MDS-based Algorithm for Nodes Localization in 3D Surface Sensor Networks

Transcription

1 MDS-based Algorthm for odes Localzaton n 3D Surface Sensor etworks Bljana Rsteska Stojkoska, Danco Davcev Faculty of Computer Scence and Engneerng Unversty Ss. Cyrl and Methodus Skopje, Macedona bljana.stojkoska@fnk.ukm.mk, danco.davcev@fnk.ukm.mk Abstract As Wreless Sensor etwork (WS) has become a key technology for dfferent types of smart envronment, nodes localzaton n WS has arsen as a very challengng problem n the research communty. Most of the applcatons for WSs necesstate a prory known nodes postons. In ths paper, we propose an algorthm for three dmensonal (3D) nodes localzaton n surface WS based on multdmensonal scalng (MDS) technque. Usng extensve smulatons, we nvestgated n detals our approach regardng dfferent network topologes, varous network parameters and performance ssues. The results from smulatons show that our algorthm produces small localzaton error and outperforms MDS-MAP n terms of accuracy. Keywords-wreless sensor networks; multdmensonal scalng; 3D surface localzaton; nodes postonng. I. ITRODUCTIO A wreless sensor network (WS) s a network of autonomous dstrbuted sensor devces that obtan varous measurements of dfferent real-lfe occurrences [][]. After takng samples of physcal or envronmental condtons at dfferent locatons (lght level, ar temperature, humdty, etc.), each sensor sends data to ts closest neghbor responsble for retransmttng the packets [3]. The fnal destnaton s the snk node responsble for storng data or for further processng. Although ntally developed for mltary applcatons, today, WSs are used n many ndustral and cvlan applcaton areas, habtat montorng, healthcare applcatons and traffc control [4]. odes localzaton s the bass for many applcatons of WS, such as event detecton and target trackng. A manual dsposton s mpossble not only for large scale WSs, but also when a WS s deployed on naccessble terran. The most straghtforward soluton to the localzaton problem s to apply Global Postonng System (GPS) recevers to each node [5]. But t s an expensve soluton and napplcable for ndoor envronments [6][7]. Fndng out accurate postons of the WS nodes wthout GPS support has been studed for many years. Many dfferent technques [6][7] have been proposed for solvng ths problem, but most of them consder only twodmensonal (D) network. In ths paper, we nvestgate multdmensonal scalng technque [8] for nodes localzaton n three dmensonal surface WSs. We also propose a heurstc approach n dstance matrx calculaton that mproves the accuracy compared wth well known MDS- MAP [9]. Henceforth, we wll refer to our approach as Improved Multdmensonal Scalng Algorthm (IMDS). The rest of ths paper s organzed as follows. The second secton refers to the multdmensonal scalng technque for nodes localzaton n 3D-WS. The thrd secton gves a detaled explanaton of our IMDS algorthm. Secton four presents the results provded from our smulatons. Fnally, we conclude ths paper n secton fve. II. THREE DIMESIOAL MDS Multdmensonal scalng (MDS) s a set of analytcal technques that has been used for reducng the dmensonalty of the data (objects), showng multdmensonal data as ponts n D or 3D space [8]. MDS algorthm uses the dstances between each par of object as nput and generates D-ponts or 3D-ponts as output. The nput requred by MDS should be presented as dstance matrx, representng the dstances between the objects that should be analyzed. The purpose of ths method s to vsualze dssmlarty data n order to better understand and comprehend t. MDS can be easly translated nto WS doman f the sensor network and dstances between neghborng nodes are represented as a graph wth ts edges respectvely. In WSs, MDS performs as centralzed, range-based localzaton algorthm. Dstance measurements between each par of neghborng nodes wll be collected at the snk node. There, all avalable nformaton wll be used n order to obtan the unknown dstances between non-neghborng nodes. There are a few well-known technques for dstance measurement between neghborng nodes [6][7][0], lke Receved Sgnal Strength Indcator (RSSI), Tme of Arrval, Tme Dfference of Arrval (TDoA) and Angle of Arrval (AoA). RSSI [0] measurement of dstances s often preferred as t does not requre addtonal hardware. RSSI s based on the phenomenon that the ntensty of emtted sgnal decreases as the dstance from the sgnal source ncreases. If the functon of the attenuaton n dependence on a dstance s known n advance, the dstance between the emsson source and the recever can be easly estmated. The tme needed for a message to travel from one node to another s used to provde range nformaton n ToA and TDoA technques, whle AOA s defned as the angle between the propagaton drecton of the wave and some reference drecton. The man advantage of usng MDS s ts ablty to reconstruct the relatve map of the network even when there Copyrght (c) IARIA, 03. ISB:

2 are no anchor nodes (nodes wth a pror known locaton). If gven suffcent porton of anchor nodes, MDS performs very accurate poston estmaton enablng local map to be transformed nto an absolute map [9][]. There are dfferent versons of MDS for nodes localzaton n a two dmensonal WS. The most popular s MDS-MAP, proposed by Y Shang and Wheeler Ruml [9], where Djkstra algorthm s used to calculate the unknown dstances from the dstance matrx. In [9] t s shown that MDS-MAP outperforms other technques, especally when appled on densty networks. Other approaches based on MDS-MAP exst [], but most of them are complex and thus more computatonally dependent. In [3], the authors ntroduce MDS-MAP(P), whch s a decentralzed verson of the MDS-MAP. MDS-MAP(P) outperforms MDS-MAP on rregular network topologes, but requres ntensve computatonal resources at each node. It computes local maps at each node n the network and then merges local maps nto a global map. Usng absolute postons of the anchors, ths global map can be easly transformed nto an absolute map. Although a lot of research has been carred out regardng MDS-MAP for WS localzaton, all of the algorthms proposed n the lterature based on MDS-MAP consder only two dmensonal networks. To the extent of our knowledge, ths s the frst research that extensvely nvestgates three dmensonal surface WS localzaton based on MDS. A. Multdmensonal scalng (MDS) for 3D-WS MDS-MAP for 3D WS conssts of 3 steps: Step : Calculate shortest dstances between every par of nodes (usng ether Djkstra s or Floyd s all pars shortest path algorthm). Ths s the dstance matrx that serves as nput to the multdmensonal scalng n step. Step : Apply classcal multdmensonal scalng to the dstance matrx. The frst 3 largest egenvalues and egenvectors gve a relatve map wth relatve locaton for each node. Step 3: Transform the relatve map nto absolute map usng suffcent number of anchor nodes (at least 4). B. Fndng optmal rotaton and translaton between correspondng 3D nodes Generatng an absolute map (step 3) of the WS requres at least four anchor nodes. Let P = p, p p } and Q = q, q q } be {,..., {,..., two sets of correspondng nodes, where s the number of anchor nodes n the WS. We wsh to fnd a transformaton that optmally algns the two sets n terms of least square errors,.e., we seek a rotaton matrx R and a translaton vector t such that R, t = ( R, t) = arg mn ( Rp + t) q. () Ths transformaton s also known as Eucldean or Rgd transformaton, because t preserves the shape and the sze. There are many algorthms purposed n the lterature that compute a rgd 3D transformaton [4]. The most explored are based on Sngular Value Decomposton (SVD), as t s known to be the most stable [5]. Fndng the optmal rgd transformaton wth SVD can be broken down nto the followng steps: Compute the weghted centrods of both pont sets p = p, = q, = Compute the centered vectors q () = p ': = p p, q ': q q, Compute the 3x3 covarance matrx = =,, (3) T H = P' Q', (4) where P and Q are the 3x matrces that have p and q as ther columns, respectvely. ' ' Compute the sngular value decomposton H = UΣV T, (5) The rotaton we are lookng for s then R = VU T, (6) Compute the optmal translaton as t = q Rp. (7) C. Tme complexty of MDS-MAP for 3D-WS In step, dstance matrx constructon usng Djkstra's or Floyd's algorthm requres O ( n 3 ), where n s the number of nodes n the network. In step, applyng MDS to the dstance matrx has complexty of O ( n 3 ) due to sngular value decomposton. In step 3, the relatve map s transformed through lnear transformatons. Computng the rgd transformaton takes O() tme for computng P and Q, whle computng SVD takes only O 3 ) tme (snce the dmenson of covarance matrx H s 3x3). Applyng the transformaton (rotaton and translaton) to the whole relatve map takes O(n-) tme, where s the number of anchors (<<n). III. ( 3 IMPROVED MDS-BASED APPROACH FOR WS POSITIOIG In ths secton, we wll explan n detals our mproved multdmensonal scalng algorthm (IMDS) for nodes localzaton n WS. MDS s very accurate technque for dmensonalty reducton. If the correct dstance matrx s gven as nput, MDS algorthm wll reconstruct the map of the network wthout error. But, calculatng dstance matrx for networks Copyrght (c) IARIA, 03. ISB:

3 where only dstances between neghborng nodes are known s not a trval task. Ths problem n MDS-MAP s solved by applyng Djkstra s (or Floyd s) all pars shortest path algorthm. Djksta s algorthm s a graph search algorthm that solves the sngle-source shortest path problem. In WS localzaton problem, the sensor network s represented as a graph wth non-negatve edge path costs, whle the real, Eucledan dstance between two non-neghborng nodes s replaced wth the dstance calculated usng Djksta algorthm. But the assumpton that Djkstra dstance between two nodes correlates wth ther Eucldean dstance s hardly true. Ths approxmaton produces an error,.e., the postons obtaned as MDS output usually dffer from the correct postons. The dfference between the real and the predcted postons s known as estmaton error. The error s bgger when the nodes are n mult-hop communcaton range, whch s a common case n obstructed envronments. It s usually caused by the presence of obstacles or terran rregulartes that can obstruct the lne-of-sght between nodes or cause sgnal reflectons. Fg. shows two examples when Djksta algorthm wll calculate much larger dstance between non-neghborng nodes. Left sde of the pcture shows an example of two nodes A and B that are far from each other. The dstance between A and B wll be calculated as AB=a+b+c+d, whch s much longer then the real Eucldan dstance. Ths scenaro s present when the network s deployed on vast regons where the rado range of the nodes s short compared wth the length of the regon. On the rght sde of Fg., there s an example where two nodes can t communcate drectly although they are very close to each other. The reason for ths s the presence of obstacle that obstructs the lne-of-sght. In ths scenaro, Djksta algorthm s completely napplcable as t calculates a few tmes longer dstance. A. Dstance matrx calculaton Consder there are three nodes n a network: A, B and C (Fg. ), wth known dstances between nodes A and B (d =AB), and between nodes B and C (d =BC). Snce dstance matrx requres the dstances between every par of nodes n the network, the dstance between nodes A and C has to be obtaned. We wll refer to ths dstance as a. If maxmum rado range of the nodes n the network s R, then, we know for sure that node C can lay anywhere on the curve C C. If Djkstra s algorthm s used for ths purpose, t wll calculate the dstance a as a=ab+bc, whch s the longest possble theoretcal dstance between nodes A and C. More precsely, C wll lay exactly on C. On the other hand, f we calculate the shortest possble theoretcal dstance between nodes A and C, t wll be very close to R. We can conclude that: R < a d + d. (8) Fgure. Dstance approxmaton Fgure. Dstance approxmaton As t can be seen from the two examples presented n Fg., the dstance calculated usng Djksta algorthm always ncrease the real dstance. In order to reduce ths dstance, n ths paper, we propose an alternatve heurstc approach. By reducng the dstance matrx error, we ntend to reduce the overall estmaton error. To mnmze the possble error, we purpose a heurstc soluton that assumes that the node C les exactly n the mddle of the curve C C. Hence, the dstance a=ac can be calculated usng cosne formula as: a = d + d d d cos ( ABC ). (9) In order to calculate the dstance a, frst, we need to fnd the angle usng cosne formula: ABC = ABC + C BC (0) The angle ABC can be calculated agan wth the cosne formula: d + d R ABC = arccos( ) () d d Snce C BC = CBC, () C BC = C BC, (3) C BC = ( ABC ), (4) Copyrght (c) IARIA, 03. ISB:

4 ABC = ABC + ( ABC ), π ABC = + ABC (5) Fnally, a = d + d d d cos ( ABC )= π = d + d d d cos ( + ABC )= = d + d + d d sn ( ABC ), (6) where d + d R ABC = arccos( ) (7) d d We note here that our algorthm preserves the tme complexty of MDS-MAP algorthm. IV. PERFORMACE EVALUATIO The performance of the algorthms for WS localzaton depends on dfferent network parameters, such as the network topology, the number of anchors (.e., the anchor-tonode rato), the rado range, the densty of nodes, etc. Hence, the locaton estmaton error s gong to be evaluated as a functon of dfferent parameters. A. etwork model We assume a typcal sensor network composed of hundreds (or thousands) of sensor nodes deployed unformly across three dmensonal montored area (valley or mountan). Each sensor s equpped wth an omn-drectonal antenna and has lmted resources (CPU, battery, memory, etc.). Snce rado sgnals are omn-drectonal, only nodes wthn certan rado range R can communcate wth each other. If two nodes are wthn each others transmsson range they are called neghbors. Further, we made followng assumptons: odes are statc and unaware of ther locaton. There s a path between every par of nodes. odes deployed n close proxmty to each other exchange messages. Each node uses RSSI (or any other) method for dstance estmaton. RSSI provde accurate neghborng sensor dstance estmaton. We smulated both technques (MDS-MAP and IMDS) on dfferent surface WSs wth Matlab. We consdered: Dfferent network topologes: o 00 nodes randomly deployed on valley terran (topology I) o 00 nodes randomly deployed on mountan terran (topology II) 4, 6, 0 and 5 anchors for absolute map constructon (for 3D rgd transformaton SVD method was used) Dfferent rado ranges (R) that lead to dfferent average connectvty (average number of neghbors). Rado range error er (from er= 0%R to er=30% R wth step 5% of R) Thus 80 dfferent networks were smulated ( x 4 x 5 x 7) and each node locaton was dscovered wth both MDS- MAP and IMDS technque. The connectvty parameter and the estmaton error for each scenaro represent average over 30 trals for both algorthms. The average estmaton error s normalzed by the rado range R: n = dstance( pos ( estmated) pos ( true) Error = 00%, (8) ( n ) R where n s the number of nodes n the network, s the number of anchor nodes, locaton and (estmated) pos ) s the estmated (true) pos s the true locaton of the -th node. B. Comparson of MDS-MAP and IMDS for 3D surface WS It s expected that MDS-based algorthms for WS localzaton wll not work well for such scenaros, bascally because of mult-hop dstance between each par of nodes. Our mproved heurstc approach presented n ths paper s expected to acheve more acceptable accuracy. Fg. 3 shows an example of two typcal 3D surfaces. On the upper pcture there s a surface, whch represents a valley, whle the lower surface represents a mountan. In our smulatons, two scenaros are constructed to emulate a terran wth a valley and a terran wth a mountan. 00 nodes are deployed randomly wth a unform dstrbuton over these two surfaces. Fgure 3. Typcal 3D surface, valley (upper) and mountan (lower) Fg. 4 and Fg. 5 compare the results of MDS-MAP and IMDS for topology I and topology II respectvely. Copyrght (c) IARIA, 03. ISB:

5 In the case of topology I (Fg. 4), when er s small, both IMDS and MDS-MAP produce very smlar estmaton error. Ths error s much more affected by the number of anchors. As er ncreases, IMDS performs much better than MDS- MAP for all connectvty levels, regardless of the number of anchors. Fgure 4. Comparson of MDS-MAP and IMDS for topology I In case of topology II, for small er MDS-MAP has smaller estmaton error than IMDS (Fg. 5). For large values of range error er, IMDS s better than MDS-MAP n terms of accuracy. The characterstcs of IMDS to produce smaller estmaton error than MDS-MAP for large range error er s very mportant, as range measurement n the real applcatons s prone to error. When adoptng dstance measurement based on RSSI, the range error measurement s at least 0%R. The results presented n [0] show average range error measurement between 5%R and 30%R for longer rado range R. Smlar research that nvestgates RSSI s conducted n [6] and [7], reportng average error around 0%R. Fgure 5. Comparson of MDS-MAP and IMDS for topology II The average performance of IMDS as a functon of connectvty for valley WS s gven on Fg. 6. IMDS s very stable and predctve. Estmaton error decreases as connectvty ncreases. The rado range error er affects the estmaton error n a way that larger er deterorates the performance of IMDS. As expected, the number of anchors affects the results,.e., havng more anchors slghtly mproves performance for all connectvty levels (Fg. 7). If we compare the results for topology I and topology II, we can notce that both MDS-MAP and IMDS show better performance for topology I (valley terran). The man reason for ths s the characterstc of the terran. Valley terran s Copyrght (c) IARIA, 03. ISB:

6 very regular because all nodes that are wthn rado range R can communcate wth each other. Mountan terran should be consdered as an rregular topology. The mountan presents an obstacle that obstruct the rado propagaton between the nodes, whch means that sometmes nodes that are very close to each other cannot communcate,.e., cannot measure the dstance between each other. For terrans wth obstacles, nodes localzaton problem should be solved dfferently. IMDS algorthm should manage herarchcal network organzaton based on cluster formaton. Ths cluster-based approach, whch s already developed and mplemented for D networks n [8], should be consdered for 3D surface networks. Fgure 6. The effect of range error on the estmaton error for topology I Fgure 7. The effect of number of anchors on the estmaton error for topology I V. COCLUSIO AD FUTURE WORK In ths paper, we mplemented mproved MDS-based algorthm (IMDS) for nodes localzaton n 3D surface WS. In IMDS, a novel technque for dstance matrx refnement was ntroduced n order to reduce the estmaton error. We nvestgated two surface network models (valley and mountan) and we showed that our approach outperforms MDS-MAP n terms of accuracy. IMDS performs much better than MDS-MAP especally when rado range error er s large. For future work, we ntend to nvestgate our algorthm on network where nodes are deployed on more complex 3D terrans. It s also a challenge to smulate rado propagaton model n such complex 3D terrans, whch s not a trval task due to the presence of obstacles. Ths way, we beleve ths work wll contrbute for future development of smart network technologes n dfferent domans, especally for context- aware applcatons. REFERECES [] I. F. Akyldz, S. Welan, Y. Sankarasubramanam, and E. Cayrc, A Survey on Sensor etwork. IEEE Communcatons Magazne 40, 00, pp [] F. L. Lews, Wreless Sensor etworks, chapter 4 n D. J. Cook and S. K. Das, edtors, Smart Envronments: Technologes, Protocols, and Applcatons, John Wley, ew York, 004. [3] K. Akkaya and M. Youns, A survey of routng protocols n wreless sensor networks, Elsever Ad Hoc etwork Journal, May 005, 3(3), pp [4] C. F. Garca-Hernandez, P. H. Ibarguengoyta-Gonzalez, and J. A. Perez-Daz, Wreless Sensor etworks and Applcatons: A Survey, IJCSS Internatonal Journal of Computer Scence and etwork Securty, 7(3), (007), pp [5] B. W. Parknson, 994. GPS eyewtness: the early years. GPS World, 5(9), pp [6] J. Wang, R. K. Ghosh, and S. K. Das, A survey on sensor localzaton, Journal of Control Theory and Applcatons, 8():-, 00. [7] A. Pal, Localzaton Algorthms n Wreless Sensor etworks: Current Approaches and Future Challenges. In etwork Protocols and Algorthms, 00, Vol., o., pp [8] T. Cox and M. Cox, Multdmensonal Scalng, Chapman & and Hall, London, 994. [9] Y. Shang, W. Ruml, Y. Zhang, and M. P. J. Fromherz, Localzaton from mere connectvty, n Proceedngs of ACM Internatonal symposum on moble ad hoc networkng and computng, June 003, pp. 0-. [0] Z. Janwu and Z. Lu, Research on dstance measurement based on RSSI of ZgBee, Computng, Communcaton, Control, and Management, 009. CCCM 009. ISECS Internatonal Colloquum on, vol.3, 8-9 Aug. 009, pp. 0-. [] X. J and H. Zha, Sensor Postonng nwreless Ad-hoc Sensor etworks usng Multdmensonal Scalng, Proceedngs of 3rd Annual Jont Conference of the IEEE Computer and Communcatons Socetes (IFOCOM 004), March 004, vol. 4, pp [] B. R. Stojkoska and V.Krandzska Improved MDS-based algorthm for nodes localzaton n wreless sensor networks, IEEE EUROCO, July 03, pp [3] Y. Shang and W. Ruml, Improved mds-based localzaton, n Twenty-thrd AnnualJont Conference of the IEEE Computer and Communcatons Socetes (IFOCOM), 004, pp [4] D. W. Eggert, A. Lorusso, and R. B. Fsher, Estmatng 3-D Rgd Body Transformatons: A Comparson of Four Major Algorthms, Machne Vson and Applcatons (997) Vol. 9, o. 5/6, pp [5] A. Lorusso, D. Eggert, and R. Fsher, A Comparson of Four Algorthms for Estmatng 3-D Rgd Transformatons, In: Proceedngs of the 4th Brtsh Machne Vson Conference (BMVC 995), Brmngham, England, (September 995), pp [6] A. Awad, T. Frunzke, and F. Dressler, Adaptve dstance estmaton and localzaton n WS usng RSSI measures, n Proc. 0th Euromcro Conf. Dgtal Syst. Des. Archtectures, Methods Tools (DSD) 007, pp [7] A. Faheem, R. Vrrankosk, and M. Elmusrat, Improvng RSSI Based Dstance Estmaton for Wreless Sensor etworks, n Conference Proceedngs of the 00 IEEE Internatonal Copyrght (c) IARIA, 03. ISB:

7 Conference on Wreless Technology and Systems (ICWIT 00), Aug. 00. [8] B. Stojkoska, D. Davcev, and A. Kulakov, Cluster-based MDS Algorthm for odes Localzaton n Wreless Sensor etworks wth Irregular Topologes, Proceedngs of The The Ffth ACM/IEEE Internatonal Conference on Soft Computng as Transdscplnary Scence and Technology (CSTST'08), Pars, France, October 6-30, 008, pp Copyrght (c) IARIA, 03. ISB: