Removing Heavily Curved Path: Improved DV-Hop Localization in Anisotropic Sensor Networks

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1 2011 Seventh International Conference on Mobile Ad-hoc and Sensor Networks Removing Heavily Curved Path: Improved DV-Hop Localization in Anisotropic Sensor Networks Ziqi Fan 1, Yuanfang Chen 1, Lei Wang 1, Lei Shu 2, Takahiro Hara 2 1 School of Software, Dalian University of Technology, China ziqi.tom.fan@gmail.com, yuanfang chen@ieee.org, lei.wang@dlut.edu.cn 2 Department of Multimedia Engineering, Osaka University, Japan lei.shu@ieee.org, hara@ist.osaka-u.ac.jp Abstract In Wireless Sensor Networks (WSNs) a multitude of location-dependent applications have been proposed recently, which is very intriguing for researchers to discover and design more accurate and cost-effective localization algorithms. In anisotropic networks, the Euclidean distance between a pair of nodes may not correlate closely with the hop count between them because the corresponding shortest path may have to curve around intermediate holes, resulting in poor distance estimation. And without the help of a large number of uniformly deployed seed nodes, those schemes fail in anisotropic WSNs. To address this issue and improve the accuracy of localization, we propose the Removing Heavily Curved Path (RHCP) scheme in this paper. RHCP takes advantage of selecting the paths which are not heavily affected by the holes to recalculate the location of each unknown node. Through simulation, the results reveal that RHCP performs better than original DV-Hop in anisotropic networks with different shape of holes. In addition, through iterations of RHCP, the results get improved for different anchor node densities. Index Terms Spline Curve, Curvature, Localization, Anisotropic Networks I. INTRODUCTION In wireless sensor networks (WSNs), it is very intriguing for researchers to discover and design more accurate and costeffective localization algorithms [1], [2]. Existing approaches fall into two categories: Range-based approaches and Rangefree approaches [3]. Range-based approaches are based on the assumption that some physical location information can be measured, such as the distance and the relative directions of neighbor nodes. Several hardware technologies provide the capability to measure the distance between two sensor nodes. Radio Signal Strength (RSS) [4] [5] based ranging techniques are based on the fact that the strength of radio signal diminishes during propagation. A more promising technique is the combined use of ultrasound/acoustic and radio signals to estimate distances by determining the Time Difference of Arrival (TDoA) of these signals [6]. The Angle of Arrival (AoA) data is typically gathered using radio or microphone arrays, which allow a receiver to determine the direction of a transmitter. Those methods can obtain accuracy within a few degrees [7]. By contrast, range-free approaches do not depend on special functionality of hardware. Approximate point in triangle Corresponding author: Lei Wang, lei.wang@dlut.edu.cn. (APIT) lets each node estimate whether it resides inside or outside several triangular regions bounded by the seeds it hears, and refines the computed location by overlapping the regions a sensor could possibly reside in. Multi-dimensional scaling (MDS) [8] is a data analysis technique used to visualize proximity of a set of objects in a low dimensional space. A percentage of seed nodes cooperate to obtain the transformation matrixes. Each node measures its proximities to the seeds and calculates its location by applying transformation on the proximity measurements. Beacon based localization approaches utilize estimates of distances to reference nodes that may be several hops away [9]. These distances are propagated from reference nodes to unknown nodes using a basic distance-vector technique. DV-Hop [14] is a scheme in this category. There are two main steps in DV-Hop algorithm. Step one: each anchor node broadcasts a beacon HopItem throughout the network which contains the anchor s location and a hopcount value initialized to one. Each receiving node maintains the HopItem which has the minimum hop-count value per anchor of all beacons it receives in its HopTable. HopItems with higher hop-count values to a particular anchor are defined as suboptimal information and will be ignored. Then those optimal HopItem are flooded outward with hop-count values incremented at every intermediate hop. Step two: once an anchor gets hop-count value to other anchors, it estimates the average distance of one hop, which is estimated by anchor i using the following formula: HopDistance i = j i (xi x j ) 2 +(y i y j ) 2 j i h, (1) ij where (x i,y i ), (x j,y j ) are coordinates of anchor i and anchor j, h ij is the hops between anchor i and anchor j. Then, average per hop distance is flooded to the entire network. After receiving hop-size, non-anchor nodes multiply the average per hop distance by the hop-count value to derive the physical distance to the anchor. Each anchor node floods its HopDistance to network. Unknown nodes receive HopDistance information, and save the first one. At the same time, they transmit the HopDistance to their neighbor nodes. Finally, unknown nodes compute the distance to the anchor nodes by multiplying HopDistance with hop countvalue to the anchor nodes. After obtaining the distances between each pair of unknown /11 $ IEEE DOI /MSN

2 node and anchor node, the unknown node uses triangulation to calculate its coordinate. Based on DV-Hop, a number of further enhanced versions were proposed in [10], [11], [12], [13]. In [10], after unknown nodes have been localized by DV-Hop, each unknown node (UN) will get any two anchors in its list each time, and localized it by trilateration together with a reference node. The reference node (RN) is the anchor node from which UN received average distance per hop firstly. Then RN is regarded as unknown, and UN regarded as anchor node which has localized by DV-Hop. Using trilateration again, comparing them with the actual coordinate, and saving the one which has the smallest error finally. However, this approach would degrade its performances in anisotropic networks, where static holes may exist in the network field, as shown in Fig. 1. In anisotropic WSNs, the Euclidean distance between a pair of nodes may not correlate closely with the shortest path between them because the shortest path may have to bypass intermediate holes. The heavily curved shortest paths caused by the holes can reduce the accuracy of using triangulation method to calculate the coordinates. Unfortunately, anisotropic networks are more likely to exist in realistic situations caused by mountains, lakes, buildings, etc. In [11], the authors propose the Rendered Path (REP) protocol, a rangefree localization scheme in anisotropic sensor networks. REP takes advantage of geometric information to render the shortest paths among nodes. By introducing the virtual hole concept, REP constructs virtual shortest paths in order to estimate the distances between node pairs. In [12], the authors propose a novel approach that uses Voronoi diagrams scale the DV- Hop localization algorithm while maintaining or even reducing its localization error. Two types of localization can result from the proposed algorithm: the physical location of the node (e.g., space, latitude, longitude), or a region limited by the node s Voronoi cell. In [13], the authors propose a new localization algorithm and improve the DV-Hop algorithm by using a differential error correction scheme that is designed to reduce the location error accumulated over multiple hops. This scheme needs no additional hardware support and can be implemented in a distributed way. The proposed method can improve location accuracy without increasing communication traffic and computing complexity. As the key differences from researches in [10], [11]: Paper [10] employs all the nodes to optimize localization result, while our RHCP relies on part of the nodes; Paper [11] is based on the detection of holes boundary, while RHCP needs not. The main contribution of this paper are as follows: We proposed an revised DV-Hop algorithm which is called Removing Heavily Curved Path (RHCP). It uses the coordinates calculated by DV-Hop and improves the result by getting rid of the heavily curved shortest paths affected by the holes in the anisotropic networks. After removing those paths, the hop count between each pair of anchor node and unknown node becomes more representative of the Euclidean distance between them. Therefore, the accuracy of using triangulation method to calculate the coordinates would be improved. We further iterate RHCP on the calculated coordinate of unknown nodes. We find that the accuracy of localization improves along with the increasing number of iterations. By simulation, the result line chart of estimated error goes down gradually along with the increasing number of iterations and reaches its limit at last. The rest of the paper is organized as follows. Network model is demonstrated in Section II. In Section III, we present our proposed algorithm RHCP. Section IV evaluates the proposed scheme through simulations. We conclude the paper in Section V. Symbol Definition S = {s 1,s 2,..., s n} the set of nodes including both anchor nodes and unknown nodes E the set of communication links r i transmission radius of sensor node i s i s j the Euclidean distance between s i and s j S a anchor nodes S u unknown nodes EEP set of Estimated Euclidean Paths from each unknown node to each anchor node EEP[i] set of Estimated Euclidean Paths from each anchor node to unknown node s i s i.p rerhcp store pre-processed coordinate of node i s i.rhcp store RHCP localized coordinate of node i x,y variables of spline curve x i,y i control points of spline curve a,b,c,d... coefficients of spline curve M 4 4 matrix C column vector storing coefficients Y column vector storing the value of y 1 to y 4 f i () function of piecewise polynomial curve slp i slope of curve at sensor node s i variable between 0 and 1 t i and depended on x i or y i α(s) unit-speed curve s arc length φ tangential angle measured counterclockwise from the x-axis to the unit tangent k curvature n total number of nodes in WSNs TABLE I MAIN NOTATION DEFINITIONS II. NETWORK MODEL AND PROBLEM STATEMENT A. Network Model In this paper, the communication undirected graph G = (S, E) is directly derived from the wireless network topology, where S = {s 1,s 2,..., s n } is the set of nodes and E is the set of communication links. Each node has a transmission radius r and the necessary condition for a successful communication between nodes s i and s j is s i s j r i, where s i s j is the Euclidean distance between s i and s j. Our network includes two types of sensor nodes S a (anchor nodes) and S u (unknown nodes). Unknown nodes are randomly deployed with a density ρs u within an area Ω, and a set of special sensor nodes S a with known location, are also randomly deployed 76

3 Fig. 1. In DV-Hop, for each unknown node, it uses all of the paths between the unknown node and anchor nodes to locate. But in RHCP, we use the paths AB, AC and AD to locate, removing the heavily curved paths AE and AF, to achieve better accuracy. with a density ρs a. Notations of this paper are shown in Table I. B. Problem Statement In anisotropic WSNs, i.e., Fig. 1, there is one static hole. Brown nodes represent unknown nodes and blue nodes represent anchor nodes. Due to the static hole, to localize unknown node A, several shortest paths between anchor nodes and A are impacted. The paths AE and AF, are heavily curved, which are not suitable for localizing by DV-Hop. So in design of RHCP, we devote to get rid of these heavily curved paths and just use those more representative of Euclidean path, i.e., AB, AC and AD. III. REMOVING HEAVILY CURVED PATH LOCALIZATION Our proposed RHCP is based on DV-Hop, so we need to use some schemes of DV-Hop to derive the needed input data of RHCP. Like DV-Hop, we get the information of HopDistance and hop countvalue of each anchor node to each unknown node. Then, every unknown node compute the distance to each anchor nodes by multiplying HopDistance with hop countvalue to the anchor nodes. The pseudo-code of RHCP is illustrated in Algorithm 1. There are four steps in our design. From the first step (Lines 2-7), which is a Preprocess of RHCP (PreRHCP), we can get all unknown nodes calculated coordinates. The second step (Lines 8-16) is essential for our design. We assume that the estimated error of calculated coordinates from DV-Hop algorithm in the first step are correctly given, based on that the error equals to 1 at most with high probability, which will not induce a huge impact. Then, we will construct a space piecewise polynomial curve [15] for each path between every unknown node and every anchor node, which will be showed in detail in Section III-A and Section III-B. Then, we calculate each path s average curvature, which will be showed in detail in Section III-C. And finally, we sum up every curvature of Algorithm 1: Removing Heavily Curved Path Algorithm input : set of Estimated Euclidean Path EEP output: s i.p rerhcp and s i.rhcp 1 begin 2 Step 1: 3 i=0 4 while i<numberofunknownnode do 5 Multilaterating EEP[i] to calculate s i.p rerhcp 6 i++ 7 Calculate the average estimated error of PreRHCP 8 Step 2: 9 i=0 10 while i<numberofunknownnode do 11 while j<numberofhopitem in HopTable do 12 Construct Space Piecewise Polynomial Curve 13 Calculate curvature based on Space Piecewise Polynomial Curve 14 j++ 15 Select the average curvature satisfied paths calculating s i.rhcp 16 i++ 17 Calculate the average estimated error of RHCP 18 Step 3: 19 i=0 20 while i<numberofunknownnode do 21 s i.p rerhcp = s i.rhcp 22 i++ 23 Step 4: 24 Repeat Step 2 and Step 3 to optimize the results every node on a specific path and average it as the mean estimated curvature of the path. Then we pick the path whose curvature is less than 0.1 to recalculate the unknown node s coordinate to improve the accuracy of localization. The third step (Lines 18-22) and fourth step (Lines 23-24) are designed for optimizing the localization accuracy. In Step 3, every unknown node s coordinate which is calculated by RHCP is assigned to the coordinate calculated by PreRHCP. And Step 4 iterates Step 2 and Step 3 to get more precise coordinates. A. Construct Piecewise Polynomial Curves To select the relatively smoother path in WSNs, we need to construct polynomial curve to represent the path. For details, we refer readers to [15] for spline curves and [16] for curvature. The cubic polynomial - y = a + bx + cx 2 + dx 3 is the one that is most typically chosen for constructing smooth curves. It is used because it is the lowest degree polynomial that can support an inflection, and it is very well behaved numerically. The cubic polynomial - y = a + bx + cx 2 + dx 3 is the one that is most typically chosen for constructing smooth curves. 77

4 Fig. 2. An example of a curve which has more than one inflection point. Black points represent sensor nodes. Red points represents inflection points. The curve between each two consecutive sensor nodes is a piecewise polynomial curve. In this curve at least four nodes, which satisfies that either they are anchor nodes or coordinates have been calculated, are included, and if this curve just includes four these nodes, this curve expression can be calculated by the spline function Formula 2 and it is a cubic spline function: a + bx + cx 2 + dx 3 = y, (2) and the constants a, b, c, d can be calculated using anchor nodes coordinates (the Formula 3): 1 x 1 x 2 1 x x 2 x 2 2 x x 3 x 2 3 x x 4 x 2 4 x 3 4 a b c d = y 1 y 2 y 3 y 4, (3) where the elements x i and y i are x-axis coordinate and y-axis coordinate of nodes s i, respectively. The Formula 3 can be denoted as: MC = Y, and then the C = M 1 Y. However, in WSNs, the paths between anchor nodes and unknown nodes have more than one inflection point, while a single cubic curve can only cope with one inflection point. Fig.2 is an example of a path, which contains n +1 sensor nodes, and each node s coordinate is whether informed by GPS or calculated by localization algorithm. Without the help of using a higher degree polynomial, for the reason that polynomials with degree higher than three tend to be very sensitive to the control points and do not make smooth shapes, a complex curve should be constructed by piecing together several cubic curves. Let each pair of sensor nodes represent one segment of the curve. Each curve segment is a cubic polynomial with its own coefficients. In Fig. 2, there are n +1 sensor nodes which have ascending values for the x coordinate, and are numbered with indices 0 through n. In general, f i (x) = a i + b i x + c i x 2 + d i x 3 is the function representing the curve between sensor nodes s i and s i+1. For each cubic polynomial function, there are four coefficients. So, to construct piecewise polynomial curves, we have 4 n coefficients to solve for. The following four steps shows how to construct the four linear equations for each curve. Step one: Each curve segment should pass through its sensor nodes. This gives two linear equations: a i + b i x i + c i x i 2 + d i x i 3 = y i and a i + b i x i+1 + c i x i d i x i 3 = y i+1. Step two: The curve segments should have the same slope where they join together. This gives the third linear equation for each segment b i +2c i x i+1 +3d i x i+1 2 b i+1 2c i +1x i+1 3d i +1x i +1 2 =0. Step three: To get the fourth equation we require that the curve segments have the same curvature where they join together. And this gives the final linear equation that needed 2c i +6d i x i+1 =2c i+1 +6d i+1 x i+1 or 2c i +6d i x i+1 2c i+1 6d i+1 x i+1 =0 Step four: At the left end of the curve, the first and second constraints equations are missing since there is no segment on the left. Several methods can get the values of the slopes and the simplest way is to input them by users. Denoting these slopes slp 0 and slp n, getting f 0(x 0 )=slp 0 and f n 1(x n )= slp n,sob 0 + c 0 x 0 +2d 0 x 0 2 = slp 0 and b n 1 + c n 1 x n + 2d n 1 x n 2 = slp n are derived. After having a set of linear equations, to solve for a set of unknown coefficients, all needed to do is the same with the case of a single cubic spline. B. Construct Space Piecewise Polynomial Curves In practice situation, the path in WSNs can shape like Fig. 3. However, piecewise polynomial curves cannot depict these curves. To depict them, we should be aided by another variable, which is denoted by t. In each pair of sensor nodes s i and s i+1, we need to construct two functions f xi = a xi +b xi t i +c xi t 2 i +d xit 3 i and f yi = a yi +b yi t i +c yi t 2 i +d yit 3 i instead of one. To determine the value of the new variable t in each space piecewise polynomial curve, we need parameterize the curve. In following part, we just give an example for the curve 78

5 (a) Circle shape hole (b) Concave shape hole (c) Face shape hole (d) Spiral shape hole Fig. 4. Anisotropic sensor networks topologies. Fig. 3. An example of space curve which needs another variable besides x and y to be depicted in a two dimension graph. parameterization of variable y, and variable x just needs the same process as y. In the parametric form on the right, we have defined parameters t 0, t 1 t 2... t i t i+1... which vary between 0 and 1 as we step along the x axis between control points. For each pair of sensor nodes s i and s i+1, t i = x x i,with dt i x i+1 x i dx = 1. (4) x i+1 x i Each curve segment is specified by a parametric cubic curve f yi (t i )=a yi + b yi t i + c yi t 2 i + d yi t 3 i. (5) At the left side of segment i, t i = 0 and at the right t i =1, so step one constraint in the above subsection is f yi (0) = a yi = y i and f yi (1) = a yi + b yi + c yi + d yi = y i+1. For step two constraint, we differentiate once with respect to x using the chain rule: D x f yi = f yi t i dt i d x = f yi 1. (6) x i+1 x i For step three constraint, we differentiate twice D 2 x f yi =( f yi dt i ) dt i = f 1 yi t i d x d x (x i+1 x i ) 2. (7) We force f yi and f y(i+1) to match slopes by (D x f yi )(1) = (D x f y(i+1) )(0). (8) We force f yi and f y(i+1) to match curvatures by (D 2 x f yi )(1) = (D 2 x f y(i+1) )(0). (9) Remember that we provided two extra equations by specifying slopes at the 2 endpoints slp 0 and slp n. So this gives f y0(0) = slp 0 and f y(n 1) (1) = slp n. Until now, we have got enough linear equations to construct a piecewise polynomial curve which has been explained well in subsection A. And the construction of our space piecewise polynomial curve completes. C. Curvature To get rid of the paths which are curved heavily by the impact of static holes in the anisotropic network, we use the measure curvature to denote how curved a curve is. The absolute value of the curvature is a measure of how sharply the curve bends. To introduce the definition of curvature, we consider that α(s) is a unit-speed curve, where s is the arc length. The tangential angle φ is measured counterclockwise from the x- axis to the unit tangent T = α (s) The curvature k of α is the rate of change of direction at that point of the tangent line with respect to arc length: k = dφ ds. (10) Back to our design, from the above subsection, we get the function between x and t with x = ϕ(t), and between y and t with y = ψ(t), so k can be calculated by Formula 11: k = ϕ (t)ψ (t) ϕ (t)ψ (t) [ϕ 2 (t)+ψ 2 (t)] 3/2. (11) 79

6 RHCP DV-Hop RHCP DV-Hop (a) Circle shape hole (b) Concave shape hole RHCP DV-Hop RHCP DV-Hop (c) Face shape hole (d) Spiral shape hole Fig. 5. Anchor Node Density vs. Error in anisotropic sensor networks. A. Simulation Setup IV. SIMULATIONS In order to check the performance of RHCP localization algorithm, we implemented it into our NetTopo WSN simulator [17], [18]. To quantify the influence made by the factors on localization accuracy, we use Error to denote it, which is defined as the average bias between an unknown node s real coordinate and calculated coordinate divided by sensor radius: Error = n Δxk +Δy k. (12) nr k=1 Our WSN s size is 500*500m 2. The transmission radius R for each node is 50m. The average node degree varies between 5 and 7. To validate RHCP, we construct four different kind of anisotropic topologies (Fig. 4): a circle shape hole, a concave shape hole, a face shape hole consisted of two eyes and one mouth and a spiral shape hole. We deploy 300 wireless sensor nodes randomly in the respect networks. Among the 300 nodes, we change the number of anchor nodes in 25, 50, 75, 100, 125 and 150 respectively to carry experiments on the impact of different anchor node densities leading to RHCP. Shape of Hole Circle Concave Face Spiral RHCP DV-Hop TABLE II AVERAGE ESTIMATED ERROR OF DIFFERENT ANCHOR NODE DENSITIES FOR EACH ANISOTROPIC NETWORK. 80

7 Estimated Error ( R ) AnchorNodeDensity(%)=8.3 AnchorNodeDensity(%)=16.6 AnchorNodeDensity(%)=25 AnchorNodeDensity(%)=33.3 AnchorNodeDensity(%)=41.6 AnchorNodeDensity(%)=50 Estimated Error ( R ) AnchorNodeDensity(%)=8.3 AnchorNodeDensity(%)=16.6 AnchorNodeDensity(%)=25 AnchorNodeDensity(%)=33.3 AnchorNodeDensity(%)=41.6 AnchorNodeDensity(%)= Time of repeating Time of repeating (a) Circle shape hole (b) Concave shape hole Estimated Error ( R ) AnchorNodeDensity(%)=8.3 AnchorNodeDensity(%)=16.6 AnchorNodeDensity(%)=25 AnchorNodeDensity(%)=33.3 AnchorNodeDensity(%)=41.6 AnchorNodeDensity(%)=50 Estimated Error ( R ) AnchorNodeDensity(%)=8.3 AnchorNodeDensity(%)=16.6 AnchorNodeDensity(%)=25 AnchorNodeDensity(%)=33.3 AnchorNodeDensity(%)=41.6 AnchorNodeDensity(%)= Time of Iterations Time of Iterations (c) Face shape hole (d) Spiral shape hole Fig. 6. Time of Iterations vs. Error in anisotropic sensor networks. B. Simulation Results and Analysis For each algorithm: DV-Hop and RHCP, we collect the average localization Error at each anchor node density, and the results are shown in Fig. 5 respectively. To give a more intuitive display, we average the estimated error of different anchor node density for each anisotropic network and show the results in Table II. From these results, we can conclude: RHCP outperforms original DV-Hop in different kind of anisotropic networks. In the following experiment, for each anchor node density, we collect the average localization Error for different time of iterations, which is the number of executing RHCP s Step 4 in Phase 2, and the results are shown in Fig. 6. From these results, we can conclude that: As the time of iterations increasing, the average localization Error for each anchor node density is going smaller. But till the fourth and fifth time, the effect of optimizing reaches its limit, and the line goes smooth. V. CONCLUSION Given the condition that many previous localization algorithms degrade their performance in anisotropic networks, in this paper, we propose the Removing Heavily Curved Path (RHCP) scheme, which works well in anisotropic scenarios. Each unknown node (or non-anchor node) removes the heavily curved paths which are affected by the holes in the network. This method improves the accuracy of using triangulation to calculate the unknown nodes coordinate. Simulation results show that our RHCP algorithm outperforms DV-Hop in different anchor node density in different anisotropic networks. ACKNOWLEDGEMENTS This work is partially supported by Natural Science Foundation of China under Grant No , the Fundamental Research Funds for the Central Universities No. DUT10ZD110, 81

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