement, the motion of hopping is more susceptibe to air disturbance. We need to estimate the impact of air disturbance so that we can adjust the hopping orientation propery when wind is detected. We aso mode the infuence of air disturbance on the hopping range of the sensors so that we can obtain an accurate estimation of the number of hops needed to traverse a certain path. We assume the hopping sensor has a fixed propeing impetus. In order for the sensor to have the ongest horizonta hopping range, the sensor shoud jump towards a tited upward direction. When there is no air disturbance, the optima upward ange that provides the ongest horizonta range is α = 45. Assuming a fat terrain, the horizonta distance traveed by the sensor in the air is (4) r = t v h = t v cos α = v 2 sin 2α. (5) g We assume that the air disturbance ony infuences the horizonta veocity of the sensor. If the veocity of the wind in the horizonta direction is v w, the rea veocity of the hopping sensor in the horizonta direction is v hrea = v h + v w. If the direction of the targeted ocation is given in v hrea, the hopping direction of the sensor can be given as v h = v hrea v w. (6) Under the infuence of air disturbance, the number of hops needed for a certain distance wi aso change. When there is no wind, the hopping range of the sensor can be given in Equation (5). Since the wind ony infuences the horizonta veocity of the sensor, the fying time of the sensor remains the same when there is air disturbance. Thus the hopping range of the sensor with wind can be represented as r w = t v hrea. (7) In practice, we use the proportion of the hopping range under air disturbance to the norma hopping range to faciitate cacuation of the number of hops needed. According to Equation (5) and Equation (7), we have r w r = v hrea. (8) v h Assuming the number of hops needed for a trip of ength under the norma condition is N, the number of hops needed under air disturbance N w can be obtained through N w N = r w r = r r w = v h v hrea = v h v h + v w. (9) 57 In concusion, when the air disturbance generates a horizonta veocity of v w, the hopping direction of the sensor can be given using Equation (6), whie the number of hops needed for a fixed range compared with the norma case can be given using Equation (9). When considering the upper bound of the number of hops needed (denoted as N wu and N u ), Equation (9) is modified to N wu N u = r w nσ r nσ = r nσ r w nσ = v h t nσ v hrea t nσ. () Simiar to Equation (4), the upper bound of the tota number of hops for a sensors H w with air disturbance can be given by H w = P Nwu (S) E t, () P (S) where E t is the number of sensors that are tasked to the current source. In order to obtain the wind information needed by the mode, anemometers are depoyed within the sensor network. The mode impied in Equation () is ony appicabe to the situations where the wind fow has a predictabe and reative stabe nature. For settings where the air fow foows random changes, a pre-depoyment can be used to measure the air fow impacts. IV. ROUTE PLANNING IN HOPPING SENSOR MIGRATIONS Route panning invoves migrating the required number of sensors from mutipe sources to the destination custer. The process incudes identifying suppying and consuming sensor custers, matching those custers and seecting the optima route to migrate the sensors. A. Matching of the Consumer and Suppiers Assume that for suppier i (i = M), the number of sensors it can provide is p i and the distance between suppier i and the consumer is i, the number of sensors that can reach the consumer can be estimated as M E est = p i P i r nσ (S). (2) i= In Equation (2), P (S) is the rate of the sensors that and in the acceptabe range, as defined in Equation (2). i r nσ is the number of hops needed between the consumer and suppier i, which is simiar to Equation (3). When considering the wind factor, the number of hops estimation can be done with the aid of Equation (8). It is noted that the number of hops estimated is based on the assumption that the sensors migrate from the suppier to the consumer directy, without using reay custers. The concept of reaying is expained in section IV-B. When reay custers are used, the number of hops needed might be arger. In the atter case, an augmenting factor shoud be appied to Equation (2). Assuming the number of sensors the consumer needs is d, we compare d and E est. If E est d the suppiers can satisfy the consumer and a matching process can be concuded. Otherwise the consumer has to wait for additiona suppiers.

4 B. Finding the Optima Migration Path In this section we discuss the agorithms to migrate the hopping sensors from mutipe suppiers to the consumer custer. We assume the routing panning process is performed at the suppier custers. Each custer has a topoogica view of the network. Source Source (a) Direct Hopping 4 3 Reay Fig. 2. Reay (b) Hopping with Reays Reay Two Hopping Strategies Destination Destination Two possibe migration strategies can be used. The first strategy is to migrate the sensors directy from the suppier to the consumer, as shown in Figure 2(a). The second strategy, however, uses intermediate custers as reay custers. Sensors can hop simutaneousy starting from the suppier and the reay custers, as iustrated by Figure 2(b). Athough sensors can be assumed to be abe to make mutipe hops, the number of hops one sensor can make (denoted as k hard ) is not unimited before refueing or recharging. For exampe, the prototype proposed by Feddema, et a [4] has a imit of hops before refueing. Direct hopping can aways guarantee a straight route but hopping with reays cannot. Furthermore, direct hopping gives sensors more freedom to adjust their hopping direction on the fy if the anding point is off the route, which wi utimatey save the number of hops. For exampe, in Figure 2(b), hop number 5 and 6 coud have been combined into one hop if the reay is not used. Thus it is desirabe to use straight hopping for reativey short migration paths but using reay hopping for onger paths. For reay hopping, the tradeoffs require us to set a imit on both the maximum and minimum distance between the reay custers. We set the upper imit k to maintain the reocation abiity of the network. A ower imit j is set to avoid unnecessary hops introduced by excessive number of reays. We use the topoogy and avaiabiity information of the custers to construct a graph. The vertices set is composed of the custers. Initiay, a the avaiabe custers, incuding the suppiers and the consumer, are connected with edges. The weight of the edges are set to be the estimated number of hops needed, as indicated in Equations (3) and (). Our objective is to find an optima path to migrate the sensors from the suppiers to the consumer. In the first route panning agorithm, we first appy the upper and ower imits of the edge weights by deeting the edges whose weights fa outside the range of [j, k]. Then Dijkstra agorithm is caed for the sources to find the shortest paths from the suppiers. If the shortest path sti cannot be found for some suppiers, we concude the suppier is separated from the consumer and the consumer is informed to search for new suppiers. The agorithm stated above tries to maintain even mobiity consumption among custers and minima number of hops using manuay set imits. However, for the custers whose edges are within the imits, the mobiity consumption and number of hops of the edges can sti vary from edge to edge. In order to migrate the sensors from the suppiers to the destination optimay, and at the same time maintain even mobiity throughout the custers, we modify the path searching agorithm by adding an additiona adjusting process. In the adjusting process, we try to minimize a fraction of the sum of the weights aong the path and a fraction of the difference of the maximum and minimum weights of the edges aong the path. The second agorithm is described in Agorithm. The agorithm takes the graph G, edge weight set W, source node set S, destination node t and the hard imit k hard as inputs. For each source node, the agorithm first runs the Dijkstra agorithm to get a shortest path soution where the ony constraint is the hard imit. After that, the soution is refined through severa iterations. The minimization target becomes a fraction of the sum of the weights aong the path and a fraction of the difference of the maximum and minimum weight of the edges aong the path, which is expressed as ( γ)w sum + γ(w max w min ) as shown in ine. Here γ ( γ ) is a coefficient to determine the fraction of the difference of the maximum and minimum weight of the edges to be considered in the agorithm. The agorithm terminates when it cannot find a better soution given the constraints and the ast known good soution is taken as the fina soution. V. PERFORMANCE EVALUATIONS In order to vaidate the proposed modes and agorithms, we simuated a hopping sensor network environment under Matab. A topoogy generator is used to generate the simuation environment in a m m rectanguar area. The simuation environment can be regarded as a sensor network where some custers are avaiabe to serve as reays, whie others are not. The avaiabe custers, as we as the source and destination custers, are potted. Two custers are connected if the number of hops needed to migrate between them is ess than k hard. The parameters regarding hopping sensor dynamics and sensor distribution are shown in Tabe I. We derive the hopping range, initia veocity and hops capacity based on the Feddema prototype minefied hopping sensor mode [4]. We assume zero correation of the X and Y direction in the anding mode. We first evauated the performance of Agorithm where the path optimaity and remaining network mobiity are controed through the setting of soft hop imits. Considering the fact that most custers may be invoved in sensing tasks and not avaiabe as reays, the simuation topoogy is generated as a sparse graph. In a sparse environment, it is safe to use a sma ower hop imit j without jeopardizing the optimaity of the path since the degree of each node is sma. 572

5 Agorithm MinHopsExt(G, W, S, t, k hard ) : p, (w sum,w max,w min ) (+, +, ) 2: W Deete edges whose weights are arger than k hard in W 3: for a s in S do 4: (G,W ) (G, W ) 5: oop 6: p Dijkstra(G,W, s, t) 7: if p then 8: w sum Get the sum of edge weights using p 9: (e max,e min ) Get the edges with maximum and minimum weight using p : (w max,w min ) Get the weights of (e max,e min ) : if ( γ)w sum + γ(w max w min ) < ( γ)w sum + γ(w max w min ) then 2: (w sum,w max,w min ) (w sum,w max,w min ) 3: p p 4: Deete edges e max and e min in W and G 5: ese 6: Use the soution in p and break oop 7: end if 8: ese 9: Use the soution in p and break oop 2: end if 2: end oop 22: end for Parameter Vaue Number of Tota Hops Capabe without Refueing (k hard ) Hops Capabe per Sensor Initiay Magnitude of Initia Horizonta Veocity ( v h ) 7m/s Hopping Range (r) 3m Sensors per Custer Initiay Depoyed 2 Acceptabe Landing Area Radius (nσ).6m Acceeration due to Gravity (g) 9.8m/s 2 TABLE I PARAMETERS USED IN HOPPING SENSOR SIMULATION In the first simuation we fixed the ower imit j to be and evauated the number of hops needed to migrate a group of sensors and the mobiity capacity of the network after the migration when the upper hop imit k changes (Figure 3). The path optimaity is measured using the average number of hops consumed by each migrated sensor (referred as H m, Figure 3(a)). In the simuation, a group of sensors (E t = ) are requested by the consumer. Thus E t Hm is the tota number of hops needed in the migration. The remaining mobiity of the network is measured using two metrics (Figure 3(b) and (c)), the standard deviation (STD) of the number of remaining hops per sensor throughout the network, and the minimum of the number of remaining hops per sensor throughout the network. In the simuation we evauated the metrics under three circumstances, the norma condition, the condition with a positive wind force and the condition with a negative wind 6 4 (a) Number of Cons umed Hops per Migrated Sens or ( H m ) (b) STD of Number of Remaining k Hops per Sens or (c) Minimum of Number of Remaining k Hops per Sens or k With Negative Wind No Wind With Positive Wind Fig. 3. Number of Consumed Hops per Migrated Sensor and Mobiity Metrics Vs. Upper Soft Limit k force. In the simuated topoogy, a positive force is modeed as a wind veocity that is 45 countercockwise of the positive X axis. Likewise, a negative force is modeed as a wind veocity that is 225 countercockwise of the positive X axis. In either case, the magnitude of the wind veocity v w is set to be / of the magnitude of the initia horizonta hopping veocity v h. The simuation resuts of Figure 3 indicates that the vaue of the upper soft imit k has a significant infuence on the path optimaity and remaining network mobiity. Wind factors aso sighty infuence the resuts, but not as dramatic as k. The infuence of wind factors on the minimum of the number of remaining hops per sensor (Figure 3(c)) is even negigibe, athough we can see that the case with positive wind has higher number of remaining hops. This is because that the impact of k is so significant that a arger range of Y axis has diminished the sight differences of the three cases. Larger k wi yied better paths with fewer hops, but the STD of the number of remaining hops per sensor is aso arger, which indicates imbaance of the mobiity distribution throughout the network. The number of remaining hops per sensor aso confirms the trend with a steep curve. When k is over a certain vaue (5 in this case), the parameters wi not change since the constraints of the topoogy is reached. When k is too sma, however, a path might not be found due to the topoogy constraints Topo Topo2 Topo3 γ =.2 γ =.5 γ =.8 Fig. 4. Number of Consumed Hops per Migrated Sensor ( H m) under Different Topoogies and γ 573

6 In the second simuation we evauated the performance of Agorithm, where a dynamic approach is used to baance the need of path optimaity and network mobiity distribution. We use scaing coefficient γ to determine the optimization target (as shown in Line of Agorithm ). In this simuation we evauated the effect of the γ coefficient over the average number of hops consumed in a path and the mobiity metrics of the network after the migration is performed Topo Topo2 Topo3 γ =.2 γ =.5 γ =.8 Fig. 5. STD of Number of Remaining Hops per Sensor under Different Topoogies and γ Topo Topo2 Topo3 γ =.2 γ =.5 γ =.8 Fig. 6. Minimum of Number of Remaining Hops per Sensor under Different Topoogies and γ Unike the method with upper and ower hopping imits, we argue that the dynamic method is more resiient to topoogica changes of the network. To vaidate this, we simuated the agorithm over three different topoogies, namey Topo (the same as the topoogy used in the first simuation), Topo2, and Topo3, as indicated in the Figures 4, 5 and 6. The resuts show that under different topoogies, the coefficient has comparabe baancing infuences. The resuts aso show that under different γ vaues, the minimum of the number of remaining hops per sensor changes more dramaticay compared with the STD of the number of remaining hops per sensor. This indicates that the minimum of the number of remaining hops per sensor is a more sensitive mobiity metric. The same observation is aso drawn from the first simuation, as we see a sharper curve in Figure 3(c) than Figure 3(b). VI. CONCLUSIONS Hopping sensors are more adaptabe to harsh terrains compared with wheeed mobie sensors. We studied the muti-hop anding of the sensors under a mutivariate norma distribution mode and obtained an upper bound of the number of hops needed given the physica distance of the source and destination. The bound is further refined by considering the aerodynamics of air disturbances. Under such premises we focused on the probem of migrating a number of hopping sensors from mutipe source custers to a destination in a sensor network. We argue that cascaded reay hopping can speed up the migration but frequent reays may introduce unnecessary hops. We devised two agorithms to find the best migration pan. One uses upper and ower reay edge hop imits, whie the other uses a baancing coefficient to construct a new optimization target dynamicay. The two schemes are simuated using the physica and dynamic parameters of the Feddema prototype minefied hopping robots [4]. Simuation resuts indicated that both agorithms are effective in baancing the requirement of path optimaity and maintaining the reocation capabiity of the network. The dynamic agorithm is aso shown to be resiient to topoogica changes of the network. REFERENCES [] M. Confente, C. Cosma, P. Fiorini, and J. Burdick, Panetary Exporation Using Hopping Robots, in 7th ESA Workshop on Advanced Space Technoogies for Robotics and Automation ASTRA 22, ESTEC, Noordwijk, The Netherands, 22. [2] E. Hae, N. Schara, J. Burdick, and P. Fiorini, A minimay actuated hopping rover for exporation of ceestiabodies, in Proceedings of IEEE Internationa Conference on Robotics and Automation, 2. ICRA, San Francisco, CA, USA, 2, pp [3] S. Bergbreiter and K. Pister, Design of an Autonomous Jumping Microrobot, in Proceedings of IEEE Internationa Conference on Robotics and Automation, 27. ICRA 7, 27. [4] J. T. Feddema and D. Schoenwad, Decentraized contro of cooperative robotic vehices, IEEE Transactions on Robotics and Automation, vo. 8, no. 5, pp , 22. [5] G. Wang, G. Cao, and T. L. Porta, Movement-Assisted Sensor Depoyment, in IEEE INFOCOM, 24. [6] J. Cortes, S. Martinez, T. Karatas, and F. Buo, Coverage contro for mobie sensing networks, in Proceedings of the 22 IEEE Internationa Conference on Robotics and Automation, Washington, DC, 22. [7] J. Wu and S. Yang, SMART: A Scan-Based Movement-Assisted Sensor Depoyment Method in Wireess Sensor Networks, in IEEE INFOCOM, 25. [8] G. Wang, G. Cao, T. L. Porta, and W. Zhang, Sensor Reocation in Mobie Sensor Networks, in IEEE INFOCOM, 25. [9] S. Cheappan, X. Bai, B. Ma, and D. Xuan, Sensor Networks Depoyment Using Fip-Based Sensors, in Proceedings of 2nd IEEE Internationa Conference on Mobie Ad-Hoc and Sensor Systems (MASS 25), Washington, DC, USA, 25. [] J. Friedman, D. C. Lee, I. Tsigkogiannis, S. Wong, D. Chao, D. Levin, W. J. Kaiser, and M. B. Srivastava, RAGOBOT: A New Patform for Wireess Mobie Sensor Networks, in Proceedings of the st IEEE Internationa Conference on Distributed Computing in Sensor Systems (DCOSS 25), 25. [] Y. Zou and K. Chakrabarty, Sensor Depoyment and Target Locaization in Distributed Sensor Networks, ACM Transaction on Embedded Computing Systems, vo. 3, pp. 6 9, 24. [2] B. Carbunar, A. Grama, J. Vitek, and O. Carbunar, Redundancy and Coverage Detection in Sensor Networks, ACM Transactions on Sensor Networks, vo. 2, no., pp , 26. [3] R. Ghrist and A. Muhammad, Coverage and Hoe-detection in Sensor Networks via Homoogy, in The Fourth Internationa Symposium on Information Processing in Sensor Networks (IPSN 5), UCLA, Los Angees, Caifornia, USA, 25. [4] Q. Fang, J. Gao, and L. J. Guibas, Locating and Bypassing Routing Hoes in Sensor Networks, in INFOCOM 24, 24. [5] A. Genz, Numerica Computation of Mutivariate Norma Probabiities, J. Comp. Graph. Stat.,