Vijayanth Vivekanandan* and Vincent W.S. Wong

Transcription

1 Int. J. Sensor Networks, Vol. 1, Nos. 3/, 19 Ordinal MDS-based localisation for wireless sensor networks Vijayanth Vivekanandan* and Vincent W.S. Wong Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, Canada, VT 1Z *Corresponding author Abstract: There are various applications in wireless sensor networks which require knowing the relative or actual position of the sensor nodes. Over the past few years, there have been different localisation algorithms proposed in the literature. The algorithms based on classical Multi-Dimensional Scaling (MDS) only require 3 or anchor nodes and can provide higher accuracy than some other schemes. In this paper, we propose and analyse another type of MDS (called ordinal MDS) for localisation in wireless sensor networks. Ordinal MDS differs from classical MDS in that it only requires a monotonicity constraint between the shortest path distance and the Euclidean distance for each pair of nodes. We conduct simulation studies under square and C-shaped topologies with different connectivity levels and number of anchors. Results show that ordinal MDS provides a lower position estimation error than classical MDS in both hop-based and range-based scenarios. Keywords: wireless sensor networks; localisation; Multi-Dimensional Scaling (MDS). Reference to this paper should be made as follows: Vivekanandan, V. and Wong, V.W.S. () Ordinal MDS-based localisation for wireless sensor networks, Int. J. Sensor Networks, Vol. 1, Nos. 3/, pp Biographical notes: Vijayanth Vivekanandan received a BASc and an MASc in Electrical and Computer Engineering from the University of British Columbia, Canada, in 3 and 5, respectively. His research interests are in localisation algorithms in wireless sensor networks. He is currently pursuing a career in industry. Vincent W.S. Wong received his BSc from the University of Manitoba in 199, his MASc from the University of Waterloo in 199 and his PhD from the University of British Columbia (UBC) in. He worked as a systems engineer at PMC-Sierra, Inc. from to 1. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering at UBC. His current research interests are in resource and mobility management for wireless mesh networks, wireless sensor networks and heterogeneous wireless networks. He received the Natural Sciences and Engineering Research Council PostGraduate Scholarship and the Fessenden PostGraduate Scholarship from Communications Research Centre, Industry Canada, during his graduate studies. He serves as a TPC member in various conferences, including the IEEE ICC and Globecom. 1 Introduction The miniaturisation of small devices capable of sensing and communicating with each other has made the possibility of deploying large-scale wireless sensor networks a reality (Akyildiz et al., ). Wireless sensor networks can be deployed in different scenarios, ranging from military applications to wildlife and environment monitoring. Hundreds or thousands of sensor nodes are randomly scattered over an area of interest for information gathering. These sensor nodes have limited processing power and battery supply, and need to configure themselves autonomously. For applications such as position-based routing, event discovery and target tracking, the geographic location of the sensor nodes need to be known. Consider the example where a sensor network is used to detect a fire event in a forest. Once a sensor node has detected that the temperature is higher than a certain threshold, it sends a message to the central authority by relaying through other nodes in a multi-hop manner. The message needs to indicate the location of the node which detected the event. Thus, localisation of sensor nodes is important in some applications. The current location technology such as Global Positioning System (GPS) does not meet the sensor nodes requirements for low cost, low energy consumption, and small size (Niculescu, ). In addition, GPS requires line-of-sight to global satellites that may not be available for certain applications in sensor networks. Over the past few years, there have been different localisation schemes proposed in the literature. These algorithms can be divided into two groups: centralised Copyright Inderscience Enterprises Ltd.

2 17 V. Vivekanandan and V.W.S. Wong (Shang et al., 3; Doherty et al., 1) and distributed (Capkun et al., 1; Savarese et al., ; Langendeon and Reijers, 3; He et al., 3; Priyantha et al., 3; Vivekanandan and Wong, ). It is generally true that distributed algorithms are more robust and energy efficient than centralised algorithms. In each group, some algorithms assume simple connectivity information between neighbouring nodes (Shang et al., 3, ) while some others need to gather the ranging information (e.g. estimated distance between two neighboring nodes) (Ji and Zha, ; Niculescu and Nath, 1; Savvides et al., 1) and angle information (Niculescu and Nath, 3; Chintalapudi et al., ). In order to determine the actual or absolute position of each sensor node, a small fraction of special nodes (called anchor nodes) with known positions are necessary. Since different localisation algorithms perform well under different assumptions, it is difficult to determine which one is the optimal scheme. For most of the performance comparisons reported to date, the common performance metric is the localisation error normalised with respect to the radio range. This metric depends on the node connectivity (density), anchor population, and the distance measurement error models. The recently proposed localisation algorithms based on classical (MDS) (Shang et al., 3, ; Ji and Zha, ) have proven to be robust with respect to both hop-based and range-based implementations. Only three or four anchor nodes are necessary to determine the absolute locations, in two or three dimensions, respectively. These MDS algorithms achieve a higher accuracy than some other schemes. By using the similar terminology in Shang et al. (3), we use the term MDS-MAP(C) for the classical MDS localisation algorithm. The original MDS-MAP(C) is a centralised algorithm. In Shang et al. (), a distributed MDS algorithm named MDS-MAP(P, C) was proposed where P indicates the use of patching of local maps and C indicates the use of classical MDS. As mentioned in Shang et al. (3, ) further work is required to study the application of other MDS techniques (e.g. probabilistic MDS, ordinal MDS) on localisation in sensor networks. In this paper, we propose the implementation of ordinal MDS for localisation in sensor networks and compare the performance with classical MDS. We call our proposed scheme MDS-MAP(P, O) where P again denotes the use of patching of local maps and O denotes the use of ordinal MDS. MDS-MAP(P, O) is also a distributed algorithm. The main difference between classical MDS and ordinal MDS is that the former assumes there is a linear equation which relates the shortest path distance and the Euclidean distance between each pair of nodes, the latter simply assumes a monotonicity constraint. That is, for ordinal MDS, given two pairs of nodes (i, j) and (k, l), if the shortest path distance of (i, j) is greater than that of (k, l), then the Euclidean distance of (i, j) is also greater than that of (k, l) and vice versa. The contributions of this paper are as follows (Vivekanandan, 5): 1 We present the implementation details of the ordinal MDS algorithm for localisation in wireless sensor networks. We conduct simulations to study the performance between ordinal MDS and classical MDS by varying the connectivity levels and number of anchors. Under square and C-shaped topologies, results show that MDS-MAP(P, O) has a lower position estimation error than MDS-MAP(P, C) in both hop-based and ranged-based scenarios. Our proposed MDS-MAP(P, O) algorithm is essential for future sensor applications which require a high accuracy of nodes position by using a small number of anchor nodes. The rest of this paper is organised as follows. The related work is summarised in Section. The MDS-MAP(P, O) algorithm is described in Section 3. Performance comparisons between classical MDS and ordinal MDS algorithms are given in Section. Conclusions are given in Section 5. Related Work In this section, we first summarise several recent papers on localisation in sensor networks. Survey papers in this area can also be found in Niculescu (), and Hightower and Boriello (1). We then review the MDS-MAP(C) and MDS-MAP(P, C) algorithms (Shang et al., ). In the APIT scheme (He et al., 3), each node first identifies if it is within a particular triangle formed by a set of anchors within radio range. The position is estimated to be the center of the intersection of all triangles in which the node has identified to be within. The drawbacks of this approach lie in the high density of anchors required in addition to the extended range required by the anchors. In the convex optimisation scheme (Doherty et al., 1), anchors have to be placed near the corners and edges of the network for optimal performance, a requirement that may not be suitable for some ad hoc deployments. Anchor information propagation methods (Saverese et al., ; Langendoen and Reijers, 3; Niculescu and Nath, 1) require each anchor to broadcast its position to the network. Nodes use this information as well as the distance or hop counts from the anchors to laterate or bound their positions. In a slightly different approach, iterative localisation (Savvides et al., 1) can be used. Nodes with sufficient neighbouring anchors can compute their positions. As more nodes obtain their positions, these nodes can also act as anchors. However, error propagation may occur throughout the network. In addition, this technique may fail when refinement schemes based on local-neighbour information exchange are used. Several direction or angle-based schemes have been proposed. In Niculescu and Nath (3), the original APS scheme (Niculescu and Nath, 1) is modified to propagate bearings to anchors. Nodes that have at least three bearings to anchors can triangulate their positions. In Chintalapudi et al. (), both range and angle information is used to determine the node s position. The advantage is that only one anchor is needed to obtain an estimate for a node. Several relative coordinate systems have also been proposed (Capkun et al., 1). Such schemes avoid the use of anchors. However, all nodes determine their positions relative to a reference group of nodes coordinate system. In Priyantha et al. (3), additional rules based on graph rigidity concepts are applied to ensure that the network topology is correct before computing the relative positions of nodes. In all these techniques, it is feasible to

3 Ordinal MDS-based localisation for wireless sensor networks 171 apply anchors in order to transform the relative points into absolute locations. In Sichitiu and Ramadurai (), a single mobile anchor is used to localise the system. The mobile anchor node traverses within the network and allows all nodes to compute the location estimate based on at least three neighbouring nodes locations. In Ssu et al. (5), multiple mobile anchors are used. A monte carlo localisation algorithm for mobile sensor networks was proposed in Hu and Evans (). In Pathirana et al. (5), mobile robots and a robust extended Kalman filter-based state estimator are used for localisation. The advantage of MDS localisation algorithms is the relatively low percentage of estimation error while using a small number of anchor nodes. The MDS-MAP(C) scheme is a centralised algorithm. The major steps are as follows (Shang et al., 3): Given the network hop-count or distance information, Dijkstra s algorithm is used to determine the shortest path between each pair of nodes. The results are stored in a distance matrix. The classical MDS algorithm is then applied on the distance matrix to create the global relative map. By using the anchor nodes positions, the global relative map is transformed into the global absolute map. The classical (or metric) MDS algorithm assumes that there exists a linear transformation which relates the shortest path distance and the Euclidean distance between each pair of nodes. For each pair of nodes (i, j), if the shortest path distance is denoted by p ij and the Euclidean distance is denoted by d ij, then d ij = mp ij + c for some constants m and c. Classical MDS uses singular value decomposition (Lay, 3) to determine the relative coordinates of the sensor nodes. Simulation results show that in a topology where the nodes are uniformly placed, MDS-MAP(C) has a lower location estimation error when compared with Doherty et al. (1) and Savarese et al. (). The MDS-MAP(P, C) is a distributed localisation algorithm (Shang et al., ). Each node first creates a local map within its two-hop neighbours by using the classical MDS algorithm. Each local map is then refined by using the least-squares minimisation. The local maps are then patched or merged to create a global relative map. Finally, by using the anchor nodes positions, the global relative map is transformed into the global absolute map. Simulation results from Shang et al. () show that MDS-MAP(P, C) has a better performance than MDS-MAP(C). Further improvement can also be achieved with a global refinement phase after obtaining the global relative map, where the least-squares minimisation is used to match the calculated distances with the measured distances between neighbouring nodes. However, this global refinement phase has a complexity of O(N 3 ) where N is the total number of sensor nodes. In addition, as stated in Shang et al. (), performance may further be improved if other MDS algorithms (e.g. weighted MDS, probabilistic MDS, ordinal MDS) are used. In the next section, we study the performance of using the ordinal (or non-metric) MDS algorithm for localisation. 3 MDS-MAP(P, O) localisation algorithm In this section, we describe our proposed MDS-MAP(P, O) localisation algorithm. MDS-MAP(P, O) is distributed and can be considered as an extension of MDS-MAP(P, C). The modification is the use of the ordinal MDS (instead of classical MDS) during the estimation phase. The major steps of the MDS-MAP(P, O) algorithm are as follows (Vivekanandan, 5): 1 Each node first gathers either the distance (for range-based) or hop count (for hop-based) information within its two-hop neighboruhood. In each node, the Dijkstra s algorithm is invoked to determine the shortest path between each pair of nodes within the two-hop neighbourhood. We use the notation p ij to denote the shortest path distance between nodes i and j. 3 The ordinal MDS algorithm is applied to create the relative local map for each node. Each local map is refined by using the least-squares minimisation between the calculated Euclidean distance and the measured distance (or hop) between each pair of neighbouring nodes. 5 The local maps are then patched (or merged) into a global map by using a predetermined initial starting node s local map and sequentially adding each neighbour that has the largest number of common nodes to the starting node. This map then grows until all nodes have been included. The global absolute map is created by using the anchors positions and the global relative map. Assume that the average number of sensor nodes in each twohop neighbourhood is M, the average number of neighbours is K, the total number of sensor nodes is N, and total number of anchors is A. In the above MDS-MAP(P, O) algorithm, Steps () and () have a complexity of O( M 3 ). Step (3) has a complexity of O( M ). Steps (5) and () have a complexity of O( K 3 N) and O(A 3 + N), respectively. We now describe the ordinal MDS algorithm (Step (3) above) in detail. The major steps of the ordinal MDS algorithm are as follows (Hardle and Simar, 3): 1 Assign arbitrary initial location estimation (x i, y i ) for i M, where M includes all the nodes within the two-hop neighbourhood. Specify ɛ> and set n =. For each i, j M, compute the Euclidean distance by d n ij = ( x n i x n j ) + ( y n i y n j ) (1) 3 By using the matrices [p ij ] and [dij n ], apply monotone regression by using the Pool-Adjacent Violators (PAV) algorithm (Hardle and Simar, 3) to determine [ ˆd ij n ]. For example, once the p ij s are ordered from the smallest to the largest, if (p ij <p kl ) and (dij n >dn kl ), then ˆd n ij = ˆd n kl = ( d n ij + dn kl) / Otherwise, ˆd n ij = dn ij and ˆd n kl = dn kl. If there are consecutive violators, they are all grouped and averaged together, as per the PAV algorithm, to maintain monotonicity throughout the entire set.

4 17 V. Vivekanandan and V.W.S. Wong Increment n by 1. For i M, compute the new relative coordinate (xi n,yn i ) for node i by x n i y n i = x n 1 α i + M 1 = y n 1 α i + M 1 j M,j =i j M,j =i ( 1 ( 1 ) ˆd n 1 ij (x n 1 d n 1 j ij ) ˆd n 1 ij (y n 1 d n 1 j ij ) x n 1 i ) y n 1 i where M denotes the number of sensor nodes within the two-hop neighbourhood. 5 For each i, j M, update the Euclidean distance d n ij by using Equation (1). Use Kruskal s Stress1 test to determine the goodness fit (Kruskal, 19; Kruskal and Wish, 197): ( ) i<j dij n Stress1 = ˆd n 1 ij ( ) () i<j dij n 7 If Stress1 <ɛ, stop. Otherwise, go to Step (3). In the above algorithm, the first two steps calculate the Euclidean distance from an arbitrary initial configuration. Step (3) determines the disparities ˆd ij n by constructing a monotone regression relationship (Borg and Groenen, 1997) between p ij s and dij n s. Step () updates the relative positions. The parameter α is the step width. We use α =. as suggested by Kruskal (195). Step (5) updates the Euclidean distance. The Stress1 measure in Step () determines whether or not the updated values dij n fit the given dissimilarities ˆd n 1 ij. Note that other goodness fit tests (e.g. Kruskal s Stress, normalised raw stress, S-Stress) can also be used; however we choose the Stress1 measure since it is the most common measure used for ordinal MDS. Step (7) determines if the derived configuration s goodness fits are close enough such that the procedure can be terminated. The MDS-MAP(P, O) algorithm assumes that there is a monotonic relationship between the shortest path distances and the actual Euclidean distances. This assumption may not be valid if the network being considered is sparse and large. However, most of the applications in wireless sensor networks require the networks to be dense (i.e. with a high connectivity or average node degree) in order to provide redundancy and robustness in case of a node s failure. In addition, in our distributed approach, only the nodes within the -hop neighbourhood are being considered. In this case, the assumption of the monotonic relationship between the shortest path distances and the actual Euclidean distances is valid. By the iterative nature of the ordinal MDS algorithm in minimising stress in Equation (), the final solution may not guarantee to be the global minimum (Groenen and VandeVelden, ). In fact, the ordinal MDS algorithm can have several local minima. However, the use of the anchors in our application of the ordinal MDS algorithm increases the likelihood of reaching the global minimum. This is due to the imposed transformation required to obtain the absolute coordinates for all of the nodes. Another way to further increase the chance of reaching the global minimum is by using the multiple starting configurations approach and retaining the configuration which results in the lowest stress value. However, this approach is inefficient due to the additional computation effort required. Performance evaluation and comparison To implement MDS-MAP(P, O) algorithm, we modified the source codes for MDS-MAP(P, C) that were written in Matlab (Shang et al., ). Two different topologies are considered as the sensor network s coverage area. The first one is a uniformly distributed square region. The second one is an irregular C-shaped topology. In both topologies, we vary the average connectivity levels (i.e. average number of neighboring sensors) and the number of anchors in the area. The average connectivity level is varied between 9 and 1 by modifying the radio range R, within the fixed coverage area. In each set of simulations, 5 trials were performed and 95% confidence intervals were plotted. We envision that there is a large number of sensors within a square coverage area to increase the robustness of the sensor network. In our simulations, we use either 1 or sensor nodes. The number of anchors is between four and ten. Although in theory only three anchors are necessary to obtain the absolute position estimates for MDS-MAP (P, O), the location of the anchors in this minimal case has an impact on the estimation error. We thus recommend the use of at least four anchors for deployment. In the hop-based scenarios, hop count is used as the distance metric between a pair of nodes. For each node to have a unique position in MDS-MAP(P, O), the hop count values are blurred with noise so that nodes with identical hop count values to neighbours are not co-located. In the range-based scenarios, the range is modelled as the actual distance combined with Gaussian noise. Thus, the range is a random value drawn from a normal distribution with actual distance as mean and variance of 5%. We also vary the amount of Gaussian error and study its effect on the average positioning error. In the ordinal MDS algorithm, we set ɛ to be 1..1 Random uniform network topology For evaluation of the random uniform deployment, a 1r by 1r square topology was used, where r represents the reference unit length. Anchor nodes are placed randomly within the coverage area, and have the same communication range (i.e. radio range denoted by R) as other nodes..1.1 Hop-based performance In Figure 1 the topologies estimated by hop-based MDS-MAP(P, C) and MDS-MAP(P, O) are shown. The

5 Ordinal MDS-based localisation for wireless sensor networks 173 hollow circles represent the actual positions. The lines indicate the amount of error of the estimated positions. The filled circles represent the location of the four anchors. The position estimation errors by MDS-MAP(P, C) and MDS-MAP(P, O) are 7% and % of the radio range, respectively. Figure 1 Topology results of hop-based (a) MDS-MAP(P, C) and (b) MDS-MAP(P, O) in a 1r 1r square network region employing uniform random placement of nodes with connectivity level of 1 and four anchors. Anchors are denoted by shaded circles. Lines represent position estimation error 1 (a) MDSMAP(P,C), (error=.7r) 1 1 (b) MDSMAP(P,O), (error=.r) 1 Figure 1 shows that the estimation error encountered by each sensor node is different. Most of them have small estimation error while a few of them may have higher estimation error. The ordinal MDS algorithm reduces the amount of estimation error for the nodes which would have a higher estimation error if classical MDS were used. This can be attributed to the fact that the ordinal algorithm iteratively improves the initial random topology estimate thereby improving the goodness fit of the ordered distances within the two-hop neighbourhood of each node. In other words, the larger error encountered by classical MDS is due to the inaccurate modelling of the shortest path distances being equal to the actual distances between nodes, which may not always be the case. The monotonic constraint in ordinal MDS provides accurate iterative minimisation of stress and hence lower position errors for most nodes, especially nodes with large estimation errors. Figure shows the position estimation errors as a function of the average connectivity level by hop-based MDS-MAP(P, C) and MDS-MAP(P, O), respectively, with different numbers of anchors deployed. Results show that MDS-MAP(P, O) outperforms MDS-MAP(P, C) by a 5% lower position estimation error. The performance improvement confirms the conjecture that in sensors localisation problem, the use of the monotonic constraints in ordinal MDS is more appropriate than the use of linear constraints in classical MDS. The estimated node positions with error less than % of the radio range have been proven to suit the applications of sensor networks (He et al., 3). When the number of anchors is greater than 3 and the connectivity level is greater than 1, the position estimation error is always less than % of the radio range (our target value). As the average connectivity level increases, the confidence intervals reduce in size. This shows that dense networks can provide more consistent average error values. This is due to the fact that dense networks have smaller two-hop regions, which in turn lead to more accurate shortest path distances. These distances therefore improve the classical MDS results as well as the ordinal MDS results, since more accurate distances translate into more accurate proximities in the ordinal case. Figure 5 3 Performance of hop-based MDS-MAP(P, C) and MDS-MAP(P, O) in a 1r 1r square network region employing uniform random placement of nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors Range-based performance 5 3 (b) anchors (d) 1 anchors Figure 3 shows the topologies estimated by range-based MDS-MAP(P, C) and MDS-MAP(P, O), respectively. The position estimation errors by MDS-MAP(P, C) and MDS-MAP(P, O) are 1% and 13% of the radio range, respectively. Figure 3 Topology results of range-based (a) MDS-MAP(P, C) and (b) MDS-MAP(P, O) in a 1r 1r square network region employing uniform random placement of nodes with connectivity level of 1 and four anchors. Anchors are denoted by shaded circles. Lines represent position estimation error 1 (a) MDSMAP(P,C), (error=.1r) 1 (b) MDSMAP(P,O), (error=.13r) 1 1

6 17 V. Vivekanandan and V.W.S. Wong Figure shows the position estimation error as a function of the average connectivity level by range-based MDS-MAP (P, C) and MDS-MAP(P, O), respectively. As the number of anchors deployed increases the additional gain in accuracy is minimal above anchors. Results show that MDS-MAP (P, O) outperforms MDS-MAP(P, C) by a % lower position estimation error. We notice that the improvement in the range-based case is only slightly better compared with the hop-based case. However, it is clear that the range-based case outperforms the hop-based scheme by a significant margin which is desirable in order for the tradeoff of additional ranging hardware to be accepted. Figure 3 1 Performance of range-based MDS-MAP(P, C) and MDS-MAP(P, O) in a 1r 1r square network region employing uniform random placement of nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors (b) anchors (d) 1 anchors Results show that in the range-based case, the position estimation error is more sensitive to the average connectivity level than the number of anchors. Again, if the target position estimation error is less than %, our proposed MDS-MAP (P, O) can achieve this value when the average node connectivity level is above nine. By comparing Figures and, it is evident that the range-based scheme outperforms the hop-based scheme in MDS-MAP(P, O). This is due to the fact that more accurate distance measurement information is available in the range-based scheme. By just using the connectivity information (i.e. hop-count), the hop-based scheme will have some nodes having the same distance measurement when in fact their actual Euclidean distances may differ. This directly affects the ordinal scheme when it comes to applying the monotonic constraint and iterative stress reduction..1.3 Sensitivity analysis of range error on positioning error In this section, we study the effect on the position estimation error when the distance between two neighbouring nodes is not determined accurately in the range-based scheme. As mentioned in Section, a Gaussian noise model is used to model the estimated range distances between two nodes. The actual distance is taken as the mean value. We change the value of the variance from % to 5%. Figure 5 shows that the position estimation error increases as the range error (i.e. variance) increases. Results also show that the range-based scheme is more sensitive to the range error than to the average connectivity level. Figure Effects of node ranging error on performance of range-based MDS-MAP(P, O) in a 1r 1r square network region employing uniform random placement of nodes, and four deployed anchors = 9 = 1 = 15 = 1 = Range Error (%) Figure shows that both MDS-MAP(P, O) and MDS-MAP(P, C) exhibit similar behaviour to the increase of range error. This is due to the fact that both algorithms need to use the same shortest-path distance matrix constructed from the measured node ranges. Figure 1 1 Performance comparison between range-based MDS-MAP(P, C) and MDS-MAP(P, O) with respect to varying node range error MDSMAP(P,C) MDSMAP(P,O) Range Error (%)

7 Ordinal MDS-based localisation for wireless sensor networks Sensitivity analysis of anchors location on positioning error In this section, we study the effect on the position estimation error when anchors are being placed at different locations. Figure 7 shows the results when the anchors are being placed 1 linearly in a rectangular manner 3 close to each other and randomly. As we expect, the position estimation error is the highest when anchors are close to each other (see Figure 7 (c)). On the other hand, the position estimation errors for the other three cases are similar (see Figure 7 (a), (b), (d)), with the rectangular case performing the best with an error of 9%. Since the ordinal MDS algorithm does not require the use of triangulation, the position estimation error is still considered to be small even when the anchors are being placed linearly (see Figure 7 (a)). As long as the anchors are not close to each other, the orientation between the anchors has a small impact on the error as seen from the performance in both linear and rectangular cases (Figure 7 (a) (b)). Figure 7 Topology results of range-based MDS-MAP(P, O) when anchors are being placed (a) linearly, (b) in a rectangular manner, (c) close to each other and (d) randomly. The topology consists of a 1r 1r square network region employing uniform random placement of nodes with connectivity level of 1 and four anchors. Anchors are denoted by shaded circles. Lines represent position estimation error 1 (a) Linearly Aligned Anchors (error=.11r) 1 (c) Closely Spaced Anchors (error=.5r) 1 (b) Rectangular Aligned Anchors (error=.9r) 1 1 (d) Randomly Chosen Anchors (error=.13r) 1 The least-squares minimisation is used for the measured and calculated distances between neighbouring nodes. This optional refinement step has a complexity of O(N 3 ) where N is the total number of sensor nodes. We use the notation MDS-MAP(P, O, R) to denote the original MDS-MAP (P, O) algorithm with global relative map refinement. Figures and 9 show the performance comparisons between MDS-MAP(P, O) and MDS-MAP(P, O, R) in hop-based and range-based scenarios, respectively. The number of anchors deployed is varied from to 1. In the hop-based case, there is significant reduction in the position estimation error when the average node connectivity level is above 9. The difference between the results is greater than 3% for high average connectivity levels. In the range-based case, there is only a slight improvement when the average connectivity level is greater than 1. Note that the global relative map refinement comes at a cost. A sensor node must process the global map and then propagate the results to all the sensors in the network (e.g. via flooding). This may cause a higher signalling overhead. Figure Performance of hop-based MDS-MAP(P, O) and MDS-MAP(P, O, R) in a 1r 1r square network region employing uniform random placement of nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors (b) anchors (d) 1 anchors Random irregular network topology Further improvement via (optional) global relative map refinement The accuracy of the MDS-MAP(P, O) localisation algorithm can further be improved by using an optional global relative map refinement (Shang et al., ). This optional step is invoked after the patching of the local maps. Whereas most papers presented have only considered uniform sensor network deployments, the method in which these networks are meant to be deployed may not guarantee uniform coverage. Wireless sensor networks may exhibit regions of sparseness once deployed. Therefore, localisation algorithms must be able to perform well under different conditions. In this section, we evaluate the performance of MDS-MAP(P, O) by using the same topology in Shang et al. (), (i.e. a C-shaped topology). In our simulations, we notice that the position estimation errors are changed when the anchors are placed at different positions. For good performance, we recommend to have at least one anchor on each wing of a C-shaped topology.

8 17 V. Vivekanandan and V.W.S. Wong Figure 9 Performance between range-based MDS-MAP(P, O) and MDS-MAP(P, O, R) in a 1r 1r square network region employing uniform random placement of nodes, for varying levels of connectivity and anchors deployed 3 1 (a) anchors (c) anchors Hop-based performance 3 1 (b) anchors (d) 1 anchors Figure 1 shows the topologies estimated by hop-based MDS-MAP(P, C) and MDS-MAP(P, O), respectively. The position estimation errors by MDS-MAP(P, C) and MDS-MAP(P, O) are 7% and 5% of the radio range, respectively. The position estimation error of each individual sensor node varies. There is no correlation for sensors that are closer to the anchors to have better position estimation. Figure 1 Topology results of hop-based (a) MDS-MAP(P, C) and (b) MDS-MAP(P, O) in a 1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes with connectivity level of 1 and four anchors. Anchors are denoted by shaded circles. Lines represent position estimation error the square topology case; however, the confidence intervals among the two algorithms show considerable overlap. This is expected since the estimated shortest path distances are more prone to errors arising from the geometry of nodes that are within the inside corners of the network. When the average connectivity levels are 1 or higher, the position estimation error is less than 5%. Figure 11 Performance between hop-based MDS-MAP(P, C) and MDS-MAP(P, O) in a 1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors (b) anchors Range-based performance 5 (d) 1 anchors Figure 1 shows the topologies estimated by range-based MDS-MAP(P, C) and MDS-MAP(P, O), respectively. The position estimation errors by MDS-MAP(P, C) and MDS-MAP(P, O) are 5% and % of the radio range, respectively. 1 (a) MDSMAP(P,C), (error=.7r) 1 (b) MDSMAP(P,O), (error=.5r) Figure 1 Topology results of range-based (a) MDS-MAP (P, C) and (b) MDS-MAP(P, O) ina1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes with connectivity level of 1 and four anchors. Anchors are denoted by shaded circles. Lines represent position estimation error 1 (a) MDSMAP(P,C), (error=.5r) (b) MDSMAP(P,O), (error=.r) Figure 11 shows the position estimation errors as a function of the average connectivity level by hop-based MDS-MAP (P, C) and MDS-MAP(P, O), in a C-shaped network topology. Results show that MDS-MAP(P, O) outperforms MDS-MAP(P, C) by a 9% lower position estimation error when the connectivity is 1. This difference is greater than 1 1

9 Ordinal MDS-based localisation for wireless sensor networks 177 Figure 13 shows the position estimation errors as a function of the average connectivity level by rangebased MDS-MAP(P, C) and MDS-MAP(P, O), for a C-shaped network topology. When the average connectivity level is 1, results show that MDS-MAP(P, O) outperforms MDS-MAP(P, C) by a % lower position estimation error. This difference is greater than the square topology case; however, once again the confidence intervals among the two algorithms show significant overlap. When the average connectivity level is 1 or higher, the estimation error is less than %. Figure Performance of range-based MDS-MAP(P, C) and MDS-MAP(P, O) in a 1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors (b) anchors (d) 1 anchors Further improvement via (optional) global relative map refinement Figures 1 and 15 show the performance comparisons between MDS-MAP(P, O) and MDS-MAP(P, O, R) in hop-based and range-based scenarios, respectively. In the hop-based case, there is significant reduction on the position estimation error when the average node connectivity level is 1 or greater. The additional refinement scheme outperforms ordinal MDS in the hop-based results on average 5%, for connectivity levels 1 and greater, which is larger than in the square topology case. In the range-based case, the refinement scheme achieves on average % better accuracy than the ordinal MDS scheme, which again is a higher gain in accuracy compared to the square topology results. Thus the benefit of the MDS-MAP(P, O, R) is greater in the irregular (Cshaped) topology than in the uniform (square) topology. This is expected, since the iterative refinement of the global map in the irregular case takes into account the topology from the result of the initial MDS-MAP(P, O) result. Thus, the refinement scheme is able to improve the result by iteratively reducing position estimation error from a global viewpoint of the network. Figure 1 Performance between hop-based MDS-MAP(P, O) and MDS-MAP(P, O, R) in a 1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors (b) anchors (d) 1 anchors Figure 15 Performance between range-based MDS-MAP(P, O) and MDS-MAP(P, O, R) in a 1r 1r irregular (C-shaped) network region employing uniform random placement of 1 nodes, for varying levels of connectivity and anchors deployed (a) anchors (c) anchors Conclusions (b) anchors (d) 1 anchors In this paper, we proposed and analysed the MDS-MAP (P, O) localisation algorithm for wireless sensor networks. The MDS-MAP(P, O) algorithm is an extension of the MDS-MAP(P, C) algorithm originally proposed in Shang et al. (3, ). We extend their work by using the ordinal MDS algorithm instead of the classical MDS algorithm. Our proposed MDS-MAP(P, O) algorithm is essential for future sensor applications which require a high accuracy of nodes position by using a smaller number of anchor nodes. The algorithm can be applied not only to the case where nodes are equipped with distance-estimation hardware (range-based), but also to the case where only connectivity information

10 17 V. Vivekanandan and V.W.S. Wong (hop-based) is available. We conducted simulation studies under both regular (square) and irregular (C-shaped) topologies. Simulation results show that MDS-MAP(P, O) provides a lower position estimation error than MDS- MAP(P, C) in both hop-based and range-based scenarios. Further work includes investigating the overhead for control packet exchange and the energy involved in each sensor node for computation. We will also study the use of MDS techniques in situations where the anchors can move within the network. Acknowledgements This work is supported by the Natural Sciences and Engineering Research Council of Canada under grant number 1-3. The authors would like to thank Yi Shang, Wheeler Ruml, Ying Zhang and Markus Fromherz for providing us with their software code for testing and modification. References Akyildiz, I., Su, W., Sankarasubramaniam, Y. and Cayirci, E. () Wireless sensor networks: a survey, Computer Networks, Vol. 3, March, pp.393. 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