IEEE TRANSACTIONS ON MOBILE COMPUTING,             VOL. 9,   NO. 6,   JUNE 2010                                           ...
898                                                                         IEEE TRANSACTIONS ON MOBILE COMPUTING,        ...
ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
900                                                                            IEEE TRANSACTIONS ON MOBILE COMPUTING,     ...
ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
902                                                                            IEEE TRANSACTIONS ON MOBILE COMPUTING,     ...
ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
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ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
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ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
908                                                                          IEEE TRANSACTIONS ON MOBILE COMPUTING,       ...
ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS                            ...
Accurate and Energy-Efficient Range-Free Localization for Mobile Sensor Networks
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Accurate and Energy-Efficient Range-Free Localization for Mobile Sensor Networks

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Accurate and Energy-Efficient Range-Free
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Accurate and Energy-Efficient Range-Free Localization for Mobile Sensor Networks

  1. 1. IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010 897 Accurate and Energy-Efficient Range-Free Localization for Mobile Sensor Networks Shigeng Zhang, Jiannong Cao, Senior Member, IEEE, Lijun Chen, and Daoxu Chen, Member, IEEE Abstract—Existing localization algorithms for mobile sensor networks are usually based on the Sequential Monte Carlo (SMC) method. They either suffer from low sampling efficiency or require high beacon density to achieve high localization accuracy. Although papers can be found for solving the above problems separately, there is no solution which addresses both issues. In this paper, we propose an energy efficient algorithm, called WMCL, which can achieve both high sampling efficiency and high localization accuracy in various scenarios. In existing algorithms, a technique called bounding-box is used to improve the sampling efficiency by reducing the scope from which the candidate samples are selected. WMCL can further reduce the size of a sensor node’s bounding-box by a factor of up to 87 percent and, consequently, improve the sampling efficiency by a factor of up to 95 percent. The improvement in sampling efficiency dramatically reduces the computational cost. Our algorithm uses the estimated position information of sensor nodes to improve localization accuracy. Compared with algorithms adopting similar methods, WMCL can achieve similar localization accuracy with less communication cost and computational cost. Our work has additional advantages. First, most existing SMC-based localization algorithms cannot be used in static sensor networks but WMCL can work well, even without the need of experimentally tuning parameters as required in existing algorithms like MSL*. Second, existing algorithms have low localization accuracy when nodes move very fast. We propose a new algorithm in which WMCL is iteratively executed with different assumptions on nodes’ speed. The new algorithm dramatically improves localization accuracy when nodes move very fast. We have evaluated the performance of our algorithm both theoretically and through extensive simulations. We have also validated the performance results of our algorithm by implementing it in real deployed static sensor networks. To the best of our knowledge, we are the first to implement SMC-based localization algorithms for wireless sensor networks in real environment. Index Terms—Mobile sensor networks, localization, sequential monte carlo methods. Ç1 INTRODUCTIONW IRELESS sensor networks (WSNs) have been used in many fields, including environmental and habitatmonitoring, precision agriculture, animal tracking, and WSNs is to use existing localization techniques, e.g., attaching a Global Positioning System (GPS) receiver on every sensor node. However, as the scale of sensor networksdisaster rescue. In many applications, it is essential for becomes larger and larger, these methods become infeasiblenodes to know their positions. For example, data should be because of their high cost or inconvenience. Many localiza-labeled with the positions where they are collected to help tion algorithms have been proposed in the past several years,the scientists perform corresponding analysis. Position [4], [5], [8], [10], [11], [12], [16], [18], [19], [20], [21], [22], [23],information of nodes are also necessary in many network [28], [30]. Depending on whether absolute range measure-protocols, e.g., clustering and routing which depend on the ments (point-to-point distances, angles, etc.) are used or not,geographical information of nodes. For example, geogra- they can be roughly classified into two categories [11]: range-phical routing protocols such as Greedy Perimeter Stateless based and range-free. Range-based algorithms usually needRouting (GPSR) [14] need to know nodes’ position some special hardware to obtain accurate absolute rangeinformation in order to select the next-hop relaying node. measurements and can achieve higher localization accuracy The procedure through which the nodes obtain their than range-free algorithms. Range-free algorithms, on thepositions is called localization. In localization, the nodes in a other hand, do not need special hardware and are low costlysensor network can be categorized into two types: beacon and more attractive in recent years.nodes which are aware of their positions and sensor nodes In some recently emerging applications such as animalwhich need to determine their positions using a localization monitoring and tracking [13], [29], sensor nodes may movealgorithm. A straightforward method for localization in after deployment. These nodes form mobile sensor networks in contrast to traditional static sensor networks in which sensor nodes remain stationary after deployment. The motion of. S. Zhang, L. Chen, and D. Chen are with the State Key Laboratory for sensor nodes makes most existing localization algorithms Novel Software Technology, Nanjing University, 22 Hankou Road, Nanjing 210093, China. designed for static sensor networks inapplicable to mobile E-mail: zsg@dislab.nju.edu.cn, {chenlj, cdx}@nju.edu.cn. sensor networks. The simple idea of executing these. J. Cao is with the Department of Computing, Hong Kong Polytechnic algorithms periodically in a mobile sensor network is University, Hung Hom, Kowloon, Hong Kong. infeasible, because this will incur high communication cost E-mail: csjcao@comp.polyu.edu.hk. and/or high computational cost [23]. There are someManuscript received 25 Dec. 2008; revised 9 June 2009; accepted 10 Sept. localization algorithms specially designed for mobile sensor2009; published online 23 Feb. 2010.For information on obtaining reprints of this article, please send e-mail to: networks, [2], [7], [12], [20], [23], [24], [27], [32]. All of them aretmc@computer.org, and reference IEEECS Log Number TMC-2008-12-0513. based on the Sequential Monte Carlo (SMC) method. This isDigital Object Identifier no. 10.1109/TMC.2010.39. because the posterior distribution of a sensor node’s position 1536-1233/10/$26.00 ß 2010 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  2. 2. 898 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010after move can be naturally formalized using a nonlinear We have evaluated the performance of the proposeddiscrete time model and the SMC method provides simple algorithms both theoretically and through extensive simula-simulation-based approaches to estimating the distribution. tions. We have validated the convergence of the proposed Previous SMC-based localization algorithms either suffer algorithms in real environments by implementing it in a realfrom low sampling efficiency or require high beacon density deployed sensor network. We also have implemented someto achieve high localization accuracy. In such localization other SMC-based localization algorithms and compared theiralgorithms, a sensor node’s position distribution is repre- performance with our proposed algorithms. The resultssented with a set of weighted samples. In order to obtain show that, even in static sensor networks, our proposedenough valid samples to accurately characterize the distribu- algorithms can effectively improve the localization accuracy.tion of its position, a sensor node needs to repeat the sampling The rest of this paper is organized as follows: In Section 2,step (generate candidate samples) and the filtering step we introduce the background of SMC-based localization(evaluate candidate samples and filter out invalid samples) algorithms and review related work. In Section 3, we describemany times. However, generating candidate samples is the WMCL algorithm in detail. WMCL-A, WMCL-B, and theusually a very costly operation. For example, in Section 6 Iterative WMCL (IWMCL) are also presented in this section.we will show that, on the Micaz [1] platform the cost of We theoretically analyze the localization accuracy of ourgenerating a candidate sample is much higher than evaluat- proposed algorithms in Section 4. Then we evaluate theing it. Because sensor nodes usually have limited computa- performance of the proposed algorithms under differenttional ability, it is necessary to improve the sampling parameter values by extensive simulations and carefullyefficiency in order to reduce the computational cost. Another analyze the results in Section 5. We also compare theproblem of most existing SMC-based localization algorithms performance of our proposed algorithms with other SMC-is that they only rely on increasing beacon density to improve based algorithms and present the results in this section. Inlocalization accuracy. However, beacon nodes are usually order to validate the performance results of SMC-basedmore expensive than sensor nodes. Because there are much localization algorithms, we implement the proposed algo-more sensor nodes than beacon nodes in a sensor network, it rithms and some other algorithms in real deployed staticwill be very beneficial if sensor nodes can be used to improve sensor networks and report the results in Section 6. Finally,the localization accuracy. we conclude this paper in Section 7. In this paper, we propose an energy efficient algorithmwhich addresses both aforementioned issues. The algorithmis based on the sequential Monte Carlo Localization (MCL) 2 BACKGROUND AND RELATED WORKalgorithm proposed in [12] and is named Weighted MCL 2.1 Background(WMCL). WMCL achieves high sampling efficiency and We consider a wireless sensor network in which both theachieves high localization accuracy even when the beacon beacon nodes and the sensor nodes can move. We assume thedensity is low. However, WMCL incurs much more commu- motion is markovian, i.e., a sensor node’s future position isnication cost than the original MCL algorithm. In order to only determined by its current position and is independentreduce the communication cost, we propose two approx-imate algorithms, WMCL-A and WMCL-B, which can with its past positions. Assume the time is divided intoachieve nearly the same localization accuracy as WMCL but discrete time units. A sensor node’s position flt ; t 2 N can be Ngincur much less communication cost. In fact, compared with modeled as a Markov process of initial distribution pðl0 Þ andMCL, WMCL-A, and WMCL-B incur only slightly additional transition equation pðlt jltÀ1 Þ. A sensor node obtains somecommunication cost (tens of bytes) but achieve much higher observations about its neighbors denoted by ot in time unit t.localization accuracy with much less computational cost. Given the process flt ; t 2 N the observations fot ; t 2 N Ã g Ng, N WMCL uses the following techniques to improve the are assumed to be conditionally independent and of marginalsampling efficiency and localization accuracy. In existing distribution pðot jlt Þ. Then, a sensor node’s position distribu-algorithms, a technique called bounding-box is used to tion is described asimprove the sampling efficiency by reducing the scope ofthe selecting of candidate samples. By using two-Hop pðl0 Þbeacon neighbors’ negative effects and sensor neighbors’ pðlt jltÀ1 Þ for t ! 1estimated position information, WMCL further reduces the pðot jlt Þ for t ! 1;average size of the sensor nodes’ bounding-boxes by afactor of up to 87 percent and, consequently, improves the and we want to compute pðlt jo1;...;t Þ.sampling efficiency by a factor of up to 95 percent. By using The SMC method provides a set of simulation-basedestimated position information of sensor nodes, WMCL approaches to computing this posterior distribution. In thisgreatly improves the localization accuracy. Compared with paper, we use an approach called particle filter [9] because itMSL* [20] which uses similar method, WMCL achieves is simple and easy to implement. In this approach, thesimilar localization accuracy but is much more efficient in distribution is represented with a set of N weightedterms of both communication cost and computational cost. samples [20]:Further more, it can be used in static sensor networks ÈÀ ðiÞ ðiÞ ÁÉwithout the need of experimentally tuning parameters as in pðlt jo1;...;t Þ % lt ; wt ; i¼1;...;Nalgorithms such as MSL*. Taking advantage of this ðiÞ ðiÞproperty, we propose a new algorithm in which WMCL is where lt is a sample of this distribution and wt is its P ðiÞiteratively executed with different assumptions on the normalized weight ( N wt ¼ 1). i¼1nodes’ speed. The new algorithm can dramatically improve Algorithm 1 shows a generic framework of SMC-basedlocalization accuracy when nodes move very fast. localization algorithms. There are three steps in the Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  3. 3. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 899algorithm: initialization, importance sampling, and filtering. In shrink its polygon. When sensor nodes are mobile, a sensorthe initialization step, N samples are randomly drawn from node dilates its polygon before broadcasting it. In thethe deployment area. In time unit t, in the importance algorithm proposed in [31], a sensor node records thesampling step, candidate samples are drown based on the history beacon information it has heard and uses a mobilitysamples in the previous time unit and their weights are pattern to predict its position.computed using the observations collected in time unit t. In There are localization algorithms specially designed forthe filtering step, samples with weight 0 are filtered out. The mobile sensor networks [2], [7], [12], [20], [23], [24], [27], [32].importance sampling step and the filtering step may repeat MCL [12], the first of such algorithms, assumes that all theseveral times in order to obtain enough number of candidate nodes are mobile and know their maximum speed. The timesamples with weight greater than 0. Then N samples are is divided into discrete time units and in each time unit all theselected from these candidate samples and their weights are sensor nodes update their positions. While achieving highnormalized. At last, the weighted average of the N samples localization accuracy, MCL left spaces for improvement inis used as the estimation of a sensor node’s position in the two aspects: Its sampling efficiency is low and it relies on highcurrent time unit. beacon density to achieve high localization accuracy. The Monte Carlo localization Boxed (MCB) algorithm [2] im-Algorithm 1. A framework of SMC-based localization proved MCL’s sampling efficiency by using bounding-boxalgorithms (called anchor box in their paper) to restrict the scope from 1: Step 1: Initialization which the candidate samples are drawn. Considering that 2: t 0 when many beacon nodes are observed a small number of 3: for i 1; N do samples will be enough to characterize the distribution of a ðiÞ 4: Sample l0 $ pðl0 Þ sensor node’s position, the authors in [27] proposed the 5: end for Sample Adaptive Monte Carlo Localization (SAMCL) algo- ð1Þ ðNÞ rithm which can adaptively determine the number of 6: L0 fðl0 ; 1=NÞ; . . . ; ðl0 ; 1=NÞg 7: t 1 samples hence can reduce the computational cost. WMCL 8: Lt ¼ fg further reduces the size of the bounding-box constructed in MCB by using two-Hop beacon neighbors’ negative effects 9: while jLt j < N do and sensor nodes’ estimated position information. The10: Step 2: Importance Sampling average size of the bounding-boxes is reduced by a factor of11: Ct ¼ fg up to 87 percent. This makes WMCL achieve higher sampling12: for i 1; N do efficiency than MCB even when the filtering conditions in ðiÞ ðiÞ13: Sample lt $ pðlt jltÀ1 Þ WMCL are much stricter than that in MCB. The techniques ðiÞ ðiÞ ðiÞ14: Evaluate the weight of lt as wt ¼ pðot jlt Þ ~ used in [27] can be used as a complementarity to WMCL to dynamically determine the number of needed valid samples. ðiÞ ðiÞ15: Ct ¼ Ct [ fðlt ; wt Þg ~ In MCL a sensor node only uses its beacon neighbors16: end for within two hops as its observation. Some works improve17: Step 3: Filtering MCL’s localization accuracy by using more information as ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ observation. The Mobile and Static sensor network Localiza-18: Ct0 ¼ fðlt ; wt Þjðlt ; wt Þ 2 Ct and wt > 0g ~ ~ ~19: Lt ¼ Lt [ Ct0 tion (MSL*) algorithm [20] uses neighboring sensor nodes’ estimated position information to improve the localization20: end while accuracy of MCL. In MSL*, a sensor node uses all its neighbors21: Lt ¼ choose(Lt , N) ==choose N valid samples wi ~ (both the beacon nodes and sensor nodes) within two hops to22: Normalize the weights of samples in Lt : wi ¼ PN t i t compute its position. In order to keep a low computational ~ w23: t t þ 1, goto line 8 i¼1 t cost, MSL* adopts a resampling strategy different from that of MCL. In MSL*, a sample in last time unit could be reserved in2.2 Related Work the current time unit with a probability proportional to itsWorks can be found on applying localization algorithms weight; new samples are drawn only when there are notdesigned for static sensor networks to mobile sensor enough samples inherited from last time unit. This makesnetworks. In [25], the authors investigated that, when such MSL* more suitable for sensor networks in which nodes movean algorithm was applied to a mobile sensor network, how slowly and perform poorly when nodes move very fast, asoften that algorithm should be executed to achieve required the simulation results in [20] show. In order to reduce thelocalization accuracy with minimum energy cost. In [3], the communication cost incurred in MSL*, the authors of [20] alsoauthors proposed an algorithm to localize mobile nodes proposed the MSL algorithm. MSL incurs much less com-with only two beacon nodes whose transmission range munication cost but achieves lower localization accuracycould cover the whole network. The distance between a than MSL*. Both MSL* and MSL suffer from low samplingsensor node and a beacon node is derived from the power efficiency and use some parameters whose values need to beof received signals and averaged on a given window to experimentally determined. In WMCL no such parameterscounteract interference and fading. Then triangulation is are used. Compared with MSL*, WMCL achieves muchused to compute the sensor node’s position. In [6], the higher sampling efficiency and similar localization accuracyauthors proposed an algorithm which could be used in both while incurring only about 2/3 communication cost. Westatic and mobile sensor networks. Each sensor node also propose two approximate algorithms, WMCL-A andconstructs a polygon that contains all its possible positions WMCL-B. They incur slightly higher communication costand broadcasts the description of this polygon to its than MSL but achieve much higher localization accuracyneighbors. A sensor node uses its received information to with much less computational cost. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  4. 4. 900 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010 There are some other variants of MCL, for example, thedual and Mixture MCL [24], Multihop-based Monte CarloLocalization (MMCL) [32], and Range-based MCL [7]. Thedual and Mixture MCL improves the localization accuracyof MCL by exchanging the probability functions used in thesampling step and in the filtering step. It incurs highercomputational cost than MCL. MMCL and Range-basedMCL use multihop sensor-beacon distances to improve thelocalization accuracy and to reduce the number of neededbeacons. Compared with them, WMCL doesn’t use multi-hop sensor-beacon distances so incurs much less commu- Fig. 1. How MCL [12] works: 1) The left figure shows the sample set LtÀ1 in time unit t À 1. 2) Sampling based on LtÀ1 and filtering usingnication cost. observations in time unit t.3 THE PROPOSED WMCL ALGORITHM model to study the performance of the proposed algorithm under irregular communication models.In this section, we describe our proposed WMCL algorithm The Q-UDG model will be introduced in Section 5.4.in detail. We first introduce the network model, then 3. Random motion of nodes. We assume that all thedescribe the three main parts of WMCL: bounding-box nodes can move after deployment and they haveconstruction, samples’ weights computing, and maximum some limited knowledge about their motion. Wepossible localization error computing. assume that every node knows its maximum speed The WMCL algorithm follows the generic framework defined by vmax .presented in Algorithm 1 and makes the following mod-ifications. Before starting sampling (line 8), each sensor For the sake of simplicity in description, we give somenode constructs a bounding-box from which the candidate definitions here.samples are drawn (Section 3.2). For each candidate sample, Definition 1 (Node/Beacon Density/Degree). The nodeits weight is computed with our proposed weight comput- density n and the beacon density s are defined asing methods (Section 3.3). Finally, before broadcasting theestimated position information to other sensor nodes, each n m n ¼ ; s ¼ ; ð1Þsensor node computes its maximum possible localization S Serror with our proposed maximum possible localization respectively. The node degree nd and the beacon degree sderror computing method (Section 3.4). represent the average number of nodes and of beacon nodes in a The iterative WMCL (IWMCL), which can dramatically node’s communication range and are defined asimprove the localization accuracy when nodes move veryfast, is described at the last of this section. nd ¼ n à r2 ; sd ¼ s à r2 ; ð2Þ3.1 Network Model respectively.Our algorithm can be applied to diverse network models. Definition 2 (SðsÞ; T ðsÞ; USðsÞ). For a sensor node s, we useHowever, in order to compare the performance of our SðsÞ to denote the set of its one-Hop beacon neighbors, T ðsÞ toalgorithm with the existing ones, we adopt a network denote the set of its two-Hop beacon neighbors, and USðsÞ tomodel consistent with them [2], [12], [20], [23], [24], [27]. Wehave the following assumptions: denote the set of its one-Hop neighboring sensor nodes. When it is clear which sensor node is referred to according to the 1. Uniform node deployment. We assume there are context, we use S; T ; US as their abbreviations. n nodes of which m are beacon nodes uniformly deployed in a planar rectangle area with size S. 3.2 Building the Bounding-Box Beacon nodes are assumed to be aware of their Fig. 1 illustrates how MCL works. There are two areas exact position information all the time. The node involved in MCL: the candidate samples area and the valid deployment strategy determines the initial distribu- samples area. The candidate samples area is used to draw tion of sensor nodes’ positions, say, pðl0 Þ. new candidate samples and the valid samples area is used 2. Unit Disk Graph (UDG) connectivity model. In the to filter out invalid samples. When the candidate samples UDG model, two nodes p and q can directly area is large and the valid samples area is small, candidate communicate with each other if and only if they samples drawn in the sampling step have high probability are within the communication range defined by a to be filtered out in the filtering step. In MCL, the possible radius r. If two nodes p and q are within r, we say q locations of a sensor node after move lie in a disk with is a one-Hop neighbor of p. If p and q are within 2r radius vmax . So the size of the candidate samples area will but not r, we say q is a two-Hop neighbor of p. The increase when vmax increases. On the other hand, when sd connectivity model determines the margin distribu- increases, the size of the valid samples areas will decrease. tion of observations, say, pðot jlt Þ. Denote by Vt the total number of candidate samples drawn The UDG model represents the ideal case. We use in the sampling step in time unit t and define the sampling this model to derive a lower bound on the localiza- tion accuracy of the proposed algorithm in Section 4. efficiency in t as However, the performance of the proposed algo- jLt j rithm will be affected by the irregularity of commu- et ¼ ; Vt nications. In our simulation, we use the Q-UDG Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  5. 5. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 901Fig. 2. Reduce the size of the bounding-box: The shadowed area shouldbe eliminated. Fig. 3. The explanation of approximate method 1. The probability of observing s at li equals to the probability of two nodes locating at Rt;sthen in MCL the sampling efficiency will decrease when t and li can communicate with each other, say, the probability of Rt;s tvmax or sd increases, which will cause high computational resides in area II. This probability can be approximated as the probabilitycost accordingly. In order to improve the sampling that RtÀ1;s resides in area I.efficiency, Baggio and Langendoen [2] use a techniquecalled bounding-box (called anchor-box in their paper) to 3.3 Weighting the Samplesreduce the size of the candidate samples area. Assuming 3.3.1 Three Approximate Methodsthat a sensor node has n one-Hop beacon neighbors, a In WMCL, ot ¼ S [ T [ US. So the weight of a candidatebounding-box ðxmin ; xmax ; ymin ; ymax Þ can be built as follows: sample is computed as À Á Y À Á Y À Á xmin ¼ maxn fxi À rg; xmax ¼ minn fxi þ rg i¼1 i¼1 wi ¼ p ot jli ¼ ~t p sjli p sjli : ð6Þ ð3Þ t t t ymin ¼ maxn fyi À rg; ymax ¼ minn fyi þ rg; i¼1 i¼1 s2S;T s2USwhere ðxi ; yi Þ means the coordinate of the ith beacon When s 2 S or s 2 T , pðsjli Þ can be easily computed [12], tneighbor. Two-Hop beacon neighbors are also used to [20]. If s 2 S, thenreduce the size of the bounding-box in [2] by replacing r À Á  À Á à p sjli ¼ d li ; s t t r : ð7Þwith 2r in (3). We further reduce the size of the bounding-box as follows: If s 2 T , thenSuppose that a bounding-box ðxmin ; xmax ; ymin ; ymax Þ has been À Á  À Á Ãbuilt as above. We use two-Hop beacon neighbors’ negative p sjli ¼ r d li ; s t t 2r : ð8Þeffects to reduce the size of the bounding-box. See Fig. 2 for However, when s 2 US, pðsjli Þ is not easy to compute. In tthe illustration. Assuming q is p’s two-Hop beacon neighbor, this case, what has been observed is not s’s real position, butthen the shadowed region doesn’t contain p; otherwise q will s’s estimated position in the last time unit. The probabilitybe p’s one-Hop neighbor. So we can eliminate the shadowed can be expressed asregion without any loss of valid samples. À Á È À Á Now suppose each sensor node knows its maximum p sjli ¼ P r d Rt;s ; li t t r ^ Rt;s $ pðRt;s jRtÀ1;s Þ É ð9Þlocalization error in x-axis ERx and maximum localization ^ RtÀ1;s % posEst ;error in y-axis ERy (how to compute them will be explainedin Section 3.4). Let us use Rt;p and Et;p to denote a sensor where % means RtÀ1;s is estimated as posEst in time unitnode p’s real position and estimated position in time unit t, t À 1. It can be explained as the probability of the followingrespectively. Assuming q 2 USðpÞ, we have: event: A sensor node can communicate with s whose position in the last time unit is RtÀ1;s and current position follows the jRt;p À Rt;q j r distribution pðRt;s jRtÀ1;s Þ and RtÀ1;s is estimated as posEst. jRt;q À RtÀ1;q j vmax According to different forms of posEst a sensor node p ð4Þ receives, we propose three approximate methods to jXðRtÀ1;q Þ À XðEtÀ1;q Þj ERx;tÀ1 ðqÞ compute the probability pðsjli Þ:t jY ðRtÀ1;q Þ À Y ðEtÀ1;q Þj ERy;tÀ1 ðqÞ; 1. (Used in WMCL) If posEst ¼ Ls , then we can usewhere XðÁÞ/Y ðÁÞ means the x/y value of Á and ERx;t ðqÞ/ tÀ1 the following formula to compute an approximationERy;t ðqÞ means q’s ERx /ERy in time unit t. The bounding- of pðsjli Þ: tbox can be further shrunk (take xmin as an example):  À iÁ à P À iÁ l2Ls d l; lt r þ vmax xmin ¼ maxfxmin ; maxq2USðpÞ fXðEtÀ1;q Þ À vmax À r p sjlt % tÀ1 : ð5Þ jLs j tÀ1 À ERx;tÀ1 ðqÞgg: See Fig. 3 for the illustration. The probability of A sensor node’s estimated position estimation in last observing s at li equals to the probability of two nodes ttime unit can also be used to shrink the bounding-box. Take locating at Rt;s and li can communicate with each txmin as an example: other, say, the probability of Rt;s resides in area II. This xmin ¼ maxfxmin ; XðEtÀ1;p Þ À vmax À ERx;tÀ1 ðpÞg: probability can be approximated as the probability that RtÀ1;s resides in area I. After the bounding-box is built, the candidate samples Note here we only need to transmit the positions ofare drawn from the bounding-box. samples in Ls without transmitting their weights, tÀ1 Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  6. 6. 902 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010 Fig. 5. (a) Computing ERx and ERy ; the biggest dot is ðxe ; ye Þ. (b) A way to reduce ERx and ERy with little risk.Fig. 4. The explanation of approximate method 2. We can use SIII =SIas an estimation of the weight computed using approximate method 1. show that WMCL achieves nearly the same accuracy as MSL* when vmax is small. while in MSL* the weights also need to be transmitted. This can greatly reduce the communication cost. 3.4 Computing ERx and ERy 2. (Used in WMCL-A) The first method needs to know After obtaining N valid samples, a sensor node computes the whole set of samples of all the sensor neighbors, the weighted average of these samples as its position which incurs high communication cost. As illu- estimation. Using the position estimation and the bounding- strated in Fig. 4, if we know s’s bounding box Bs , we box, a sensor node can compute its ERx and ERy , as can approximately compute the probability as illustrated in Fig. 5. À Á SB B Assume p’s position estimation is ðxe ; ye Þ and its bound- p sjli % s ; t ing-box is ðxmin ; xmax ; ymin ; ymax Þ. It is obvious that ERx is SBs maxfxe Àxmin ; xmax Àxe g and ERy is maxfye Àymin ; ymax Àye g. where B represents the square centering at li with t We can reduce ERx and ERy with some risks. Assume p’s side-length r þ vmax . sample set is fðx1; y1Þ; . . . ; ðxN ; yNÞg. Let x0min ¼ minfx1; . . . ; xNg 3. (Used in WMCL-B) In our previous work [23], we use and we can further refine xmin to be ðxmin þ x0min Þ=2. Do the the following formula to compute the value of pðsjli Þ: t same to xmax , ymin and ymax we can get reduced ERx and ERy . À iÁ Â À Á Ã p sjlt ¼ d EtÀ1;s ; li vmax þ r þ ERtÀ1 ðsÞ ; A more riskily method is to use the smallest rectangle t enclosing all of p’s valid samples to compute ERx and ERy . where ERtÀ1 ðsÞ means the maximum localization This method can improve localization accuracy a lot. error of s in time unit t À 1 and is computed as However, when using this method the procedure of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi constructing the bounding-box should be carefully manipu- ER2 ðsÞ þ ER2 ðsÞ: x;tÀ1 y;tÀ1 lated. In this case the inequality presented in (4) may not hold and some inconsistence may happen. For example, it is We call the algorithms using the three different weight possible that xmin is larger than xmax and consequently thecomputing methods WMCL, WMCL-A, and WMCL-B, bounding-box cannot be built.respectively. After p gets ðxe ; ye Þ and ERx ; ERy , it broadcasts them to its neighbors. Its neighbors will use these information to3.3.2 Approximation Analysis compute their position estimation in the next time unit.In this section, we analyze the effects of different approx-imate methods. The accurate value of pðsjli Þ defined in (9) is 3.5 Iterative WMCL tdifficult to compute. A good approximation method is The localization accuracy of WMCL will degrade when vmaxproposed in [20] and is used in the MSL* algorithm. As increases. This is because when vmax increases, theillustrated in Fig. 3, this probability can be estimated as constraints introduced by sensor nodes’ estimated position À Á È É information become very weak and cannot be used to p sjli ¼ P r l 2 Ls ^ l 2 I t tÀ1 improve the localization accuracy effectively. X Â À Á Ã % wi d l; li r þ vmax : In WMCL the candidate samples are directly drawn from t t l2Ls tÀ1 the bounding-box so it is also applicable even when vmax ¼ 0. Here, we propose IWMCL, the Iterative WMCL, The weight computing method used in WMCL computes which can alleviate the negative effects of vmax on WMCL’san approximate value of this probability. Assume there are localization accuracy. The idea is simple: After all sensorM valid samples in Ls and M 0 of them reside in area I. tÀ1 nodes obtain their position estimation in time unit t, theyThen in WMCL pðsjli Þ is approximated as t broadcast their position estimations to their neighbors. À Á M0 Upon receiving the new position estimations, the WMCL is p sjli % t : executed with the assumption that vmax equals to 0. M The IWMCL algorithm is described as follows: There areIn other words, when computing the value of pðsjli Þ in t two phases in IWMCL:WMCL, we assume that s’s position in last time unit followsa uniform distribution. So the difference between the real 1. Phase 1: normal run of WMCL. Every sensor nodevalue of pðsjli Þ and the approximated value computed in t collects its sensor neighbors’ position estimations inWMCL depends on the real distribution of s’s position in last time unit and computes its position estimationlast time unit. Our simulation results presented in Section 5 using WMCL. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  7. 7. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 903 Z 1 À Á EðzÞ ¼ z d 1 À eÀðs 8rzþn 4rzÞ 0 r ¼ : 8sd þ 4nd However, because the sensor nodes’ estimated position information is inaccurate so we can speculate that it will be very difficult to achieve this lower bound in a real implementation of WMCL.Fig. 6. A lower bound on localization error for connectivity-based Rudafshani and Datta [20] have argued that this lowerlocalization algorithms [17]. The summation of the area of I and the areaof II is about 4rz. bound is not rigorous when node degree is low (nd 10). However, as the results in their paper showed, when node degree is high this lower bound can still give some insights 2. Phase 2: special run of WMCL with the assumption about what is the best localization accuracy an algorithm vmax ¼ 0. After every sensor node gets its position which depends on only local connectivity status can achieve. estimation in the current time unit, it broadcasts this information to its neighbors. Then every sensor node runs WMCL with the assumption that vmax ¼ 0. This 5 PERFORMANCE EVALUATION THROUGH phase can repeat several times in order to improve SIMULATION the localization accuracy. In this section, we evaluate the performance of the proposed By replacing WMCL with WMCL-A the communication algorithms through extensive simulations. We modify thecost can be dramatically reduced. widely used simulator developed by Hu and Evans [12] to implement the proposed algorithms. In order to evaluate4 THEORETICAL ANALYSIS ON THE LOWER BOUND our contributions in improving sampling efficiency, we also OF THE LOCALIZATION ACCURACY implement the MCL algorithm and the MCB [2] algorithm and compare their sampling efficiency with our algorithms.In this section, we will derive a lower bound on the In order to evaluate our contribution in improvinglocalization accuracy of our algorithm. We use the method localization accuracy, we compare our methods with theproposed by Nagpal et al. [17]. The main idea of this method methods proposed in MSL [20]. Because MSL* and MSL useis described as follows: The sensor network can be different resampling strategy from that of MCL [12], forrepresented as a graph in which each vertex corresponds consistency in comparison, here we implement variants ofto a sensor node in the network. There is an edge between MSL* and MSL with the same resampling strategy as intwo vertices if and only if their corresponding nodes in the MCL and call them VMSL* and VMSL, respectively.1 Thesensor network can communicate with each other. Because communication cost of VMSL* and VMSL are the same aseach sensor node only uses its neighbors’ position informa- that of MSL* and MSL.tion to estimate its position, when a sensor node move from We consider the following metrics: 1) localizationone position to another position, the two positions will be accuracy, 2) computational cost, and 3) communication cost. The main parameters we consider are nodes’ max-indistinguishable if the set of the sensor’s neighborhood imum speed vmax , node degree nd , and beacon degree sd , fordoesn’t change. Then the average distance a sensor node can they are the main factors affecting the algorithms’ perfor-move without changing its connectivity status gives a lower mance. When considering the impact of communicationbound on the expected localization accuracy. irregularity on the algorithms’ performance we use the Q- Fig. 6 shows the case in which only one-Hop neighbors UDG model [15]. We tune the value of d in the Q-UDGare used in the localization. Assuming a UDG connectivity model to simulate different irregularity of communications.model, the probability that a sensor node can move z We assume that all the nodes are uniformly deployed in awithout affecting its connectivity status equals to the 500 units  500 units square region and the communicationprobability that at least one other nodes reside in area I range r is set to be 100 units. We use a modified Randomand II. When the nodes are uniformly distributed in the Waypoint mobility model proposed in the MCL algorithmdeployment region, the probability that k nodes reside in a [12] as the motion model. Unless otherwise specified, thegiven region R with size a follows a binomial distribution default values of the parameters are: nd ¼ 10, sd ¼ 1,Bðn; a=SÞ and can be approximated using a Poisson vmax ¼ :2r. The number of effective samples is set todistribution because n is very large: 50 according to [12], [20]. The value of necessary parameters needed in VMSL* and VMSL are set to the same value as ðn aÞk Àn a used in the original MSL* and MSL algorithm [20]. The P rðk nodes in RÞ ¼ e : ð10Þ localization error is measured as a multiple in r. For different k! parameter values, we randomly generate 30 networks and inThe summation of the area of I and the area of II is about 4rz each network we run 1,000 time units and average the[12]. So we have: metrics between time unit 600 and time unit 1,000. As we have studied in [33], with this setting the obtained data are F ðzÞ ¼ 1 À eÀn 4rz : stable in the sense that the difference between the results of In WMCL, each sensor node uses its beacon neighbors different simulation procedures will be less than 0:01r. Allwithin two hops and sensor neighbors within one hop to 1. We replaced the weight computing function used in MSL* with ourestimate its position. So for WMCL, the localization proposed methods in the simulation code provided by the authors of [20]accuracy is bounded below by and found that they achieve similar localization accuracy. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  8. 8. 904 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010Fig. 7. How sd , nd , and vmax affects the localization accuracy. (a) sd ; (b) nd ; (c) vmax .the data points presented here are the average of these position of node s. The value of b is experimentally set to 730 independent experiment results. in VMSL. From (6), we can conclude that when closenesss is large or nd is large, wi ¼ pðot jli Þ will approach to 0. This ~t t5.1 Localization Accuracy means that in VMSL a sensor node may not obtain enoughLocalization accuracy is the most important metric in valid samples to represent its position distribution when ndevaluating localization algorithms. we study how the increases. In fact, in our simulation the number of validlocalization accuracy of different algorithms varies when samples in VMSL drops to 38 when nd is 26, while in allsd , nd , and vmax varies. other algorithms the number of valid samples always Fig. 7a shows how the localization accuracy of different remain about 50.algorithms varies when sd increases. When sd increases, in Fig. 7c shows how the localization accuracy varies whenall algorithms the localization accuracy improves. However, vmax increases. From our analysis in Section 4, the localiza-VMSL* and our proposed algorithms perform much better tion accuracy should not be affected by vmax if enough validthan other algorithms. This is because VMSL* and our samples can be obtained. However, in Fig. 7c we can findproposed algorithms use sensor nodes’ estimated position that localization accuracy first improves then degrades.information to improve localization. We can also see that We explain this as follows: As illustrated in Figs. 8 and 9,WMCL-A and WMCL-B achieve nearly the same localiza- vmax can impact the “quality” of the generated candidatetion accuracy as WMCL and VMSL*. samples in the sampling step. In the two figures, the disk Fig. 7b shows how the localization accuracy varies when denoted by I represents the whole valid samples area of and increases. We can see that the localization accuracy of sensor node in last time unit, and the ellipse denoted by IIVMSL* and our proposed algorithms improve when nd represents the valid samples area covered by the sensorincreases. We can also see that, when nd 15, the accuracy of node’s sample set in last time unit. PtÀ1 and Pt are the realMCL and MCB is also improved slightly when nd increases. positions of the sensor node in last time unit and in currentThe reason is as follows: When nd is small, the network is time unit, respectively. We can see that when vmax is small,very sparse and maybe some beacon nodes within 2r of a only a small part of the valid samples area in current timesensor node cannot be found as two-Hop beacon neighbors. unit can be covered by the candidate samples area. BecauseWhen nd 15, almost all the beacon nodes within 2r of a all the valid samples are drawn from this part, the positionsensor node can be found and the increase in nd will not estimation of a sensor node will be inaccurate. When vmax isaffect the localization accuracy of MCL and MCB any more. large, the candidate samples area can cover a larger part of It is strange to see that for VMSL the localization the valid samples area. So when vmax increases, theaccuracy first improves then degrades when nd increases. localization accuracy can be improved. However, afterIn fact, in VMSL the value of pðsjli Þ is approximated as [20] t vmax exceeds a threshold, the candidate samples area can À iÁ cover the whole valid samples area and the localization p sjlt ¼ bÀclosenesss ; accuracy should not be affected by vmax any more.where closenesss is a metric representing the averagedistance between all the valid samples and the estimatedFig. 8. When vmax is small, the candidate samples area can only contain Fig. 9. When vmax is large, the candidate samples area can contain aa small part of the valid samples area. large part of the valid samples area. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  9. 9. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 905Fig. 10. When vmax increases, the real beacon degree decreases Fig. 11. IWMCL can effectively improve localization accuracy when vmaxslightly. is large (iterate four times). On the other hand, when vmax increases, the average much higher than evaluating it. So the sampling efficiency isnumber of beacon neighbors a sensor node can hear a very import metric in SMC-based localization algorithmsdecreases,2 as shown in Fig. 10. Combing these two factors because higher sampling means less candidate sampleswe can explain why vmax affects different algorithms in generation and consequently less computational cost.different manners. In our proposed algorithms, the bound- Fig. 12a shows how the number of candidate samplesing-box always contains the whole valid samples area, so it varies when sd increases in different algorithms. As we haveis only affected by the second factor. Other algorithms are pointed out in Section 3.2, the sampling efficiency degradesaffected by both factors so their accuracy first improves fast in MCL and VMSL* when sd increases. On the otherbecause more possible samples are obtained and then hand, the sampling efficiency in our proposed algorithmdegrades after vmax exceeds a threshold. The reason that and MCB only degrades slightly when sd increases. TheMCB is also affected by the first factor is that in its sampling efficiency in our proposed algorithms is slightlyimplementation vmax is used to construct a smaller rectangle higher than that in MCB but the accuracy in our algorithmsfrom which the candidate samples are drawn. is much higher than that in MCB. From Fig. 10, we can see that when vmax becomes larger, In Fig. 12a, we can see that the sampling efficiency inthe localization accuracy of our proposed algorithms VMSL* and VMSL is higher than that in MCL. However, indegrade fast (even worse than MCL when vmax 1:2r). This VMSL* and VMSL a parameter call (set to be .1r in theis because when vmax is large, the constraints introduced by simulation, the same as in [20]) is added to vmax insensor neighbors become very weak so cannot effectively generating candidate samples. This parameter enlarges theimprove the localization accuracy. The IWMCL algorithm candidate samples area so intuitively the sampling effi-we proposed in Section 3.5 can achieve high localization ciency in VMSL* and VMSL should be lower than that inaccuracy even when vmax is large. In Fig. 11, we can see that MCL. It can be explained as follows: The parameter hasIWMCL and IWMCL-A can effectively migrate the negative two impacts on the sampling efficiency. First it mayeffects of vmax on the localization accuracy of WMCL. More degrade the sampling efficiency because it enlarges theimportantly, the computational cost and the communication candidate samples area. Second it may improve thecost of IWMCL-A is much lower than that of VMSL*. So we sampling efficiency because by adding it the candidatecan use IWMCL-A instead of VMSL* in a network in which samples area can cover more valid samples area, as shownnodes move very fast. in Figs. 15 and 16. When vmax is small, the second impact5.2 Computational Cost dominates the first impact, so VMSL* and VMSL achieve higher sampling efficiency than MCL. When vmax is large,In SMC-based localization algorithms, the computational the second impact becomes small and VMSL* and VMSLcost consists of two parts: the cost in generating candidate should achieve similar sampling efficiency as MCL, assamples and the cost in evaluating the candidate samples. shown in Fig. 12b. To verify our conclusion, we modifyThere are two key operations in evaluating the candidate VMSL* and VMSL with the same transition equation ofsamples: computing the distance between two samples and MCL (without adding ). The results are shown in Fig. 12c.comparing the distance with a predefined value (e.g., the We can see that VMSL* and VMSL achieve similar samplingcommunication radius r). In VMSL, there are another key efficiency as MCL. This also explains why in MCL theoperation used in evaluating a candidate samples: comput- sampling efficiency first improves then degrades when vmaxing the value of 7x where x is a negative. In this section, we increases, as shown in Fig. 12b.first compare the cost in generating candidate samples in Fig. 12b shows how the number of generated candidatedifferent algorithms, then compare the cost in evaluating samples varies in different algorithms when vmax increases.candidate samples in different algorithms. We can find that the sampling efficiency in MCL, VMSL*,5.2.1 Generating Candidate Samples and VMSL degrades dramatically when vmax increases. This is because when vmax increases, the size of the candidateAccording to the cost of different operations listed in samples area increases dramatically in these algorithms. OnSection 6.1, the cost of generating a candidate sample is the other hand, in our proposed algorithms and MCB, the 2. The decrease is only observed from the simulation. We cannot explain candidate samples area is bounded by the bounding-boxit theoretically now and consider this as a future work. and so is not effected by vmax . Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  10. 10. 906 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010Fig. 12. How sd , nd , and vmax affects the sampling efficiency. The sampling efficiency is improved by a factor of up to 95 percent (compared withMCL). (a) sd ; (b) vmax ; and (c) sd without . Figs. 13 and 14 shows the size of the bounding-box builtin MCB and in our algorithms. We can see that ouralgorithms effectively reduce the size of bounding-box. Theimprovement factor is about 63 percent when sd varies andfrom 4 percent to 87 percent when vmax varies. In Fig. 14 wecan also find that when vmax increases, the improvementbecomes smaller and smaller. This means that the localiza-tion accuracy of the proposed algorithms degrades to thelocalization accuracy in MCB when vmax increases. In thiscase, IWMCL and IWMCL-A can be used to improve thelocalization accuracy.5.2.2 Evaluating Candidate Samples Fig. 13. The average size of bounding-boxes when sd varies.There are two types of operations in evaluating a candidatesample: computing the distance3 between two points andcomparing the distance with a predefined value. As listedin Section 6.1, the cost of the first type of operation is nearly100 times higher than the cost of the second type ofoperation. So in this section, we only compare the numberof distance computing in evaluating candidate samples indifferent algorithms. Some algorithms need some otherspecial type of operations. For example, in the bounding-box building phase of our proposed algorithms, it needs tocompute the square root of a number; in the VMSLalgorithm, it needs to compute a power of a number. Wealso compare such computational costs. Fig. 17 shows how the computational cost in evaluating Fig. 14. The average size of bounding-boxes when vmax varies.candidate samples varies when sd , nd , and vmax increases.We can see that when sd and vmax varies, the number ofdistance computing operations is approximately propor-tional to the sampling efficiency showed in Fig. 12. FromFig. 17b, we can see that the computational cost inevaluating candidate samples in WMCL-A and WMCL-Bincreases proportional to nd . However, the cost in these twoalgorithms is still always (except in WMCL-A when nd is Fig. 15. When vmax is small, adding can greatly improve the samplingvery large) less than MCL. This is due to the high sampling efficiency.efficiency in these two algorithms. Table 1 lists the cost in evaluating candidate samples inWMCL and VMSL*. We can see that the difference betweenthe cost in the two algorithms is not very large although thesampling efficiency in WMCL is much higher than that inVMSL*. This is because in these two algorithms, the majorsource of distance computing is in the computing of pðot jlt Þand this is not effected much by the sampling efficiency. 3. Here, we define the distance between two points p and q as kp À qk, Fig. 16. When vmax is large, the improvement in sampling efficiency isbecause jp À qj r is equivalent to kp À qk r2 . small. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  11. 11. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 907Fig. 17. Number of distance computing in different algorithms. (a) sd ; (b) nd ; and (c) vmax .However, WMCL still incurs less computational cost in needs to broadcast its position information and each nodeevaluating candidate samples and incurs much less needs to rebroadcast the information it received from itscomputational cost in generating candidate samples. beacon neighbors, so the communication cost is Oðsd þ 1Þ. For WMCL, WMCL-A, and WMCL-B, there are some In WMCL and VMSL*, the communication cost isadditional computational cost incurred in the phase of OðN þ sd þ 1Þ, where N is the number of samples a sensorbounding-box construction. This is because we need to node reserves in each time unit. This is because in WMCLcompute the intersection points of a circle and a rectangle. and VMSL*, every sensor node also needs to broadcast isThis needs to compute the square roots of some number. estimated position information obtained in last time unit. InHowever, the number of this type of operations is only WMCL-A, WMCL-B, and VMSL, the communication cost ofapproximately equal to the number of two-Hop beacon each node is Oðsd þ 1 þ kÞ, where k represents the addi-neighbors and is very small. For the VMSL algorithm the tional information a sensor node broadcasts. In WMCL-Aoperation of computing the power of a given number also and WMCL-B, it is the estimated position in last time unit and ERx and ERy . In WMSL, it is the closeness of the sensorincurs large computational cost. The number of this node in last time unit.operation is proportional to the number of generated Figs. 18 and 19 shows the communication cost of eachcandidate samples in VMSL. node in different algorithms when sd and nd increase. We5.3 Communication Cost assume each point is represented with two 4-byte integer numbers and the closeness is represented with one 4-byteFirst we theoretically analyze the communication cost of float number. We can see that the additional communica-different algorithms. In MCL and MCB, each beacon node tion cost incurred in WMCL-A and WMCL-B is very limited, but is slightly higher than that in VMSL. TABLE 1 Table 2 lists the communication cost for each node in Number of Distance Computing in WMCL and VMSL* (Â104 ) WMCL and VMSL* when sd and nd increases. We can find that the communication cost in these two algorithms is dominated by the cost in transmitting the whole set of the samples and is not affected a lot by sd and nd . However, VMSL* always incurs about 50 percent more communica- tion cost than WMCL. 5.4 The Impact of Irregular Communications In this section, we study the performance of the proposed algorithms under irregular communications. We use the Quasi Unit Disk Graph (Q-UDG) model presented in [15] toFig. 18. Number of bytes transmitted when sd varies. Fig. 19. Number of bytes transmitted when nd varies. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  12. 12. 908 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010 TABLE 2 TABLE 3 Number of Bytes Transmitted in WMCL and VMSL* (Bytes) Cost of Different Operations (50,000 Times) Fig. 21. Localization accuracy of WMCL-A and Centroid in one randomly selected network. ATmega128L microcontroller. The operating system used is TinyOS 2.1. For each type of operation, we execute that operation 50,000 times and measure the time used to complete the execution. The results are listed in Table 3. We use two different random generators provided inFig. 20. Localization accuracy versus communication irregularity. TinyOS 2.1 (implemented in component RandomC and RandomLfsrC, respectively) to measure the cost of generat-study the the impact of irregular communications on the ing candidate samples. The letter one is much faster butlocalization accuracy of different algorithms. with less randomness. We can see that generating a In the Q-UDG model, there is a parameter d 2 ½0; 1Š candidate sample is much costlier than computing thewhich is used to control the irregularity of the communica- distance between two samples. We also measured the costtions. Two nodes are connected by an edge if their distance of computing the distance between two points representedis at most dÃr where r is the maximum communication using two float numbers. The cost is 0.706 s and is still muchrange as defined in the UDG model. If the distance between lower than the cost of generating a candidate samples.two nodes is greater than r, then no edge exists between The power operation needed in the VMSL algorithm andthem. When the distance is between dÃr and r, we assume the sqrt operation needed in our proposed algorithms is alsothat the probability that there is an edge between the two very costly. However, in our proposed algorithms, thenodes follows a uniform distribution. number of the sqrt operations is very small (usually less Fig. 20 shows the localization accuracy of different than sd ) but in VMSL the number of power operations isalgorithms when the value of the parameter d varies from much larger (proportional to the number of generated1 to 0.5. We can see that when d is larger than 0.8, the local- candidate samples). So it is necessary to improve samplingization accuracy of the proposed algorithms only degrades efficiency in SMC-based algorithms because this canslightly. However, when d is smaller than 0.8, the localization dramatically reduce the computational cost.accuracy degrades dramatically. However, the proposedalgorithms always perform better than other algorithms. 6.2 Validate the Convergence of the Proposed The irregular communication not only degrades the Algorithmslocalization accuracy, but also incurs more computational In the first set of experiments, we validate the convergencecost. In our simulation, in all the algorithms the computa- of our proposed SMC-based algorithms in a real deployedtional cost increases by a factor of at least 2 when d changes sensor network. We deployed 40 Micaz motes in a 2:2 m Âfrom 1 to 0.5. 2:2 m square region on the ground of our laboratory (see Fig. 22a). Twelve of them were deployed on the perimeter6 EXPERIMENTS of the region as beacon nodes, and others were randomlyWe have implemented our algorithms in real deployed deployed inside the deployment area. We carefully adjustedsensor networks to validate the obtained performance results. the transmitting power of the Micaz motes such that theWe first measured the cost of different operations in real transmission range is about 60 cm.hardware to justify our efforts in reducing the computational We run the WMCL-A algorithm on the sensor nodes andcost. After that, we validate the convergence of the proposed the result is shown in Fig. 21. From Fig. 21, we can see thatalgorithms in real environments. At last, we implement five WMCL-A converges fast in a static sensor network. ThisSMC-based localization algorithms: MCL, MCB, VMSL, validates the effectiveness of our proposed algorithms. ForWMCL-A, and WMCL-B in a sensor network consists of comparison, we also implemented the Centroid algorithm23 Micaz motes and compare their localization accuracy. [5] which is often used as a comparison algorithm in real implementation. We run the two algorithms in five6.1 Cost of Key Operations randomly generated networks and found that the localiza-We have measured the cost of different operations on the tion accuracy of WMCL-A was about 30 percent higher thanMicaz platform [1]. The Micaz mote equips with an Atmel that of Centroid. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.
  13. 13. ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 909Fig. 22. The deployment of sensors. (a) The indoor scenario, and (b) the outdoor scenario.6.3 Real Implementation in a Outdoor Network algorithms on the improvement of localization accuracy andWe have implemented the aforementioned SMC-based reduction of energy consumption.localization algorithms and evaluated their localization In our simulations, we find that irregular communica-accuracy in a static sensor network. We deployed 23 Micaz tions will not only degrade the localization accuracy of themotes in a 6 M Â 6 M square region on a volleyball court proposed algorithms but also will incur much more(see Fig. 22b). We set the transmitting power of the sensor computational cost. In our implementation of the proposednodes to be about À10 dBm so that the transmission range is algorithms in real environment, we find it is difficult toabout 3 m. Six of the twenty-three nodes were placed on the determine a maximum transmission range between twoperimeter of the deployment region and others were nodes. The connectivity between two sensor nodes is veryrandomly deployed inside the region. unstable. The connectivity status may vary greatly in Fig. 23 shows the localization error of every sensor node different times, in different places, or even betweenin different algorithms. The average localization error in different sensor nodes. In this case, the performance in realdifferent algorithms is listed in Table 4. We can see that our implementation will be much worse than the performanceproposed algorithms outperform other algorithms. The obtained from simulations.improvement factor is from 23 percent to 37 percent. ACKNOWLEDGMENTS7 CONCLUSION The authors would like to thank the anonymous reviewersIn this paper, we present an accurate and energy efficient and the editors for their invaluable suggestions thatrange-free localization algorithm for mobile sensor net- improved the quality of this paper. They also thank Chaoworks. We measured the cost of key operations in SMC- Yang and Yingpei Zeng for the help in collecting experi-based localization algorithms on real hardware and found mental data. The authors gratefully acknowledge the codethat a high sampling efficiency is necessary to reduce the sent by Hu and Evans and the code sent by Masoomehcomputational cost. Then, we propose a set of algorithms Rudafshani. This work has been supported in part by thewhich achieve high sampling efficiency and high localiza- National Natural Science Foundation of China under Granttion accuracy. The results from our simulations and No. 60873026, 60573132, 60673154, 90718031, and 60721002,experiments validate the effectiveness of our proposed the National Grand Fundamental Research 973 Program of China under Grant No. 2006CB303000 and 2009CB320705, HK RGC under GRF grant PolyU 5102/08E and HK PolyU under grant 1-BB6C. REFERENCES [1] http://www.xbow.com/products/product_pdf_files/wireless_ pdf/6020-0060-01_a_mic az.pdf, 2008. [2] A. Baggio and K. Langendoen, “Monte-Carlo Localization for Mobile Wireless Sensor Networks,” Proc. Conf. Mobile Ad-Hoc andFig. 23. Localization accuracy of different algorithms in the outdoor Sensor Networks (MSN ’06), pp. 317-328, 2006.deployment. [3] P. Bergamo and G. Mazzini, “Localization in Sensor Networks with Fading and Mobility,” Proc. 13th IEEE Int’l Symp. Personal, TABLE 4 Indoor and Mobile Radio Comm. (PIMRC ’02), pp. 75-754, 2002. Average Localization Error in Different Algorithms [4] P. Biswas and Y. Ye, “Semidefinite Programming for Ad Hoc Wireless Sensor Network Localization,” Proc. Third Int’l Symp. Processing in Sensor Networks (IPSN ’04), 2004. [5] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-Less Low Cost Outdoor Localization for Very Small Devices,” IEEE Personal Comm., vol. 7, no. 5, pp. 28-34, Oct. 2000. Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.

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